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<TITLE>Complex sinusoids</TITLE>
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<H2><A NAME="SECTION001111000000000000000">
Complex sinusoids</A>
</H2>
<P>
Recall the formula for a (real-valued) sinusoid from Page
<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = a \cos (\omega n + \phi )
\end{displaymath}
-->
<IMG
WIDTH="140" HEIGHT="28" BORDER="0"
SRC="img76.png"
ALT="\begin{displaymath}
x[n] = a \cos (\omega n + \phi )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This is a sequence of cosines of angles (called phases) which increase
arithmetically
with the sample number <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">. The cosines are all adjusted by the factor <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">.
We can now rewrite this as the real part of a much simpler and easier to
manipulate sequence of complex numbers, by using the properties of their
arguments and magnitudes.
<P>
<DIV ALIGN="CENTER"><A NAME="fig07.02"></A><A NAME="7829"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.2:</STRONG>
The powers of a complex number <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$"> with <IMG
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img22.png"
ALT="$\vert Z\vert=1$">, and the same
sequence multiplied by a constant <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$A$">.</CAPTION>
<TR><TD><IMG
WIDTH="205" HEIGHT="204" BORDER="0"
SRC="img651.png"
ALT="\begin{figure}\psfig{file=figs/fig07.02.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
Suppose that a complex number <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$"> happens to have magnitude one and
argument <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">, so that
it can be written as:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
Z = \cos(\omega) + i \sin(\omega)
\end{displaymath}
-->
<IMG
WIDTH="145" HEIGHT="28" BORDER="0"
SRC="img652.png"
ALT="\begin{displaymath}
Z = \cos(\omega) + i \sin(\omega)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Then for any integer <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">, the number <IMG
WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img653.png"
ALT="$Z^n$"> must have magnitude one as well
(because magnitudes multiply) and argument <IMG
WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img654.png"
ALT="$n\omega$"> (because arguments add).
So,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{Z^n} = \cos(n\omega) + i \sin(n \omega)
\end{displaymath}
-->
<IMG
WIDTH="173" HEIGHT="28" BORDER="0"
SRC="img655.png"
ALT="\begin{displaymath}
{Z^n} = \cos(n\omega) + i \sin(n \omega)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This is also true for negative values of <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">, so for example,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
\end{displaymath}
-->
<IMG
WIDTH="199" HEIGHT="38" BORDER="0"
SRC="img656.png"
ALT="\begin{displaymath}
{1 \over Z} = {Z^{-1}} = cos(\omega) - i \sin(\omega)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Figure <A HREF="#fig07.02">7.2</A> shows graphically how the powers of <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$"> wrap around
the unit circle, which is the set of all complex numbers of magnitude one.
They form a geometric sequence:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\ldots, {Z^0}, {Z^1}, {Z^2}, \ldots
\end{displaymath}
-->
<IMG
WIDTH="123" HEIGHT="27" BORDER="0"
SRC="img657.png"
ALT="\begin{displaymath}
\ldots, {Z^0}, {Z^1}, {Z^2}, \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and taking the real part of each term we get a real sinusoid with
initial phase zero and amplitude one:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\ldots, \cos(0), \cos(\omega), \cos(2 \omega), \ldots
\end{displaymath}
-->
<IMG
WIDTH="203" HEIGHT="28" BORDER="0"
SRC="img658.png"
ALT="\begin{displaymath}
\ldots, \cos(0), \cos(\omega), \cos(2 \omega), \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Furthermore, suppose we multiply the elements of the sequence by some (complex)
constant <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$A$"> with magnitude <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> and argument <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img77.png"
ALT="$\phi$">. This gives
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\ldots, A, AZ, A{Z^2}, \ldots
\end{displaymath}
-->
<IMG
WIDTH="131" HEIGHT="27" BORDER="0"
SRC="img659.png"
ALT="\begin{displaymath}
\ldots, A, AZ, A{Z^2}, \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The magnitudes are all <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> and the argument of the <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">th term is
<!-- MATH
$n \omega + \phi$
-->
<IMG
WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img660.png"
ALT="$n \omega + \phi$">, so the sequence is equal to
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{AZ^n} = a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)]
\end{displaymath}
-->
<IMG
WIDTH="272" HEIGHT="28" BORDER="0"
SRC="img661.png"
ALT="\begin{displaymath}
{AZ^n} = a \cdot [\cos(n \omega + \phi) + i \sin(n \omega + \phi)]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and the real part is just the real-valued sinusoid:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\mathrm{re}(A{Z^n}) = a \cdot \cos(n \omega + \phi)
\end{displaymath}
-->
<IMG
WIDTH="181" HEIGHT="28" BORDER="0"
SRC="img662.png"
ALT="\begin{displaymath}
\mathrm{re}(A{Z^n}) = a \cdot \cos(n \omega + \phi)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The complex number <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$A$"> encodes both the real amplitude <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">
and the initial phase <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img77.png"
ALT="$\phi$">; the unit-magnitude complex
number <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$"> controls the frequency which is just its argument <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">.
<P>
Figure <A HREF="#fig07.02">7.2</A> also shows the sequence <!-- MATH
$A, AZ, A{Z^2}, \ldots$
-->
<IMG
WIDTH="109" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img663.png"
ALT="$A, AZ, A{Z^2}, \ldots$">;
in effect this is the same sequence as <!-- MATH
$1, Z, {Z^2}, \ldots$
-->
<IMG
WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img664.png"
ALT="$1, Z, {Z^2}, \ldots$">, but amplified and
rotated according to the amplitude and initial phase. In a complex
sinusoid of this form, <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$A$"> is called the
<A NAME="7846"></A>
<I>complex amplitude</I>.
<P>
Using complex numbers to represent the amplitudes and phases of sinusoids can
clarify manipulations that otherwise might seem unmotivated. For instance,
suppose we want to know the amplitude and phase of the sum of two sinusoids
with the same frequency. In the language of this chapter, we let the two
sinusoids be written as:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
X[n] = A {Z^n} , \ Y[n] = B {Z^n}
\end{displaymath}
-->
<IMG
WIDTH="185" HEIGHT="28" BORDER="0"
SRC="img665.png"
ALT="\begin{displaymath}
X[n] = A {Z^n} , \ Y[n] = B {Z^n}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$A$"> and <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img311.png"
ALT="$B$"> encode the phases and amplitudes of the two signals.
The sum is then equal to:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
X[n] + Y[n] = (A+B) {Z^n}
\end{displaymath}
-->
<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"
ALT="$\vert A+B\vert$"> and whose phase equals
<IMG
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.
<P>
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Miller Puckette
2006-12-30
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