NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION: "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
2017-09-28 by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University
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Earlier, we used Euler's
Identity to show
Setting 
, we see that both sine and cosine (and hence
all real sinusoids)
consist of a sum of equal and opposite circular motion. Phrased
differently, every real sinusoid consists of an equal contribution
of positive and
negative frequency components. This is true of all real
signals. When we get to spectrum analysis,
we will find that every real signal contains equal amounts of
positive and
negative frequencies, i.e., if 
denotes the spectrum of the real signal
, we
will always have 
.
Note that, mathematically, the complex sinusoid 
is really simpler and more basic than
the real sinusoid 
because 
consists of one frequency
while 
really consists of two frequencies and 
.
We may think of a real sinusoid as being the sum of a
positive-frequency and a negative-frequency complex sinusoid, so in
that sense real sinusoids are ''twice as complicated'' as complex
sinusoids. Complex sinusoids are also nicer because they have a
constant modulus. ''Amplitude
envelope detectors'' for complex sinusoids are trivial: just
compute the square root of the sum of the squares of the real and
imaginary parts to obtain the instantaneous peak amplitude
at any time. Frequency demodulators are similarly trivial: just
differentiate the phase of the complex sinusoid to obtain its
instantaneous frequency. It should therefore come as no
surprise that signal processing engineers often prefer to convert
real sinusoids into complex sinusoids before processing them
further.
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