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<H2><A ID="SECTION001243000000000000000"></A>
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<A ID="sect8.singlesideband"></A>
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<BR>
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Single Sideband Modulation
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</H2>
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<P>
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As we saw in Chapter 5, multiplying two real sinusoids together results
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in a signal with two new components at the sum and difference of the
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original frequencies. If we carry out the same operation with complex
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sinusoids, we get only one new resultant frequency; this is one result of
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the greater mathematical simplicity of complex sinusoids as compared to
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real ones. If we multiply a complex sinusoid <!-- MATH
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$1, Z, {Z^2}, \ldots$
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-->
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<IMG
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WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img664.png"
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ALT="$1, Z, {Z^2}, \ldots$">
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with another one, <!-- MATH
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$1, W, {W^2}, \ldots$
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-->
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<IMG
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WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img998.png"
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ALT="$1, W, {W^2}, \ldots$"> the result is
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<!-- MATH
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$1, WZ, {{(WZ)}^2}, \ldots$
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-->
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<IMG
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WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img999.png"
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ALT="$1, WZ, {{(WZ)}^2}, \ldots$">, which is another complex sinusoid whose
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frequency, <IMG
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WIDTH="55" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1000.png"
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ALT="$\angle(ZW)$">, is the sum of the two original frequencies.
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<P>
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In general, since complex sinusoids have simpler properties than real ones, it
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is often useful to be able to convert from real sinusoids to complex ones. In
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other words, from the real sinusoid:
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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 \cdot \cos (\omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="120" HEIGHT="28" BORDER="0"
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SRC="img1001.png"
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ALT="\begin{displaymath}
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x[n] = a \cdot \cos (\omega 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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(with a spectral peak of amplitude <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img409.png"
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ALT="$a/2$"> and 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 $">) we would like
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a way of computing the complex sinusoid:
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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 \left ( \cos (\omega n) + i \sin (\omega n) \right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="210" HEIGHT="28" BORDER="0"
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SRC="img1002.png"
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ALT="\begin{displaymath}
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X[n] = a \left ( \cos (\omega n) + i \sin (\omega n) \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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so that
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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] = \mathrm{re} (X[n])
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\end{displaymath}
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-->
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<IMG
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WIDTH="105" HEIGHT="28" BORDER="0"
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SRC="img1003.png"
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ALT="\begin{displaymath}
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x[n] = \mathrm{re} (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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We would like a linear process for doing this, so that superpositions of
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sinusoids get treated as if their components were dealt with separately.
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<P>
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Of course we could equally well have chosen the complex sinusoid with
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frequency <IMG
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WIDTH="26" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1004.png"
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ALT="$-\omega$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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X'[n] = a \left ( \cos (\omega n) - i \sin (\omega n) \right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="214" HEIGHT="28" BORDER="0"
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SRC="img1005.png"
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ALT="\begin{displaymath}
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X'[n] = a \left ( \cos (\omega n) - i \sin (\omega n) \right )
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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and in fact <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> is just half the sum of the two. In essence we need a filter
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that will pass through positive frequencies (actually frequencies between
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0 and <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">, corresponding to values of <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$"> on the top half of the complex
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unit circle) from negative values
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(from <IMG
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WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img576.png"
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ALT="$-\pi$"> to 0, or equivalently, from <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $"> to <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 $">--the bottom
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half of the unit circle).
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<P>
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One can design such a filter by designing a low-pass filter with cutoff
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frequency <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$\pi /2$">, and then performing a rotation by <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$\pi /2$"> radians using the
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technique of Section <A HREF="node143.html#sect8.twopolebandpass">8.3.4</A>. However, it turns out to be
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easier to do it using two specially designed networks of all-pass filters
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with real coefficients.
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<P>
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Calling the transfer functions of the two filters <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1006.png"
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ALT="$H_1$"> and <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1007.png"
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ALT="$H_2$">, we design
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the filters so that
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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({H_1}(Z)) - \angle({H_2}(Z)) \approx
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\left \{
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\begin{array}{ll}
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\pi/2 & {0 < \angle(Z) < \pi} \\
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-\pi/2 & {-\pi < \angle(Z) < 0}
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\end{array}
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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="353" HEIGHT="45" BORDER="0"
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SRC="img1008.png"
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ALT="\begin{displaymath}
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\angle({H_1}(Z)) - \angle({H_2}(Z)) \approx
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\left \{
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\be...
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...pi} \\
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-\pi/2 & {-\pi < \angle(Z) < 0}
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\end{array} \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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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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{H_1}(Z) \approx i {H_2}(Z) , \; 0 < \angle(Z) < \pi
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\end{displaymath}
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-->
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<IMG
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WIDTH="221" HEIGHT="28" BORDER="0"
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SRC="img1009.png"
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ALT="\begin{displaymath}
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{H_1}(Z) \approx i {H_2}(Z) , \; 0 < \angle(Z) < \pi
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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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{H_1}(Z) \approx -i {H_2}(Z) , \; -\pi < \angle(Z) < 0
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\end{displaymath}
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-->
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<IMG
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WIDTH="246" HEIGHT="28" BORDER="0"
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SRC="img1010.png"
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ALT="\begin{displaymath}
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{H_1}(Z) \approx -i {H_2}(Z) , \; -\pi < \angle(Z) < 0
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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 for any incoming real-valued signal <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> we simply form a complex number
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<IMG
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WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1011.png"
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ALT="$a[n] + i b[n]$"> where <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img503.png"
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ALT="$a[n]$"> is the output of the first filter and <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1012.png"
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ALT="$b[n]$"> is
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the output of the second. Any complex sinusoidal component of <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$">
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(call it <IMG
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WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img653.png"
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ALT="$Z^n$">) will be transformed to
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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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{H_1}(Z) + i {H_2}(Z) \approx
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\left \{
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\begin{array}{ll}
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2 {H_1}(Z) & {0 < \angle(Z) < \pi} \\
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0 & \mbox{otherwise}
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\end{array}
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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="318" HEIGHT="45" BORDER="0"
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SRC="img1013.png"
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ALT="\begin{displaymath}
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|
{H_1}(Z) + i {H_2}(Z) \approx
|
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|
\left \{
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|
\begin{array}{ll}
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...
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...ngle(Z) < \pi} \\
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0 & \mbox{otherwise}
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\end{array} \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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<P>
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Having started with a real-valued signal, whose energy is split equally into
|
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positive and negative frequencies, we end up with a complex-valued one with
|
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only positive frequencies.
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<P>
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