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.