[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.63,0:00:04.38,Default,,0000,0000,0000,,Hi, and welcome to module 9.5 of digital \Nsignal processing.
Dialogue: 0,0:00:04.38,0:00:08.87,Default,,0000,0000,0000,,In this module, we will touch briefly on \Nsome topics in receiver design.
Dialogue: 0,0:00:08.87,0:00:12.25,Default,,0000,0000,0000,,A lot of things, unfortunately, happen to \Nthe signal while it's traveling through
Dialogue: 0,0:00:12.25,0:00:15.24,Default,,0000,0000,0000,,the channel. \NThe signal picks up noise, we have seen
Dialogue: 0,0:00:15.24,0:00:18.42,Default,,0000,0000,0000,,that already. \NIt also gets distorted because the
Dialogue: 0,0:00:18.42,0:00:21.70,Default,,0000,0000,0000,,channel will act as some sort of filter \Nthat is not necessarily all pass and
Dialogue: 0,0:00:21.70,0:00:25.65,Default,,0000,0000,0000,,linear phase. \NInterference happens too, that might be
Dialogue: 0,0:00:25.65,0:00:29.86,Default,,0000,0000,0000,,parts of the channel that we thought were \Nusable, and they're actually not, so the
Dialogue: 0,0:00:29.86,0:00:33.89,Default,,0000,0000,0000,,receiver really has to deal with a copy \Nof the transmitted signal that is very,
Dialogue: 0,0:00:33.89,0:00:40.46,Default,,0000,0000,0000,,very far from the idealized version we \Nhave seen so far.
Dialogue: 0,0:00:40.46,0:00:44.74,Default,,0000,0000,0000,,The way receivers, especially digital \Nreceivers, can cope with the distortions
Dialogue: 0,0:00:44.74,0:00:48.71,Default,,0000,0000,0000,,and noise introduced by the channel, is \Nby implemented adaptive filtering
Dialogue: 0,0:00:48.71,0:00:52.62,Default,,0000,0000,0000,,techniques. \NNow we will not have the time to go into
Dialogue: 0,0:00:52.62,0:00:56.36,Default,,0000,0000,0000,,very many details about adaptive signal \Nprocessing, again these are topics that
Dialogue: 0,0:00:56.36,0:01:01.38,Default,,0000,0000,0000,,you will be able to study in more \Nadvanced signal processing classes.
Dialogue: 0,0:01:01.38,0:01:05.54,Default,,0000,0000,0000,,But I think it is important to give you \Nan overview of things that have to happen
Dialogue: 0,0:01:05.54,0:01:10.39,Default,,0000,0000,0000,,inside a receiver, inside your ADSL \Nreceiver for instance.
Dialogue: 0,0:01:10.39,0:01:13.84,Default,,0000,0000,0000,,So you can enjoy this high data rates \Nthat are available today.
Dialogue: 0,0:01:13.84,0:01:18.33,Default,,0000,0000,0000,,And the first technique that we will look \Nat is adaptive equalization and then we
Dialogue: 0,0:01:18.33,0:01:22.35,Default,,0000,0000,0000,,will look at some very simple timing \Nrecovery that it used in practice in
Dialogue: 0,0:01:22.35,0:01:27.94,Default,,0000,0000,0000,,receivers. \NLet's begin with a blast from the past.
Dialogue: 0,0:01:27.94,0:01:33.22,Default,,0000,0000,0000,,[NOISE] Those of you that are a little \Nbit older will certainly have recognized
Dialogue: 0,0:01:33.22,0:01:38.34,Default,,0000,0000,0000,,this sound as the obligatory soundtrack \Nevery time you used to connect to the
Dialogue: 0,0:01:38.34,0:01:43.85,Default,,0000,0000,0000,,internet. \NAnd indeed this is the sound made by a
Dialogue: 0,0:01:43.85,0:01:49.47,Default,,0000,0000,0000,,V34 modem that was the standard dial up \Nconnection device seen in the 90s until
Dialogue: 0,0:01:49.47,0:01:54.89,Default,,0000,0000,0000,,the early 2000. \NNow if you have ever used a modem, you've
Dialogue: 0,0:01:54.89,0:02:00.90,Default,,0000,0000,0000,,heard the sound and you probably wondered \Nwhat was going on, so we're going to
Dialogue: 0,0:02:00.90,0:02:06.84,Default,,0000,0000,0000,,analyze what we just heard from the \Ngraphical point of view.
Dialogue: 0,0:02:06.84,0:02:11.13,Default,,0000,0000,0000,,If we look at the block diagram for the \Nreceiver once again, what we're going to
Dialogue: 0,0:02:11.13,0:02:15.42,Default,,0000,0000,0000,,do is we're going to plot the base band \Ncomplex samples as points on the complex
Dialogue: 0,0:02:15.42,0:02:20.27,Default,,0000,0000,0000,,blank. \NSo we're going to take B, r of n has the
Dialogue: 0,0:02:20.27,0:02:26.23,Default,,0000,0000,0000,,as the horizontal coordinate and b, i of \Nn as the vertical coordinate.
Dialogue: 0,0:02:26.23,0:02:30.43,Default,,0000,0000,0000,,And before we do so let's just look for a \Nsecond at what happens inside the
Dialogue: 0,0:02:30.43,0:02:35.12,Default,,0000,0000,0000,,receiver when the signal at the input is \Na simple sinusoid, like cosine of omega c
Dialogue: 0,0:02:35.12,0:02:41.30,Default,,0000,0000,0000,,plus omega zero n. \NWe are demodulating this very simple
Dialogue: 0,0:02:41.30,0:02:45.96,Default,,0000,0000,0000,,signal with the two carriers, the cosine \Nof omega c n and sine of omega c n.
Dialogue: 0,0:02:45.96,0:02:51.10,Default,,0000,0000,0000,,And then we're filtering the result with \Na low pass field.
Dialogue: 0,0:02:51.10,0:02:54.48,Default,,0000,0000,0000,,So if we work out this formula with \Nstandard trigonometric identities, we can
Dialogue: 0,0:02:54.48,0:02:59.54,Default,,0000,0000,0000,,always express for instance the product \Nof two cosine functions as the sum.
