463 lines
12 KiB
HTML
463 lines
12 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
<TITLE>Complex sinusoids</TITLE>
|
|
<META NAME="description" CONTENT="Complex sinusoids">
|
|
<META NAME="keywords" CONTENT="book">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
|
|
|
<LINK REL="previous" HREF="node105.html">
|
|
<LINK REL="up" HREF="node105.html">
|
|
<LINK REL="next" HREF="node107.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html2143"
|
|
HREF="node107.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html2137"
|
|
HREF="node105.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html2133"
|
|
HREF="node105.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html2139"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
|
<A NAME="tex2html2141"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html2144"
|
|
HREF="node107.html">Time shifts and phase</A>
|
|
<B> Up:</B> <A NAME="tex2html2138"
|
|
HREF="node105.html">Complex numbers</A>
|
|
<B> Previous:</B> <A NAME="tex2html2134"
|
|
HREF="node105.html">Complex numbers</A>
|
|
<B> <A NAME="tex2html2140"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html2142"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION001111000000000000000">
|
|
Complex sinusoids</A>
|
|
</H2>
|
|
|
|
<P>
|
|
Recall the formula for a (real-valued) sinusoid from Page
|
|
<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[n] = a \cos (\omega n + \phi )
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="140" HEIGHT="28" BORDER="0"
|
|
SRC="img76.png"
|
|
ALT="\begin{displaymath}
|
|
x[n] = a \cos (\omega n + \phi )
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
This is a sequence of cosines of angles (called phases) which increase
|
|
arithmetically
|
|
with the sample number <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$n$">. The cosines are all adjusted by the factor <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img4.png"
|
|
ALT="$a$">.
|
|
We can now rewrite this as the real part of a much simpler and easier to
|
|
manipulate sequence of complex numbers, by using the properties of their
|
|
arguments and magnitudes.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig07.02"></A><A NAME="7829"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.2:</STRONG>
|
|
The powers of a complex number <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img20.png"
|
|
ALT="$Z$"> with <IMG
|
|
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img22.png"
|
|
ALT="$\vert Z\vert=1$">, and the same
|
|
sequence multiplied by a constant <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img23.png"
|
|
ALT="$A$">.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="205" HEIGHT="204" BORDER="0"
|
|
SRC="img651.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig07.02.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
Suppose that a complex number <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img20.png"
|
|
ALT="$Z$"> happens to have magnitude one and
|
|
argument <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $">, so that
|
|
it can be written as:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
Z = \cos(\omega) + i \sin(\omega)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="145" HEIGHT="28" BORDER="0"
|
|
SRC="img652.png"
|
|
ALT="\begin{displaymath}
|
|
Z = \cos(\omega) + i \sin(\omega)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Then for any integer <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$n$">, the number <IMG
|
|
WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img653.png"
|
|
ALT="$Z^n$"> must have magnitude one as well
|
|
(because magnitudes multiply) and argument <IMG
|
|
WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img654.png"
|
|
ALT="$n\omega$"> (because arguments add).
|
|
So,
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{Z^n} = \cos(n\omega) + i \sin(n \omega)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="173" HEIGHT="28" BORDER="0"
|
|
SRC="img655.png"
|
|
ALT="\begin{displaymath}
|
|
{Z^n} = \cos(n\omega) + i \sin(n \omega)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
This is also true for negative values of <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$n$">, so for example,
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="199" HEIGHT="38" BORDER="0"
|
|
SRC="img656.png"
|
|
ALT="\begin{displaymath}
|
|
{1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Figure <A HREF="#fig07.02">7.2</A> shows graphically how the powers of <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img20.png"
|
|
ALT="$Z$"> wrap around
|
|
the unit circle, which is the set of all complex numbers of magnitude one.
