miller-book/images.tex

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% !!! IMAGES START HERE !!!

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$N$%
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$\pi /2$%
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$3\pi /2$%
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$f()$%
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$a=2$%
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$Z$%
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$A$%
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$H$%
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$\omega $%
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$1.5\omega $%
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{\newpage\clearpage
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure2372}%
\begin{figure}\psfig{file=figs/fig02.11.ps}

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\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2682}%
\begin{displaymath}
    {y_{\mathrm{INT}}}(x) = 
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath2683}%
\begin{displaymath}
       -f (f-1)(f-2)/6 \cdot y[{x_0}-1] 
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2684}%
\begin{displaymath}
    - (f+1) f (f-2) / 2 \cdot y[{x_0}+1] 
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3000}%
$f = x - {x_0}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3006}%
$N + 1 - k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3008}%
$k=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3020}%
$N-1/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3032}%
$N-2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3036}%
$1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3040}%
$N-3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2685}%
\begin{displaymath}
    x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N-1 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3044}%
$N+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2686}%
\begin{displaymath}
    x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N 
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3052}%
$n=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3064}%
$N+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3068}%
$2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3070}%
$N+3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2687}%
\begin{displaymath}
    x[n] = \cos(2 \pi (n-1) / N) ,\, n = 0, \ldots, N+2 
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
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\lthtmlfigureA{figure2398}%
\begin{figure}\psfig{file=figs/fig02.12.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3366}%
\fbox{ $ \mathrm{tabosc4}\sim $}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3076}%
${2^m} + 3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure2410}%
\begin{figure}\psfig{file=figs/fig02.13.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3386}%
\fbox{ $\mathrm{tabread4}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3392}%
\fbox{ $\mathrm{tabwrite}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3402}%
\fbox{ $\mathrm{pack}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure2428}%
\begin{figure}\psfig{file=figs/fig02.14.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3424}%
\fbox{ $\mathrm{hip}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure2441}%
\begin{figure}\psfig{file=figs/fig02.15.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3444}%
\fbox{ $\mathrm{cos}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3100}%
$-\pi/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3458}%
\fbox{ $\mathrm{samphold}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure2464}%
\begin{figure}\psfig{file=figs/fig02.16.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3478}%
\fbox{ $\mathrm{loadbang}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3484}%
\fbox{ $\mathrm{expr}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3490}%
\fbox{$\mathrm{wrap}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3496}%
\fbox{ $\mathrm{send}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3498}%
\fbox{ $\mathrm{s}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3500}%
\fbox{ $\mathrm{receive}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3502}%
\fbox{ $\mathrm{r}\sim$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3140}%
$c$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2688}%
\begin{displaymath}
    t = p + cf
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2689}%
\begin{displaymath}
    f = {{t - p} \over c} = {{{{2 ^ {h/12}}} - p} \over c}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3148}%
$1000$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3150}%
$R=44100$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath3946}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3972}%
$f(t)=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3984}%
$R/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3992}%
$\pi + \omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3947}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3948}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3949}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4006}%
$\infty$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3562}%
\begin{figure}\psfig{file=figs/fig03.01.ps}

\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4024}%
$f(t)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3950}%
\begin{displaymath}
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\end{displaymath}%
\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4030}%
$1/n$%
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\lthtmlcheckvsize\clearpage}

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{\newpage\clearpage
\lthtmlfigureA{figure3582}%
\begin{figure}\psfig{file=figs/fig03.02.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4036}%
$B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4038}%
$B=4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4040}%
$0, 1, ... B-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure3590}%
\begin{figure}\psfig{file=figs/fig03.03.ps}

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{\newpage\clearpage
\lthtmldisplayA{displaymath3951}%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmldisplayA{displaymath3953}%
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4048}%
$t[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3954}%
\begin{displaymath}
    (2, 1), (4.75, 0), (7.5, 1), (10.25, 0), (13, 1), \ldots
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4054}%
$R/B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3607}%
\begin{figure}\psfig{file=figs/fig03.04.ps}

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\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4058}%
$\lfloor t \rfloor$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4064}%
$(n+f, y)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4070}%
$0 \le f < 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3955}%
\begin{displaymath}
    fx + (1-f)y,
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath3956}%
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath3957}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3614}%
\begin{figure}\psfig{file=figs/fig03.05.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4084}%
$n-B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure3625}%
\begin{figure}\psfig{file=figs/fig03.06.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure3640}%
\begin{figure}\psfig{file=figs/fig03.07.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4092}%
$3.999\ldots$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3654}%
\begin{figure}\psfig{file=figs/fig03.08.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3663}%
\begin{figure}\psfig{file=figs/fig03.09.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure3678}%
\begin{figure}\psfig{file=figs/fig03.10.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3688}%
\begin{figure}\psfig{file=figs/fig03.11.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure3703}%
\begin{figure}\psfig{file=figs/fig03.12.ps}

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\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4370}%
\fbox{ $\mathrm{del}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4372}%
\fbox{ $\mathrm{delay}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4380}%
\fbox{ $\mathrm{pipe}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4390}%
\fbox{ $\mathrm{moses}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4396}%
\fbox{ $\mathrm{select}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4398}%
\fbox{ $\mathrm{sel}$\  }%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure3739}%
\begin{figure}\psfig{file=figs/fig03.13.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4428}%
\fbox{ $\mathrm{line}$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4442}%
\fbox{ $\mathrm{vline}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3760}%
\begin{figure}\psfig{file=figs/fig03.14.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4476}%
\fbox{ $\mathrm{snapshot}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4484}%
\fbox{ $\mathrm{env}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3775}%
\begin{figure}\psfig{file=figs/fig03.15.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure3785}%
\begin{figure}\psfig{file=figs/fig03.16.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4514}%
\fbox{ $\mathrm{notein}$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4520}%
\fbox{ $\mathrm{stripnote}$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4526}%
\fbox{ $\mathrm{trigger}$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4528}%
\fbox{ $\mathrm{t}$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure4581}%
\begin{figure}\psfig{file=figs/fig04.01.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure4591}%
\begin{figure}\psfig{file=figs/fig04.02.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5047}%
$10R$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5051}%
$y[n] = n/(10R)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5035}%
\begin{displaymath}
    y[n] = {{ \left ( {{n} \over {N}} \right ) } ^ 4} ,
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5036}%
\begin{displaymath}
    {f_1} (x) = x \hspace{0.2in} \mathrm{(linear),}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5037}%
\begin{displaymath}
    \ \ \ \ \ \ \ %
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5038}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline5063}%
$1/100$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

