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In real life not all signals are
band-limited.

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And if we sample non-band-limited signals
there

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will be a nasty effect called aliasing.

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You may remember this effect in Module
Two.

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We had seen the wagon wheel effect where
wagon wheels would go backwards.

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Here we are going to explore this in
detail, specifically with sinusoids.

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And we are going to see that if a sinusoid
is

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not sampled fast enough, then it will fall
back into seemingly a

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lower-frequency sinusoid.

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So there is an infinite set of sinusoids
that

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when sampled, all looks the same in their
sampled version.

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The

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[INAUDIBLE]

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phenomenon is very fundamental.
And we need to understand it in detail.

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>> Module 6.4.
Sampling and Aliasing - Introduction.

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We will see what happens if we do
so-called Raw sampling.

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So we take any signal, we sample it.

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When it's band-limited, everything goes
fine.

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When it's not band-limited, things can go
wrong

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and we'll see specifically this with
Sinusoidal aliasing.

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Which is a phenomenon when, sampled

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sinusoid actually looks like another
sinusoid.

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Remember how we do sinc sampling.
So we take a sinc function.

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We scale it by Ts, we shift it by n times
Ts.

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And we take the inner product with x of t
to get the sample, xn.

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This can be written as the convolution
between sinc Ts.

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So we denote scaling of the sinc function
by Ts at location nTs.

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In a block diagram we take

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x of T, we go through a low pass filter,
an ideal low pass filter of bandwidth

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omega N, and we take samples every Ts
seconds to derive a sequence xn.

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Raw sampling is when we don't care about

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first taking the inner product with the
sinc function.

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So we just take xt.

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And every Ts seconds we take a sample xn.

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So you must remember the wagonwheel
effect.

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We had seen simulated movie how a wheel
could go backwards,

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if we were not careful about having a high
sampling rate.

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The complex exponential is a familiar
character.

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Now we look at the continuous time complex
exponential, xt is e to the j.

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Capital omega naught times t.

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It's always periodic, it has a period 2 pi
over

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capital omega.

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Note all angle speeds are allowed, so
fourier transform of this complex

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exponential is a delta, that is seated at
omega is equal to omega naught.

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It is obviously band-limited to omega
naught.

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If we look at this continuous time complex
exponential on the unit

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circle, then it is a phaser that runs
around the unit circle.

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If we take samples of this continuous time
complex exponential, so

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xn is e to the j, on omega naught, Ts
times n.

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Then they're all samples or snapshots at
regular intervals of this rotating point.

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So resulting digital frequency is small
omega naught,

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which is equal to omega naught times Ts.

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Okay.

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So please note the difference between
small omega and capital omega.

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When Ts is smaller than pi over capital
omega 0, or small omega 0 is smaller than

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pi, then the phaser will advance in small
steps as we can see here on this figure.

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When Ts is become pi omega 0 and 2 pi
omega 0, that is

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small omega 0 is between pi and 2 pi, then
the phaser advances in big steps.

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However this looks as if it were going in

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a negative direction, as you can see here,
it steps

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in small steps, and it looks like it's
going

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in the opposite directions than what it is
actually going.

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Finally, when Ts

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is bigger than 2 pi over omega naught, or
small omega

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naught is bigger than 2 pi, than again we
think it

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goes in the positive direction.
But it does actually don't full circle.

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Plus the little step.

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So we see that is large frequency looks
actually like a small frequency.

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Okay, we see x1,

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x2, x3, x4 and so on, as if it were going
in small steps.

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And so this large frequency actually
mimics like a

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small frequency, and that's a phenomenon
we call aliasing.

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Let's look what happens when we
reconstruct or

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interpolate based on the samples we have
just seen.

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Let's look at this in a blog diagram.

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So we have x of t.
We sample with a sampling period Ts.

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Then we interpolate again with a spacing
of Ts.

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And we get x-hat of t.

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What is x-hat of t, with respect to x of
t?

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Now let's look at the output, x-hat of t,
and what frequency

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will be reproduced.
So the first case is easy.

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That's when the sampling period Ts is more
than pi over omega naught.

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Why?

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Because we are meeting the sampling
theorem and so the

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same frequency that went in should
actually also come out.

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So the digital frequency in this case,
digital frequency small omega

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naught is between 0 and pi and the output
is indeed

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what went in, e to the j omega naught t.
The second case is

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when Ts is in intermediate range, so
between

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pi over omega naught and 2-pi or omega
naught.