Dialogue: 0,0:02:59.54,0:03:03.68,Default,,0000,0000,0000,,Of the cosine of the sum of the angles \Nplus the cosine of the difference of the
Dialogue: 0,0:03:03.68,0:03:06.44,Default,,0000,0000,0000,,angles. \NAnd same for the product of cosine and
Dialogue: 0,0:03:06.44,0:03:09.61,Default,,0000,0000,0000,,sine. \NSo if we do that, we get four terms, two
Dialogue: 0,0:03:09.61,0:03:16.52,Default,,0000,0000,0000,,of which have a frequency that will fall \Noutside of the pass band of the filter h.
Dialogue: 0,0:03:16.52,0:03:20.36,Default,,0000,0000,0000,,So when we apply the filter to these \Nterms, we're left only with cosine of
Dialogue: 0,0:03:20.36,0:03:26.70,Default,,0000,0000,0000,,omega 0 n plus j sine of omega 0 n. \NWhich is of course e to the j omega 0 n.
Dialogue: 0,0:03:26.70,0:03:30.93,Default,,0000,0000,0000,,So when the input to the receiver is a \Ncosine, the points in the complex
Dialogue: 0,0:03:30.93,0:03:35.82,Default,,0000,0000,0000,,baseband sequence will be points around \Nthe circle and a difference between two
Dialogue: 0,0:03:35.82,0:03:42.71,Default,,0000,0000,0000,,successive points is the angle omega 0. \NThe reason why we might be called to
Dialogue: 0,0:03:42.71,0:03:46.41,Default,,0000,0000,0000,,demodulate a simple sinusoid is because \Nthe receiver will send what are called
Dialogue: 0,0:03:46.41,0:03:49.83,Default,,0000,0000,0000,,pile tones, simple sinusoids that are \Nused to probe the line and gauge the
Dialogue: 0,0:03:49.83,0:03:54.73,Default,,0000,0000,0000,,response of the channel at particular \Nfrequencies.
Dialogue: 0,0:03:54.73,0:03:58.13,Default,,0000,0000,0000,,So with this in mind, let's look at the \Nslow-motion analysis of the base band
Dialogue: 0,0:03:58.13,0:04:01.74,Default,,0000,0000,0000,,signals. \NSamples, when the input is the audio file
Dialogue: 0,0:04:01.74,0:04:06.44,Default,,0000,0000,0000,,we just heard before. \NSo lets start with a part that goes like
Dialogue: 0,0:04:06.44,0:04:09.71,Default,,0000,0000,0000,,this[SOUND]. \NThis signal contains several sinusoids,
Dialogue: 0,0:04:09.71,0:04:14.78,Default,,0000,0000,0000,,that we can see here in the plot. \NAnd the sinusoids also contain abrupt
Dialogue: 0,0:04:14.78,0:04:18.46,Default,,0000,0000,0000,,phase reversal, meaning that, at some \Ngiven points in time The phase of the
Dialogue: 0,0:04:18.46,0:04:23.32,Default,,0000,0000,0000,,sinusoid is augmented by pi. \NYou can see this as this small explosions
Dialogue: 0,0:04:23.32,0:04:27.81,Default,,0000,0000,0000,,in the circular pattern in the plot. \NThese phase reversals, are used by the
Dialogue: 0,0:04:27.81,0:04:31.33,Default,,0000,0000,0000,,transmitter and the receiver, as time \Nmarkers, to estimate the propagation
Dialogue: 0,0:04:31.33,0:04:35.45,Default,,0000,0000,0000,,delay of the signal, from source to \Ndestination.
Dialogue: 0,0:04:35.45,0:04:38.84,Default,,0000,0000,0000,,The next part goes like this. \N[NOISE].
Dialogue: 0,0:04:38.84,0:04:42.61,Default,,0000,0000,0000,,And this is a training sequence. \NThe transmitter sends a sequence of known
Dialogue: 0,0:04:42.61,0:04:46.67,Default,,0000,0000,0000,,symbols, namely the receiver knows the \Nsymbol that are transmitted.
Dialogue: 0,0:04:46.67,0:04:49.80,Default,,0000,0000,0000,,And so the receiver can use this \Nknowledge to train an equalizer to undo
Dialogue: 0,0:04:49.80,0:04:53.88,Default,,0000,0000,0000,,the affects of the channel. \NThe last part is the data transmission
Dialogue: 0,0:04:53.88,0:04:57.45,Default,,0000,0000,0000,,proper, the noisy part if you want, of \Nthe audio file.
Dialogue: 0,0:04:57.45,0:05:00.99,Default,,0000,0000,0000,,And the interesting thing is that the \Ntransmitter and receiver perform a
Dialogue: 0,0:05:00.99,0:05:04.88,Default,,0000,0000,0000,,handshake procedure, using a very low bit \Nrate QAM transmission using only four
Dialogue: 0,0:05:04.88,0:05:10.63,Default,,0000,0000,0000,,points, therefore two bits per symbol. \NTo exchange the parameters of the real
Dialogue: 0,0:05:10.63,0:05:13.78,Default,,0000,0000,0000,,data transmission that is going to \Nfollow.
Dialogue: 0,0:05:13.78,0:05:16.39,Default,,0000,0000,0000,,The speed, the constellation size, and so \Non.
Dialogue: 0,0:05:16.39,0:05:20.25,Default,,0000,0000,0000,,Using the four point QAM constellation in \Nthe beginning ensures that, even in very
Dialogue: 0,0:05:20.25,0:05:23.85,Default,,0000,0000,0000,,noisy conditions, transmitter and \Nreceiver can exchange their vital
Dialogue: 0,0:05:23.85,0:05:29.50,Default,,0000,0000,0000,,information. \NSo even from this simple qualitative
Dialogue: 0,0:05:29.50,0:05:33.27,Default,,0000,0000,0000,,description of what happens in a real \Ncommunication scenario, we can see that
Dialogue: 0,0:05:33.27,0:05:38.13,Default,,0000,0000,0000,,the task that the receiver is saddled \Nwith is very complicated.
