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, that 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 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 things that have to happen

inside a receiver, inside your ADSL 
receiver for instance.

So you can enjoy this 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 it used in practice in

receivers. 
Let's begin with a blast from the past.

[NOISE] 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

V34 modem that was the standard dial up 
connection device seen in the 90s until

the early 2000. 
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 base band 
complex samples as points on the complex

blank. 
So we're going to take B, r of n has the

as the horizontal coordinate and b, i of 
n as the vertical coordinate.

And before we do so let's just look for a 
second at what happens inside the

receiver when the signal at the input is 
a simple sinusoid, like cosine of omega c

plus omega zero n. 
We are demodulating this very simple

signal with the two carriers, the cosine 
of omega c n and sine of omega c n.

And then we're filtering the result with 
a low pass field.

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 cosine and

sine. 
So if we do that, we get four terms, two

of which have a frequency that will fall 
outside of the pass band of the filter h.

So when we apply the filter to these 
terms, we're left only with cosine of

omega 0 n plus j sine of 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 a difference between two

successive points is the angle omega 0. 
The reason why we might be called to

demodulate a simple sinusoid is because 
the receiver will send what are called

pile tones, simple sinusoids 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 the 
slow-motion analysis of the base band

signals. 
Samples, when the input is the audio file

we just heard before. 
So lets start with a part that goes like

this[SOUND]. 
This signal contains several sinusoids,

that we can see here in the plot. 
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. 
These 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. 
[NOISE].

And this is a training sequence. 
The transmitter sends a sequence of known

symbols, namely the receiver knows the 
symbol that are transmitted.

And so the receiver can use this 
knowledge to train an equalizer to undo

the affects 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 the 
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 a real 
communication 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 a 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 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 example 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 that in and of

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 
transmissions of the original digital

signal and the beginning of the 
demodulation of the received signal.

We have a digital to analog converter and 
a transmitter, this is transmitter part

of the chain that operates with a given 
sample period Ts.

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 of j omega.

At the input of the receiver, we have a 
continuous time signal, s hat of 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 to 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're used in the A 
to D and D to A converters are

synchronized or run exactly at the same 
frequency.

Let's start with problem of delay 
compensation.

To simplify the analysis we'll assume 
that the clocks at transmitter and

receiver are synchronized and 
synchronous.

So T prime of s is equal to T s, 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.

We can express this in samples in the 
following way.

We write d as the product of the sampling 
period, times 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 discreet time signal is this one,

it's just a delta and a 0. 
It gets sent to D-to-A converter.

And the converter will output a 
continuous time signal that looks like an

interpolation function like the sinc. 
And like all interpolation functions, it

will have a maximum peak at zero that 
corresponds to the known 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 interpolation function.

The receiver converts this into a 
discrete time sequence.

Now in the figure here it looks as if the 
sample incidence at the transmitter and

receiver are perfectly aligned. 
Now this is not necessarily the case

because the starting time for the 
interpolator at the transmitter and the

sampler at the receiver are not 
necessarily synchronous.

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 at 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, add the receiver to offset the bulk 
delay, we will just set the nominal time

n equal to 0, 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 tau.

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

passband characteristic of the channel. 
The trick is to embed discontinuities in

pilot tones and to recognize this 
discontinuities at 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 bass band signal, which is simply a

complex exponential at a known frequency, 
omega 0.

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 
frequency omega 0, times n.

The receiver will receive a delayed 
version of this, which contains the delay

now in sample, and fraction of sample, b 
plus tau.

After we demodulate this cosine you 
remember we got a complex exponential and

we can also compensate already for the 
bulk delay which we know.

So for an integer number of sample b we 
obtain a base band signal at b of n which

is e to the j omega and minus tau. 
Since we know the frequency of omega 0 we

can just multiply this quantity by e to 
the minus j omega 0 n and obtain e to the

minus j omega 0 tau, which is a constant. 
And which we can invert given that we

know the frequency omega 0. 
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 subsample values

and in theory to do that we should use a 
sinc fractional delay namely a filter

with impulse response sinc of n plus 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 subsample value with a distance of

tau from the nearest sample 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 1, 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 x L of n and 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 equal to 1 and we will have

three Lagrange polynomials. 
And so, we will need to use three samples

of the sequence to compute interpolation. 
These three polynomials will be centered

in n minus 1 and in n plus 1 and scaled 
by the values of the samples at these

locations. 
And finally, we will sum the poll numbers

together and compute their value in n 
plus tau.

So, we start with the first one, which is 
centered in n minus 1.

And, like all interpolation polynomials, 
its value is 1 in n minus 1, and 0, at

other integer values of the argument. 
The second polynomial will be centered in

n, and the third polynomial will be 
centered in n plus 1.

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 plus tau.

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 values of each

Lagrange polynomial in tau. 
So d tau of k, are 2 N plus 1 values, the

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 equal to 1, we 
can compute these polynomials for t equal

to 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 2 N plus 1 Lagrangian

coefficients and filter it with the 
resulting FIR.

The added advantage of this 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 go see the rechannel it with an

arbitrary frequency response D j of 
omega.

And the transmission chain goes from the 
pass band signal s of n, discreet time,

into a D-to-A converter, analog signal s 
of t, it gets filtered by the channel,

gives us hat s of t, which is sampled at 
the receiver to give us.

A received fast band signal, hat s of n. 
But now we have seen in the previous

module that this block diagram can be 
converted into an all digital scheme

where our band pass signal s of n gets 
filtered by the discrete time equivalent

of the channel. 
And, gives us a filtered version of the

bandpass signal, as 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 of 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

s e of 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 the 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. 
So the filter that compensates for the

distortion introduced by the 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 DPS 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 boot strapping.

The transmitter will send a prearranged 
sequence of symbols to the receiver.

So let's call the sequence of symbols a t 
of n.

This gets modulated and generates a pass 
band signal s of n.

Now at the receiver the sequence a t of 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 pass band signal s 
of n.

And so, for the bootstrapping part of the 
adaptation, we actually have an exact

copy of the transmitted pass band signal 
that we can use to drive the adaptation

of the coefficients. 
The training 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're 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'll be the topic of your next signal

processing class.