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\lthtmldisplayA{displaymath6041}%
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\lthtmlinlinemathA{tex2html_wrap_inline6349}%
$k \tau$%
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{\newpage\clearpage
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$m \tau$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6357}%
$km\tau$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6379}%
$300m+225n$%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6387}%
$f(x) = x^5$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath6051}%
\begin{displaymath}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6389}%
$x^3$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\begin{displaymath}
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmldisplayA{displaymath6055}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6395}%
$a[n]$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6403}%
$f(x) = x^k$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath6056}%
\begin{displaymath}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6409}%
$x^k$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6411}%
$f_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6413}%
$af_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6415}%
${a^2}{f_2}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6421}%
$0<a<1$%
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{\newpage\clearpage
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$1<a<2$%
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{\newpage\clearpage
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$2<a<3$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6435}%
${f_0}=0$%
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{\newpage\clearpage
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${f_1}=1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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${f_2}=1/2$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6441}%
${f_3}=1/6$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6058}%
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{\newpage\clearpage
\lthtmldisplayA{displaymath6059}%
\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6443}%
$e \approx 2.7$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6445}%
$e^x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6453}%
$e^a$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6060}%
\begin{displaymath}
    f(a \cos(\omega n)) = 
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6457}%
$a=$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure5874}%
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{\newpage\clearpage
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6461}%
$\phi=-\pi/2$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6062}%
\begin{displaymath}
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6463}%
${J_k}(a)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6064}%
\begin{displaymath}
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{\newpage\clearpage
\lthtmlfigureA{figure5892}%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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\lthtmldisplayZ
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{\newpage\clearpage
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath6068}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6069}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
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{\newpage\clearpage
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$\omega_p$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\begin{displaymath}
    x[n] = \cos(\omega_c n + a \cos(\omega_m n) + b \cos(\omega_p n))
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$\omega_c n$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6493}%
$\omega_c n +  b \cos(\omega_p n)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
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\lthtmldisplayZ
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{\newpage\clearpage
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${J_3}(a) {J_k}(b)$%
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{\newpage\clearpage
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$(3+k)\pi / 2$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6499}%
$\approx$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\begin{displaymath}
    \cos( \omega_c n + a \cos( \omega_m n + b \cos( \omega_p n )))
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$-\pi \le \phi \le \pi$%
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{\newpage\clearpage
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$T$%
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{\newpage\clearpage
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{\newpage\clearpage
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7185}%
${m_a}$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{section}
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{\newpage\clearpage
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7195}%
$f(0)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7203}%
$f(x) = {e^x}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath7119}%
\begin{displaymath}
    {m_a}[n] = {e^{a \cdot (\cos(\omega n) - 1))}}
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{\newpage\clearpage
\lthtmldisplayA{displaymath7120}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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${b^2}=2a$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
\lthtmldisplayA{displaymath7122}%
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{\newpage\clearpage
\lthtmldisplayA{displaymath7123}%
\begin{displaymath}
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{\newpage\clearpage
\lthtmldisplayA{displaymath7124}%
\begin{displaymath}
    h(b \sin({\omega \over 2} n)) =
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7211}%
$G$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7221}%
$1/(1+b)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7229}%
${m_a}(\omega n)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath7125}%
\begin{displaymath}
    {M_a}(\phi) = W (a \phi),
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$-\pi$%
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{\newpage\clearpage
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{\newpage\clearpage
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${\omega_c}/\omega$%
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{\newpage\clearpage
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{\newpage\clearpage
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$c(\phi)$%
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{\newpage\clearpage
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{\newpage\clearpage
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$k+1$%
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{\newpage\clearpage
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$1-q$%
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{\newpage\clearpage
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$(k+q)\omega$%
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{\newpage\clearpage
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$1+b$%
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{\newpage\clearpage
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$0.5$%
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$p
{c_1} + q {c_2}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7431}%
$c_2$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7433}%
$p=1-q$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7435}%
${c_1} + q ({c_2} - {c_1})$%
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{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline7715}%
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{\newpage\clearpage
\lthtmldisplayA{displaymath8550}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8722}%
$i=\sqrt{-1}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8551}%
\begin{displaymath}
    a = \mathrm{re}(Z)
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{\newpage\clearpage
\lthtmldisplayA{displaymath8552}%
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{\newpage\clearpage
\lthtmlfigureA{figure7770}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8744}%
$|Z|$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8746}%
$(a,b)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8553}%
\begin{displaymath}
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8748}%
$\angle(Z)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8554}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8756}%
$\theta$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8555}%
\begin{displaymath}
    a = r \cos (\theta)
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath8556}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8762}%
$Z=a+bi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8557}%
\begin{displaymath}
    Z = r \cdot \left[ \cos(\theta) + i \sin(\theta) \right ]
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8776}%
$Z_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8778}%
$Z_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8558}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8559}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8560}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8561}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8780}%
$\sin({\theta_1}) \sin({\theta_2})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8782}%
$i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8562}%
\begin{displaymath}
     {Z_1}{Z_2} = {r_1}{r_2}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8786}%
${Z_1}{Z_2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8788}%
${r_1}{r_2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8790}%
${\theta_1}+{\theta_2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{figure7827}%
\begin{figure}\psfig{file=figs/fig07.02.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8564}%
\begin{displaymath}
    Z = \cos(\omega) + i \sin(\omega)
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8808}%
$Z^n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8810}%
$n\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8565}%
\begin{displaymath}
    {Z^n} = \cos(n\omega) + i \sin(n \omega)
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8566}%
\begin{displaymath}
    {1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8567}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8568}%
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath8569}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8826}%
$n \omega + \phi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8570}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8571}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8838}%
$A, AZ, A{Z^2}, \ldots$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8840}%
$1, Z, {Z^2}, \ldots$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8572}%
\begin{displaymath}
    X[n] = A {Z^n} , \ Y[n] = B {Z^n}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8573}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8848}%
$|A+B|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8850}%
$\angle(A+B)$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8852}%
$X[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8854}%
$X$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8574}%
\begin{displaymath}
    Y[n] = X[n-d]
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8860}%
$Y$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8878}%
${X_1}+{X_2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8575}%
\begin{displaymath}
    X[n] = A {Z^n}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8576}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8884}%
${Z^{-d}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8888}%
$- d \omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8890}%
$\omega = \angle(Z)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure7865}%
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\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8900}%
$d/R$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure7880}%
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\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8910}%
$x[0]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8912}%
$x[1]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8577}%
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{\newpage\clearpage
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${Z^{d/2}}$%
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{\newpage\clearpage
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${Z^{-d/2}}$%
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{\newpage\clearpage
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$Z^d$%
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{\newpage\clearpage
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$Z^{-d}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8966}%
${Z^{d}}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8970}%
$-d \omega / 2$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\lthtmlinlinemathA{tex2html_wrap_inline8988}%
$2\pi/d$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9014}%
$g{Z^{-d}}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8589}%
\begin{displaymath}
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9022}%
$Y[n]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
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$H=Y[n]/X[n]$%
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{\newpage\clearpage
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\begin{displaymath}
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{\newpage\clearpage
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9064}%
$W^n$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$\angle(W)$%
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{\newpage\clearpage
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$gW^{-d}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$\sqrt{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$1-g$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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${x_r}[n]$%
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{\newpage\clearpage
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$(0, \ldots, 0)$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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${x_2}[n]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$c=s$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$j$%
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{\newpage\clearpage
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$w$%
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{\newpage\clearpage
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$j=k=1$%
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{\newpage\clearpage
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{\newpage\clearpage
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$g \le 1$%
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{\newpage\clearpage
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$t/d$%
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{\newpage\clearpage
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$20 {\log_{10}} (g)$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$x[n-d]$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9262}%
$d=1.5$%
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{\newpage\clearpage
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$x[n-1.5]$%
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{\newpage\clearpage
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$x[n+1]$%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9278}%
$n - d[n]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9280}%
$d[n]$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9312}%
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{\newpage\clearpage
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{\newpage\clearpage
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${d_0}+s$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$R/10$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9924}%
\fbox{ \texttt{block\~}}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9926}%
\fbox{ \texttt{switch\~}}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\fbox{$\mathrm{block}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$1000/(2f)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$1/(2f)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$2f, 4f, 6f\cdots$%
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{\newpage\clearpage
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$f, 3f, \ldots$%
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{\newpage\clearpage
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$2f$%
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{\newpage\clearpage
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$\sqrt{1/2}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$H(\omega)$%
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{\newpage\clearpage
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$\angle( H(\omega))$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$Q{Z^{-1}}$%
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{\newpage\clearpage
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$|1 - Q{Z^{-1}}|$%
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{\newpage\clearpage
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{\newpage\clearpage
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$\omega = \alpha$%
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{\newpage\clearpage
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$1-r$%
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{\newpage\clearpage
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$\overline{Q}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$\overline{Z} = {Z^{-1}}$%
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{\newpage\clearpage
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$Z\overline{Z}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
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$P$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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{\newpage\clearpage
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$|P| < 1$%
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{\newpage\clearpage
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$|P|> 1$%
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{\newpage\clearpage
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{\newpage\clearpage
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$P=Q$%
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${Q_1}, \ldots, {Q_j}$%
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{\newpage\clearpage
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${P_1}, \ldots, {P_k}$%
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{\newpage\clearpage
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$\overline{P}$%
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{\newpage\clearpage
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    H(Z) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11207}%
$Q_i$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11209}%
$P_i$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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        {A{Z^n}}
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmldisplayA{displaymath10975}%
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        {H_{\mathrm{im}}}(Z) = {{
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11235}%
$1/\overline{Q}$%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11243}%
$1/(1-p)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11247}%
$1-p$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11255}%
$p = 1-\omega$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath10976}%
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{\newpage\clearpage
\lthtmlfigureA{figure10336}%
\begin{figure}\psfig{file=figs/fig08.13.ps}

