Of'course not all signals are[INAUDIBLE].
Signals have channel spectras, but these
are not band-limited, and they are
sampled.
The out of band spectrum will fall back
exactly in an alias manner as we have seen
with Sinusoids.
Of'course this is a nasty phenomena
because suspect from that we've seen.
The base band does not correspond to the
proper spectrum of the signal and so this
has to be avoided.
What you have to do is, you have to apply
low pass filtering, mainly low pass
filters that cut off frequencies beyond
half of the sampling frequency and then we
can properly sample a signal.
This is what we are going to see in this
module.
Module 6.5, sampling and aliasing.
We've seen aliasing for very particular
signals.
Sinus wave to complex exponentials.
Now we are going to look at aliasing for
arbitrary spectra's and look at a few
examples.
We'll finish by rubbing up sampling of
arbitrary signals using some projection
theorem onto the space of unlimited
signals.
Remember the row sampling of an arbitrary
signal, so you have a continuous time
signal xc of t samples every ts seconds to
give a sequence xn which is equal to the
continuous time signals at multiples of
the sampling interval ts.
In fully transformed domain we have A
spectra of the continuous time signal
capital xc j capital omega and at the
output we have a discrete time Fourier
transform of the sequence that's capital X
e to the j omega.
What is that going to be in general and
how is it going to be related to the input
spectrum.
The key idea is the following.
Pick a sampling interval TS or a frequency
omega N equal to pi over Ts.
Start with omega 0 which is smaller than
omega N, the Niquest frequency which is
the maximum frequency that can be
faithfully represented with this sampling
system.
So the input is e to the J omega 0t, it's
sampled with Ts, it gives an output e to
the J omega nought Ts times N, that's a
sequence resulting from this sampling
operation.
Then let's add 2 omega n, to the input
frequency.
So now the frequency only gots 0 plus 2
omega n.
It is sampled.
And the output, it is only got 0 plus 2
omega n times ts times n.
We expand this product, then we see that
the second term becomes 2 pi n, but 2 pi n
raised to the power e to the j, is equal
to one, and therefore, we have the same
discrete time sequence as before.
So, we do not see this higher frequency
complex exponential, it simply looks like
the lower frequency exponential omega 0.
So in general if we have 2 frequencies A
times complex exponential omega 0 plus B
times the complex exponential omega 0 plus
2 omega N sampled it gives you A plus B e
to the j omega 0 Tsn.
So the upper frequency 1 has been folded
back or aliased back.
On to the lower frequency 1 and we simply
see that the sum of these 2 and so the
alias part has been shifted back to the
non-aliased one.
So the what is the spectrum of a
raw-sampled signal in general.
Well lets look at this sample x of n, it
is xc at location nTs, we can write this
as the inverse fourier transform of the
spectrum capital Xc multiplied by e to the
j.
Omega nTs because that's the occasion we
want to evaluate in this fourier
transform.
Frequencies that are 2 omega N apart will
be aliased.
We have seen this specifically on an
example just a couple of slides before.
So we can split the integration interval
into pieces of size 2 omega N and this we
do here.
So x(n) this inverse fourier transform is
now.
There's a sum of the pieces from 2k minus
1 omega n, to 2k plus 1 omega n, of the
fully transformed here, and multiplied by
the complex exponential at nTs.
Lets look at this on a picture.
So everything that these at location to
omega n will be folding back to the
origin, what is that, 2 omega n plus
epsilon will be folded back to epsilon.
What is at 2 omega n minus epsilon gets
folded back to minus epsilon.
The same symmetrically around minus 2
omega n.
It's the same from 4 omega n and minus 4
omega n et cetera, et cetera.
So, we can go back to our formula.
We do a change of variables so that the
integral is between minus omega n and
omega n.
This change is the frequency variable
inside of xc.
Then we also note that adding multiples of
2k omega n to the frequency doesn't change
the frequency of that c.
[unknown] effect, we can interchange
summation and integral to end up this one
or two pi's, the integral, the summation.
Of the spectrum xc and its shifts by
multiples of 2k omega m n.
And then at the right end we still have
the integration variable for the inverse
fully transform.
Lets define a periodized spectrum, so
capital X tilde c is the summation of Xc
and its shifted versions by even multiples
of omega n.
Therefore, the sample xn, is integral
between minus omega n and omega n, of the
periodized spectrum x tilde multiplied by
e to the j omega nts.
Let's use a falling change of variable.
Small omega is equal to capital omega
times Ts.
This will change the integration
boundaries from minus omega, omega n, into
minus pi to pi, it changes the frequencies
inside the x tilled, and it changes it's
exponent to j o Omega N, which is more
convenient.
So, we recognize now that xn is the
inverse discrete time Fourier Transform of
1 over ts[INAUDIBLE] spectrum X til c
Rescaled by TS.
An important point to note is that
this[UNKNOWN] spectrum, rescaled, is
actually 2 pi periodic.
So it's a valid DTFT.
