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:
Of course we could equally well have chosen the complex sinusoid with
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
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.