SIGNAL PROCESSING &
SIMULATION NEWSLETTER
Note: This is not a
particularly interesting topic for anyone other than those who are involved in
simulation. So if you have difficulty with this issue, you may safely stop
reading after page 5 without feeling guilty.
Hilbert Transform, Analytic
Signal and the Complex Envelope
In
Digital Signal Processing we often need to look at relationships between real
and imaginary parts of a complex signal. These relationships are generally
described by Hilbert transforms.
Hilbert transform not only helps us relate the I and Q components but it
is also used to create a special class of causal signals called analytic which
are especially important in simulation. The analytic signals help us to
represent bandpass signals as complex signals which have specially attractive
properties for signal processing.
Hilbert
Transform is not a particularly complex concept and can be much better
understood if we take an intuitive approach first before delving into its
formula which is related to convolution and is hard to grasp. The following diagram that is often seen in
text books describing modulation gives us a clue as to what a Hilbert Transform
does.
Figure 1 - Role of Hilbert Transform in modulation
The
role of Hilbert transform as we can guess here is to take the carrier which is
a cosine wave and create a sine wave out of it. So let’s take a closer look at
a cosine wave to see how this is done by the Hilbert transformer. Figure 2a
shows the amplitude and the phase spectrum of a cosine wave. Now recall that
the Fourier Series is written as
where
and 
and
An and Bn are the spectral amplitudes of cosine and sine
waves. Now take a look at the phase spectrum. The phase spectrum is computed by
Cosine wave has no sine spectral content, so Bn
is zero. The phase calculated is 90° for both positive and
negative frequency from above formula. The wave has two spectral components
each of magnitude 1/2A, both positive and lying in the real plane. (the real
plane is described as that passing vertically (R-V plane) and the Imaginary
plane as one horizontally (R-I plane) through the Imaginary axis)
Figure
2b shows the same two spectrums for a sine wave. The sine wave phase is not
symmetric because the amplitude spectrum is not symmetric. The quantity An
is zero and Bn has either a positive or negative value. The phase is
+90° for the positive frequency and -90° for the negative frequency.
Now
we wish to convert the cosine wave to a sine wave. There are two ways of doing
that, one in time domain and the other in frequency domain.
Hilbert Transform in
Frequency Domain
Now
compare Figure 2a and 2b, in particular the spectral amplitudes. The cosine
spectral amplitudes are both positive and lie in the real plane. The sine wave
has spectral components that lie in the Imaginary plane and are of opposite
sign.
To
turn cosine into sine, as shown in Figure 3 below, we need to rotate the
negative frequency component of the cosine by +90° and the positive frequency
component by -90°. We will need to rotate the +Q
phasor by -90° or in other words multiply
it by -j. We also need to rotate the -Q phasor by +90° or multiply it by j.
Figure 3 - Rotating phasors to create a sine wave out of a cosine
We
can describe this transformation process called the Hilbert Transform as
follows:
All negative frequencies of
a signal get a +90° phase shift and all positive frequencies get a -90° phase shift.
If
we put a cosine wave through this transformer, we get a sine wave. This phase
rotation process is true for all signals put through the Hilbert transform and
not just the cosine.
For
any signal g(t), its Hilbert Transform has the following property
(Putting a little hat
over the capital letter representing the time domain signal is the typical way
a Hilbert Transform is written.)
A
sine wave through a Hilbert Transformer will come out as a negative cosine. A
negative cosine will come out a negative sine wave and one more transformation
will return it to the original cosine wave, each time its phase being changed
by 90°.
For
this reason Hilbert transform is also called a “quadrature filter”. We can draw
this filter as shown below in Figure 4.
Figure 4 - Hilbert Transform shifts the phase of positive frequencies
by -90° and negative frequencies
by +90°.
So
here are two things we can say about the Hilbert Transform.
1.
It
is a peculiar sort of filter that changes the phase of the spectral components
depending on the sign of their frequency.
2.
It
only effects the phase of the signal. It has no effect on the amplitude at all.
Hilbert transform in Time
Domain
Now
look at the signal in time domain. Given a signal g(t), Hilbert Transform of
this signal is defined as
(1)
Another way to write this definition is to recognize
that Hilbert Transform is also the convolution of function 1/ pt with the signal g(t). So
we can write the above equation as
(2)
Achieving a Hilbert Transform in time domain means
convolving the signal with the function 1/ pt. Why the function 1/ pt, what is its significance?
