[NOISE] Those of you that are older will certainly have recognized the 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 in the 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've 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 pilot the base band complex samples as points on the complex plane. So we're going to take b r of n 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 0 n. We are demodulating this very simple signal with the two carriers, the cosine of omega cn, and sine of omega cn. And then we're filtering the result with a lowpass filter. So, if we work out this formula of standard trigonometric identities. We can always express the product of two cosin functions as the sum of the cosin of the sum of the angles plus the cosin of the difference of the angles, and same for the product cosin and sin. 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 these terms, we're left only with cosine of omega zero n, plus j sin of omega zero n, which is of course e to the j omega zero n. So when the input to the receiver is a cosine. The points in the complex base band sequence will be points around the circle and the difference between two successive points is the angle omega zero. The reason why we might be called to demodulate simple sinusoids is because the receiver will send what are called pilot 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 signal samples when the input is the audio file we just heard before. So let's start with the part that goes like this. [NOISE] This signal contains several sinusoids that we can see here in the bot. 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 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 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 the transmitter and receiver perform a handshake procedure using a very low bit rate QEM transmission 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 tasks that the receiver is settled 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 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 display the channel is compensated using adaptive equalization techniques. And we'll see some examples of that as well. And clock drifts are tackled by timing recovering 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 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 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 G omega. At the input of the receiver we have a continuous time signal as 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 channel. 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 the D to A converters are synchronized or run at exactly the same frequency