So let's review where we stand in terms of transmitter design. We have the user's bitstream that comes into the system. This is sent through a scrambler that makes sure that the resulting bitstream is equiprobable. The mapper will split the bitstream into m-bit chunks. And each chunk will be associated to a complex-valued symbol. This will create a complex value sequence a[n]. And to fit that over the bandwidth prescribed by the channel, we have to upsample it, which means inserting K minus 1 0's after each symbol of the sequence and then low passing the sequence with a filter with cutoff frequency pi over K. Now, the sequence b[n], the upsample sequence that now fits the bandwidth constraint, is a complex-valued baseband signal. Graphically, we can show this. If this is the bandwidth constraint of the channel, the positive and negative frequencies, the sequence b[n] is a baseband sequence whose bandwidth fits the bandwidth prescribed by the channel. So now the question is, how do we modulate this while also making the transmitted signal real, and making it real in a way that we can retrieve it and reconstruct the baseband complex signal at the receiver? So the way it's done is very straightforward. We just take the real part of the baseband signal multiplied by e to the j omega c n, where omega c is the center frequency of the bandwidth of the channel. If we work it out, the math is very simple. So we have the product, of course, of the real part and imaginary part of the baseband signal multiplied by cosine of omega c n + j sine of omega c n. And the real part of this equation is simply the real part of the baseband signal multiplied by the cosine of omega c n minus the imaginary part of the baseband signal multiplied by sine of omega c n. Now, you see here, we're multiplying each component, the real and imaginary part of the baseband signal, by a carrier. And the two carriers are orthogonal to each other. So we have a cosine carrier, and a sine carrier. Now, these two carriers are orthogonal because they're shifted by a phase of 90 degrees. Now, when two things are 90 degrees apart, they're said to be in quadrature. And so the first component of the signal will be called the in-phase part, and the second component, the one modulated by the sine, will be the quadrature part. And this is the nomenclature behind the acronym QAM. Okay, so now this signal s[n] is clearly real. And our next step will be to show that if we receive this at the receiver, we will be able to recover b[n], the complex baseband signal. But before we do that, let's look at the modulation process in the frequency domain, because the intuition will help us understand why we can recover the baseband signal exactly at the receiver. So in the next few diagrams, we will show the spectra of br and bi, the in-phase and quadrature components of the baseband signal. Let's assume for the sake of convenience that these spectra are purely real. And we will indicate the real quantities with shades of blue and purely imaginary quantities with shades of pink. The math would stay the same for arbitrary spectra, but this assumption will allow us to draw a simpler picture. So if we start by plotting the spectrum of the real part of the baseband signal, let's assume it has a shape like this. Then we plot the spectrum of the imaginary part of the baseband signal, and let's assume that, again, it's a purely real spectrum, except that it has a slightly different shape. And these shapes are completely immaterial, they just help us see what happens during the modulation process. The real part is multiplied by the cosine of omega c n, and therefore a cosine modulation takes place. It is shifted left and right and centered in omega c. For the imaginary part of the baseband signal, the modulation takes place with a sine and there is a change of sign involved. So the resulting spectrum will be purely imaginary and it will be shifted at omega c and minus omega c with a change of sign in the negative part of the spectrum. And so this is the spectrum of the signal that we actually send over the real channel. As you can see, the signal is real and, indeed, the spectrum has a Hermitian symmetry in the sense that its real part is symmetric and the imaginary part is antisymmetric. Okay, so now let's assume that the transmission goes well, and our job, the receiver is to recover the complex baseband signal. A naive approach, we try the usual method, which is multiplying the signal by the carrier. Well, in this case, we have two carriers, the cosine and sine, so let's start by multiplying by the cosine. And so if we take s[n], which is the transmitted and then received signal, and we multiply it by cosine of omega c n, what we obtain is br, the real part of the baseband signal, multiplied by cosine squared of omega c n minus bi, the imaginary part of the baseband signal, multiplied by sine of omega c n times cosine of omega c n. Now we use some really basic trigonometry to express these functions as functions of 2 times omega c n. So cosine squared of omega c n becomes 1 + cosine of 2 omega c n over 2 and sine of omega c n cosine of omega c n becomes sine of 2 omega c n over 2. And if we rearrange the terms of this equation, we obtain one-half times the real part of the baseband signal plus one-half of a term that only contains carriers at twice omega c n, which is really similar to the situation we had in the simple cosine modulation of the modulation for a real signal. Again, it's much more intuitive if we look at this in the frequency domain. And so, this was the signal that was transmitted and then received. If we multiply this by cosine of omega c n, we obtain one-half of the spectrum of the real part of the baseband signal plus spurious components at twice the modulation frequency. Now, we know what to do to get rid of these spurious components, we just use a low-pass filter. And in this case we can use, again, a raised cosine. When we use this, we have what is called a matched filter configuration, where we use the same filter at the receiver that we used as a transmitter. This filter will eliminate the components out of the baseband and we will recover the original real part of the baseband signal. We can do exactly the same for the quadrature component for the complex part of the baseband signal. And in this case, we multiply the received signal by sine of omega c n. And if we work out the math with simple trigonometry once again, then we obtain minus one-half times the imaginary part of the baseband signal plus out-of-band components at twice the modulation frequency, which we can filter out with a low-pass filter.
