2022-04-12 22:02:59 -03:00
|
|
|
<!DOCTYPE html>
|
2022-04-12 21:54:18 -03:00
|
|
|
|
|
|
|
|
<!--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>
|
2022-04-12 22:02:59 -03:00
|
|
|
|
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
|
|
|
|
|
2022-04-12 21:54:18 -03:00
|
|
|
<TITLE>Shifts and phase changes</TITLE>
|
|
|
|
|
<META NAME="description" CONTENT="Shifts and phase changes">
|
|
|
|
|
<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="next" HREF="node170.html">
|
|
|
|
|
<LINK REL="previous" HREF="node168.html">
|
|
|
|
|
<LINK REL="up" HREF="node167.html">
|
|
|
|
|
<LINK REL="next" HREF="node170.html">
|
|
|
|
|
</HEAD>
|
|
|
|
|
|
|
|
|
|
<BODY >
|
|
|
|
|
<!--Navigation Panel-->
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3104"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node170.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3098"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node167.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3092"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node168.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3100"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3102"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Next:</B> <A ID="tex2html3105"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node170.html">Fourier transform of a</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Up:</B> <A ID="tex2html3099"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node167.html">Properties of Fourier transforms</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Previous:</B> <A ID="tex2html3093"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node168.html">Fourier transform of DC</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html3101"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">Contents</A></B>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html3103"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<BR>
|
|
|
|
|
<BR>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
|
2022-04-12 23:32:40 -03:00
|
|
|
<H2><A ID="SECTION001322000000000000000"></A>
|
|
|
|
|
<A ID="sect9.shift"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
|
|
|
|
Shifts and phase changes
|
|
|
|
|
</H2>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Section <A HREF="node107.html#sect7.phase">7.2</A> showed how time-shifting a signal changes the
|
|
|
|
|
phases of its sinusoidal components, and Section <A HREF="node154.html#sect8.singlesideband">8.4.3</A>
|
|
|
|
|
showed how multiplying a signal by a complex sinusoid shifts its component
|
|
|
|
|
frequencies. These two effects have corresponding identities
|
|
|
|
|
involving the Fourier transform.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
First we consider a time shift. If <IMG
|
|
|
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img669.png"
|
|
|
|
|
ALT="$X[n]$">, as usual, is a complex-valued
|
|
|
|
|
signal that repeats every <IMG
|
|
|
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img3.png"
|
|
|
|
|
ALT="$N$"> samples, let <IMG
|
|
|
|
|
WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img717.png"
|
|
|
|
|
ALT="$Y[n]$"> be <IMG
|
|
|
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img669.png"
|
|
|
|
|
ALT="$X[n]$"> delayed <IMG
|
|
|
|
|
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img28.png"
|
|
|
|
|
ALT="$d$">
|
|
|
|
|
samples:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
Y[n] = X[n-d]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="112" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img671.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
Y[n] = X[n-d]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
which also repeats every <IMG
|
|
|
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img3.png"
|
|
|
|
|
ALT="$N$"> samples since <IMG
|
|
|
|
|
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img670.png"
|
|
|
|
|
ALT="$X$"> does. We can reduce the Fourier
|
|
|
|
|
transform of <IMG
|
|
|
|
|
WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img717.png"
|
|
|
|
|
ALT="$Y[n]$"> this way:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\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}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="400" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1101.png"
|
|
|
|
|
ALT="\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}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{V ^ {0}} X[-d] +
|
|
|
|
|
{V ^ {1}} X[-d+1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} X[-d+N-1]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="393" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1102.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{V ^ {0}} X[-d] +
|
|
|
|
|
{V ^ {1}} X[-d+1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} X[-d+N-1]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{V ^ {d}} X[0] +
|
|
|
|
|
{V ^ {d+1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {d+N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="333" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1103.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{V ^ {d}} X[0] +
|
|
|
|
|
{V ^ {d+1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {d+N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
= {V^d} \left (
|
|
|
|
|
{V ^ {0}} X[0] +
|
|
|
|
|
{V ^ {1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} X[N-1]
|
|
|
|
|
\right )
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="336" HEIGHT="30" BORDER="0"
|
|
|
|
|
SRC="img1104.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
= {V^d} \left (
|
|
|
|
|
{V ^ {0}} X[0] +
|
|
|
|
|
{V ^ {1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} X[N-1]
|
|
|
|
|
\right )
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
= {V^d} {\cal FT} \left \{ X[n] \right \} (k)
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="136" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1105.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
= {V^d} {\cal FT} \left \{ X[n] \right \} (k)
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
(The third line is just the second one with the terms summed in a
|
|
|
|
|
different order). We therefore get the Time Shift Formula for Fourier
|
2022-04-12 23:32:40 -03:00
|
|
|
Transforms: <A ID="12441"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\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}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="447" HEIGHT="35" BORDER="0"
|
|
|
|
|
SRC="img1106.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{\cal FT} \left \{ X[n-d] \right \} (k) =
|
|
|
|
|
\left ( {
|
|
|
|
|
\par...
