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274 lines
7.7 KiB
274 lines
7.7 KiB
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<H1><A ID="SECTION001120000000000000000"></A>
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<A ID="sect7.phase"></A>
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<BR>
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Time shifts and phase changes
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</H1>
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<P>
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Starting from any (real or complex) signal <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$">, we can make other signals by
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time shifting the signal <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img670.png"
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ALT="$X$"> by a (positive or negative) integer <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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Y[n] = X[n-d]
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\end{displaymath}
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-->
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<IMG
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WIDTH="112" HEIGHT="28" BORDER="0"
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SRC="img671.png"
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ALT="\begin{displaymath}
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Y[n] = X[n-d]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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so that the <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$">th sample of <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img672.png"
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ALT="$Y$"> is the 0th sample of <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img670.png"
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ALT="$X$"> and so on. If the
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integer <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> is positive, then <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img672.png"
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ALT="$Y$"> is a delayed copy of <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img670.png"
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ALT="$X$">. If
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<IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> is negative, then <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img672.png"
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ALT="$Y$"> anticipates <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img670.png"
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ALT="$X$">; this can be done to a recorded
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sound but isn't practical as a real-time operation.
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<P>
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Time shifting is a linear operation (considered as a function of the input
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signal <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img670.png"
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ALT="$X$">); if you time shift a sum <IMG
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WIDTH="63" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img673.png"
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ALT="${X_1}+{X_2}$"> you get the same result as
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if you time shift them separately and add afterward.
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<P>
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Time shifting has the
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further property that, if you time shift a sinusoid of frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">, the
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result is another sinusoid of the same frequency; time shifting never
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introduces frequencies that weren't present in the signal before it was
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shifted. This property, called
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<A ID="7855"></A><I>time invariance</I>,
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makes it easy to analyze the effects of time shifts--and linear combinations
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of them--by considering separately what the operations do on individual
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sinusoids.
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<P>
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Furthermore, the effect of a time shift on a sinusoid is simple: it just
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changes the phase. If we use a complex sinusoid, the effect is even simpler.
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If for instance
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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X[n] = A {Z^n}
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\end{displaymath}
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-->
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<IMG
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WIDTH="86" HEIGHT="28" BORDER="0"
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SRC="img674.png"
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ALT="\begin{displaymath}
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X[n] = A {Z^n}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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then
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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Y[n] = X[n-d] = A {Z^{(n-d)}} = {Z^{-d}} A {Z^n} = {Z^{-d}} X[n]
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\end{displaymath}
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-->
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<IMG
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WIDTH="358" HEIGHT="28" BORDER="0"
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SRC="img675.png"
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ALT="\begin{displaymath}
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Y[n] = X[n-d] = A {Z^{(n-d)}} = {Z^{-d}} A {Z^n} = {Z^{-d}} X[n]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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so time shifting a complex sinusoid by <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> samples is the same thing as
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scaling it by <IMG
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WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img676.png"
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ALT="${Z^{-d}}$">--it's just an amplitude change by a particular
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complex number. Since <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img22.png"
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ALT="$\vert Z\vert=1$"> for a sinusoid, the amplitude change does not
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change the magnitude of the sinusoid, only its phase.
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<P>
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The phase change is equal to <IMG
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WIDTH="34" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img677.png"
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ALT="$- d \omega$">, where <!-- MATH
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$\omega = \angle(Z)$
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-->
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<IMG
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WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img678.png"
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ALT="$\omega = \angle(Z)$"> is
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the angular frequency of the sinusoid. This is exactly what we should expect
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since the sinusoid advances <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> radians per sample and it is offset
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(i.e., delayed) by <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> samples.
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<P>
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<B> Next:</B> <A ID="tex2html2158"
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HREF="node108.html">Delay networks</A>
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<B> Up:</B> <A ID="tex2html2152"
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HREF="node104.html">Time shifts and delays</A>
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<B> Previous:</B> <A ID="tex2html2146"
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HREF="node106.html">Complex sinusoids</A>
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<B> <A ID="tex2html2154"
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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