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\lthtmlfigureA{figure10344}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11311}%
$d = \omega$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11313}%
$p = 1 - d / \sqrt{g}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11315}%
$q = 1 - d \sqrt{g}$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath10977}%
\begin{displaymath}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11329}%
$p = 1 - \beta$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11333}%
$\cos \omega + i \sin \omega$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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{\newpage\clearpage
\lthtmldisplayA{displaymath10979}%
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{\newpage\clearpage
\lthtmlfigureA{figure10359}%
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\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath10980}%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmldisplayA{displaymath10982}%
\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlfigureA{figure10369}%
\begin{figure}\psfig{file=figs/fig08.16.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
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\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11361}%
$r=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11365}%
$r=\infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10984}%
\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath10985}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11381}%
$\alpha = \pi/2$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10986}%
\begin{displaymath}
    { (1 - {r^2}) - 2 r i }
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{\newpage\clearpage
\lthtmldisplayA{displaymath10987}%
\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlfigureA{figure10391}%
\begin{figure}\psfig{file=figs/fig08.17.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlfigureA{figure10397}%
\begin{figure}\psfig{file=figs/fig08.18.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath10989}%
\begin{displaymath}
    | H(\cos(\omega) + i \sin(\omega)) | 
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11405}%
$R(Z)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11413}%
$|R(Z)| = 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11415}%
$Z=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10990}%
\begin{displaymath}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11417}%
$|U|=1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10991}%
\begin{displaymath}
    J(Z) = H(R(Z))
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11427}%
$J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10992}%
\begin{displaymath}
    \cos(\phi) + i \sin(\phi) = R(\cos(\omega) + i \sin(\omega))
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10993}%
\begin{displaymath}
    H(Z) =
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10994}%
\begin{displaymath}
    R(Z) = -{Z^2} = -
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11453}%
$-i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10995}%
\begin{displaymath}
    J(Z) = 
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlfigureA{figure10438}%
\begin{figure}\psfig{file=figs/fig08.19.ps}

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\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11473}%
$R(Z) = -Z^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10996}%
\begin{displaymath}
    S(Z) =
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11485}%
$S(1) = 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11487}%
$S(-1) = -1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11493}%
$\overline{Z}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11495}%
$\overline{W}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11497}%
$S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10997}%
\begin{displaymath}
    S(\cos(\omega) + i \sin(\omega)) = i
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11507}%
$R = - {Z^2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11511}%
$H(R(S(Z)))$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10998}%
\begin{displaymath}
    a = \cos({{\pi}\over 4} - {{\omega} \over 2}) , \;
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11521}%
$2n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11531}%
$J(Z) = H(R(S(Z)))$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11533}%
$J(Z)=\infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11535}%
$R(S(Z))$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11541}%
$R(S(Z)) = W$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath10999}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11000}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11001}%
\begin{displaymath}
    Z = { { 
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11553}%
${a^2}+{b^2}=1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10463}%
\begin{figure}\psfig{file=figs/fig08.20.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlfigureA{figure10470}%
\begin{figure}\psfig{file=figs/fig08.21.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11561}%
$1-|P|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11563}%
$1-1/n$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath11002}%
\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11571}%
$e$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11575}%
$n=5$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10485}%
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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11591}%
$1/e$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11003}%
\begin{displaymath}
    {{n} \over {2\pi/\omega}} = {{1} \over {2\pi}} {{\omega} \over {b}}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11615}%
$P<1$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath11004}%
\begin{displaymath}
    H(Z) = {{
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11621}%
$1/\overline{P}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10509}%
\begin{figure}\psfig{file=figs/fig08.23.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlfigureA{figure10526}%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11645}%
$x[n]^2$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10537}%
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\lthtmlfigureZ
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{\newpage\clearpage
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{\newpage\clearpage
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11647}%
$2 \alpha$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11649}%
${{a^2} / 2}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11008}%
\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11651}%
$({a^2} +
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10549}%
\begin{figure}\psfig{file=figs/fig08.27.ps}