Dialogue: 0,0:05:38.13,0:05:40.76,Default,,0000,0000,0000,,So it's a dirty job, but a receiver has \Nto do it.
Dialogue: 0,0:05:40.76,0:05:45.45,Default,,0000,0000,0000,,And a receiver has to cope with four \Npotential sources of problem.
Dialogue: 0,0:05:45.45,0:05:49.90,Default,,0000,0000,0000,,Interference. \NThe propagation delay, so the delay
Dialogue: 0,0:05:49.90,0:05:52.82,Default,,0000,0000,0000,,introduced by the channel. \NThe linear distortion introduced by the
Dialogue: 0,0:05:52.82,0:05:55.60,Default,,0000,0000,0000,,channel. \NAnd drifts in internal clocks between the
Dialogue: 0,0:05:55.60,0:06:00.94,Default,,0000,0000,0000,,digital system inside the transmitter and \Nthe digital system inside the receiver.
Dialogue: 0,0:06:00.94,0:06:04.83,Default,,0000,0000,0000,,So when it comes to interference the \Nhandshake procedure, and the line probing
Dialogue: 0,0:06:04.83,0:06:08.26,Default,,0000,0000,0000,,pilot tones are used in clever ways to \Ncircumvent the major sources of
Dialogue: 0,0:06:08.26,0:06:13.00,Default,,0000,0000,0000,,interference. \NWe will see some example later on when we
Dialogue: 0,0:06:13.00,0:06:16.41,Default,,0000,0000,0000,,discuss ADSL. \NThe propagation delay, is tackled by a
Dialogue: 0,0:06:16.41,0:06:20.99,Default,,0000,0000,0000,,delay estimation procedure, that we will \Nlook at in just a second.
Dialogue: 0,0:06:20.99,0:06:24.99,Default,,0000,0000,0000,,The distortion to this by the channel is \Ncompensated using adaptive equalization
Dialogue: 0,0:06:24.99,0:06:29.13,Default,,0000,0000,0000,,techniques, and we will see some examples \Nof that as well.
Dialogue: 0,0:06:29.13,0:06:33.16,Default,,0000,0000,0000,,And clock drifts are tackled by timing \Nrecovery techniques that in and of
Dialogue: 0,0:06:33.16,0:06:37.64,Default,,0000,0000,0000,,themselves are quite sophisticated and \Ntherefore we leave them to more advanced
Dialogue: 0,0:06:37.64,0:06:41.92,Default,,0000,0000,0000,,classes. \NGraphically, if we sum up the chain of
Dialogue: 0,0:06:41.92,0:06:45.74,Default,,0000,0000,0000,,events that occur between the \Ntransmissions of the original digital
Dialogue: 0,0:06:45.74,0:06:51.44,Default,,0000,0000,0000,,signal and the beginning of the \Ndemodulation of the received signal.
Dialogue: 0,0:06:51.44,0:06:56.66,Default,,0000,0000,0000,,We have a digital to analog converter and \Na transmitter, this is transmitter part
Dialogue: 0,0:06:56.66,0:07:01.30,Default,,0000,0000,0000,,of the chain that operates with a given \Nsample period Ts.
Dialogue: 0,0:07:01.30,0:07:05.50,Default,,0000,0000,0000,,This generates an analog signal which is \Nsent over a channel.
Dialogue: 0,0:07:05.50,0:07:08.47,Default,,0000,0000,0000,,We can represent the channel for the time \Nbeing as a linear filter in the
Dialogue: 0,0:07:08.47,0:07:13.19,Default,,0000,0000,0000,,continuous time domain. \NWith frequency response d of j omega.
Dialogue: 0,0:07:13.19,0:07:17.28,Default,,0000,0000,0000,,At the input of the receiver, we have a \Ncontinuous time signal, s hat of t.
Dialogue: 0,0:07:17.28,0:07:22.69,Default,,0000,0000,0000,,Which is a distorted and delayed version \Nof the original analog signal.
Dialogue: 0,0:07:22.69,0:07:27.60,Default,,0000,0000,0000,,We will neglect noise for the time being. \NThis signal is sampled by an a to d
Dialogue: 0,0:07:27.60,0:07:32.11,Default,,0000,0000,0000,,converter that operates at a period t \Nprime of s.
Dialogue: 0,0:07:32.11,0:07:36.10,Default,,0000,0000,0000,,And we obtain the sequence of samples \Nthat will be input to the modulator.
Dialogue: 0,0:07:36.10,0:07:38.21,Default,,0000,0000,0000,,So this is the receiver part of the \Nchain.
Dialogue: 0,0:07:38.21,0:07:41.62,Default,,0000,0000,0000,,We have to take into account the \Ndistortion introduced by the channel, and
Dialogue: 0,0:07:41.62,0:07:45.85,Default,,0000,0000,0000,,we have to take into account the \Npotentially time varying discrepancies in
Dialogue: 0,0:07:45.85,0:07:49.48,Default,,0000,0000,0000,,the clocks between the transmitter and \Nthe receiver.
Dialogue: 0,0:07:49.48,0:07:52.98,Default,,0000,0000,0000,,These two systems are geographically \Nremote and there is no guarantee that the
Dialogue: 0,0:07:52.98,0:07:56.20,Default,,0000,0000,0000,,two internal clocks that're used in the A \Nto D and D to A converters are
Dialogue: 0,0:07:56.20,0:08:00.31,Default,,0000,0000,0000,,synchronized or run exactly at the same \Nfrequency.
Dialogue: 0,0:08:00.31,0:08:03.28,Default,,0000,0000,0000,,Let's start with problem of delay \Ncompensation.
Dialogue: 0,0:08:03.28,0:08:07.10,Default,,0000,0000,0000,,To simplify the analysis we'll assume \Nthat the clocks at transmitter and
Dialogue: 0,0:08:07.10,0:08:10.35,Default,,0000,0000,0000,,receiver are synchronized and \Nsynchronous.
Dialogue: 0,0:08:10.35,0:08:14.47,Default,,0000,0000,0000,,So T prime of s is equal to T s, and the \Nchannel acts as a simple delay.