Let’s look at the Fourier Transform of this function. What does that tell us?
Given in Eq 3, the transform looks a lot like the Hilbert transform we talked
about before.
(3)
The term sgn in Eq 3 above , called signum is
simpler than it seems. Here is the way we could have written it which would
have been more understandable.
(4)
In Figure 5 we show the signum function and its
decomposition into two familiar functions.
Figure 5 - Signum Function decomposed into a
unit function and a constant
For shortcut, writing sgn is useful but it is better
if it is understood as a sum of the above two much simpler functions. (We will
use this relationship later.)
(5)
We see in 6 figure that although 1/pt is a real function, is has a Fourier
transform that lies strictly in the imaginary plane. Do you recall what this
means in terms of Fourier Series coefficients? What does it tell us about a
function if it has no real components in its Fourier transform? It says that
this function can be represented completely by a sum of sine waves. It has no
cosine component at all.
In Figure 7, we see a function composed of a sum of
50 sine waves. We see the similarity of this function with that of 1/pt. Now you can see that
although the function 1/pt looks nothing at all a sinusoid, we can still approximate it with a
sum of sinusoids.
The function f(t) = 1/pt gives us a spectrum that
explains the Hilbert Transform in time domain, albeit this way of looking at
the Hilbert Transform is indeed very hard to grasp.
We limit our discussion of Hilbert transform to
Frequency domain due to this difficulty.
Figure 7 -
Approximating function f(t) = 1/p t with a sum of 50 sine wave
We
can add the following to our list of observations about the Hilbert Transform.
3.
The
signal and its Hilbert Transform are orthogonal. This is because by rotating
the signal 90° we have now made it
orthogonal to the original signal, that being the definition of orthogonality.
4.
The
signal and its Hilbert Transform have identical energy because phase shift do
not change the energy of the signal only amplitude changes can do that.
Analytic Signal
Hilbert
Transform has other interesting properties. One of these comes in handy in the
formulation of an Analytic signal. Analytic signals are used in Double and
Single side-band processing (about SSB and DSB later) as well as in creating
the I and Q components of a real signal.
An
analytic signal is defined as follows.
(6)
An
analytic signal is a complex signal created by taking a signal and then adding
in quadrature its Hilbert Transform. It is also called the pre-envelope of the
real signal.
So what is the analytic
signal of a cosine?
Substitute
cos wt for g(t) in Eq 6, knowing that its Hilbert transform is a sine, we get
The
analytic function of a cosine is the now familiar phasor or the complex exponential,
ejwt.
What is the analytic signal
of a sine?
Now
substitute sin wt for g(t) in Eq 6, knowing that its Hilbert transform is a
-cos, we get once again a complex exponential.
Do you
remember what the spectrum of a complex exponential looks like? To remind you,
I repeat here the figure from Tutorial 6.
f 
Figure 8 - Fourier transform of a complex exponential
We can
see from the figure above, that whereas the spectrum of a sine and cosine spans
both the negative and positive frequencies, the spectrum of the analytic
signal, in this case the complex exponential, is in fact present only in the
positive domain. This is true for both sine and cosine and in fact for all real
signals.
Restating the results: the
Analytic signal for both and sine and cosine is the complex exponential. Even
though both sine and cosine have a two sided spectrum as we see in figures
above, the complex exponential which is the analytic signal of a sinusoid has a
one-sided spectrum.
We
can generalize from this: An analytic signal (composed of a real signal and its
Hilbert transform) has a spectrum that exists only in the positive frequency domain.
Let’s
take at a look at the analytic signal again.
(7)
The
conjugate of this signal is also a useful quantity.
(8)
This
signal has components only in the negative frequencies and can be used to
separate out the lower side-bands.
Now
back to the analytic signal. Let’s extend our understanding by taking Fourier
Transform of both sides of Eq 7. We get
(9)
The
first term is the Fourier transform of the signal g(t), and the second term is
the inverse Hilbert Transform. We can rewrite by use of property 
Eq 9 as
(10)
One
more simplification gives us
(11)
This
is a very important result and is applicable to both lowpass and modulated
signals. For modulated or bandpass signals, its net effect is to translate the
signal down to baseband, double the spectral magnitudes and then chop-off all
negative components.
