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335 lines
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HREF="node164.html">Fourier analysis of periodic</A>
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HREF="node165.html">Periodicity of the Fourier</A>
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<H2><A NAME="SECTION001312000000000000000"></A>
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<A NAME="sect9-IFT"></A>
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<BR>
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Fourier transform as additive synthesis
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</H2>
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<P>
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Now consider an arbitrary 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="$X[n]$"> 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="$N$">
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samples. (Previously we
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had assumed 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="$X[n]$"> could be obtained as a sum of sinusoids, and we haven't
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yet found out whether every periodic <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 obtained that way.) Let
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<IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1062.png"
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ALT="$Y[k]$"> denote the Fourier transform 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="$X$"> for <!-- MATH
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$k = 0, ..., N-1$
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-->
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<IMG
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WIDTH="110" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1063.png"
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ALT="$k = 0, ..., N-1$">:
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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[k] = {\cal FT}\left \{ X[n] \right \} (k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="151" HEIGHT="28" BORDER="0"
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SRC="img1064.png"
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ALT="\begin{displaymath}
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Y[k] = {\cal FT}\left \{ X[n] \right \} (k)
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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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<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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= {{\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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\end{displaymath}
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-->
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<IMG
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WIDTH="393" HEIGHT="30" BORDER="0"
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SRC="img1065.png"
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ALT="\begin{displaymath}
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= {{\left [ {U^{-k}} \right ]} ^ {0}} X[0] +
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{{\left [ {U^...
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...X[1] +
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\cdots +
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{{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
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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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<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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= {{\left [ {U^{0}} \right ]} ^ {k}} X[0] +
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{{\left [ {U^{-1}} \right ]} ^ {k}} X[1] +
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\cdots +
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{{\left [ {U^{-(N-1)}} \right ]} ^ {k}} X[N-1]
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\end{displaymath}
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-->
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<IMG
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WIDTH="395" HEIGHT="39" BORDER="0"
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SRC="img1066.png"
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ALT="\begin{displaymath}
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= {{\left [ {U^{0}} \right ]} ^ {k}} X[0] +
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{{\left [ {U^{...
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...1] +
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\cdots +
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{{\left [ {U^{-(N-1)}} \right ]} ^ {k}} X[N-1]
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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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In the second version we rearranged the exponents to show that <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1062.png"
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ALT="$Y[k]$"> is a sum
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of complex sinusoids, with complex amplitudes <IMG
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WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1067.png"
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ALT="$X[m]$"> and frequencies <IMG
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WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1068.png"
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ALT="$-m\omega$">
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for <!-- MATH
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$m = 0, \ldots, N-1$
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-->
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<IMG
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WIDTH="123" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1069.png"
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ALT="$m = 0, \ldots, N-1$">. In other words, <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1062.png"
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ALT="$Y[k]$"> can be considered as a
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Fourier series in its own right, whose <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$">th component has strength <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1070.png"
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ALT="$X[-m]$">.
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(The expression <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1070.png"
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ALT="$X[-m]$"> makes sense because <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="$X$"> is a periodic signal).
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We can also express the amplitude of the partials of <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1062.png"
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ALT="$Y[k]$"> in terms of its own
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Fourier transform. Equating the two gives:
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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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{1 \over N} {\cal FT} \left \{ Y[k] \right \} (m) = X[-m]
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\end{displaymath}
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-->
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<IMG
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WIDTH="190" HEIGHT="38" BORDER="0"
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SRC="img1071.png"
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ALT="\begin{displaymath}
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{1 \over N} {\cal FT} \left \{ Y[k] \right \} (m) = X[-m]
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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 means in turn that <IMG
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WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1070.png"
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ALT="$X[-m]$"> can be obtained by summing sinusoids with
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amplitudes <IMG
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WIDTH="55" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1072.png"
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ALT="$Y[k]/N$">. Setting <IMG
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WIDTH="60" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1073.png"
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ALT="$n = -m$"> gives:
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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] = {1 \over N} {\cal FT} \left \{ Y[k] \right \} (-n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="183" HEIGHT="38" BORDER="0"
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SRC="img1074.png"
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ALT="\begin{displaymath}
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X[n] = {1 \over N} {\cal FT} \left \{ Y[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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<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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= {{\left [ {U^{0}} \right ]} ^ {n}} Y[0] +
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{{\left [ {U^{1}} \right ]} ^ {n}} Y[1] +
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\cdots +
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{{\left [ {U^{N-1}} \right ]} ^ {n}} Y[N-1]
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\end{displaymath}
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-->
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<IMG
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WIDTH="360" HEIGHT="30" BORDER="0"
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SRC="img1075.png"
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ALT="\begin{displaymath}
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= {{\left [ {U^{0}} \right ]} ^ {n}} Y[0] +
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{{\left [ {U^{...
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...Y[1] +
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\cdots +
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{{\left [ {U^{N-1}} \right ]} ^ {n}} Y[N-1]
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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 shows that any periodic <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 indeed be obtained as a sum of
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sinusoids. Further, the formula explicitly shows how to reconstruct <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="$X[n]$">
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from its Fourier transform <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1062.png"
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ALT="$Y[k]$">, if we know its value for the integers <!-- MATH
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$k=0,
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\ldots, N-1$
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-->
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<IMG
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WIDTH="118" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1076.png"
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ALT="$k=0,
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\ldots, N-1$">.
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<P>
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<HR>
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<BR>
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<B> Next:</B> <A NAME="tex2html3060"
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HREF="node167.html">Properties of Fourier transforms</A>
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<B> Up:</B> <A NAME="tex2html3054"
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HREF="node164.html">Fourier analysis of periodic</A>
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<B> Previous:</B> <A NAME="tex2html3050"
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HREF="node165.html">Periodicity of the Fourier</A>
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<B> <A NAME="tex2html3056"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html3058"
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<ADDRESS>
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Miller Puckette
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2006-12-30
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</ADDRESS>
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</BODY>
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</HTML>
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