Hi, and welcome to module 9.5
of Digital Signal Processing.
In this module, we will touch briefly
on some topics in receiver design.
A lot of things unfortunately
happen to the signal while it's
traveling through the channel.
The signal picks up noise,
we have seen that already.
It also gets distorted because the channel
will act as some sort of filter
that is not necessarily all-pass and
linear phase.
Interference happens, too.
There might be parts of the channel
that we thought were usable and
they're actually not.
So the receiver really has to deal with
a copy of the transmitted signal that is
very, very far from the idealized
version we have seen so far.
The way receivers,
especially digital receivers,
can cope with the distortions and
the noise introduced by the channel is by
implemented adaptive filtering techniques.
Now we will not have
the time to go into very
many details about adaptive
signal processing.
Again, these are topics that
you will be able to study in
more advanced signal processing classes.
But I think it is important
to give you an overview
of the things that have to happen inside
a receiver, inside your ADSL receiver for
instance, so that you can enjoy these
high data rates that are available today.
And the first technique that we will
look at is adaptive equalization and
then we will look at some very simple
timing recovery that is used in
practice in receivers.
Let's begin with a blast form the past.
[SOUND] Those of you that are a little bit
older will certainly have recognized this
sound as the obligatory soundtrack every
time you used to connect to the Internet.
And indeed,
this is the sound made by a V.34 modem,
that was the standard dial-up connection
device in 90s until the early 2000s.
Now, if you have ever used a modem,
you've heard the sound and
you probably wondered what was going on.
So we're going to analyze what we just
heard from the graphical point of view.
If we look at the block diagram for
the receiver once again,
what we're going to do is we're going
to plot the baseband complex samples
as points on the complex plane.
So we going to take br(n) as
the horizontal coordinate, and
bi(n) as the vertical coordinate.
And before we do so, let's just look for
second at what happens inside
the receiver when the signal at
the input is a simple sinusoid,
like cosine (omega c + omega 0) n.
We are demodulating this very simple
signal with the two carriers,
the cosine (omega c n) and
sine (omega c n), and
then we're filtering the result
with a low pass filter.
So if we work out this formula with
standard trigonometric identities,
we can always express, for instance,
the product of two cosine
functions as the sum of the cosine
of the sum of the angles plus the cosine
of the difference of the angles.
And same for
the product of the cosine and sine.
So if we do that,
we get four terms, two of which
have a frequency that will fall outside
of the past band of the filter H.
So when we apply the filter to this terms,
we're left only with cosine (omega
0 n) + j sine (omega 0 n), which is,
of course, e to the j omega 0 n.
So when the input to
the receiver is a cosine,
the points in the complex baseband
sequence will be points around the circle
and the difference between two
successive points is the angle, omega 0.
The reason why we might be called to
demodulate the simple sinusoid is
because the receiver will send
what are called pilot tones.
Simple sinusoid that are used
to probe the line and
gauge the response of the channel
at particular frequencies.
So with this in mind, let's look at this
slow motion analysis of the baseband
signal samples when the input is
the audio file which is turned before.
So let's start with the part
that goes like this.
[SOUND] This signal contains several
sinusoids that we can see here in the pod,
and the sinusoids also contain abrupt
phase reversal, meaning that at
some given points in time, the phase
of the sinusoid is augmented by pi.
You can see this as this small explosions
in the circular pattern in the plot.
This phase reversals are used by
the transmitter and the receiver as time
markers to estimate the propagation delay
of the signal from source to destination.
The next part goes like this.
[SOUND] And this is a training sequence.
The transmitter sends a sequence of
known symbols, namely the receiver knows
the symbols that are being transmitted and
so the receiver can use this knowledge to
train an equalizer to undo
the effects of the channel.
The last part is the data transmission
proper, the noisy part if you want,
of the audio file.
And the interesting thing
is that transmitter and
receiver perform
a handshake procedure using
a very low bit rate QAM transmission
using only four points.
Therefore, two bits per symbol,
to exchange the parameters of the real
data transmission that is going to follow.
The speed,
the constellation size and so on.
Using the four point QAM constellation in
the beginning ensures that even in very
noisy conditions, transmitter and receiver
can exchange their vital information.
So even from this simple qualitative
description of what happens in the real
communications scenario,
we can see that the task that the receiver
is saddled with is very complicated.