Dialogue: 0,0:08:14.47,0:08:17.37,Default,,0000,0000,0000,,So the received signal is simply a \Ndelayed version of the transmitted
Dialogue: 0,0:08:17.37,0:08:20.82,Default,,0000,0000,0000,,signal. \NWhich implies that the frequency response
Dialogue: 0,0:08:20.82,0:08:24.11,Default,,0000,0000,0000,,of the channel is simply e to the minus j \Nomega d.
Dialogue: 0,0:08:24.11,0:08:27.00,Default,,0000,0000,0000,,So, the channel introduces a delay of d \Nseconds.
Dialogue: 0,0:08:27.00,0:08:30.80,Default,,0000,0000,0000,,We can express this in samples in the \Nfollowing way.
Dialogue: 0,0:08:30.80,0:08:34.79,Default,,0000,0000,0000,,We write d as the product of the sampling \Nperiod, times b plus tau.
Dialogue: 0,0:08:34.79,0:08:38.41,Default,,0000,0000,0000,,Where b is an integer. \NAnd tau is strictly less than one-half in
Dialogue: 0,0:08:38.41,0:08:41.38,Default,,0000,0000,0000,,magnitude. \NSo b is called the bulk delay because it
Dialogue: 0,0:08:41.38,0:08:45.28,Default,,0000,0000,0000,,gives us an integer number of samples of \Ndelay at the receiver and tau is the
Dialogue: 0,0:08:45.28,0:08:50.85,Default,,0000,0000,0000,,fractional delay. \NSo the fraction of samples introduced by
Dialogue: 0,0:08:50.85,0:08:56.48,Default,,0000,0000,0000,,the continuous time delay of d. \NSo, how do we compensate for this delay?
Dialogue: 0,0:08:56.48,0:08:59.56,Default,,0000,0000,0000,,Well, the bulk delay is rather easy to \Ntackle.
Dialogue: 0,0:08:59.56,0:09:02.79,Default,,0000,0000,0000,,Imagine the transmitter begins \Ntransmission by sending just an impulse
Dialogue: 0,0:09:02.79,0:09:06.21,Default,,0000,0000,0000,,over the channel. \NSo, the discreet time signal is this one,
Dialogue: 0,0:09:06.21,0:09:10.29,Default,,0000,0000,0000,,it's just a delta and a 0. \NIt gets sent to D-to-A converter.
Dialogue: 0,0:09:10.29,0:09:13.16,Default,,0000,0000,0000,,And the converter will output a \Ncontinuous time signal that looks like an
Dialogue: 0,0:09:13.16,0:09:17.37,Default,,0000,0000,0000,,interpolation function like the sinc. \NAnd like all interpolation functions, it
Dialogue: 0,0:09:17.37,0:09:21.42,Default,,0000,0000,0000,,will have a maximum peak at zero that \Ncorresponds to the known zero sample.
Dialogue: 0,0:09:21.42,0:09:23.54,Default,,0000,0000,0000,,This signal gets transmitted over the \Nchannel.
Dialogue: 0,0:09:23.54,0:09:27.10,Default,,0000,0000,0000,,And it gets to the receiver after a delay \Nd that we can estimate for instance by
Dialogue: 0,0:09:27.10,0:09:31.87,Default,,0000,0000,0000,,looking at the displacement of the peak \Nof the interpolation function.
Dialogue: 0,0:09:31.87,0:09:34.67,Default,,0000,0000,0000,,The receiver converts this into a \Ndiscrete time sequence.
Dialogue: 0,0:09:34.67,0:09:38.36,Default,,0000,0000,0000,,Now in the figure here it looks as if the \Nsample incidence at the transmitter and
Dialogue: 0,0:09:38.36,0:09:41.92,Default,,0000,0000,0000,,receiver are perfectly aligned. \NNow this is not necessarily the case
Dialogue: 0,0:09:41.92,0:09:44.96,Default,,0000,0000,0000,,because the starting time for the \Ninterpolator at the transmitter and the
Dialogue: 0,0:09:44.96,0:09:48.71,Default,,0000,0000,0000,,sampler at the receiver are not \Nnecessarily synchronous.
Dialogue: 0,0:09:48.71,0:09:52.79,Default,,0000,0000,0000,,But any difference in starting time can \Nbe integrated into the propagation delay
Dialogue: 0,0:09:52.79,0:09:56.31,Default,,0000,0000,0000,,as long as the sampling periods are the \Nsame.
Dialogue: 0,0:09:56.31,0:10:00.53,Default,,0000,0000,0000,,So with this, all we need to do at the \Nreceiver is to look for the maximum value
Dialogue: 0,0:10:00.53,0:10:05.56,Default,,0000,0000,0000,,in the sequence of samples. \NBecause of the shape of the interpolating
Dialogue: 0,0:10:05.56,0:10:09.85,Default,,0000,0000,0000,,function, we know that the real maximum \Nwill be at most half a sample in either
Dialogue: 0,0:10:09.85,0:10:13.55,Default,,0000,0000,0000,,direction of the location, of the maximum \Nsample value.
Dialogue: 0,0:10:13.55,0:10:17.39,Default,,0000,0000,0000,,So, add the receiver to offset the bulk \Ndelay, we will just set the nominal time
Dialogue: 0,0:10:17.39,0:10:20.75,Default,,0000,0000,0000,,n equal to 0, to coincide with the \Nlocation of the maximum value of the
Dialogue: 0,0:10:20.75,0:10:25.11,Default,,0000,0000,0000,,sample sequence. \NNow of course we need to compensate for
Dialogue: 0,0:10:25.11,0:10:28.72,Default,,0000,0000,0000,,the fractional delay, so we need to \Nestimate tau.
Dialogue: 0,0:10:28.72,0:10:31.23,Default,,0000,0000,0000,,And to do that we'll use a different \Ntechnique.
Dialogue: 0,0:10:31.23,0:10:34.58,Default,,0000,0000,0000,,Let me add in passing, that in real \Ncommunication devices, of course we're
Dialogue: 0,0:10:34.58,0:10:37.73,Default,,0000,0000,0000,,not using impulses to offset the bulk \Ndelay.