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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11655}%
$1, W, {W^2}, \ldots$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11657}%
$1, WZ, {{(WZ)}^2}, \ldots$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11659}%
$\angle(ZW)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11009}%
\begin{displaymath}
    x[n] = a \cdot \cos (\omega n)
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\lthtmldisplayZ
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{\newpage\clearpage
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11011}%
\begin{displaymath}
    x[n] = \mathrm{re} (X[n])
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11665}%
$-\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11012}%
\begin{displaymath}
    X'[n] = a \left ( \cos (\omega n) - i \sin (\omega n) \right )
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11683}%
$H_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11685}%
$H_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11013}%
\begin{displaymath}
    \angle({H_1}(Z)) - \angle({H_2}(Z)) \approx 
    \left \{
    \begin{array}{ll}
        \pi/2 & {0 < \angle(Z) < \pi} \\
        -\pi/2 & {-\pi < \angle(Z) < 0}
    \end{array}
    \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11014}%
\begin{displaymath}
    {H_1}(Z) \approx i {H_2}(Z) , \; 0 < \angle(Z) < \pi
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11015}%
\begin{displaymath}
    {H_1}(Z) \approx -i {H_2}(Z) , \; -\pi < \angle(Z) < 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11689}%
$a[n] + i b[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11693}%
$b[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath11016}%
\begin{displaymath}
    {H_1}(Z) + i {H_2}(Z) \approx
    \left \{
    \begin{array}{ll}
        2 {H_1}(Z) & {0 < \angle(Z) < \pi} \\
        0 & \mbox{otherwise}
    \end{array}
    \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12147}%
\fbox{\texttt{lop\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12149}%
\fbox{$\mathrm{lop}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12153}%
\fbox{ \texttt{hip\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12159}%
\fbox{ \texttt{bp\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12161}%
\fbox{$\mathrm{bp}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12165}%
\fbox{ \texttt{noise\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12167}%
\fbox{$\mathrm{noise}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10598}%
\begin{figure}\psfig{file=figs/fig08.28.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12183}%
\fbox{ \texttt{vcf\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12185}%
\fbox{$\mathrm{vcf}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10610}%
\begin{figure}\psfig{file=figs/fig08.29.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10620}%
\begin{figure}\psfig{file=figs/fig08.30.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10626}%
\begin{figure}\psfig{file=figs/fig08.31.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10639}%
\begin{figure}\psfig{file=figs/fig08.32.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12231}%
\fbox{ \texttt{rzero\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12233}%
\fbox{ \texttt{rzero\_rev\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12235}%
\fbox{ \texttt{rpole\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12237}%
\fbox{$\mathrm{rzero}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12239}%
\fbox{$\mathrm{rzero\_rev}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12241}%
\fbox{$\mathrm{rpole}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12245}%
\fbox{ \texttt{czero\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12247}%
\fbox{ \texttt{czero\_rev\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12249}%
\fbox{ \texttt{cpole\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12251}%
\fbox{$\mathrm{czero}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12253}%
\fbox{$\mathrm{czero\_rev}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12255}%
\fbox{$\mathrm{cpole}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure10670}%
\begin{figure}\psfig{file=figs/fig08.33.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11727}%
$i/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13143}%
$n=0,\ldots,N-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13155}%
$4\pi/N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13159}%
$2(N-1)\pi/N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13173}%
$0 \leq k < N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13175}%
$2\pi k / N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13187}%
$\omega=2\pi/N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13191}%
$U$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12979}%
\begin{displaymath}
    U = \cos(\omega) + i \sin(\omega)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12980}%
\begin{displaymath}
    {P_k}[n] = {A_k}{{\left [ {U^k} \right ]} ^ {n}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13199}%
${A_k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12981}%
\begin{displaymath}
    \angle({U^k}) = k \angle(U) = k\omega
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12982}%
\begin{displaymath}
    X[n] = 
    {A_0}{{\left [ {U^0} \right ]} ^ {n}}
    +  {A_1}{{\left [ {U^1} \right ]} ^ {n}}
    + \cdots
    + {A_{N-1}}{{\left [ {U^{N-1}} \right ]} ^ {n}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13205}%
$A_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13207}%
$-k\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12983}%
\begin{displaymath}
    {A_k} = {1\over N} \left (
    {{\left [ {U^{-k}} \right ]} ^ {0}} X[0] +
    {{\left [ {U^{-k}} \right ]} ^ {1}} X[1] +
    \cdots +
    {{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
    \right )
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12984}%
\begin{displaymath}
    {\cal FT}\left \{ X[n] \right \} (k) = 
    {V ^ {0}} X[0] +
    {V ^ {1}} X[1] +
    \cdots +
    {V ^ {N-1}} X[N-1]
 \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13213}%
$V = {U^{-k}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13225}%
$V$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12985}%
\begin{displaymath}
     V = \cos(-k\omega) + i\sin(-k\omega)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath12986}%
\begin{displaymath}
    {\cal FT}\left \{ X[n] \right \} (k+N) =
    {\cal FT}\left \{ X[n] \right \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13251}%
$Y[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13255}%
$k = 0, ..., N-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12988}%
\begin{displaymath}
    Y[k] = {\cal FT}\left \{ X[n] \right \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12989}%
\begin{displaymath}
    = {{\left [ {U^{-k}} \right ]} ^ {0}} X[0] +
    {{\left [ {U^{-k}} \right ]} ^ {1}} X[1] +
    \cdots +
    {{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12990}%
\begin{displaymath}
    = {{\left [ {U^{0}} \right ]} ^ {k}} X[0] +
    {{\left [ {U^{-1}} \right ]} ^ {k}} X[1] +
    \cdots +
    {{\left [ {U^{-(N-1)}} \right ]} ^ {k}} X[N-1]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13259}%
$X[m]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13261}%
$-m\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13263}%
$m = 0, \ldots, N-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13269}%
$X[-m]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12991}%
\begin{displaymath}
    {1 \over N} {\cal FT} \left \{ Y[k] \right \} (m) = X[-m]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13279}%
$Y[k]/N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13281}%
$n = -m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12992}%
\begin{displaymath}
    X[n] = {1 \over N} {\cal FT} \left \{ Y[k] \right \} (-n) 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12993}%
\begin{displaymath}
    = {{\left [ {U^{0}} \right ]} ^ {n}} Y[0] +
    {{\left [ {U^{1}} \right ]} ^ {n}} Y[1] +
    \cdots +
    {{\left [ {U^{N-1}} \right ]} ^ {n}} Y[N-1] 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13289}%
$k=0,
\ldots, N-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13291}%
$X[n]=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13295}%
$N>1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12994}%
\begin{displaymath}
    {\cal FT} \left \{ X[n] \right \} (k) = 
    \left \{
    \begin{array}{ll}
        N & {k=0} \\
        0 & {k=1, \ldots, N-1}
    \end{array}
    \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13303}%
$V \not= 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12996}%
\begin{displaymath}
    {\cal FT} \left \{ X[n] \right \} (k) = 
    {{
        {V^N} - 1
    } \over {
        V - 1
    }}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12997}%
\begin{displaymath}
    \xi = \cos(\pi k / N) - i \sin(\pi k / N)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13305}%
${\xi^2} = V$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13307}%
$\xi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12998}%
\begin{displaymath}
    {\cal FT} \left \{ X[n] \right \} (k) = 
        {\xi^{N-1}}
    {{
        {\xi^N} - {\xi^{-N}}
    } \over {
        \xi - {\xi^{-1}}
    }}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath12999}%
\begin{displaymath}
    {\xi^N} - {\xi^{-N}} =
        \left (\cos(\pi k) - i \sin(\pi k) \right ) - 
        \left (\cos(\pi k) + i \sin(\pi k) \right )
        = - 2 i \sin(\pi k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13000}%
\begin{displaymath}
    {\cal FT} \left \{ X[n] \right \} (k) = 
    \left ( {
        \parbox[t][0.1in]{0in}{\mbox{}}
        \cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
    } \right ) 
    {{
        \sin(\pi k)
    } \over {
        \sin(\pi k / N)
    }}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13309}%
$V=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13001}%
\begin{displaymath}
    {\cal FT} \left \{ X[n] \right \} (k) = 
    \left ( {
        \parbox[t][0.1in]{0in}{\mbox{}}
        \cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N) 
    } \right )
        {D_N}(k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13311}%
${D_N}(k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13002}%
\begin{displaymath}
    {D_N}(k) = 
    \left \{
    \begin{array}{ll}
        N & {k= 0} \\
            {{
                \sin(\pi k)
            } \over {
                \sin(\pi k / N)
            }}
        & {k\not=0,\; -N < k < N}
    \end{array}
    \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13317}%
$k=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13319}%
$k=100$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13003}%
\begin{displaymath}
    \cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13321}%
${\cal FT} \left \{ X[n] \right \} (k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13335}%
$k>1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13337}%
$k < -1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13341}%
$k=-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12411}%
\begin{figure}\psfig{file=figs/fig09.01.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12416}%
\begin{figure}\psfig{file=figs/fig09.02.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath13005}%
\begin{displaymath}
    {\cal FT} \left \{ Y[n] \right \} (k) = 
    {V ^ {0}} Y[0] +
    {V ^ {1}} Y[1] +
    \cdots +
    {V ^ {N-1}} Y[N-1]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13006}%
\begin{displaymath}
    =
    {V ^ {0}} X[-d] +
    {V ^ {1}} X[-d+1] +
    \cdots +
    {V ^ {N-1}} X[-d+N-1]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13007}%
\begin{displaymath}
    =
    {V ^ {d}} X[0] +
    {V ^ {d+1}} X[1] +
    \cdots +
    {V ^ {d+N-1}} X[N-1]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13008}%
\begin{displaymath}
    = {V^d} \left (
    {V ^ {0}} X[0] +
    {V ^ {1}} X[1] +
    \cdots +
