For more complicated signals which are expressible as a sum of many sinusoids, a filter can be constructed which shifts each sinusoidal component by a quarter cycle. This is called a Hilbert transform filter. Let denote the output at time of the Hilbert-transform filter applied to the signal . Ideally, this filter has magnitude at all frequencies and introduces a phase shift of at each positive frequency and at each negative frequency. When a real signal and its Hilbert transform are used to form a new complex signal , the signal is the (complex) analytic signal corresponding to the real signal . In other words, for any real signal , the corresponding analytic signal has the property that all ``negative frequencies'' of have been ``filtered out.''
To see how this works, recall that these phase shifts can be impressed on a complex sinusoid by multiplying it by . Consider the positive and negative frequency components at the particular frequency :
Now let's apply a degrees phase shift to the positive-frequency component, and a degrees phase shift to the negative-frequency component:
Adding them together gives
and sure enough, the negative frequency component is filtered out. (There is also a gain of 2 at positive frequencies.)
For a concrete example, let's start with the real sinusoid
The analytic signal is then
Figure 4.16 illustrates what is going on in the frequency domain. At the top is a graph of the spectrum of the sinusoid consisting of impulses at frequencies and zero at all other frequencies (since ). Each impulse amplitude is equal to . (The amplitude of an impulse is its algebraic area.) Similarly, since , the spectrum of is an impulse of amplitude at and amplitude at . Multiplying by results in which is shown in the third plot, Fig.4.16c. Finally, adding together the first and third plots, corresponding to , we see that the two positive-frequency impulses add in phase to give a unit impulse (corresponding to ), and at frequency , the two impulses, having opposite sign, cancel in the sum, thus creating an analytic signal , as shown in Fig.4.16d. This sequence of operations illustrates how the negative-frequency component gets filtered out by summing with to produce the analytic signal corresponding to the real signal .
As a final example (and application), let , where is a slowly varying amplitude envelope (slow compared with ). This is an example of amplitude modulation applied to a sinusoid at ``carrier frequency'' (which is where you tune your AM radio). The Hilbert transform is very close to (if were constant, this would be exact), and the analytic signal is . Note that AM demodulation4.14is now nothing more than the absolute value. I.e., . Due to this simplicity, Hilbert transforms are sometimes used in making amplitude envelope followers for narrowband signals (i.e., signals with all energy centered about a single ``carrier'' frequency). AM demodulation is one application of a narrowband envelope follower.
Generalized Complex Sinusoids
Sinusoidal Frequency Modulation (FM)