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

where is the complex coefficient (setting the amplitude and phase) of the positive-frequency complex sinusoid at frequency .

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'':

The phase-quadrature component can be generated from the in-phase component by a simple quarter-cycle time shift.

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

Applying the ideal phase shifts, the Hilbert transform is

The analytic signal is then

by Euler's identity. Thus, in the sum , the negative-frequency components of and cancel out, leaving only the positive-frequency component. This happens for any real signal , not just for sinusoids as in our example.

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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