    {V ^ {N-1}} X[N-1]
    \right )
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13009}%
\begin{displaymath}
    = {V^d}  {\cal FT} \left \{ X[n] \right \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13010}%
\begin{displaymath}
    {\cal FT} \left \{ X[n-d] \right \} (k) = 
    \left ( {
        \parbox[t][0.1in]{0in}{\mbox{}}
        \cos(-dk\omega) + i\sin(-dk\omega) 
    } \right )
    {\cal FT} \left \{ X[n] \right \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13365}%
$X[n-d]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13369}%
$-dk\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13011}%
\begin{displaymath}
    Y[n] = {Z^n} X[n]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13012}%
\begin{displaymath}
    Z = \cos(\alpha) + i \sin(\alpha)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13014}%
\begin{displaymath}
    =
    {V ^ {0}} X[0] +
    {V ^ {1}} Z X[1] +
    \cdots +
    {V ^ {N-1}} {Z^{N-1}} X[N-1]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13015}%
\begin{displaymath}
    =
    {{(VZ)} ^ {0}} X[0] +
    {{(VZ)} ^ {1}} X[1] +
    \cdots +
    {{(VZ)} ^ {N-1}} X[N-1]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13016}%
\begin{displaymath}
    = {\cal FT} \left \{ X[n] \right \} (k - {{\alpha } \over {\omega}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13017}%
\begin{displaymath}
    {\cal FT} \left \{ (\cos(\alpha) + i \sin(\alpha)) X[n] \right \} (k) =
    {\cal FT} \left \{ X[n] \right \} (k - {{\alpha N} \over {2 \pi}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13379}%
${Z^n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13018}%
\begin{displaymath}
    {\cal FT} \left \{ {Z^n} \right \} (k) =
    {\cal FT} \left \{ 1 \right \}(k - {{\alpha } \over {\omega}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13019}%
\begin{displaymath}
    = \left [ \cos(\Phi(k)) + i \sin(\Phi(k))\right ]
    {D_N}(k - {{\alpha } \over {\omega}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13385}%
${D_N}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13387}%
$\Phi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13020}%
\begin{displaymath}
    \Phi(k) = - \pi \cdot (k - {{\alpha } \over {\omega}}) \cdot (N-1)/N
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12482}%
\begin{figure}\psfig{file=figs/fig09.03.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12488}%
\begin{figure}\psfig{file=figs/fig09.04.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13427}%
$w[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13021}%
\begin{displaymath}
    {\cal FT} \left \{ w[n] X[n] \right \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13022}%
\begin{displaymath}
    w[n] = {1\over 2} - {1\over 2} \cos(2\pi n / N)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13023}%
\begin{displaymath}
      w[n] = {1\over 2} - {1\over 4}  {U^n} - {1\over 4}  {U^{-n}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13024}%
\begin{displaymath}
     {\cal FT} \left \{ w[n] {Z^n} \right \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13025}%
\begin{displaymath}
     = {\cal FT} \left \{ {1\over 2} {Z^n} - {1\over 4} {(UZ)^n} 
         - {1\over 4} {({U^{-1}}Z)^n}\right \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13026}%
\begin{displaymath}
    \approx \left [ \cos(\Phi(k)) + i \sin(\Phi(k))\right ]
    M(k - {{\alpha } \over {\omega}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13027}%
\begin{displaymath}
    \Phi(k) = - \pi \cdot (k - {{\alpha } \over {\omega}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13028}%
\begin{displaymath}
    M(k) = 
    {\left [ {
        {1\over 2}{D_N}(k)
        + {1\over 4}{D_N}(k + 1)
        + {1\over 4}{D_N}(k - 1)
    } \right ] }
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13437}%
$M(k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12534}%
\begin{figure}\psfig{file=figs/fig09.05.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13447}%
${D_n}(k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13451}%
$1/4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13029}%
\begin{displaymath}
    {D_N}(k) \approx {
    	{N \sin(\pi k) } \over {\pi k}
    }
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13453}%
$k = 3/2, 5/2, \ldots$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13030}%
\begin{displaymath}
    {2 \over {3 \pi}} \approx -13 \mathrm{dB} , \;
    {2 \over {5 \pi}} \approx -18 \mathrm{dB} , \;
    {2 \over {7 \pi}} \approx -21 \mathrm{dB} , \;
    {2 \over {9 \pi}} \approx -23 \mathrm{dB} ,
    \ldots
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13031}%
\begin{displaymath}
    {2 \over {5 \pi}} - {1\over 2} [ {2 \over {3 \pi}} + {2 \over {7 \pi}} ]
    	\approx -32.30 \mathrm{dB}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13457}%
$-42$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13459}%
$-49$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13461}%
$-54$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13463}%
$-59$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12556}%
\begin{figure}\psfig{file=figs/fig09.06.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13481}%
$k\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12563}%
\begin{figure}\psfig{file=figs/fig09.07.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13487}%
$k = 0, 1, \ldots, N-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13493}%
$m = \ldots, 0, 1, \ldots$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13499}%
$mH$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13505}%
$mH+n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13032}%
\begin{displaymath}
    S[m, k] = {\cal FT}\{w(n)X[n-mH]\} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13033}%
\begin{displaymath}
    S[k] = S[m, k]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13034}%
\begin{displaymath}
    C[m] = S[m, k]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13533}%
$H=N/4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13535}%
$FT$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13537}%
$3/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{figure12577}%
\begin{figure}\psfig{file=figs/fig09.08.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13539}%
$C[m]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13541}%
$g[m]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13543}%
$g[m]C[m]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13549}%
$C[m-1]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13551}%
$C[m-2]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13553}%
$|C[m]|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13565}%
$g[m, k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13567}%
$S[m, k] = C[m]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13035}%
\begin{displaymath}
    g[m, k] =  
    \left \{
    \begin{array}{ll}
        {1 - f[k]/|S[m, k]|} & {|S[m, k]| > f[k]} \\
        0 & \mbox{otherwise}
    \end{array}
    \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13571}%
$f[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13577}%
$|S[m, k]|-f[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13579}%
$x = |S[m, k]|/f[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13581}%
$g(x) = 1-1/x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13583}%
$x<1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{figure12596}%
\begin{figure}\psfig{file=figs/fig09.09.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13589}%
$|S[m, k]|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13591}%
$|T[m, k]|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13036}%
\begin{displaymath}
    g[m, k] = {{|T[m, k]|}\over{|S[m, k]|}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure12610}%
\begin{figure}\psfig{file=figs/fig09.10.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13593}%
$\alpha = 3\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13595}%
$k=3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13037}%
\begin{displaymath}
    \begin{array}{lll}
        {\angle S[0, 2] = \phi + \pi} &
            {\angle S[0, 3] = \phi} &
            {\angle S[0, 4] = \phi + \pi}\\
        {\angle S[1, 2] = \phi + H\alpha + \pi } &
            {\angle S[1, 3] = \phi + H\alpha} &
            {\angle S[1, 4] = \phi + H\alpha + \pi}\\
        {\angle S[2, 2] = \phi + 2H\alpha + \pi } &
            {\angle S[2, 3] = \phi + 2H\alpha} &
            {\angle S[2, 4] = \phi + 2H\alpha + \pi}\\
    \end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13601}%
$H\alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13038}%
\begin{displaymath}
    H \alpha = \angle S[1, 3] - \angle S[0, 3]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13039}%
\begin{displaymath}
    \alpha = {{\angle S[1, 3] - \angle S[0, 3] + 2 p \pi} \over H}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13607}%
$2\pi/H$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13611}%
$8\pi/N = 4 \omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13623}%
$m-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12629}%
\begin{figure}\psfig{file=figs/fig09.11.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12634}%
\begin{figure}\psfig{file=figs/fig09.12.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13040}%
\begin{displaymath}
    T[k] = {\cal FT}(W(n)X[n]) (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13041}%
\begin{displaymath}
    T'[k] = {\cal FT}(W(n)X[n+H]) (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13042}%
\begin{displaymath}
    \angle S[m, k] = \angle S[m-1, k] +
    \left ( \angle T'[k] -  \angle T[k] \right )
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13043}%
\begin{displaymath}
    = \angle \left ( 
        {{S[m-1, k] T'[k]}
            \over
        {T[k]}}
    \right )
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13044}%
\begin{displaymath}
    S[m, k] \; = \; 
    a
    \; \cdot \;
    {
        {  \left | 
            {{S[m-1, k] T'[k]}
            \over
            {T[k]}}
        \right |}
        ^
        {-1}
    }
    \; \cdot \;
    {
        {{S[m-1, k] T'[k]}
        \over
        {T[k]}}
    }
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13651}%
$a = |T'[k]|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13045}%
\begin{displaymath}
    S[m, k] \; = \; 
    {
        {  \left | 
            {{S[m-1, k]}
            \over
            {T[k]}}
        \right |}
        ^
        {-1}
    }
    \; \cdot \;
    {
        {{S[m-1, k] T'[k]}
        \over
        {T[k]}}
    }
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13657}%
$S[m,k+1]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13659}%
$T'[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13661}%
$T'[k+1]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13665}%
$S[m-1,k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13669}%
$T[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13671}%
$2N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13046}%
\begin{displaymath}
    \angle T[k+1] - \angle T[k] = \angle S[m-1, k+1] - \angle S[m-1, k]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13047}%
\begin{displaymath}
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13679}%
$S/T$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13048}%
\begin{displaymath}
    S[m, k] \; = \; 
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13049}%
\begin{displaymath}
    R[k] \; = \; 
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13685}%
$R[k]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13050}%
\begin{displaymath}
    S[m, k] \; = \; 
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\end{displaymath}%
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{\newpage\clearpage
\lthtmldisplayA{displaymath13051}%
\begin{displaymath}
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13052}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13695}%
$R'$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13703}%
$T'$%
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\lthtmlcheckvsize\clearpage}