So it's a dirty job but
a receiver has to do it.
And the receiver has to cope with
four potential sources of problem.
Interference, the propagation delay.
So, the delay introduced by the channel.
The linear distortion
introduced by the channel.
And drifts in the internal clocks between
the digital system inside the transmitter,
and the digital system
inside the receiver.
So, when it comes to interference,
the handshake procedure and
the line probing pilot tones,
are used in clever ways to circumvent
the major sources of interference.
We will see some examples
later on when we discuss ADSL.
The propagation delay is tackled
by a delay estimation procedure,
that we will look at in just a second.
The distortion to this by
the channel is compensated using
adaptive equalization techniques, and
we will see some examples of that as well.
And clock drifts are tackled by timing
recovery techniques, then in and
off themselves are quite sophisticated,
and
therefore we leave them
to more advanced classes.
Graphically, if we sum up the chain
of events that occur between
the transmission of the original
digital signal, and
the beginning of the demodulation
of the received signal.
We have a digital to analog
converter at the transmitter.
This is the transmitter part of the chain
that operates with a given
sampling period T s.
This generates an analog signal
which is sent over a channel.
We can represent the channel, for
the time being, as a linear filter
in the continuous time domain,
with frequency response D( j omega).
At the input of the receiver, we have
a continuous time signal s hat (t),
which is a distorted and delayed
version of the original analog signal.
We will neglect noise for the time being.
This signal is sampled by an A/D converter
that operates at a period T prime of s.
And we obtain the sequence of samples
that will be input to the modulator.
So this is the receiver part of the chain.
We have to take into account
the distortion introduced by the channel.
And we have to take into account the
potentially time varying discrepancies in
the clocks between the transmitter and
the receiver.
These two systems are Geographically
remote, and there is no guarantee that
the two internal clocks that
are used in the A to D and
D to A converters are synchronised or
run exactly at the same frequency.
Let's start with problem
of delay compensation.
To simplify the analysis we will assume
that the clocks that transmitter and
receiver are synchronized and synchronous.
So T prime of S is equal to TS.
And the channel acts as a simple delay.
So the received signal is simply a delayed
version of the transmitted signal
which implies that the frequency
response of the channel is simply
E to the minus J omega D.
So the channel introduces
a delay of d seconds.
You can express this in samples,
in the following way.
We write d as the product of the sampling
period time time b plus tau where
b is an integer and tau is strictly
less than one-half in magnitude.
So b is called the bulk delay because it
gives us an integer number of samples
of delay at the receiver and
tau is the fractional delay.
So the fraction of samples introduced
by the continuous time delay of d.
So how do we compensate for this delay?
Well, the bulk delay is
rather easy to tackle.
Imagine the transmitter
begins transmission
by sending just an impulse
over the channel.
So the discrete time signal is this one,
it's just a delta and
zero, it gets sent the d2a converter, and
the converter will output a continuous
time signal that looks like
an interpolation function, like a sink.
And like all interpolation functions,
it will have a maximum peak in zero
that corresponds to the non-zero sample.
This signal gets transmitted
over the channel and
it gets to the receiver after a delay,
D, that we can estimate, for instance,
by looking at the displacement of
the peak of the intervalation function.
The receiver converts this
into discrete time sequence.
Now in the figure here it look
as if the sampling instance and
the transmitter and
receiver are perfectly aligned.
Now this is not necessarily the case
because the starting time for
the interpolator, and
the transmitter, and the sampler, and
the receiver are not
necessarily synchronized.
But any difference in starting time can
be integrated into the propagation delay
as long as the sampling
periods are the same.
So with this, all we need to do
in the receiver, is to look for
the maximum value in
the sequence of samples.
Because of the shape of
the interpolating function,
we know that the real maximum will be at
most half a sample in either direction
of the location of
the maximum sample value.
So, at the receiver to
offset the bulk delay,
we will just set the nominal time,
n equal to zero,
to coincide with the location of
the maximum value of the sample sequence.
Now of course, we need to compensate for
the fractional delay, so
we need to estimate tao.
And to do that,
we'll use a different technique.
Let me add in passing that in real
communication devices, of course we're not
using impulses to offset the bulk delay
because impulses are full band signals and
so they would be filtered out by the pass
band characteristic of the channel.