|
|
|
|
|
...-dk\omega)
|
|
|
|
|
} \right )
|
|
|
|
|
{\cal FT} \left \{ X[n] \right \} (k)
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
The Fourier transform of <IMG
|
|
|
|
|
WIDTH="64" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img1107.png"
|
|
|
|
|
ALT="$X[n-d]$"> is a phase term times the Fourier transform
|
|
|
|
|
of <IMG
|
|
|
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img669.png"
|
|
|
|
|
ALT="$X[n]$">. The phase is changed by <IMG
|
|
|
|
|
WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img1108.png"
|
|
|
|
|
ALT="$-dk\omega$">, a
|
|
|
|
|
linear function of the frequency <IMG
|
|
|
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$k$">.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Now suppose instead that we change our starting signal <IMG
|
|
|
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img669.png"
|
|
|
|
|
ALT="$X[n]$"> by multiplying
|
|
|
|
|
it by a complex exponential <IMG
|
|
|
|
|
WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img653.png"
|
|
|
|
|
ALT="$Z^n$"> with angular frequency <IMG
|
|
|
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img7.png"
|
|
|
|
|
ALT="$\alpha $">:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
Y[n] = {Z^n} X[n]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="104" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1109.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
Y[n] = {Z^n} X[n]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
Z = \cos(\alpha) + i \sin(\alpha)
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="145" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1110.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
Z = \cos(\alpha) + i \sin(\alpha)
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
The Fourier transform is:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\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}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="400" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1101.png"
|
|
|
|
|
ALT="\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}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{V ^ {0}} X[0] +
|
|
|
|
|
{V ^ {1}} Z X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} {Z^{N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="353" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1111.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{V ^ {0}} X[0] +
|
|
|
|
|
{V ^ {1}} Z X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} {Z^{N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{{(VZ)} ^ {0}} X[0] +
|
|
|
|
|
{{(VZ)} ^ {1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{{(VZ)} ^ {N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="373" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1112.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
=
|
|
|
|
|
{{(VZ)} ^ {0}} X[0] +
|
|
|
|
|
{{(VZ)} ^ {1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{{(VZ)} ^ {N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
= {\cal FT} \left \{ X[n] \right \} (k - {{\alpha } \over {\omega}})
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="149" HEIGHT="35" BORDER="0"
|
|
|
|
|
SRC="img1113.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
= {\cal FT} \left \{ X[n] \right \} (k - {{\alpha } \over {\omega}})
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
We therefore get the Phase Shift Formula for Fourier Transforms:
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="12464"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\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}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="396" HEIGHT="39" BORDER="0"
|
|
|
|
|
SRC="img1114.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{\cal FT} \left \{ (\cos(\alpha) + i \sin(\alpha)) X[n] \ri...
|
|
|
|
|
...l FT} \left \{ X[n] \right \} (k - {{\alpha N} \over {2 \pi}})
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
<HR>
|
|
|
|
|
<!--Navigation Panel-->
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3104"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node170.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3098"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node167.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3092"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node168.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3100"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html3102"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Next:</B> <A ID="tex2html3105"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node170.html">Fourier transform of a</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Up:</B> <A ID="tex2html3099"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node167.html">Properties of Fourier transforms</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Previous:</B> <A ID="tex2html3093"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node168.html">Fourier transform of DC</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html3101"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">Contents</A></B>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html3103"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
<ADDRESS>
|
|
|
|
|
Miller Puckette
|
|
|
|
|
2006-12-30
|
|
|
|
|
</ADDRESS>
|
|
|
|
|
</BODY>
|
|
|
|
|
</HTML>
|