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\lthtmlfigureA{figure12678}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13711}%
$(-1)^k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13713}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12688}%
\begin{figure}\psfig{file=figs/fig09.14.ps}

\end{figure}%
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\lthtmlinlinemathA{tex2html_wrap_inline14083}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14085}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14093}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14095}%
\fbox{$\mathrm{rfft}\sim$}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13723}%
$N/2+1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14099}%
\fbox{\texttt{tabreceive\~}}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\fbox{$\mathrm{tabreceive}\sim$}%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14115}%
\fbox{\texttt{rifft\~}}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14117}%
\fbox{$\mathrm{rifft}\sim$}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13735}%
$N=512$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12717}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13739}%
$s[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13743}%
$m[k]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath13053}%
\begin{displaymath}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14143}%
\fbox{\texttt{bang\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14145}%
\fbox{$\mathrm{bang}\sim$}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14149}%
\fbox{\texttt{tabsend\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14151}%
\fbox{$\mathrm{tabsend}\sim$}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13761}%
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{\newpage\clearpage
\lthtmlfigureA{figure12740}%
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\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13771}%
$c[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13773}%
$1/f[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13779}%
$c[k]/f[k]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure12748}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14183}%
\fbox{\texttt{lrshift\~}}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14185}%
\fbox{$\mathrm{lrshift}\sim$}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14189}%
\fbox{\texttt{q8\_rsqrt\~}}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14191}%
\fbox{$\mathrm{q8\_rsqrt}\sim$}%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13799}%
$R'[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13829}%
$n=N/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13839}%
$2.5\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline13843}%
$k=2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
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\lthtmlfigureA{figure14230}%
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\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14745}%
$N/R$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14757}%
$A[k]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14633}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14634}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14635}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14765}%
${U^{N/2}} = -1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14637}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14638}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14767}%
$X'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14639}%
\begin{displaymath}
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14769}%
$X''$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14640}%
\begin{displaymath}
    X''[n] = {{X[n] - X[n+N/2]}\over 2} =
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14777}%
$X[n] = X[n+N/2]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14783}%
$X'[n] = X[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14785}%
$X''[n] = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14791}%
$N/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14795}%
$X[n] = -X[n+N/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14801}%
$X[-n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14807}%
$X[N-n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14641}%
\begin{displaymath}
    X[-n] = 
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14642}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14809}%
$X'[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14643}%
\begin{displaymath}
      X'[n] =  {{X[n] + X[-n]}\over 2} =
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14811}%
$X''[n]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14644}%
\begin{displaymath}
      X''[n] = {{X[n] - X[-n]}\over {2i}} =
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14817}%
$X[-n] = X[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14819}%
$X[-n] = -X[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{figure14263}%
\begin{figure}\psfig{file=figs/fig10.02.ps}

