And then, if we change to s sub 1 and we do the scalar product,

we get the same result, zero.

Which, again, means that this frequency is not present in this x signal.

But when we change to s sub 2, and we do the scalar product, the result is 4.

Which means that basically this x signal is this sinusoid.

It's completely present in this sinusoid, and

we get the result of 4 which is the sum of all the samples.

And then, by S sub 3, again is equal to zero.

So that means that we have computed the DFT of X sub N,

and we have obtained that is equal to 4 for

K equal to 2, and is equal to zero for the rest.

Meaning that we have the presence of the frequency, K equal to 2.

Let's do that with bigger signal.

So, this is an example of the scalar operation

of a simple signal that has all 1's.

It has 8 samples,

the first four are 1s and the last four are minus 1s.

So this would be like a rectangular kind of signal.

If we compute the DFT, or the scalar product between this x signal and

the eight complex sine waves that we had seen in the previous slides,

the result will be this one.

We'll get the magnitude and the phase, so it's going to be a complex

spectrum, and we can display it as polar coordinates.

And here we can see that in this signal there is some frequencies present.

The frequency k equals 0 is not present.

Frequency k equals 1 is very much present, but

it also present k equals 3, and it's also present k equal 5.

And again, is also present k equals 7.

And the phases mean how this sine waves are located in,

sort of in the time location.