The trick is to embed
discontinuities in pilot tones and
to recognize those
discontinuities in the receiver.
As we have seen in the animation
at the beginning of this module,
we use phase reversals, which are abrupt
discontinuities in sinusoids,
to provide a recognizable instant in
time for the receiver to latch on.
Okay, so what about the fractional delay?
Well for the fractional delay,
we use a sinusoid instead of a delta.
So we build a base band signal,
which is simply a complex exponential
at a known frequency omega zero.
This will be converted to a real signal
before being sent to the D to A converter.
And so what we transmit
actually is cosine of omega c,
the carrier frequency, plus the pilot's
frequency omega zero times n.
The receiver will receive
a delayed version of this
which contains a delay now in samples and
fraction of sample, b + tau.
After we demodulate this cosine,
you remember we get a complex exponential.
And we can also compensate already for
the bulk delay, which we know.
So, for an integer number of sample b.
And we obtain a base band signal,
half b of n, which is e to the j omega,
m minus tau.
Since we know the frequency omega zero,
we can just multiple this quantity
by e to the minus j omega zero n.
And obtain e to the minus j omega
zero tau, which is a constant and
which we can invert, given that
we know the frequency omega zero.
And so now we have an estimate for both
the bulk delay and the fractional delay.
Now we have to bring back
the signal to the original timing.
The bulk delay is really no problem.
It's just an integer number of samples.
What creates a problem is the fractional
delay because that will shift
the peaks with respect to
the sampling intervals.
So if we want to compensate for
the bulk delay we need to
compute sub-sample values.
And in theory to do that we should
use a sinc fractional delay,
namely a filter with impulse
response sinc(n + tau).
In practice however,
we will use a local interpolation, and
this is a very practical application
of the Lagrange interpolation technique
that we saw in module 6.2.
So graphically the situation is like so.
We have a stream of samples coming in.
And for each sample we want to
compute the sub-sample value
with a distance of tau from
the nearest sampling interval.
And we want to only use a local
neighborhood of samples to estimate this.
Now you remember from module 6.2
the Lagrange approximation works
by building a linear combination
of Lagrange polynomials
weighed by the samples of the function.
So as per usual we choose
the sampling interval equal to one so
that we lighten the notation.
We have a continuous
time function x of T and
we want to compute x of n plus tau with
tau less than one half in magnitude.
So we have samples of this function
at integers n and the local
Lagrange approximation around n is given
by this linear combination of Lagrange
polynomials weighted by the samples of the
functions around the approximation point.
So, we use the notation XL(n;t).
n is the center point and t is the value
from the center point at which we
want to compute the approximation.
And the Lagrange polynomials
are given by this formula here
which is the same as in module 6.2.
So the delayed compensated
input signal will be set
equal to the Lagrange
approximation at tau.
So let's look at an example.
Assume that we want a second-order
approximation, so we pick N = 1, and
we will have three Lagrange polynomials.
And so we will need to use three samples
of the sequence to compute interpolation.
The three polynomials will be
centered in n- 1, n, and n + 1,
and scaled by the values of
the samples at these locations.
And finally we will sum the pool numbers
together and computer value in n + tau.
So, we start with the first one,
which is centered in n- 1.
And like all interpolation
polynomials its value is one and
n- 1 and zero at other integer
values of the argument.
The second polynomial
will be centered in n and
the third polynomial will
be centered in n + one.
When we sum them together
we obtain a second order
curve that goes through the points
that interpolates the three points and
then we can compute the approximation
as the value of this curve in n + 10.
Now the nice thing about this approach
is that if we look at the approximation,
if we take the Lagrange
approximation around n,
we can define a set of
coefficients d tau of k,
which are the the values of each
Lagrange polynomial in tau.
So d tau of k are 2N+1 values
that form the coefficients
of an FIR filter and we can compute
the value of the Lagrange approximation
simply as the convolution of the incoming
sequence with this interpolation filter.
So for example, if these are the three
Lagrange polynomials for
N = 1, we can compute this polynomials for
t = tau,
where tau is the fractional
delay that we estimated before.
And we will obtain three coefficients,
like here, for instance,
is an example for tau equal to 0.2.
Three coefficients that
give us an FIR filter, and
then we can just simply filter the samples
coming into the receiver with this filter
to compensate for the fractional delay.