Dialogue: 0,0:10:37.73,0:10:41.29,Default,,0000,0000,0000,,Because impulses are full-band signals \Nand so they would be filtered out by the
Dialogue: 0,0:10:41.29,0:10:46.41,Default,,0000,0000,0000,,passband characteristic of the channel. \NThe trick is to embed discontinuities in
Dialogue: 0,0:10:46.41,0:10:51.31,Default,,0000,0000,0000,,pilot tones and to recognize this \Ndiscontinuities at the receiver.
Dialogue: 0,0:10:51.31,0:10:54.78,Default,,0000,0000,0000,,As we have seen in the animation at the \Nbeginning of this module, we use phase
Dialogue: 0,0:10:54.78,0:10:58.75,Default,,0000,0000,0000,,reversals, which are abrupt \Ndiscontinuities in sinusoids, to provide
Dialogue: 0,0:10:58.75,0:11:02.95,Default,,0000,0000,0000,,a recognizable instant in time for the \Nreceiver to latch on.
Dialogue: 0,0:11:02.95,0:11:07.70,Default,,0000,0000,0000,,Okay, so what about the fractional delay? \NWell, for the fractional delay, we use a
Dialogue: 0,0:11:07.70,0:11:11.86,Default,,0000,0000,0000,,sinusoid instead of a delta, so we build \Na bass band signal, which is simply a
Dialogue: 0,0:11:11.86,0:11:16.89,Default,,0000,0000,0000,,complex exponential at a known frequency, \Nomega 0.
Dialogue: 0,0:11:16.89,0:11:20.66,Default,,0000,0000,0000,,This will be converted to a real signal \Nbefore being sent to the D-to-A
Dialogue: 0,0:11:20.66,0:11:24.37,Default,,0000,0000,0000,,converter, and so what we transmit \Nactually is cosine of omega c, the
Dialogue: 0,0:11:24.37,0:11:30.27,Default,,0000,0000,0000,,carrier frequency, plus the pilot \Nfrequency omega 0, times n.
Dialogue: 0,0:11:30.27,0:11:34.56,Default,,0000,0000,0000,,The receiver will receive a delayed \Nversion of this, which contains the delay
Dialogue: 0,0:11:34.56,0:11:39.19,Default,,0000,0000,0000,,now in sample, and fraction of sample, b \Nplus tau.
Dialogue: 0,0:11:39.19,0:11:43.16,Default,,0000,0000,0000,,After we demodulate this cosine you \Nremember we got a complex exponential and
Dialogue: 0,0:11:43.16,0:11:47.90,Default,,0000,0000,0000,,we can also compensate already for the \Nbulk delay which we know.
Dialogue: 0,0:11:47.90,0:11:52.37,Default,,0000,0000,0000,,So for an integer number of sample b we \Nobtain a base band signal at b of n which
Dialogue: 0,0:11:52.37,0:11:58.32,Default,,0000,0000,0000,,is e to the j omega and minus tau. \NSince we know the frequency of omega 0 we
Dialogue: 0,0:11:58.32,0:12:02.42,Default,,0000,0000,0000,,can just multiply this quantity by e to \Nthe minus j omega 0 n and obtain e to the
Dialogue: 0,0:12:02.42,0:12:09.20,Default,,0000,0000,0000,,minus j omega 0 tau, which is a constant. \NAnd which we can invert given that we
Dialogue: 0,0:12:09.20,0:12:12.47,Default,,0000,0000,0000,,know the frequency omega 0. \NAnd so now we have an estimate for both
Dialogue: 0,0:12:12.47,0:12:16.53,Default,,0000,0000,0000,,the bulk delay and the fractional delay. \NNow we have to bring back the signal to
Dialogue: 0,0:12:16.53,0:12:20.14,Default,,0000,0000,0000,,the original timing. \NThe bulk delay is really no problem.
Dialogue: 0,0:12:20.14,0:12:23.55,Default,,0000,0000,0000,,It's just an integer number of samples. \NWhat creates a problem is the fractional
Dialogue: 0,0:12:23.55,0:12:28.38,Default,,0000,0000,0000,,delay because that will shift the peaks \Nwith respect to the sampling intervals.
Dialogue: 0,0:12:28.38,0:12:33.27,Default,,0000,0000,0000,,So if we want to compensate for the bulk \Ndelay we need to compute subsample values
Dialogue: 0,0:12:33.27,0:12:37.86,Default,,0000,0000,0000,,and in theory to do that we should use a \Nsinc fractional delay namely a filter
Dialogue: 0,0:12:37.86,0:12:45.20,Default,,0000,0000,0000,,with impulse response sinc of n plus tau. \NIn practice however, we will use a local
Dialogue: 0,0:12:45.20,0:12:48.56,Default,,0000,0000,0000,,interpolation and this is a very \Npractical application of the Lagrange
Dialogue: 0,0:12:48.56,0:12:52.56,Default,,0000,0000,0000,,interpolation technique that we saw in \Nmodule 6.2.
Dialogue: 0,0:12:52.56,0:12:56.22,Default,,0000,0000,0000,,So graphically the situation is like so, \Nwe have a stream of samples coming in.
Dialogue: 0,0:12:56.22,0:13:00.83,Default,,0000,0000,0000,,And for each sample, we want to compute \Nthe subsample value with a distance of
Dialogue: 0,0:13:00.83,0:13:06.11,Default,,0000,0000,0000,,tau from the nearest sample interval. \NAnd we want to only use a local
Dialogue: 0,0:13:06.11,0:13:11.51,Default,,0000,0000,0000,,neighborhood of samples to estimate this. \NNow, you remember from module 6.2.
Dialogue: 0,0:13:11.51,0:13:14.99,Default,,0000,0000,0000,,The Lagrange approximation works by \Nbuilding a linear combination of Lagrange
Dialogue: 0,0:13:14.99,0:13:18.31,Default,,0000,0000,0000,,polynomials weighed by the samples of the \Nfunction.
