Now, omega, in all cases, involves pi therefore,

this cosine is periodic in the time domain.

Time is the horizontal axis.

So discrete time, m, right?

So, for example, here, for omega equals pi over 8, the period is

2pi over omega, which is equal to 16.

The period here is equal to 8, equal to 4, equal to 2 and so on.

So the period keeps decreasing as you move

to the right, therefore the frequency keeps increasing.

So for omega equals 0.

Cosine of 0 is equal to 1.

This is the signal that does not have any other frequency other than the 0

frequency, the busy signal where omega equals

pi over 8 we that the frequency increases.

We see, right from here to here, is one period of the signal.

5 over 4 keep increasing and omega equals pi this is the

highest possible frequency of the discrete cosine.

And as a matter of fact cosine pi m equals to minus 1 to the n.

So the values of the signal keep alternating.

It switches from 1 to minus 1 and back to 1 and so on.

So this is the highest possible variation of the signal.

Now as the frequency keeps incre, increasing

from omega plus pi to 2 pi, right?

We see that the frequency of the variation of the cosine keeps decreasing.

As a matter of fact this and this signal is identical,

because 3 pi over 2, plus pi over 2, equals 2 pi.

So these are two complementary angles and cosine of

pi over 2 equals cosine of 2 pi minus pi over 2 which equals 3 pi over 2, right?

So generally I have cosine a equals cosine of 2 pi minus a.

Alright?

And similarly, these two are the same

signals and these two are the same signals.