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
![$ A\exp[j(\omega t +
\phi)]$](http://www.dsprelated.com/josimages_new/mdft/img554.png)
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.
4.13
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:
Creation of the analytic signal
from the real sinusoid
and the derived phase-quadrature sinusoid
, viewed in the frequency domain. a) Spectrum of
. b) Spectrum
of
. c) Spectrum of
. d) Spectrum of
.
![\includegraphics[width=2.8in]{eps/sineFD}](http://www.dsprelated.com/josimages_new/mdft/img576.png) |
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.
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