Dialogue: 0,0:13:18.31,0:13:21.82,Default,,0000,0000,0000,,So, as per usual, we choose the sampling \Ninterval equal to 1, so that we lighten
Dialogue: 0,0:13:21.82,0:13:25.88,Default,,0000,0000,0000,,the notation. \NWe have a continuous time function x of
Dialogue: 0,0:13:25.88,0:13:29.86,Default,,0000,0000,0000,,t, and we want to compute x of n plus \Ntau, with tau less than one half in
Dialogue: 0,0:13:29.86,0:13:34.76,Default,,0000,0000,0000,,magnitude. \NSo we have samples of this function at
Dialogue: 0,0:13:34.76,0:13:39.50,Default,,0000,0000,0000,,integers, n, and the local Lagrange \Napproximation around n, is given by this
Dialogue: 0,0:13:39.50,0:13:44.67,Default,,0000,0000,0000,,linear combination of Lagrange \Npolynomials.
Dialogue: 0,0:13:44.67,0:13:48.56,Default,,0000,0000,0000,,Weighted by the samples of the functions \Naround the approximation point.
Dialogue: 0,0:13:48.56,0:13:54.10,Default,,0000,0000,0000,,So we use the notation x L of n and t. \NN is the center point and t is the value
Dialogue: 0,0:13:54.10,0:13:59.76,Default,,0000,0000,0000,,from the center point at which we want to \Ncompute the approximation.
Dialogue: 0,0:13:59.76,0:14:03.24,Default,,0000,0000,0000,,And the Lagrange polynomials are given by \Nthis formula here, which is the same as
Dialogue: 0,0:14:03.24,0:14:07.52,Default,,0000,0000,0000,,in module 6.2. \NSo the delayed compensated input signal
Dialogue: 0,0:14:07.52,0:14:11.30,Default,,0000,0000,0000,,will be set equal to the Lagrange \Napproximation at tau.
Dialogue: 0,0:14:11.30,0:14:14.12,Default,,0000,0000,0000,,So let's look at an example. \NAssume that we want a second order
Dialogue: 0,0:14:14.12,0:14:17.47,Default,,0000,0000,0000,,approximation. \NSo we pick N equal to 1 and we will have
Dialogue: 0,0:14:17.47,0:14:22.85,Default,,0000,0000,0000,,three Lagrange polynomials. \NAnd so, we will need to use three samples
Dialogue: 0,0:14:22.85,0:14:28.14,Default,,0000,0000,0000,,of the sequence to compute interpolation. \NThese three polynomials will be centered
Dialogue: 0,0:14:28.14,0:14:31.85,Default,,0000,0000,0000,,in n minus 1 and in n plus 1 and scaled \Nby the values of the samples at these
Dialogue: 0,0:14:31.85,0:14:35.83,Default,,0000,0000,0000,,locations. \NAnd finally, we will sum the poll numbers
Dialogue: 0,0:14:35.83,0:14:38.61,Default,,0000,0000,0000,,together and compute their value in n \Nplus tau.
Dialogue: 0,0:14:38.61,0:14:42.50,Default,,0000,0000,0000,,So, we start with the first one, which is \Ncentered in n minus 1.
Dialogue: 0,0:14:42.50,0:14:47.11,Default,,0000,0000,0000,,And, like all interpolation polynomials, \Nits value is 1 in n minus 1, and 0, at
Dialogue: 0,0:14:47.11,0:14:52.99,Default,,0000,0000,0000,,other integer values of the argument. \NThe second polynomial will be centered in
Dialogue: 0,0:14:52.99,0:14:56.76,Default,,0000,0000,0000,,n, and the third polynomial will be \Ncentered in n plus 1.
Dialogue: 0,0:14:56.76,0:14:59.88,Default,,0000,0000,0000,,When we sum them together, we obtain a \Nsecond order curve that goes through the
Dialogue: 0,0:14:59.88,0:15:02.76,Default,,0000,0000,0000,,points, that interpolates the three \Npoints, and then we can compute the
Dialogue: 0,0:15:02.76,0:15:07.68,Default,,0000,0000,0000,,approximation as the value of this curve \Nin n plus tau.
Dialogue: 0,0:15:07.68,0:15:12.30,Default,,0000,0000,0000,,Now the nice thing about this approach, \Nis that if we look at the approximation,
Dialogue: 0,0:15:12.30,0:15:16.69,Default,,0000,0000,0000,,if we take the Lagrange approximation \Naround n.
Dialogue: 0,0:15:16.69,0:15:20.96,Default,,0000,0000,0000,,We can define a set of coefficients, d \Ntau of k, which are the values of each
Dialogue: 0,0:15:20.96,0:15:26.61,Default,,0000,0000,0000,,Lagrange polynomial in tau. \NSo d tau of k, are 2 N plus 1 values, the
Dialogue: 0,0:15:26.61,0:15:32.87,Default,,0000,0000,0000,,form, the coefficients, of an FIR filter. \NAnd we can compute the value of the
Dialogue: 0,0:15:32.87,0:15:37.22,Default,,0000,0000,0000,,Lagrange approximation simply as the \Nconvolution of the incoming sequence with
Dialogue: 0,0:15:37.22,0:15:42.48,Default,,0000,0000,0000,,this interpolation filter. \NSo for example, if these are the three
Dialogue: 0,0:15:42.48,0:15:47.45,Default,,0000,0000,0000,,Lagrange polynomials for n equal to 1, we \Ncan compute these polynomials for t equal
Dialogue: 0,0:15:47.45,0:15:54.44,Default,,0000,0000,0000,,to tau, where tau is the fractional delay \Nthat we estimated before.
Dialogue: 0,0:15:54.44,0:15:58.73,Default,,0000,0000,0000,,And we will obtain three coefficients, \Nlike here, for instance, is an example
Dialogue: 0,0:15:58.73,0:16:02.74,Default,,0000,0000,0000,,for tau equal to 0.2. \NThree coefficients that give us an FIR
Dialogue: 0,0:16:02.74,0:16:05.81,Default,,0000,0000,0000,,filter, and then we can just simply \Nfilter the samples coming into the
Dialogue: 0,0:16:05.81,0:16:10.53,Default,,0000,0000,0000,,receiver with this filter, to compensate \Nfor the fractional delay.
Dialogue: 0,0:16:10.53,0:16:13.13,Default,,0000,0000,0000,,So again, the algorithm is, estimate the \Nfractional delay.
