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<H1><A ID="SECTION001310000000000000000">
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Fourier analysis of periodic signals</A>
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</H1>
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<P>
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Suppose <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="$X[n]$"> is a complex-valued signal that repeats every <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples. (We
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are continuing to use complex-valued signals rather than real-valued ones
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to simplify the mathematics.) Because of the period <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">, the
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values of <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="$X[n]$"> for <!-- MATH
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$n=0,\ldots,N-1$
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-->
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<IMG
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WIDTH="119" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1042.png"
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ALT="$n=0,\ldots,N-1$"> determine <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="$X[n]$"> for all integer values
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of <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">.
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<P>
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Suppose further that <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="$X[n]$"> can be written as a sum of complex sinusoids of
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frequency <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img179.png"
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ALT="$0$">, <IMG
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WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img184.png"
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ALT="$2\pi/N$">, <IMG
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WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1043.png"
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ALT="$4\pi/N$">, <IMG
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WIDTH="22" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img239.png"
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ALT="$\ldots$">, <IMG
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WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1044.png"
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ALT="$2(N-1)\pi/N$">. These are the
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partials, starting with the zeroth, for a signal of period <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">. We stop at
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the <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">th term because the next one would have frequency <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $">, equivalent
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to frequency <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img179.png"
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ALT="$0$">, which is already on the list.
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<P>
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Given the values 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="$X$">, we wish to find the complex amplitudes of the
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partials. Suppose we want the <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">th partial, where <IMG
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WIDTH="77" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1045.png"
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ALT="$0 \leq k < N$">. The
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frequency of this partial is <IMG
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WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1046.png"
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ALT="$2\pi k / N$">. We can find its complex amplitude
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by modulating <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="$X$"> downward <IMG
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WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1046.png"
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ALT="$2\pi k / N$"> radians per sample in frequency, so
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that the <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">th partial is modulated to frequency zero. Then we pass the signal
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through a low-pass filter with such a low cutoff frequency that nothing but the
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zero-frequency partial remains. We can do this in effect by averaging over a
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huge number of samples; but since the signal repeats every <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples, this
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huge average is the same as the average of the first <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples. In short, to
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measure a sinusoidal component of a periodic signal, modulate it down to DC and
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then average over one period.
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<P>
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Let <IMG
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WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1047.png"
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ALT="$\omega=2\pi/N$"> be the fundamental frequency for the period <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">, and
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let <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1048.png"
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ALT="$U$"> be the unit-magnitude complex number with argument <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 $">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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U = \cos(\omega) + i \sin(\omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="146" HEIGHT="28" BORDER="0"
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SRC="img1049.png"
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ALT="\begin{displaymath}
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U = \cos(\omega) + i \sin(\omega)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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The <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">th partial of the signal <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$"> is of the form:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{P_k}[n] = {A_k}{{\left [ {U^k} \right ]} ^ {n}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="119" HEIGHT="30" BORDER="0"
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SRC="img1050.png"
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ALT="\begin{displaymath}
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{P_k}[n] = {A_k}{{\left [ {U^k} \right ]} ^ {n}}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1051.png"
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ALT="${A_k}$"> is the complex amplitude of the partial, and the frequency
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of the partial is:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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\angle({U^k}) = k \angle(U) = k\omega
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\end{displaymath}
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-->
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<IMG
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WIDTH="146" HEIGHT="28" BORDER="0"
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SRC="img1052.png"
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ALT="\begin{displaymath}
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\angle({U^k}) = k \angle(U) = k\omega
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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We're assuming for the moment that the signal <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$"> can actually be written
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as a sum of the <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$"> partials, or in other words:
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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] =
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{A_0}{{\left [ {U^0} \right ]} ^ {n}}
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+ {A_1}{{\left [ {U^1} \right ]} ^ {n}}
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+ \cdots
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+ {A_{N-1}}{{\left [ {U^{N-1}} \right ]} ^ {n}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="355" HEIGHT="30" BORDER="0"
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SRC="img1053.png"
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ALT="\begin{displaymath}
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X[n] =
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{A_0}{{\left [ {U^0} \right ]} ^ {n}}
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+ {A_1}{{\l...