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The additional frequency now is beyond pi
but smaller than 2-pi.

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So we see this in this formula here.
And then the output.

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We'll have a different frequency than the
input.

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Then if we can see omega one, which is
equal to omega naught minus 2-pi over Ts.

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Okay.

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So there's alrea, already aliasing
happening.

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Because you have this shift in frequency
here that

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transforms the original omega naught into
the output omega one.

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Finally, the red case, the third

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one here when Ts is bigger than 2 pi over

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omega naught, then the digital frequency
would be beyond 2pi.

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Okay, now you know that frequencies are
between 0 and 2 pi, so that doesn't really

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have a meaning except that they chose that

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this frequency will be folded back through
aliasing.

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And it results in an output here, e to the
j omega

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2t where omega 2 is simply omega naught
now taken module 2

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pi over Ts.
Okay.

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And this is definitely a different
frequency than what went in.

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This point is so important that we are
going to go through it one more time.

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Now, with a sinusoid, and using
frequencies in hertz.

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Remember that we usually take frequencies
in radiance.

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So frequency in radiance is simply
frequency in hertz times 2 pi.

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Our signal, xt, is cosine of

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2 pi F0 of t.
So, F0 is a frequency in hertz.

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The samples xn are simply x at multiples
of Ts.

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So that's cos of omega 0 n.

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Omega 0 is 2 pi.

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F0 over Fs, where Fs is simply 1 over the
sampling period.

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Okay, we often go back between frequencies

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and sampling periods and hertzes and
radians.

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So, you have to be used to jump

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back and forth with these notions.

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Now we are going to sample the sinusoid.

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When the sampling frequency Fs is bigger
than

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2F0, so F0 is the maximum frequency, we're

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on a sinusoid, so that's a good case

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when we sample twice above the Nyquist
frequency.

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Then the digital frequency omega naught is
between 0 and pi.

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Everything is okay.

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When Fs is exactly equal to 2F0, then
omega naught is equal to pi.

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And the signal xn

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is simply minus 1 to the n.

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And so that's a maximal digital frequency
that we're going to see.

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Then when Fs is between F0 and 2F0, so it
is

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sub Nyquist sampling, then omega 0 is
between pi and 2pi.

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And we actually see a negative frequency
corresponding to pi over omega 0.

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Finally, when Fs is smaller than F0, which
amounts to omega 0 being bigger than 2pi,

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then we have full aliasing.

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So the digital frequency is simply omega 0
mod 2pi, and this can be a much different

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frequency from the input frequency of the
sinusoid, as we shall see.

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Let us see this sampling in action, namely
on cosine 6 pi of t.

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So, F0 is 3Hz, the correct sampling
frequency Fs has to be 6Hz or higher.

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So lets' start with Fs 100Hz, so that's

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highly oversampled, you can see the red
dots.

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Are very close approximation to the
continuous time function.

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Take Fs 50Hz.

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That's still much

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bigger than the, limit frequency 6Hz.
So, still a very good approximation.

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Fs10Hz.

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This is still on the right side of the
Nyquist sampling.

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And we still see the sinusoid, but it
starts

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to be quite a sparse sampling of the
sinusoid.

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F 6Hz is exactly the limit.

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So the samples alternate between plus 1
and minus 1, which is the maximum

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digital frequency.
Fs 2.9Hz.

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Now we are actually below F0 and clearly
writing into F aliasing.

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And we can see we have a few points here.

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But it doesn't really look like it's
adequate representation of the sine wave.

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So we rescale, we take an interval of 2,
instead of the interval of 1 from before.

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We still see that these points you know,
do not give a good

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impression of the sine wave.
So let's extend to an

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interval of 4, to an interval of 10.
And now we see

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a sinusoid appearing.
But this sinusoid actually has a frequency

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of 0.1Hz, as you can see.
It has a full cycle here between 0 and 10,

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and because of aliasing, so the resulting
frequency is actually 2.9Hz modulo 3Hz.

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And that is minus 0.1Hz, or because it is
symmetric

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it is equal to a frequency of 0.1Hz.
So it's a totally aliased version

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of the initial cosine function that we
had, and we see another cosine

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which happens to be of 0.1Hz rather than
of 3Hz.

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So that's the phenomenon of aliasing, very
nicely graphically shown here.