In real life not all signals are
band-limited.
And if we sample non-band-limited signals
there
will be a nasty effect called aliasing.
You may remember this effect in Module
Two.
We had seen the wagon wheel effect where
wagon wheels would go backwards.
Here we are going to explore this in
detail, specifically with sinusoids.
And we are going to see that if a sinusoid
is
not sampled fast enough, then it will fall
back into seemingly a
lower-frequency sinusoid.
So there is an infinite set of sinusoids
that
when sampled, all looks the same in their
sampled version.
The
[INAUDIBLE]
phenomenon is very fundamental.
And we need to understand it in detail.
>> Module 6.4.
Sampling and Aliasing - Introduction.
We will see what happens if we do
so-called Raw sampling.
So we take any signal, we sample it.
When it's band-limited, everything goes
fine.
When it's not band-limited, things can go
wrong
and we'll see specifically this with
Sinusoidal aliasing.
Which is a phenomenon when, sampled
sinusoid actually looks like another
sinusoid.
Remember how we do sinc sampling.
So we take a sinc function.
We scale it by Ts, we shift it by n times
Ts.
And we take the inner product with x of t
to get the sample, xn.
This can be written as the convolution
between sinc Ts.
So we denote scaling of the sinc function
by Ts at location nTs.
In a block diagram we take
x of T, we go through a low pass filter,
an ideal low pass filter of bandwidth
omega N, and we take samples every Ts
seconds to derive a sequence xn.
Raw sampling is when we don't care about
first taking the inner product with the
sinc function.
So we just take xt.
And every Ts seconds we take a sample xn.
So you must remember the wagonwheel
effect.
We had seen simulated movie how a wheel
could go backwards,
if we were not careful about having a high
sampling rate.
The complex exponential is a familiar
character.
Now we look at the continuous time complex
exponential, xt is e to the j.
Capital omega naught times t.
It's always periodic, it has a period 2 pi
over
capital omega.
Note all angle speeds are allowed, so
fourier transform of this complex
exponential is a delta, that is seated at
omega is equal to omega naught.
It is obviously band-limited to omega
naught.
If we look at this continuous time complex
exponential on the unit
circle, then it is a phaser that runs
around the unit circle.
If we take samples of this continuous time
complex exponential, so
xn is e to the j, on omega naught, Ts
times n.
Then they're all samples or snapshots at
regular intervals of this rotating point.
So resulting digital frequency is small
omega naught,
which is equal to omega naught times Ts.
Okay.
So please note the difference between
small omega and capital omega.
When Ts is smaller than pi over capital
omega 0, or small omega 0 is smaller than
pi, then the phaser will advance in small
steps as we can see here on this figure.
When Ts is become pi omega 0 and 2 pi
omega 0, that is
small omega 0 is between pi and 2 pi, then
the phaser advances in big steps.
However this looks as if it were going in
a negative direction, as you can see here,
it steps
in small steps, and it looks like it's
going
in the opposite directions than what it is
actually going.
Finally, when Ts
is bigger than 2 pi over omega naught, or
small omega
naught is bigger than 2 pi, than again we
think it
goes in the positive direction.
But it does actually don't full circle.
Plus the little step.
So we see that is large frequency looks
actually like a small frequency.
Okay, we see x1,
x2, x3, x4 and so on, as if it were going
in small steps.
And so this large frequency actually
mimics like a
small frequency, and that's a phenomenon
we call aliasing.
Let's look what happens when we
reconstruct or
interpolate based on the samples we have
just seen.
Let's look at this in a blog diagram.
So we have x of t.
We sample with a sampling period Ts.
Then we interpolate again with a spacing
of Ts.
And we get x-hat of t.
What is x-hat of t, with respect to x of
t?
Now let's look at the output, x-hat of t,
and what frequency
will be reproduced.
So the first case is easy.
That's when the sampling period Ts is more
than pi over omega naught.
Why?
Because we are meeting the sampling
theorem and so the
same frequency that went in should
actually also come out.
So the digital frequency in this case,
digital frequency small omega
naught is between 0 and pi and the output
is indeed
what went in, e to the j omega naught t.
The second case is
when Ts is in intermediate range, so
between
pi over omega naught and 2-pi or omega
naught.
The additional frequency now is beyond pi
but smaller than 2-pi.
So we see this in this formula here.
And then the output.
We'll have a different frequency than the
input.
Then if we can see omega one, which is
equal to omega naught minus 2-pi over Ts.
Okay.
So there's alrea, already aliasing
happening.
Because you have this shift in frequency
here that
transforms the original omega naught into
the output omega one.
Finally, the red case, the third
one here when Ts is bigger than 2 pi over
omega naught, then the digital frequency
would be beyond 2pi.
Okay, now you know that frequencies are
between 0 and 2 pi, so that doesn't really
have a meaning except that they chose that
this frequency will be folded back through
aliasing.
And it results in an output here, e to the
j omega
2t where omega 2 is simply omega naught
now taken module 2
pi over Ts.
Okay.
And this is definitely a different
frequency than what went in.
This point is so important that we are
going to go through it one more time.
Now, with a sinusoid, and using
frequencies in hertz.
Remember that we usually take frequencies
in radiance.
So frequency in radiance is simply
frequency in hertz times 2 pi.
Our signal, xt, is cosine of
2 pi F0 of t.
So, F0 is a frequency in hertz.
The samples xn are simply x at multiples
of Ts.
So that's cos of omega 0 n.
Omega 0 is 2 pi.
F0 over Fs, where Fs is simply 1 over the
sampling period.
Okay, we often go back between frequencies
and sampling periods and hertzes 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're
on a sinusoid, so that's a good case
when we sample twice above the Nyquist
frequency.
Then the digital frequency omega naught is
between 0 and pi.
Everything is okay.
When Fs is exactly equal to 2F0, then
omega naught is equal to pi.
And the signal xn
is simply minus 1 to the n.
And so that's a maximal digital frequency
that we're going to see.
Then when Fs is between F0 and 2F0, so it
is
sub Nyquist sampling, then omega 0 is
between pi and 2pi.
And we actually see a negative frequency
corresponding to pi over omega 0.
Finally, when Fs is smaller than F0, which
amounts to omega 0 being bigger than 2pi,
then we have full aliasing.
So the digital frequency is simply omega 0
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 6 pi of t.
So, F0 is 3Hz, the correct sampling
frequency Fs has to be 6Hz or higher.
So lets' start with Fs 100Hz, so that's
highly oversampled, you can see the red
dots.
Are 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.
Fs10Hz.
This is still on the right side of the
Nyquist sampling.
And we still see the sinusoid, but it
starts
to be quite a sparse sampling of the
sinusoid.
F 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
writing into F aliasing.
And we can see we have a few points here.
But it doesn't really look like it's
adequate representation of the sine wave.
So we rescale, we take an interval of 2,
instead of the interval of 1 from before.
We still see that these points you know,
do not give a good
impression of the sine wave.
So let's extend to an
interval of 4, to an interval of 10.
And now we see
a sinusoid appearing.
But this sinusoid 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 minus 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
of 3Hz.
So that's the phenomenon of aliasing, very
nicely graphically shown here.