Dialogue: 0,0:16:13.13,0:16:16.95,Default,,0000,0000,0000,,The bulk delay is no problem, again. \NCompute the 2 N plus 1 Lagrangian
Dialogue: 0,0:16:16.95,0:16:19.98,Default,,0000,0000,0000,,coefficients and filter it with the \Nresulting FIR.
Dialogue: 0,0:16:19.98,0:16:23.69,Default,,0000,0000,0000,,The added advantage of this strategy is \Nthat if the delay changes over time for
Dialogue: 0,0:16:23.69,0:16:27.45,Default,,0000,0000,0000,,any reason, all we need to do is to keep \Nthe estimation running and update the FIR
Dialogue: 0,0:16:27.45,0:16:32.14,Default,,0000,0000,0000,,coefficients as the estimation changes \Nover time.
Dialogue: 0,0:16:32.14,0:16:35.82,Default,,0000,0000,0000,,Okay, now that we know how to compensate \Nfor the propagation delay introduced by
Dialogue: 0,0:16:35.82,0:16:39.28,Default,,0000,0000,0000,,the channel. \NLet's go see the rechannel it with an
Dialogue: 0,0:16:39.28,0:16:41.94,Default,,0000,0000,0000,,arbitrary frequency response D j of \Nomega.
Dialogue: 0,0:16:41.94,0:16:46.95,Default,,0000,0000,0000,,And the transmission chain goes from the \Npass band signal s of n, discreet time,
Dialogue: 0,0:16:46.95,0:16:51.89,Default,,0000,0000,0000,,into a D-to-A converter, analog signal s \Nof t, it gets filtered by the channel,
Dialogue: 0,0:16:51.89,0:16:59.55,Default,,0000,0000,0000,,gives us hat s of t, which is sampled at \Nthe receiver to give us.
Dialogue: 0,0:16:59.55,0:17:04.60,Default,,0000,0000,0000,,A received fast band signal, hat s of n. \NBut now we have seen in the previous
Dialogue: 0,0:17:04.60,0:17:07.78,Default,,0000,0000,0000,,module that this block diagram can be \Nconverted into an all digital scheme
Dialogue: 0,0:17:07.78,0:17:11.68,Default,,0000,0000,0000,,where our band pass signal s of n gets \Nfiltered by the discrete time equivalent
Dialogue: 0,0:17:11.68,0:17:17.45,Default,,0000,0000,0000,,of the channel. \NAnd, gives us a filtered version of the
Dialogue: 0,0:17:17.45,0:17:22.33,Default,,0000,0000,0000,,bandpass signal, as would appear inside \Nthe receiver.
Dialogue: 0,0:17:22.33,0:17:25.98,Default,,0000,0000,0000,,So the problem now, is that we would like \Nto undo the effects of the channel, on
Dialogue: 0,0:17:25.98,0:17:30.36,Default,,0000,0000,0000,,the transmitted signal. \NAnd the classic way to do that, is to
Dialogue: 0,0:17:30.36,0:17:34.71,Default,,0000,0000,0000,,filter the received signal hat s of n, by \Na filter E, that compensates for the
Dialogue: 0,0:17:34.71,0:17:40.23,Default,,0000,0000,0000,,distortion or the filtering introduced by \Nthe channel.
Dialogue: 0,0:17:40.23,0:17:45.25,Default,,0000,0000,0000,,So the target is that the output of the \Nfiltering operation gives us a signal hat
Dialogue: 0,0:17:45.25,0:17:50.38,Default,,0000,0000,0000,,s e of n, which is equal to the \Ntransmitted signal.
Dialogue: 0,0:17:50.38,0:17:53.13,Default,,0000,0000,0000,,How do we do that? \NIn theory, it would be enough to pick a
Dialogue: 0,0:17:53.13,0:17:56.92,Default,,0000,0000,0000,,transfer function for the filter E, which \Nis just the reciprocal of the equivalent
Dialogue: 0,0:17:56.92,0:18:01.95,Default,,0000,0000,0000,,transfer function of the channel. \NBut the problem is that we don't know the
Dialogue: 0,0:18:01.95,0:18:04.68,Default,,0000,0000,0000,,transfer function of the channel in \Nadvance because each time we transmit
Dialogue: 0,0:18:04.68,0:18:08.27,Default,,0000,0000,0000,,data over the channel, this transfer \Nfunction may change.
Dialogue: 0,0:18:08.27,0:18:12.65,Default,,0000,0000,0000,,And also, even while we're transmitting \Ndata, the transfer function might change
Dialogue: 0,0:18:12.65,0:18:16.20,Default,,0000,0000,0000,,because it is a physical system that \Nmight be subject to.
Dialogue: 0,0:18:16.20,0:18:19.74,Default,,0000,0000,0000,,Drifts and modifications. \NSo what do we do?
Dialogue: 0,0:18:19.74,0:18:25.56,Default,,0000,0000,0000,,We need to use adaptive equalization. \NSo the filter that compensates for the
Dialogue: 0,0:18:25.56,0:18:30.41,Default,,0000,0000,0000,,distortion introduced by the channel is \Ncalled an equalizer.
Dialogue: 0,0:18:30.41,0:18:34.67,Default,,0000,0000,0000,,And what we want to do Is to change the \Nfilter in time, so change the filter
Dialogue: 0,0:18:34.67,0:18:39.36,Default,,0000,0000,0000,,coefficients in a DPS realization as a \Nfunction of the error that we obtain when
Dialogue: 0,0:18:39.36,0:18:47.61,Default,,0000,0000,0000,,we compare the output of the filter with \Nthe signal that we would like to obtain.
Dialogue: 0,0:18:47.61,0:18:52.26,Default,,0000,0000,0000,,In our case the signal that we would like \Nto obtain is the transmitted signal.
Dialogue: 0,0:18:52.26,0:18:56.66,Default,,0000,0000,0000,,And so we take the received signal, we \Nfilter it with the equalizer.
Dialogue: 0,0:18:56.66,0:19:00.72,Default,,0000,0000,0000,,We look at the result. \NWe take the difference, with respect to
Dialogue: 0,0:19:00.72,0:19:04.44,Default,,0000,0000,0000,,the original signal, and we use the \Nerror, which should be zero in the ideal
Dialogue: 0,0:19:04.44,0:19:08.58,Default,,0000,0000,0000,,case, to drive the adaptation of the \Nequalizer.