So again, the algorithm is
estimate the fractional delay,
the bulk delay is no problem again.
Compute the 2N+1 Lagrangian coefficients
and filter with the resulting FIR.
The added advantage of the strategy is
that if the delay changes over time for
any reason, all we need to do is
to keep the estimation running and
update the FIR coefficients as
the estimation changes over time.
Okay, now that we know
how to compensate for
the propagation delay
introduced by the channel,
let's consider a channel with an arbitrary
frequency response D(j omega).
And the transmission chain goes
from the passband signal s(n),
discrete time, into a D/A converter,
analog signal s(t) that gets filtered
by the channel, gives us hat s(t),
which is sampled at the receiver to give
us a received passband signal hat s(n).
But now we have seen in the previous
module that this block diagram
can be converted into an old digital
scheme where our bandpass signal
s[n] gets filtered by the discrete time
equivalent of the channel and gives us
a filtered version of the bandpass signal
as it would appear inside the receiver.
So, the problem now is
that we would like to undo
the effects of the channel
on the transmitted signal.
And the classic way to do that is to
filter the received signal hat s(n)
by a filter E that compensates for
the distortion, or
the filtering, introduced by the channel.
So the target is that the output of
the filtering operation gives us a signal,
hat se(n),
which is equal to the transmitted signal.
How do we do that?
In theory, it would be enough to pick a
transfer function for the filter E, which
is just a reciprocal of the equivalent
transfer function of the channel.
But the problem is that we don't know
the transfer function of the channel in
advance because each time we
transmit data over the channel,
this transfer function may change.
And also, even while we're transmitting
data, the transfer function might change
because it is a physical system that might
be subject to drifts and modifications.
So what do we do?
We need to use adaptive equalization.
For the filter to compensate for
distortion introduced by a channel is
called an equalizer and what we want
to do is to change the filter in time.
So change the filter coefficients
in a DSP realization
as a function of the error that we
obtain when we compare the output
of the filter with the signal
that we would like to obtain.
In our case, the signal that we would
like to obtain is the transmitted signal.
And so we take the received signal,
we filter it with the equalizer, we
look at the result, we take the difference
with respect to the original signal.
And we use the error,
which should be zero in the ideal case,
to drive the adaptation of the equalizer.
But wait.
How do we get the exact transmitted
signal at the receiver?
Well, we use two tricks.
The first one is bootstrapping.
The transmitter will send a pre-arranged
sequence of symbols to the receiver.
So let's call this
sequence of symbols at(n).
This gets modulated and
generates a passband signal s(n).
Now, at the receiver,
the sequence at(n) is known.
And the receiver has an exact
copy of the modulator,
of the transmitter inside of itself.
So the transmitter can generate locally
an exact copy of the passband signal s(n).
And so for
the bootstrapping part of the adaptation,
we actually have an exact
copy of the transmitted
passband signal that we can use to drive
the adaptation of the coefficients.
The train of sequence is just long
enough to bring the equalizer to
a workable state.
For the handshake procedure that
we saw in the video before, for
instance, this would correspond
to the moment where the receiver
starts demodulating the four-point QIM.
At that moment,
the receiver will switch strategy and
implement a data driven adaptation.
The thing works like this.
The received signal gets equalized,
gets demodulated,
and then the slicer will recover
the sequence of transmitted symbols.
Since the receiver has a copy of
the transmitter inside of itself,
it can use the sequence
of transmitted symbol
to build a local copy of
the transmitted signal.
Now, of course, errors might
happen in the slicing process, and
so this local copy is not
completely error free.
But the assumption is that the equalizer
is doing already enough of a good job to
keep the number of errors in
this sequence sufficiently low.
So that the difference with respect to
the received signal is enough to refine
the adaptation of the equalizer and
especially to track the time
varying conditions of the channel.
What we have seen is just
a qualitative overview of
what happens inside of a receiver.
And there are still so many questions that
we would have to answer to be thorough.
For instance, how do we carry out
the adaptation of the coefficients in
the equalizer?
How do we compensate for
different clock rates in geographically
diverse receivers and transmitters?
How do we recover from the interference
from other transmission devices?
And how do we improve
the resilience to noise?
The answers to all those questions
require a much deeper understanding
of adaptive signal processing.
And hopefully that will be the topic
of your next signal processing class.