Since this is the IDTFT, it means that the
DTFT of the sequence, capital X of e to
the j omega, is simply up to a scale
factor of 1 over ts.
The summation of xe, and these shifts by
multiples of 2 pi over ts.
A picture is worth 1,000 equations, maybe.
So let us look at this spectrum and how it
evolves.
So we start with capital XC.
It's a spectrum, which is inside the
boundary minus omega n to omega n.
It doesn't cover the entire interval.
Then the periodized version puts
repetition spectras of 2 omega n, 4 omega
n, et cetera.
This is now a periodic spectrum of period
2 capital omega n, so spectrum of x of e
to the j omega is A scaled version of
this.
It is 2 pi periodic.
And it is shown here at the bottom in
blue.
Let's take a band limited signal that is
band limited to omega zero, equal to omega
n.
This is the limit of the Nyquist sampling
frequency.
It will be periodic of period 2 omega n,
which is equal to 2 omega zero.
So the repetitions will just.
Touch each other, it is therefore the
usual periodic spectrum and now we look at
the spectrum of x of e to the j omega.
It fills the entire interval minus pi to
pi.
What happens when omega 0 is bigger than
omega N.
Well we know trouble is going to come
because aliasing will happen.
So, we see that the spectrum is beyond the
central interval of minus omega N to omega
N.
So the repetition will overlap with each
other.
The result is still 2 omega N periodic
spectrum, we see it has a funny shape now,
it's not the original shape because of the
overlaps and when we look at the spectrum
of x of e to the j omega.
So blue spectrum, it indeed does not
resembles the original spectrum because an
entire interval has been messed up by
aliasing.
Let us look at the non-band limited
signal.
So this is a goshan type of single, which
has decayed but goes well beyond the
central interval minus omega m, 2 omega m.
It is after sampling the periodized
version is to omega n periodic of course,
but there are overlaps all over, and it
has in the center of the interval shapes
that resemble the original spectrum, but
then it has overlap And so the blue
spectrum, which is between minus pi and pi
resembles the original spectrum around the
origin 0.
But is different from the original at the
boundary of the Intro.
Since it leads us to sampling strategies
for different signal.
So if you are given a sampling period Ts
and the nyquist frequency omega n is equal
to pi over Ts, if the signal is band
limited to omega n.
Then row sampling is fine.
As I pull in to sink sampling up to a
scaling factor.
If the signal is not band limited, we have
two choices.
Either we band limit it first with a low
pass filter that will cut off all
frequencies beyond omega n in the
continuous time domain before we do
sampling.
Then, everything will work just fine as we
have seen or we roll sample the signal,
and we have aliasing.
Now let's face it, aliasing usually sounds
horrible, so usually it's not the first
choice.
So let's investigate a little bit how we
sample signals that are not strictly band
limited to begin with.
So how do we do sync sampling and
interpolation.
So sample x of n is simply the inner
product of sync function shifting to the
location n times Ts was x(t) so in put
single, this can be written simply as the
combination of the sync with x at location
nTs.
Then the interpolation, x hat of t, is the
sum of the samples multiplied by the same
functions shifted to the location n t s.
We see this in this block diagram from
left to right we have a continuous time
signal x of t.
It goes through an ideal low pass filter
with the cut off frequency capital omega
N, it is sampled every Ts seconds.
So samples are interpolated by sync
interpolation and the result is x hat of
t.
It is interesting to look at this scheme
in purely diametrical terms.
So, we have an input signal x, it's a
vector in a Hilbert's space.
We have a subspace of band limited
signals, BL.
The first thing we do is we have an
orthonormal basis for BL, it's given by
the sync function and its shifts by
integer multiples of capital Ts.
Based on this orthonormal basis, we write
an orthonormal expansion, which is the
orthonormal project of x onto the space of
band limited signal.
So it's written as a sum of the sync in a
product with x times the sinc functions.
And so we see that the scheme we had on
the previous slide is simply the projector
of x on to the subspace of band limited
signal, that projection of course can be
written in terms of the sampling theorem.
But we can not write x we can only write x
hat, the projection.
Now, let's look at the concrete example,
one that we have encountered before, which
is a non band limited signal.
Here's is its quotient shaped signal.
And we are going to do least squares
approximation on the space of band limited
signals using sync sampling and
interpolation.
So, we start with capital Xc as a
continuous time spectrum, non band
limited.
We are going to band limit it to minus
omega N to omega N.
So this is given by this green ideal
filter.
After filtering, the spectrum is strictly
band limited.
It's zero outside this interval of size 2
omega n.
Then we do the sampling and this reads to
a periodic spectrum X tilde c.
Given in these pink shape here and it
repeats at multiplies of 2 omega N as we
have seen before then we look at the DTFT
of this sequence and the DTFT is 2 pi
periodic.
It's given by this blue function and
finally we re-interpolate a continuous
time spectrum using sync interpolations
that gives us x hat c which now is exactly
the reconstruction of the band limited
version of the initial x c.
This band limited version, of course
easier orthogonal projection of the
initial spectrum onto the space of band
limited signals.