|
|
They form a geometric sequence:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\ldots, {Z^0}, {Z^1}, {Z^2}, \ldots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="123" HEIGHT="27" BORDER="0"
|
|
SRC="img657.png"
|
|
ALT="\begin{displaymath}
|
|
\ldots, {Z^0}, {Z^1}, {Z^2}, \ldots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and taking the real part of each term we get a real sinusoid with
|
|
initial phase zero and amplitude one:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\ldots, \cos(0), \cos(\omega), \cos(2 \omega), \ldots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="203" HEIGHT="28" BORDER="0"
|
|
SRC="img658.png"
|
|
ALT="\begin{displaymath}
|
|
\ldots, \cos(0), \cos(\omega), \cos(2 \omega), \ldots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Furthermore, suppose we multiply the elements of the sequence by some (complex)
|
|
constant <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img23.png"
|
|
ALT="$A$"> with magnitude <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img4.png"
|
|
ALT="$a$"> and argument <IMG
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img77.png"
|
|
ALT="$\phi$">. This gives
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\ldots, A, AZ, A{Z^2}, \ldots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="131" HEIGHT="27" BORDER="0"
|
|
SRC="img659.png"
|
|
ALT="\begin{displaymath}
|
|
\ldots, A, AZ, A{Z^2}, \ldots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
The magnitudes are all <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img4.png"
|
|
ALT="$a$"> and the argument of the <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$n$">th term is
|
|
<!-- MATH
|
|
$n \omega + \phi$
|
|
-->
|
|
<IMG
|
|
WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img660.png"
|
|
ALT="$n \omega + \phi$">, so the sequence is equal to
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{AZ^n} = a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)]
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="272" HEIGHT="28" BORDER="0"
|
|
SRC="img661.png"
|
|
ALT="\begin{displaymath}
|
|
{AZ^n} = a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)]
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and the real part is just the real-valued sinusoid:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\mathrm{re}(A{Z^n}) = a \cdot \cos(n \omega + \phi)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="181" HEIGHT="28" BORDER="0"
|
|
SRC="img662.png"
|
|
ALT="\begin{displaymath}
|
|
\mathrm{re}(A{Z^n}) = a \cdot \cos(n \omega + \phi)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
The complex number <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img23.png"
|
|
ALT="$A$"> encodes both the real amplitude <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img4.png"
|
|
ALT="$a$">
|
|
and the initial phase <IMG
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img77.png"
|
|
ALT="$\phi$">; the unit-magnitude complex
|
|
number <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img20.png"
|
|
ALT="$Z$"> controls the frequency which is just its argument <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $">.
|
|
|
|
<P>
|
|
Figure <A HREF="#fig07.02">7.2</A> also shows the sequence <!-- MATH
|
|
$A, AZ, A{Z^2}, \ldots$
|
|
-->
|
|
<IMG
|
|
WIDTH="109" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img663.png"
|
|
ALT="$A, AZ, A{Z^2}, \ldots$">;
|
|
in effect this is the same sequence as <!-- MATH
|
|
$1, Z, {Z^2}, \ldots$
|
|
-->
|
|
<IMG
|
|
WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img664.png"
|
|
ALT="$1, Z, {Z^2}, \ldots$">, but amplified and
|
|
rotated according to the amplitude and initial phase. In a complex
|
|
sinusoid of this form, <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img23.png"
|
|
ALT="$A$"> is called the
|
|
<A NAME="7846"></A>
|
|
<I>complex amplitude</I>.
|
|
|
|
<P>
|
|
Using complex numbers to represent the amplitudes and phases of sinusoids can
|
|
clarify manipulations that otherwise might seem unmotivated. For instance,
|
|
suppose we want to know the amplitude and phase of the sum of two sinusoids
|
|
with the same frequency. In the language of this chapter, we let the two
|
|
sinusoids be written as:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
X[n] = A {Z^n} , \ Y[n] = B {Z^n}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="185" HEIGHT="28" BORDER="0"
|
|
SRC="img665.png"
|
|
ALT="\begin{displaymath}
|
|
X[n] = A {Z^n} , \ Y[n] = B {Z^n}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img23.png"
|
|
ALT="$A$"> and <IMG
|
|
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img311.png"
|
|
ALT="$B$"> encode the phases and amplitudes of the two signals.
|
|
The sum is then equal to:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
X[n] + Y[n] = (A+B) {Z^n}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="181" HEIGHT="28" BORDER="0"
|
|
SRC="img666.png"
|
|
ALT="\begin{displaymath}
|
|
X[n] + Y[n] = (A+B) {Z^n}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
which is a sinusoid whose amplitude equals <IMG
|
|
WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img667.png"
|
|
ALT="$\vert A+B\vert$"> and whose phase equals
|
|
<IMG
|
|
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img668.png"
|
|
ALT="$\angle(A+B)$">. This is clearly a much easier way to manipulate amplitudes
|
|
and phases than using properties of sines and cosines. Eventually, of course,
|
|
we will take the real part of the result; this can usually be left to the
|
|
end of whatever we're doing.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html2143"
|
|
HREF="node107.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html2137"
|
|
HREF="node105.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html2133"
|
|
HREF="node105.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html2139"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
|
<A NAME="tex2html2141"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html2144"
|
|
HREF="node107.html">Time shifts and phase</A>
|
|
<B> Up:</B> <A NAME="tex2html2138"
|
|
HREF="node105.html">Complex numbers</A>
|
|
<B> Previous:</B> <A NAME="tex2html2134"
|
|
HREF="node105.html">Complex numbers</A>
|
|
<B> <A NAME="tex2html2140"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html2142"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|