So let's go back to the bandwidth constraint. As you remember, the channel imposes that we only use frequencies between Fmin and Fmax. We also want to translate these requirements into the digital domain. So we choose a sampling frequency and Fs/2, half of our sampling frequency, will be our niqueous frequency in the analog domain. Now here's a neat trick. Compute the positive bandwidth, namely the width of the channel bandwidth on the positive axis, and call that W. Now pick the sample frequency, so that two things happen. First of all, the sample frequency will have to be at least twice the maximum frequency that we can use in the channel to avoid aliasing. But then we'll choose a sample frequency as an integer multiple of the positive bandwidth. So we say that Fs = KW for K, a positive integer. With this choice, when we translate the analog specifications into the digital domain and remember the formula is always the same 2pi F/Fs. What happens is that the bandwidth in the digital domain will be 2pi W/Fs, which is equal to 2pi/K. And so we can simply upsample the sample sequence by K, so that its bandwidth will move from 2pi to 2pi/K, and therefore, its width will fit on the band allowed for by the channel. Now upsampling does not change the data rate because we're creating a sequence of symbols from the user data bitstream, and then we're introducing zeroes between samples, so we're not introducing extra information. So we produce and transmit W symbols per second. And then we upsample that by K to achieve a sample rate, which is equal to the sampling frequency. W therefore is the fundamental data rate of the system, and sometimes it's called the Baud rate of the system. The golden rule for digital communication systems is that the Baud rate will be equal to the positive bandwidth allowed by the channel. So here is our revised block diagram for the transmitter. User data comes in as a bitstream. Scrambler makes sure that we have a random sequence of symbols. The mapper will create the random sequence of symbols, we upsample this sequence by K, we filter this with a low pass filter with cutoff frequency, pi/K. And we obtain baseband signal b(n),which is centered in 0 and extends from -pi/K to pi/K. Now we need to move this baseband signal to the pass band of the channel. And to do so, we modulated with a cosine carrier whose frequency is the center frequency of the channel's band. This pass band signal s(n) can now be converted to the analog domain before being transmitted over the channel. Graphically assume these are the specifications dictated by the channel and translated to the digital domain. So here we have the positive bandwidth and the negative bandwidth. The sequence of symbols generated by the mapper is a wide sequence, and therefore its power spectral density, Pa(e to the j omega) is a full band signal. Now when we upsample the sequence by a factor of K, we reduce its spectral support to a baseband signal that goes from -pi/k to pi/k. And then we modulate this with a cosine carrier to fit it onto the bands that are available on the channel. As a final note, since we are developing a completely digital transmission system, we will probably want to use FIR filters in its implementation. Now we know that the sync filter that we have used in the upsampling operator will be a notoriously difficult filter to approximate with an FIR. So what is used in practice is another type of filter called the raised cosine. The frequency response of the raised cosine is shown here in this picture, and you can see that the transition band is no longer a discontinuity. But it is actually a smooth transition from passband to stopband. As a matter of fact, the raised cosine has a parameter that you can tune to have an even gentler transition band. Now the raised cosine remains an ideal filter because you can see it is constant over the passband and the stopband. But it is much easier to approximate than a sync. Another good property of the raised cosine is that it fulfills the interpolation property that we need to do upsampling. And the final selling point is that the impulse response, it can be shown, decays as 1/n cubed. So even short FIR approximations can get a very good response.