Analytic Signals and Hilbert Transform Filters

A signal which has no negative-frequency components is called an
*analytic signal*.^{4.12} Therefore, in continuous time, every analytic signal
can be represented as

Any real sinusoid may be converted to a positive-frequency complex sinusoid by simply generating a phase-quadrature component to serve as the ``imaginary part'':

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 which we can remove by defining the Hilbert transform filter to have magnitude 1/2 at all 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 *demodulation*^{4.14}is 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.

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