The filter will introduce a phase shift in the signal and

different phase shift for the positive and negative frequencies.

To understand the behavior of the filter, let's look at the representation of

the input spectrum by displaying both the real and

the imaginary part on a three dimensional plot.

So here suppose that the input is real valued so we have a classic pattern, where

the real part of the spectrum is symmetric and the imaginary part is antisymmetric.

We will plot the real part here on the vertical plane and

the imaginary part on the horizontal plane.

And this is the frequency axis.

The Hilbert filter will introduce a 90 degree clockwise rotation of the spectrum

for the positive frequencies, and

a 90 degree counter-clockwise rotation for the negative frequencies.

Let's look at the real part first, so

imagine that the real part of the spectrum has this triangular shape, when we apply

the Hilbert filter this part will be rotated by 90 degrees in this direction.

It will become imaginary.

And this part here corresponding to the negative frequencies

will be rotated 90 degrees in this direction and will become imaginary.

Graphically, if we were to show this rotation as it unfolds we start

with a triangular shape and then we rotate it until it becomes like so.

So the real part of the spectrum has now become the imaginary part of the spectrum.

And from symmetric it will become antisymmetric.

Similarly the imaginary part of the spectrum will be rotated in the same way,

and from antisymmetric here, will become real and symmetric like so.

So if we look at the effect on the combined spectrum,

we start with this real and imaginary part.

And after applying the Hilbert filter to this input, we end up with this

spectrum here, where the imaginary part and the real part have been exchanged and

modified so that they preserve their symmetry and antisymmetry.

So let's see how we can use the Hilbert filter to effectively perform

demodulation.

This here is a Hilbert demodulator.

The input signal is supposed to be a modulated signal.

So, an original signal x(n) multiplied by cosine at omega 0n,

where this is the carrier at frequency omega 0.

So this signal is split into two identical parts.

One is passed through as is, and the other copy is passed through the Hilbert filter.

Then it's multiplied by j and summed back to original input.

Finally, the result of the sum is multiplied by a complex exponential

at a frequency equal to the frequency of the carrier.

In the end, what we get is the demodulated signal.

So, assume this is the original signal before modulation.

When we modulate the signal, remember you get two copies at positive omega C and

minus omega C, so demodulated signal spectrum looks like so.

Okay, two copies of the original signal.

Here again we show the real part on the vertical plane, and

the imaginary part on the horizontal plane.

The top branch of the demodulator remember, here is the signal.

And the top branch will have a Hilbert filter and then a multiplication by j.

We can interchange Hilbert filter and multiplication.

So, how does jy of n look in the frequency domain?

Well, multiplication by j is just counterclockwise rotation by 90 degrees,

and it's the same for positive and negative frequencies.

So we take this spectrum here, and we just rotate this by 90 degrees, so

that the imaginary part becomes real and the real part becomes imaginary.

But there is no change in symmetry or antisymmetry of the components.

So, when we do that we're just flipping the thing and now it'll look like so.