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
![$\displaystyle z(t) = \frac{1}{2\pi}\int_0^{\infty} Z(\omega)e^{j\omega t}d\omega
$](http://www.dsprelated.com/josimages_new/mdft/img550.png)
![$ Z(\omega)$](http://www.dsprelated.com/josimages_new/mdft/img551.png)
![$ \exp(j\omega t)$](http://www.dsprelated.com/josimages_new/mdft/img552.png)
![$ \omega$](http://www.dsprelated.com/josimages_new/mdft/img368.png)
Any real sinusoid
may be converted to a
positive-frequency complex sinusoid
by simply generating a phase-quadrature component
to serve as the ``imaginary part'':
![$\displaystyle A e^{j(\omega t + \phi)} = A\cos(\omega t + \phi) + j A\sin(\omega t + \phi)
$](http://www.dsprelated.com/josimages_new/mdft/img555.png)
For more complicated signals which are expressible as a sum of many
sinusoids, a filter can be constructed which shifts each
sinusoidal component by a quarter cycle. This is called a
Hilbert transform filter. Let
denote the output
at time
of the Hilbert-transform filter applied to the signal
.
Ideally, this filter has magnitude
at all frequencies and
introduces a phase shift of
at each positive frequency and
at each negative frequency. When a real signal
and
its Hilbert transform
are used to form a new complex signal
,
the signal
is the (complex) analytic signal corresponding to
the real signal
. In other words, for any real signal
, the
corresponding analytic signal
has the property
that all ``negative frequencies'' of
have been ``filtered out.''
To see how this works, recall that these phase shifts can be impressed on a
complex sinusoid by multiplying it by
. Consider
the positive and negative frequency components at the particular frequency
:
![\begin{eqnarray*}
x_+(t) &\isdef & e^{j\omega_0 t} \\
x_-(t) &\isdef & e^{-j\omega_0 t}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img564.png)
Now let's apply a degrees phase shift to the positive-frequency
component, and a
degrees phase shift to the negative-frequency
component:
![\begin{eqnarray*}
y_+(t) &=& e^{-j\pi/2} e^{j\omega_0 t} = -j e^{j\omega_0 t} \\
y_-(t) &=& e^{j\pi/2} e^{-j\omega_0 t} = j e^{-j\omega_0 t}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img567.png)
Adding them together gives
![\begin{eqnarray*}
z_+(t) &\isdef & x_+(t) + j y_+(t) = e^{j\omega_0 t} - j^2 e^{...
... x_-(t) + j y_-(t) = e^{-j\omega_0 t} + j^2 e^{-j\omega_0 t} = 0
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img568.png)
and sure enough, the negative frequency component is filtered out. (There is also a gain of 2 at positive frequencies.)
For a concrete example, let's start with the real sinusoid
![$\displaystyle x(t)=2\cos(\omega_0 t) = e^{j\omega_0 t} + e^{-j\omega_0 t}.
$](http://www.dsprelated.com/josimages_new/mdft/img569.png)
![\begin{eqnarray*}
y(t) &=& e^{j\omega_0 t-j\pi/2} + e^{-j\omega_0 t + j\pi/2}\\
&=& -je^{j\omega_0 t} + je^{-j\omega_0 t} = 2\sin(\omega_0 t).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img570.png)
The analytic signal is then
![$\displaystyle z(t) = x(t) + j y(t) = 2\cos(\omega_0 t) + j 2\sin(\omega_0 t) = 2 e^{j\omega_0 t},
$](http://www.dsprelated.com/josimages_new/mdft/img571.png)
![$ x(t) + j y(t)$](http://www.dsprelated.com/josimages_new/mdft/img572.png)
![$ x(t)$](http://www.dsprelated.com/josimages_new/mdft/img4.png)
![$ jy(t)$](http://www.dsprelated.com/josimages_new/mdft/img573.png)
![$ x(t)$](http://www.dsprelated.com/josimages_new/mdft/img4.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.
Next Section:
Generalized Complex Sinusoids
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Sinusoidal Frequency Modulation (FM)