Complex Envelope
We
can now define a new quantity based on the analytic signal, called the Complex
Envelope. The Complex Envelope is defined as
The
part 
is called the Complex
Envelope of the signal g(t).
Let’s
rewrite it and take its Fourier Transform.
We
now see clearly that the Complex Envelope is just the frequency shifted version
of the analytic signal. Recognizing that multiplication with the complex
exponential in time domain results in frequency shift in the Frequency domain,
using the Fourier Transform results for the analytic signal above, we get
(12)
So here
is what we have been trying to get at all this time. This result says that the
Fourier Transform of the analytic signal is just the one-sided spectrum. The
carrier signal drops out entirely and the spectrum is no longer symmetrical.
This
property is very valuable in simulation. We no longer have to do simulation at
carrier frequencies but only at the highest frequency of the baseband signal.
The process applies equally to other transformation such as filters etc. which
are also down shifted. It even works when non-linearities are present in the
channel and result in additional frequencies.
There
are other uses of complex representation which we will discuss as we explore
these topics however its main use is in simulation.
__________________________________
Example
Let’s
do an example. Here is a real baseband signal.
(I
have left out the factor 2p for purposes of
simplification)
Figure 9 - A Baseband Signal
The
spectrum of this signal is shown below, both its individual spectral amplitudes
and its magnitude spectrum. The magnitude spectrum shows one spectral component
of magnitude 2 at f = 2 and -2 and an
another one of magnitude 3 at f = 3 and -3.
Figure 10a - Spectral amplitudes Figure
10b- The Magnitude Spectrum
Now
let’s multiply it with a carrier signal of cos(100t) to modulate it and to
create a bandpass signal,
Figure 11 - The modulated signal and its envelope
Let’s
take the Hilbert Transform of this signal. But before we do that we need to
simplify the above so we only have sinusoids and not their products. This
step will make it easy to compute the Hilbert Transform. By using these
trigonometric relationships,
we
rewrite the above signal as
Now
we take the Hilbert Transform of each term and get
Now
create the analytic signal by adding the original signal and its Hilbert
Transform.
Let’s
once again rearrange the terms in the above signal
Recognizing
that each pair of terms is the Euler’s representation of a sinusoid, we can now
rewrite the analytic signal as
But
wait a minute, isn’t this the original signal and the carrier written in the complex
exponential? So why all the calculations just to get the original signal back?
Now let’s take the Fourier Transform of the analytic
signal and the complex envelope we have computed to show the real advantage of
the complex envelope representation of signals.
Spectrum of the Complex Envelope Spectrum
of the Analytic Signal
Figure 12 - The Magnitude Spectrum of the Complex Envelope vs. The
Analytic Signal
Although
this was a passband signal, we see that its complex envelope spectrum is centered
around zero and not the carrier frequency. Also the spectral components are
double those in figure 10b and they are only on the positive side. If you think
the result looks suspiciously like a one-sided Fourier transform, then you
would be right.
We do
all this because of something Nyquist said. He said that in order to properly
reconstruct a signal, any signal, baseband or passband, needs to be sampled at
least two times its highest spectral frequency. That requires that we sample at
frequency of 200.
But
we just showed that if we take a modulated signal and go through all this math
and create an analytic signal (which by the way does not require any knowledge
of the original signal) we can separate the information signal the baseband
signal s(t)) from the carrier. We do this by dividing the analytic signal by
the carrier. Now all we have left is the baseband signal. All processing can be
done at a sampling frequency which is 6 (two times the maximum frequency of 3)
instead of 200.
The
point here is that this mathematical concept help us get around the signal
processing requirements by Nyquist for sampling of bandpass systems.
The
complex envelope is useful primarily for passband signals. In a lowpass signal
the complex envelope of the signal is the signal itself. But in passband
signal, the complex envelope representation allows us to easily separate out
the carrier.
Take
a look at the complex envelope again for this signal
the analytic signal
the complex envelope
We
see the advantage of this form right away. The complex envelope is just the
low-pass part of the analytic signal. The analytic signal low-pass signal has
been multiplied by the complex exponential at the carrier frequency. The
Fourier transform of this representation will lead to the signal translated
back down the baseband (and doubled with no negative frequency components)
making it possible to get around the Nyquist sampling requirement and reduce
computational load.
Copyright
1999 Charan Langton, All Rights Reserved
mntcastle@earthlink.net