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<H1><A ID="SECTION001120000000000000000"></A>
<A ID="sect7.phase"></A>
<BR>
Time shifts and phase changes
</H1>
<P>
Starting from any (real or complex) signal <IMG
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img669.png"
ALT="$X[n]$">, we can make other signals by
time shifting the signal <IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img670.png"
ALT="$X$"> by a (positive or negative) integer <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$">:
<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>
so that the <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$">th sample of <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img672.png"
ALT="$Y$"> is the 0th sample of <IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img670.png"
ALT="$X$"> and so on. If the
integer <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$"> is positive, then <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img672.png"
ALT="$Y$"> is a delayed copy of <IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img670.png"
ALT="$X$">. If
<IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$"> is negative, then <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img672.png"
ALT="$Y$"> anticipates <IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img670.png"
ALT="$X$">; this can be done to a recorded
sound but isn't practical as a real-time operation.
<P>
Time shifting is a linear operation (considered as a function of the input
signal <IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img670.png"
ALT="$X$">); if you time shift a sum <IMG
WIDTH="63" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img673.png"
ALT="${X_1}+{X_2}$"> you get the same result as
if you time shift them separately and add afterward.
<P>
Time shifting has the
further property that, if you time shift a sinusoid of frequency <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">, the
result is another sinusoid of the same frequency; time shifting never
introduces frequencies that weren't present in the signal before it was
shifted. This property, called
<A ID="7855"></A><I>time invariance</I>,
makes it easy to analyze the effects of time shifts--and linear combinations
of them--by considering separately what the operations do on individual
sinusoids.
<P>
Furthermore, the effect of a time shift on a sinusoid is simple: it just
changes the phase. If we use a complex sinusoid, the effect is even simpler.
If for instance
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
X[n] = A {Z^n}
\end{displaymath}
-->
<IMG
WIDTH="86" HEIGHT="28" BORDER="0"
SRC="img674.png"
ALT="\begin{displaymath}
X[n] = A {Z^n}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
Y[n] = X[n-d] = A {Z^{(n-d)}} = {Z^{-d}} A {Z^n} = {Z^{-d}} X[n]
\end{displaymath}
-->
<IMG
WIDTH="358" HEIGHT="28" BORDER="0"
SRC="img675.png"
ALT="\begin{displaymath}
Y[n] = X[n-d] = A {Z^{(n-d)}} = {Z^{-d}} A {Z^n} = {Z^{-d}} X[n]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
so time shifting a complex sinusoid by <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$"> samples is the same thing as
scaling it by <IMG
WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img676.png"
ALT="${Z^{-d}}$">--it's just an amplitude change by a particular
complex number. Since <IMG
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img22.png"
ALT="$\vert Z\vert=1$"> for a sinusoid, the amplitude change does not
change the magnitude of the sinusoid, only its phase.
<P>
The phase change is equal to <IMG
WIDTH="34" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img677.png"
ALT="$- d \omega$">, where <!-- MATH
$\omega = \angle(Z)$
-->
<IMG
WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img678.png"
ALT="$\omega = \angle(Z)$"> is
the angular frequency of the sinusoid. This is exactly what we should expect
since the sinusoid advances <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $"> radians per sample and it is offset
(i.e., delayed) by <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$"> samples.
<P>
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<B> Next:</B> <A ID="tex2html2158"
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<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
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