Dialogue: 0,0:19:08.58,0:19:13.90,Default,,0000,0000,0000,,But wait, how do we get the exact \Ntransmitted signal at the receiver?
Dialogue: 0,0:19:13.90,0:19:17.34,Default,,0000,0000,0000,,Well, we use two tricks. \NThe first one is boot strapping.
Dialogue: 0,0:19:17.34,0:19:22.73,Default,,0000,0000,0000,,The transmitter will send a prearranged \Nsequence of symbols to the receiver.
Dialogue: 0,0:19:22.73,0:19:26.47,Default,,0000,0000,0000,,So let's call the sequence of symbols a t \Nof n.
Dialogue: 0,0:19:26.47,0:19:31.72,Default,,0000,0000,0000,,This gets modulated and generates a pass \Nband signal s of n.
Dialogue: 0,0:19:31.72,0:19:35.99,Default,,0000,0000,0000,,Now at the receiver the sequence a t of n \Nis known.
Dialogue: 0,0:19:35.99,0:19:41.97,Default,,0000,0000,0000,,And the receiver has an exact copy of the \Nmodulator, of the transmitter, inside of
Dialogue: 0,0:19:41.97,0:19:46.48,Default,,0000,0000,0000,,itself. \NSo the transmitter can generate locally,
Dialogue: 0,0:19:46.48,0:19:50.14,Default,,0000,0000,0000,,an exact copy of the pass band signal s \Nof n.
Dialogue: 0,0:19:50.14,0:19:54.43,Default,,0000,0000,0000,,And so, for the bootstrapping part of the \Nadaptation, we actually have an exact
Dialogue: 0,0:19:54.43,0:19:58.72,Default,,0000,0000,0000,,copy of the transmitted pass band signal \Nthat we can use to drive the adaptation
Dialogue: 0,0:19:58.72,0:20:03.89,Default,,0000,0000,0000,,of the coefficients. \NThe training sequence is just long enough
Dialogue: 0,0:20:03.89,0:20:06.62,Default,,0000,0000,0000,,to bring the equalizer to a workable \Nstate.
Dialogue: 0,0:20:06.62,0:20:10.29,Default,,0000,0000,0000,,For the handshake procedure that we saw \Nin the video before, for instance.
Dialogue: 0,0:20:10.29,0:20:13.91,Default,,0000,0000,0000,,This would correspond to the moment where \Nthe receiver starts demodulating the four
Dialogue: 0,0:20:13.91,0:20:17.55,Default,,0000,0000,0000,,point QIM. \NAt that moment, the receiver will switch
Dialogue: 0,0:20:17.55,0:20:20.89,Default,,0000,0000,0000,,strategy and implement a data driven \Nadaptation.
Dialogue: 0,0:20:20.89,0:20:25.77,Default,,0000,0000,0000,,The thing works like this. \NThe received signal gets equalized, gets
Dialogue: 0,0:20:25.77,0:20:30.61,Default,,0000,0000,0000,,demodulated and then the slicer will \Nrecover the sequence of transmitted
Dialogue: 0,0:20:30.61,0:20:34.11,Default,,0000,0000,0000,,symbols. \NSince the receiver has a copy of the
Dialogue: 0,0:20:34.11,0:20:38.88,Default,,0000,0000,0000,,transmitter inside of itself, it can use \Nthe sequence of transmitted symbol.
Dialogue: 0,0:20:38.88,0:20:42.43,Default,,0000,0000,0000,,To build a local copy of the transmitted \Nsignal.
Dialogue: 0,0:20:42.43,0:20:47.12,Default,,0000,0000,0000,,Now of course, errors might happen in the \Nslicing process, and so this local copy
Dialogue: 0,0:20:47.12,0:20:52.50,Default,,0000,0000,0000,,is not completely error-free. \NBut the assumption is that the equalizer
Dialogue: 0,0:20:52.50,0:20:56.27,Default,,0000,0000,0000,,is doing already enough of a good job to \Nkeep the number of errors in this
Dialogue: 0,0:20:56.27,0:21:01.55,Default,,0000,0000,0000,,sequence sufficiently low. \NSo that the difference, with respect to
Dialogue: 0,0:21:01.55,0:21:05.71,Default,,0000,0000,0000,,the received signal, is enough to refine \Nthe adaptation of the equalizer, and
Dialogue: 0,0:21:05.71,0:21:11.80,Default,,0000,0000,0000,,especially to track the time varying \Nconditions of the channel.
Dialogue: 0,0:21:11.80,0:21:14.29,Default,,0000,0000,0000,,What we have seen, is just a qualitative \Noverview of what happens inside of a
Dialogue: 0,0:21:14.29,0:21:17.76,Default,,0000,0000,0000,,receiver. \NAnd there're still so many questions that
Dialogue: 0,0:21:17.76,0:21:21.94,Default,,0000,0000,0000,,we would have to answer to be thorough. \NFor instance, how do we carry out the
Dialogue: 0,0:21:21.94,0:21:25.14,Default,,0000,0000,0000,,adaptation of the coefficients in the \Nequalizer?
Dialogue: 0,0:21:25.14,0:21:30.39,Default,,0000,0000,0000,,How do we compensate for different clock \Nrates in geographically diverse receivers
Dialogue: 0,0:21:30.39,0:21:34.54,Default,,0000,0000,0000,,and transmitters? \NHow do we recover from the interference
Dialogue: 0,0:21:34.54,0:21:40.90,Default,,0000,0000,0000,,from other transmission devices, and how \Ndo we improve the resilience to noise?
Dialogue: 0,0:21:40.90,0:21:45.50,Default,,0000,0000,0000,,The answers to all those questions \Nrequire a much deeper understanding of
Dialogue: 0,0:21:45.50,0:21:50.65,Default,,0000,0000,0000,,adaptive signal processing, and hopefully \Nthat'll be the topic of your next signal
Dialogue: 0,0:21:50.65,0:21:54.12,Default,,0000,0000,0000,,processing class.