This point is so important that we're going to go through it one more time
now with the sinusoid and using frequencies in hertz.
Remember that we usually take frequencies in radians.
So frequency in radians is simply frequency in hertz times 2Pi.
Our signal x(t) is cosine of 2Pi F0t.
So F0 is a frequency in hertzs.
So sample x(n) or simply x at multiples of Ts, so that's cosine of omega0 n.
Omega0 is 2Pi F0 over Fs where Fs is simply 1 over the sampling period.
Okay, we often go back between frequencies and
sampling periods and hertzs and radians.
So you have to be used to jump back and forth with these notions.
Now we are going to sample the sinusoid.
When the sampling frequency Fs is bigger than 2F0, so
F0 is the maximum frequency, we are in the sinusoid, so
that's a good case when we sample twice above the Nyquist frequency.
Then the digital frequency omega0 is between 0 and Pi, everything is okay.
When Fs is exactly equal to 2F0, then omega0 is equal to Pi and
the signal x[n] is simply -1 to the n.
So that's the maximal digital frequency that we're going to see.
Then when Fs is between F0 and 2F0, so it is sub Nyquist sampling,
then omega0 is between Pi and 2Pi, and we actually see a negative frequency.
Finally, when Fs is smaller than F0,
which amounts to omega0 being bigger than 2Pi, then we have full aliasing.
So the digital frequency is simply omega0 mod 2Pi, and this can be a much different
frequency from the input frequency of the sinusoid, as we shall see.
Let us see this sampling in action, namely on cosine 6Pi of t.
So, F0 is 3Hz, the correct sampling frequency Fs has to be 6Hz or higher.
So, let's start with Fs 100Hz, so that's highly over sampled, you can see the red
dots are a very close approximation to the continuous time function.
Take Fs 50Hz, that's still much bigger than the limit frequency 6Hz,
so still a very good approximation.
Fs 10Hz, this is still on the right side of the Nyquist sampling.
And we still see the sinusoid,
but it starts to be a quite sparse sampling of the sinusoid.
Fs 6Hz, is exactly the limit.
So the samples alternate between plus 1 and minus 1,
which is the maximum digital frequency.
Fs 2.9Hz, now we are actually below F0 and clearly running towards aliasing.
And we can see, we have few points here, but
it doesn't really look like it's a adequate representation of the sinusoid.
So, we rescale, we take an interval of two instead of the interval of one from
before, we still see that these points do not give a good impression of a sinusoid.
So let's extend to an interval of 4, to an interval of 10.
And now we see a sinusiod appearing.
But this sinusiod actually has a frequency of 0.1Hz.
As you can see, it has a full cycle here between 0 and 10.
And because of aliasing, so the resulting frequency is actually,
2.9Hz modulo 3Hz, and that is -0.1Hz or
because it is symmetric it is equal to a frequency of 0.1Hz.
So it's a totally aliased version of the initial cosine function that we had.
And we see another cosine which happens to be of 0.1Hz rather than 3Hz.
So that's the phenomenon of aliasing, very nicely graphically shown here.
Paolo PrandoniLecturer
Martin VetterliProfessor
