
Positive and Negative Frequencies
In §
2.9, 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 (by
filtering out the negative-frequency component) before processing them
further.
Next Section: Plotting Complex Sinusoids versus FrequencyPrevious Section: Projection of Circular Motion