** Next:** Examples
** Up:** Applications
** Previous:** Envelope following
** Contents**
** Index**

##

Single Sideband Modulation

As we saw in Chapter 5, multiplying two real sinusoids together results
in a signal with two new components at the sum and difference of the
original frequencies. If we carry out the same operation with complex
sinusoids, we get only one new resultant frequency; this is one result of
the greater mathematical simplicity of complex sinusoids as compared to
real ones. If we multiply a complex sinusoid
with another one,
the result is
, which is another complex sinusoid whose
frequency, , is the sum of the two original frequencies.

In general, since complex sinusoids have simpler properties than real ones, it
is often useful to be able to convert from real sinusoids to complex ones. In
other words, from the real sinusoid:

(with a spectral peak of amplitude and frequency ) we would like
a way of computing the complex sinusoid:

so that

We would like a linear process for doing this, so that superpositions of
sinusoids get treated as if their components were dealt with separately.
Of course we could equally well have chosen the complex sinusoid with
frequency :

and in fact is just half the sum of the two. In essence we need a filter
that will pass through positive frequencies (actually frequencies between
0 and , corresponding to values of on the top half of the complex
unit circle) from negative values
(from to 0, or equivalently, from to --the bottom
half of the unit circle).
One can design such a filter by designing a low-pass filter with cutoff
frequency , and then performing a rotation by radians using the
technique of Section 8.3.4. However, it turns out to be
easier to do it using two specially designed networks of all-pass filters
with real coefficients.

Calling the transfer functions of the two filters and , we design
the filters so that

or in other words,

Then for any incoming real-valued signal we simply form a complex number
where is the output of the first filter and is
the output of the second. Any complex sinusoidal component of
(call it ) will be transformed to

Having started with a real-valued signal, whose energy is split equally into
positive and negative frequencies, we end up with a complex-valued one with
only positive frequencies.

** Next:** Examples
** Up:** Applications
** Previous:** Envelope following
** Contents**
** Index**
Miller Puckette
2006-12-30