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...n}}
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+ \cdots
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+ {A_{N-1}}{{\left [ {U^{N-1}} \right ]} ^ {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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By the heterodyne-filtering argument above, we expect to be able to measure
|
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each <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1054.png"
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ALT="$A_k$"> by multiplying by the sinusoid of frequency <IMG
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WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1055.png"
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ALT="$-k\omega$"> and
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averaging over a period:
|
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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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|
{A_k} = {1\over N} \left (
|
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{{\left [ {U^{-k}} \right ]} ^ {0}} X[0] +
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{{\left [ {U^{-k}} \right ]} ^ {1}} X[1] +
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\cdots +
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{{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
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\right )
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\end{displaymath}
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|
-->
|
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|
<IMG
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|
WIDTH="457" HEIGHT="38" BORDER="0"
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|
SRC="img1056.png"
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|
ALT="\begin{displaymath}
|
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|
{A_k} = {1\over N} \left (
|
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|
{{\left [ {U^{-k}} \right ]} ^ ...
|
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|
...dots +
|
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|
{{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
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|
\right )
|
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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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|
This is such a useful formula that it gets its own notation. The
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="12325"></A><I>Fourier transform</I>
|
2022-04-12 21:54:18 -03:00
|
|
|
of a signal <IMG
|
|
|
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|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img669.png"
|
|
|
|
|
ALT="$X[n]$">, over <IMG
|
|
|
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img3.png"
|
|
|
|
|
ALT="$N$"> samples, is defined as:
|
|
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|
<BR><P></P>
|
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|
|
<DIV ALIGN="CENTER">
|
|
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|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{\cal FT}\left \{ X[n] \right \} (k) =
|
|
|
|
|
{V ^ {0}} X[0] +
|
|
|
|
|
{V ^ {1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="406" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1057.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{\cal FT}\left \{ X[n] \right \} (k) =
|
|
|
|
|
{V ^ {0}} X[0] +
|
|
|
|
|
{V ^ {1}} X[1] +
|
|
|
|
|
\cdots +
|
|
|
|
|
{V ^ {N-1}} X[N-1]
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
where <IMG
|
|
|
|
|
WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img1058.png"
|
|
|
|
|
ALT="$V = {U^{-k}}$">. The Fourier transform is a function of the variable <IMG
|
|
|
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$k$">,
|
|
|
|
|
equal to <IMG
|
|
|
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img3.png"
|
|
|
|
|
ALT="$N$"> times the amplitude of the input's <IMG
|
|
|
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$k$">th partial. So far <IMG
|
|
|
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$k$">
|
|
|
|
|
has taken integer values but the formula makes sense for any value of <IMG
|
|
|
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img58.png"
|
|
|
|
|
ALT="$k$"> if we
|
|
|
|
|
define <IMG
|
|
|
|
|
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img1059.png"
|
|
|
|
|
ALT="$V$"> more generally as:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
V = \cos(-k\omega) + i\sin(-k\omega)
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="188" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img1060.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
V = \cos(-k\omega) + i\sin(-k\omega)
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
where, as before, <IMG
|
|
|
|
|
WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img1047.png"
|
|
|
|
|
ALT="$\omega=2\pi/N$"> is the (angular) fundamental
|
|
|
|
|
frequency associated with the period <IMG
|
|
|
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img3.png"
|
|
|
|
|
ALT="$N$">.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
<BR><HR>
|
|
|
|
|
<!--Table of Child-Links-->
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2022-04-12 23:32:40 -03:00
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
|
2022-04-12 23:32:40 -03:00
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<LI><A ID="tex2html3033"
|
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HREF="node165.html">Periodicity of the Fourier transform</A>
|
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|
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|
<LI><A ID="tex2html3034"
|
2022-04-12 21:54:18 -03:00
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|
HREF="node166.html">Fourier transform as additive synthesis</A>
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HREF="node165.html">Periodicity of the Fourier</A>
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Miller Puckette
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2006-12-30
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