\end{figure}%
\lthtmlfigureZ
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\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14829}%
${L_1}, \ldots, {L_j}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14833}%
${d_1}, \ldots, {d_j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14835}%
$d_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14837}%
${L_1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure14280}%
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\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14841}%
$({L_1}, {d_1}) = (0.3N, -0.3)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14843}%
$({L_2}, {d_2}) = (0.6N, 1.3)$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath14645}%
\begin{displaymath}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14845}%
$0 \le n \le N-1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14849}%
$(L, d)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14851}%
$s'[n] = d s[n-L]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14646}%
\begin{displaymath}
    x[n] = {d_1} s[n - {L_1}] + \cdots + {d_j} s[n - {L_j}]
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14647}%
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14853}%
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{\newpage\clearpage
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\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath14648}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14861}%
$-1/N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14863}%
${M_i}, \ldots, {M_l}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14865}%
${c_1}, \ldots, {c_l}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14649}%
\begin{displaymath}
    x[n] = {-N c_1} p[n - {M_1}] - \cdots - {N c_l} p[n - {M_l}]
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure14318}%
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\end{figure}%
\lthtmlfigureZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14869}%
$n^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14873}%
${c_1} + \cdots + {c_l} = 0$%
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\lthtmlcheckvsize\clearpage}

