Okay, now that we know how to compensate for the propagation delay introduced by the channel, let's go see the rechannel with an arbitrary figure's response, d j of omega. And the transmission chain goes from the pass band signal s of n, discrete time, into a D/A converter. Analog signal s of t that gets filtered by the channel gives us hat s of t. Which is sampled at the receiver to give us a received pass band signal hat s of n. But now we have seen in the previous model 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 band pass 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 a 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. 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 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 transmitting signal at the receiver? Well we use two tricks, the first one is bootstrapping. The transmitter will send a prearranged sequence of symbols to the receiver. So let's call this 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 symbol. 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.
