If we just take an abstract view of the telephone network today, we see that we have a link that goes from the home to the central office. And in the central office, a fundamental split takes place. The voice communication, when you talk on the the phone, is sent to the voice network and then relayed to what is called the plain old telephone system. P-O-T-S. The data part of your communication, when you use the ADSL, is separated from the voice content and sent to a DSLAM. DSLAM stands for Digital Subscriber Line Access Multiplier. And it's fundamentally a bank of modems that manage to handle multiple communications at the same time. And the data here then goes on to the Internet in digital format. So, if you want what we're really interested in is how to send the data from your home to the central office because what happens afterwards is already entirely in the digital domain. But here we have what is called the last mile, a piece of copper wire, namely an analog channel, that connects your home to the central office. Now a copper wire has naturally a very large bandwidth, in excess of 1 megahertz. But because of the width of the bandwidth and because the wire is not shielded, it is actually likely to pick up a lot of interference and noise. If we look at how the ADSL channel is organized, and we're showing here just the positive frequencies, we can see three distinct regions. The first one is the part reserved to the telephone conversation. This is a baseband part of the channel up to about 4 kilohertz. Then we have a region that is devoted to the upstream part of the data communication, the data that you send up to the Internet, and then a downstream part that is much larger, that is used for data download. This asymmetry between upstream and downstream is actually the reason why the communication protocol is called ADSL. ADSL stands for asymmetric digital subscriber line. If we now look at all the nasty things that can happen on the channel when we send data, we can identify three fundamental sources of worry. The first one is an attenuation curve for the channel that is completely uneven. This could be due to imperfection in the wire, parasitic capacitance,and so on and so forth. Then we might have very large noise or interference in certain regions of the spectrum. For instance, you turn on your vacuum cleaner and that raises the noise floor in the certain frequency band. And thirdly, we have very localized interference from radio communication. Now, the radio band starts well within the bandwidth of the ADSL channel. For instance, from 15 to 100 kilohertz here, you have ship-to-shore communication. Up to 500 kilohertz, you have airplane communication. And over 500 kilohertz, you have the AM radio band. So if you live near a radio station, for instance, tough luck, you have a lot of interference in the upper regions of the ADSL band. Since the channel is so wide and the type of the service is so diverse, it would be extremely difficult to try to equalize and compensate for these problems on a global scale. So the idea, instead, is to divide the channel into independent subchannels and treat each subchannel separately. So here, for instance, this channel that contains highly localized radio frequency interference would probably not even be used because it would be too difficult to compensate for this. Here on this channels, the noise level is different and so, for instance, we would use different signaling strategy according to the local signal to noise ratio. And similarly here, for channels that have a very large attenuation, probably wouldn't be worthwhile to try and send data over these things. But on the other hand we will try to exploit the cleanest sub-channels to send a maximum amount of data. Now to formalize the subchannel structure, suppose that we want to allocate N subchannels over the total positive bandwidth. We want the subchannels to have equal bandwidth. So, their bandwidth will be Fmax over N, where Fmax is the maximum frequency allowed for by the channel. And we equally space the subchannels by centering them over k Fmax over N, with k that goes from 0 to big N minus 1. This means that the first channel, K equal to zero, will be baseband and then the subsequent channels will be passband with center frequencies given by this formula. Now we want to translate this design to the digital domain, so we pick a sample and frequency that is at least twice the maximum frequency in the channel. But careful now, because Fmax is quite high. The center frequency for each subchannel will be omega k equal to two pi k Fmax over N divided by the sample frequency. And if we sample at the Nyquist frequency, so Fs is equal to twice Fmax, then omega k becomes simply two pi over two N times k. We will not simplify the 2s in the fraction because they will be useful later. The bandwidth of each sub channel is also 2 pi over 2 N. And so if we want to send symbols over any of these subchannels, remember the modulation scheme that we've seen in the previous modules, then we'll have to use an upsampling factor K that is at least 2N. If we plot the result in the digital domain, let's suppose that we just want to have three subchannels, we have something that looks like this. The center frequencies will be multiples of 2pi over 6, so we'll have 0, 2pi over 6 and 4pi over 6. And we will center channels over these frequencies. And the bandwidth of each channel will be 2pi over 6. So the first channel is the baseband channel. And then we have two passband channels with of course their negative frequency counterpart. The next step in ADSL communication is to put a QAM modem on each subchannel independently. And we will decide on the data rate for each modem based on the signal to noise ratio of each subchannel. So if the noise floor is low, then we will have a large constellation for that subchannel. And vice versa. On channels that are unusable because of noise of interference, we will just send zeros and we will not care about that. The structure of the signal and the scheme is, of course, going to be communicated from the transmitter to the receiver so that the receiver knows where to expect data. This is part of the handshaking procedure between transmitter and receiver. Now let's look in detail at the structure of the modem that we use on each subchannel. This is a classic modulation scheme, where we start with a sequence of symbols as produced by the mapper. Then we have an upsampling by a factor of 2N. So inserting 2N minus 1 zeros every other sample. And then filtering the sequence with a low pass, usually a raised-cosine, with a cutoff frequency 2pi over 2 N in this case. This produces the complex baseband signal bk of n and this complex baseband signal gets modulated with a complex exponential whose frequency is indexed by the channel number. And this is the center frequency of each channel. Omega Ck is equal to 2pi over 2N times k. And here we have, finally, the passband signal that fits the prescribed bandwidth of the cave channel. And normally if we just had one channel, we would put here a block that computes the real part and then our D2I converter. But here we have several modems in parallel, and so we have a structure that looks like this. Each channel will have two things that vary with respect to the others. The frequency of the modulation, and of course, the series of symbols produced by the mapper. We sum all the complex baseband signals together, before taking the real part and then sending the signal to the D2I converter.