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<H2><A NAME="SECTION001111000000000000000">
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Complex sinusoids</A>
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</H2>
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<P>
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Recall the formula for a (real-valued) sinusoid from Page
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<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>:
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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 \cos (\omega n + \phi )
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\end{displaymath}
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-->
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<IMG
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WIDTH="140" HEIGHT="28" BORDER="0"
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SRC="img76.png"
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ALT="\begin{displaymath}
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x[n] = a \cos (\omega n + \phi )
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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 a sequence of cosines of angles (called phases) which increase
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arithmetically
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with the sample number <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="$n$">. The cosines are all adjusted by the factor <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$">.
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We can now rewrite this as the real part of a much simpler and easier to
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manipulate sequence of complex numbers, by using the properties of their
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arguments and magnitudes.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.02"></A><A NAME="7829"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.2:</STRONG>
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The powers of a complex number <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="$Z$"> with <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="$\vert Z\vert=1$">, and the same
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sequence multiplied by a constant <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.png"
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ALT="$A$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="205" HEIGHT="204" BORDER="0"
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SRC="img651.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.02.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Suppose that a complex number <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="$Z$"> happens to have magnitude one and
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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 $">, so that
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it can be written as:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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Z = \cos(\omega) + i \sin(\omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="145" HEIGHT="28" BORDER="0"
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SRC="img652.png"
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ALT="\begin{displaymath}
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Z = \cos(\omega) + i \sin(\omega)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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Then for any integer <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">, the number <IMG
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WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img653.png"
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ALT="$Z^n$"> must have magnitude one as well
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(because magnitudes multiply) and argument <IMG
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WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img654.png"
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ALT="$n\omega$"> (because arguments add).
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So,
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{Z^n} = \cos(n\omega) + i \sin(n \omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="173" HEIGHT="28" BORDER="0"
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SRC="img655.png"
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ALT="\begin{displaymath}
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{Z^n} = \cos(n\omega) + i \sin(n \omega)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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This is also true for negative values of <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">, so for example,
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="199" HEIGHT="38" BORDER="0"
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SRC="img656.png"
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ALT="\begin{displaymath}
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{1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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Figure <A HREF="#fig07.02">7.2</A> shows graphically how the powers of <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img20.png"
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ALT="$Z$"> wrap around
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the unit circle, which is the set of all complex numbers of magnitude one.
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They form a geometric sequence:
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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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\ldots, {Z^0}, {Z^1}, {Z^2}, \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="123" HEIGHT="27" BORDER="0"
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SRC="img657.png"
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ALT="\begin{displaymath}
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\ldots, {Z^0}, {Z^1}, {Z^2}, \ldots
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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 taking the real part of each term we get a real sinusoid with
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initial phase zero and amplitude one:
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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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\ldots, \cos(0), \cos(\omega), \cos(2 \omega), \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="203" HEIGHT="28" BORDER="0"
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SRC="img658.png"
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ALT="\begin{displaymath}
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\ldots, \cos(0), \cos(\omega), \cos(2 \omega), \ldots
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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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Furthermore, suppose we multiply the elements of the sequence by some (complex)
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constant <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.png"
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ALT="$A$"> with magnitude <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> and argument <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">. This gives
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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\ldots, A, AZ, A{Z^2}, \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="131" HEIGHT="27" BORDER="0"
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SRC="img659.png"
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ALT="\begin{displaymath}
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\ldots, A, AZ, A{Z^2}, \ldots
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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The magnitudes are all <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> and the argument of the <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">th term is
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<!-- MATH
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$n \omega + \phi$
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-->
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<IMG
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WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img660.png"
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ALT="$n \omega + \phi$">, so the sequence is equal 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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{AZ^n} = a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)]
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\end{displaymath}
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-->
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<IMG
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WIDTH="272" HEIGHT="28" BORDER="0"
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SRC="img661.png"
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ALT="\begin{displaymath}
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{AZ^n} = a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)]
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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 the real part is just the real-valued 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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\mathrm{re}(A{Z^n}) = a \cdot \cos(n \omega + \phi)
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\end{displaymath}
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-->
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<IMG
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WIDTH="181" HEIGHT="28" BORDER="0"
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SRC="img662.png"
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ALT="\begin{displaymath}
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\mathrm{re}(A{Z^n}) = a \cdot \cos(n \omega + \phi)
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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 complex number <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.png"
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ALT="$A$"> encodes both the real amplitude <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$">
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and the initial phase <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">; the unit-magnitude complex
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number <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="$Z$"> controls the frequency which is just its 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 $">.
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<P>
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Figure <A HREF="#fig07.02">7.2</A> also shows the sequence <!-- MATH
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$A, AZ, A{Z^2}, \ldots$
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-->
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<IMG
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WIDTH="109" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img663.png"
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ALT="$A, AZ, A{Z^2}, \ldots$">;
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in effect this is the same sequence as <!-- 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="$1, Z, {Z^2}, \ldots$">, but amplified and
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|
rotated according to the amplitude and initial phase. In a complex
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sinusoid of this form, <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.png"
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ALT="$A$"> is called the
|
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<A NAME="7846"></A>
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<I>complex amplitude</I>.
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<P>
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Using complex numbers to represent the amplitudes and phases of sinusoids can
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|
clarify manipulations that otherwise might seem unmotivated. For instance,
|
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|
suppose we want to know the amplitude and phase of the sum of two sinusoids
|
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|
|
with the same frequency. In the language of this chapter, we let the two
|
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|
sinusoids be written as:
|
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<BR><P></P>
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<DIV ALIGN="CENTER">
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|
<!-- MATH
|
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|
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|
\begin{displaymath}
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|
X[n] = A {Z^n} , \ Y[n] = B {Z^n}
|
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|
\end{displaymath}
|
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|
-->
|
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|
<IMG
|
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|
WIDTH="185" HEIGHT="28" BORDER="0"
|
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|
SRC="img665.png"
|
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|
ALT="\begin{displaymath}
|
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|
X[n] = A {Z^n} , \ Y[n] = B {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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|
where <IMG
|
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|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img23.png"
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ALT="$A$"> and <IMG
|
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|
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
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|
SRC="img311.png"
|
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|
ALT="$B$"> encode the phases and amplitudes of the two signals.
|
|
|
|
|
The sum is then equal to:
|
|
|
|
|
<BR><P></P>
|
|
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|
|
<DIV ALIGN="CENTER">
|
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|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
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|
X[n] + Y[n] = (A+B) {Z^n}
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
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|
<IMG
|
|
|
|
|
WIDTH="181" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img666.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
X[n] + Y[n] = (A+B) {Z^n}
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
which is a sinusoid whose amplitude equals <IMG
|
|
|
|
|
WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img667.png"
|
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|
ALT="$\vert A+B\vert$"> and whose phase equals
|
|
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|
|
<IMG
|
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|
|
|
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img668.png"
|
|
|
|
|
ALT="$\angle(A+B)$">. This is clearly a much easier way to manipulate amplitudes
|
|
|
|
|
and phases than using properties of sines and cosines. Eventually, of course,
|
|
|
|
|
we will take the real part of the result; this can usually be left to the
|
|
|
|
|
end of whatever we're doing.
|
|
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|
<P>
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<HR>
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