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\lthtmldisplayA{displaymath14650}%
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    A[k] = {1 \over N} {\cal FT}\{X[n]\}(k)
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14651}%
\begin{displaymath}
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14881}%
$x[n] - x[n-1]$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14889}%
$k/N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14652}%
\begin{displaymath}
    {\cal FT}\{ x[n-1] \} =
    \left [ \cos(k \omega) - i \sin (k \omega) \right ]
    {\cal FT}\{ x[n] \}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14653}%
\begin{displaymath}
    \approx (1 - i \omega k) {\cal FT}\{ x[n] \}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14895}%
$k \omega = 2\pi k / N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14654}%
\begin{displaymath}
    \cos(k \omega) \approx 1 \; , \; \sin(k \omega) \approx k \omega
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14897}%
$(k/N)^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14655}%
\begin{displaymath}
    {\cal FT}\{ x[n] - x[n-1] \} \approx i \omega k {\cal FT}\{ x[n] \}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14899}%
$s[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14901}%
$0 \le n < N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14656}%
\begin{displaymath}
    s[n] - s[n-1] = -{1 \over N} + 
    \left \{
    \begin{array}{ll}
        {1} & {n = 0} \\
        0 & \mbox{otherwise}
    \end{array}
    \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14903}%
$-{1 \over N}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14657}%
\begin{displaymath}
    {\cal FT}\{ s[n] - s[n-1] \} (k) = 1 , \; k \neq 0, \; -N < k < N\
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14658}%
\begin{displaymath}
    {\cal FT}\{ s[n] \} (k) \approx {1 \over {i \omega k}} 
    = {{-iN} \over {2 \pi k}} 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14909}%
$k \neq 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14913}%
$1/i = -i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14915}%
${\cal FT}\{ s[n] \}(0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14921}%
$k = \pm 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14659}%
\begin{displaymath}
    {1 \over N}\left [
        {\cal FT}\{ s[n] \} (1) \cdot {U^n} + 
        {\cal FT}\{ s[n] \} (-1) \cdot {U^{-n}}
    \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14660}%
\begin{displaymath}
    \approx {{-i} \over {2 \pi}} \left [ U^n - U^{-n} \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14661}%
\begin{displaymath}
    = {{\sin ( \omega n)} \over {\pi}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14662}%
\begin{displaymath}
    {{\sin ( k \omega n)} \over {k \pi}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14663}%
\begin{displaymath}
    s[n] \approx {1 \over \pi} \left [
        {\sin ( \omega n )}
        + {{\sin ( 2 \omega n)} \over 2}
        + {{\sin ( 3 \omega n)} \over 3}
        + \cdots
    \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath14664}%
\begin{displaymath}
    p[n] - p[n-1] = {
        {
            {{({n\over N} - {1\over 2})}^2} -
            {{({{n-1}\over N} - {1\over 2})}^2}
    } \over {
        2
    }}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14665}%
\begin{displaymath}
    = {
        {
            {{({n\over N} - {N\over {2N}})}^2} -
            {{({{n}\over N} - {{N - 2}\over {2N}})}^2}
    } \over {
        2
    }}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14666}%
\begin{displaymath}
    = {
        {
            {{{2n}\over {N^2}} - {1\over {N}}} + {1\over {N^2}}
    } \over {
        2
    }}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14667}%
\begin{displaymath}
    \approx - s[n] / N .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14668}%
\begin{displaymath}
    {\cal FT}\{ p[n] \} (k) \approx
    {{-1} \over N} \cdot {{-iN} \over {2 \pi k}} \cdot {\cal FT}\{ s[n] \} (k)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14669}%
\begin{displaymath}
    \approx {{-1} \over N} \cdot {{-iN} \over {2 \pi k}}
    \cdot {{-iN} \over {2 \pi k}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14670}%
\begin{displaymath}
    = {N \over {4 {\pi ^2} {k^2}}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14671}%
\begin{displaymath}
    p[n] \approx {1 \over {2 {\pi^2}}} \left [
        {\cos ( \omega n )}
        + {{\cos ( 2 \omega n)} \over 4}
        + {{\cos ( 3 \omega n)} \over 9}
        + \cdots
    \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{figure14407}%
\begin{figure}\psfig{file=figs/fig10.06.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14672}%
\begin{displaymath}
    x[n] = s[n] - s[n-{N \over 2}]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14937}%
$0 <= n < N/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14673}%
\begin{displaymath}
    x[n] \approx {1 \over \pi} \left [
        {\sin ( \omega n )}
        + {{\sin ( 2 \omega n)} \over 2}
        + {{\sin ( 3 \omega n)} \over 3}
        + \cdots
    \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14674}%
\begin{displaymath}
    \left .
        -{\sin ( \omega n )}
        + {{\sin ( 2 \omega n)} \over 2}
        - {{\sin ( 3 \omega n)} \over 3}
        \pm \cdots
    \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14675}%
\begin{displaymath}
    = {2 \over \pi} \left [
        {\sin ( \omega n )}
        + {{\sin ( 3 \omega n)} \over 3}
        + {{\sin ( 5 \omega n)} \over 5}
        + \cdots
    \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14676}%
\begin{displaymath}
    x[n] = 8 p[n] - 8 p[n-{N \over 2}]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14677}%
\begin{displaymath}
    x[n] \approx {8 \over {{\pi^2}}} \left [
        {\cos ( \omega n )}
        + {{\cos ( 3 \omega n)} \over 9}
        + {{\cos ( 5 \omega n)} \over 25}
        + \cdots
    \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{figure14432}%
\begin{figure}\psfig{file=figs/fig10.07.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14947}%
$(0,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14949}%
$1/M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14951}%
$-2/(N-2M)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14678}%
\begin{displaymath}
    x[n] = {{N^2} \over {MN - 2{M^2}}} (p[n-M] - p[n+M])
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14953}%
$p[n]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14955}%
$p[n-(N-M)]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14957}%
$p[n+M]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14679}%
\begin{displaymath}
     x[n] = {{N^2} \over {2{\pi ^ 2} (MN - 2{M^2})}}  \left [
        \parbox[t][0.12in]{0in}{\mbox{}}
        {\cos ( \omega (n-M))} - {\cos ( \omega (n+M))}
        \right .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14680}%
\begin{displaymath}
        \left .
        + {{\cos ( 2 \omega (n-M)) - \cos ( 2 \omega (n+M))} \over 4}
        + \cdots
    \right ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14681}%
\begin{displaymath}
    \cos(a) - \cos(b) = 2  \sin({{b-a}\over 2}) \sin({{a+b}\over 2})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14682}%
\begin{displaymath}
    {\cos ( \omega (n-M))} - {\cos ( \omega (n+M))} =
    2 \sin (2 \pi M / N) \sin ( \omega n)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14683}%
\begin{displaymath}
    x[n] = a[1] \sin(\omega n) + a[2]  \sin(2 \omega n) + \cdots
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14684}%
\begin{displaymath}
    a[k] = {1 \over {{\pi ^ 2} (M/N - 2{{(M/N)}^2})}} 
    \cdot {{\sin (2 \pi k M / N) } \over {k^2}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14969}%
$M/N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath14685}%
\begin{displaymath}
    k < {{1} \over {4 M/N}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14975}%
$\sin(\theta) \approx \theta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14977}%
$a[k]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14989}%
$k \approx 1/(4\cdot 0.03) \approx 8.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14997}%
$2M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline14999}%
$N/4M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15001}%
$N/2M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15005}%
$N/2M \approx 1/17$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15007}%
$1/50$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15009}%
$16\cdot 44100 =
705600$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15011}%
$705600-20000=685600$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15013}%
$685600/440=1558$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure14466}%
\begin{figure}\psfig{file=figs/fig10.08.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{figure14476}%
\begin{figure}\psfig{file=figs/fig10.09.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15031}%
$12\omega$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure14483}%
\begin{figure}\psfig{file=figs/fig10.10.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure14488}%
\begin{figure}\psfig{file=figs/fig10.11.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15041}%
$1/b$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlfigureA{figure14496}%
\begin{figure}\psfig{file=figs/fig10.12.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure14503}%
\begin{figure}\psfig{file=figs/fig10.13.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure14510}%
\begin{figure}\psfig{file=figs/fig10.14.ps}

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline15045}%
$1 + 1/9 + 1/25 + 1/49 + 1/81 + \cdots$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}


\end{document}