1
00:00:00,012 --> 00:00:05,114
Of'course not all signals are[INAUDIBLE].
Signals have channel spectras, but these

2
00:00:05,114 --> 00:00:07,775
are not band-limited, and they are
sampled.

3
00:00:07,775 --> 00:00:12,665
The out of band spectrum will fall back
exactly in an alias manner as we have seen

4
00:00:12,665 --> 00:00:15,774
with Sinusoids.
Of'course this is a nasty phenomena

5
00:00:15,774 --> 00:00:20,635
because suspect from that we've seen.
The base band does not correspond to the

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00:00:20,635 --> 00:00:24,317
proper spectrum of the signal and so this
has to be avoided.

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00:00:24,317 --> 00:00:29,724
What you have to do is, you have to apply
low pass filtering, mainly low pass

8
00:00:29,724 --> 00:00:35,658
filters that cut off frequencies beyond
half of the sampling frequency and then we

9
00:00:35,658 --> 00:00:40,519
can properly sample a signal.
This is what we are going to see in this

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00:00:40,519 --> 00:00:43,783
module.
Module 6.5, sampling and aliasing.

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00:00:43,784 --> 00:00:47,396
We've seen aliasing for very particular
signals.

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00:00:47,396 --> 00:00:52,864
Sinus wave to complex exponentials.
Now we are going to look at aliasing for

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00:00:52,864 --> 00:00:56,316
arbitrary spectra's and look at a few
examples.

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00:00:56,316 --> 00:01:02,285
We'll finish by rubbing up sampling of
arbitrary signals using some projection

15
00:01:02,285 --> 00:01:05,691
theorem onto the space of unlimited
signals.

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00:01:05,691 --> 00:01:11,967
Remember the row sampling of an arbitrary
signal, so you have a continuous time

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00:01:11,967 --> 00:01:18,237
signal xc of t samples every ts seconds to
give a sequence xn which is equal to the

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00:01:18,237 --> 00:01:23,711
continuous time signals at multiples of
the sampling interval ts.

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00:01:23,711 --> 00:01:29,198
In fully transformed domain we have A
spectra of the continuous time signal

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00:01:29,198 --> 00:01:34,592
capital xc j capital omega and at the
output we have a discrete time Fourier

21
00:01:34,592 --> 00:01:39,016
transform of the sequence that's capital X
e to the j omega.

22
00:01:39,016 --> 00:01:44,590
What is that going to be in general and
how is it going to be related to the input

23
00:01:44,590 --> 00:01:47,638
spectrum.
The key idea is the following.

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00:01:47,638 --> 00:01:52,581
Pick a sampling interval TS or a frequency
omega N equal to pi over Ts.

25
00:01:52,581 --> 00:01:58,461
Start with omega 0 which is smaller than
omega N, the Niquest frequency which is

26
00:01:58,461 --> 00:02:04,246
the maximum frequency that can be
faithfully represented with this sampling

27
00:02:04,246 --> 00:02:07,900
system.
So the input is e to the J omega 0t, it's

28
00:02:07,900 --> 00:02:13,759
sampled with Ts, it gives an output e to
the J omega nought Ts times N, that's a

29
00:02:13,759 --> 00:02:17,796
sequence resulting from this sampling
operation.

30
00:02:17,796 --> 00:02:21,656
Then let's add 2 omega n, to the input
frequency.

31
00:02:21,656 --> 00:02:25,338
So now the frequency only gots 0 plus 2
omega n.

32
00:02:25,338 --> 00:02:29,422
It is sampled.
And the output, it is only got 0 plus 2

33
00:02:29,422 --> 00:02:34,712
omega n times ts times n.
We expand this product, then we see that

34
00:02:34,712 --> 00:02:41,012
the second term becomes 2 pi n, but 2 pi n
raised to the power e to the j, is equal

35
00:02:41,012 --> 00:02:47,186
to one, and therefore, we have the same
discrete time sequence as before.

36
00:02:47,186 --> 00:02:53,584
So, we do not see this higher frequency
complex exponential, it simply looks like

37
00:02:53,584 --> 00:02:59,941
the lower frequency exponential omega 0.
So in general if we have 2 frequencies A

38
00:02:59,941 --> 00:03:06,591
times complex exponential omega 0 plus B
times the complex exponential omega 0 plus

39
00:03:06,591 --> 00:03:11,181
2 omega N sampled it gives you A plus B e
to the j omega 0 Tsn.

40
00:03:11,181 --> 00:03:16,130
So the upper frequency 1 has been folded
back or aliased back.

41
00:03:16,131 --> 00:03:21,926
On to the lower frequency 1 and we simply
see that the sum of these 2 and so the

42
00:03:21,926 --> 00:03:26,340
alias part has been shifted back to the
non-aliased one.

43
00:03:26,340 --> 00:03:31,137
So the what is the spectrum of a
raw-sampled signal in general.

44
00:03:31,137 --> 00:03:36,939
Well lets look at this sample x of n, it
is xc at location nTs, we can write this

45
00:03:36,939 --> 00:03:43,195
as the inverse fourier transform of the
spectrum capital Xc multiplied by e to the

46
00:03:43,195 --> 00:03:46,866
j.
Omega nTs because that's the occasion we

47
00:03:46,866 --> 00:03:50,247
want to evaluate in this fourier
transform.

48
00:03:50,247 --> 00:03:54,478
Frequencies that are 2 omega N apart will
be aliased.

49
00:03:54,478 --> 00:04:00,271
We have seen this specifically on an
example just a couple of slides before.

50
00:04:00,271 --> 00:04:06,349
So we can split the integration interval
into pieces of size 2 omega N and this we

51
00:04:06,349 --> 00:04:10,240
do here.
So x(n) this inverse fourier transform is

52
00:04:10,240 --> 00:04:13,886
now.
There's a sum of the pieces from 2k minus

53
00:04:13,886 --> 00:04:20,256
1 omega n, to 2k plus 1 omega n, of the
fully transformed here, and multiplied by

54
00:04:20,256 --> 00:04:25,646
the complex exponential at nTs.
Lets look at this on a picture.

55
00:04:25,646 --> 00:04:31,935
So everything that these at location to
omega n will be folding back to the

56
00:04:31,935 --> 00:04:38,354
origin, what is that, 2 omega n plus
epsilon will be folded back to epsilon.

57
00:04:38,354 --> 00:04:43,861
What is at 2 omega n minus epsilon gets
folded back to minus epsilon.

58
00:04:43,861 --> 00:04:47,241
The same symmetrically around minus 2
omega n.

59
00:04:47,241 --> 00:04:52,311
It's the same from 4 omega n and minus 4
omega n et cetera, et cetera.

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00:04:52,311 --> 00:04:57,567
So, we can go back to our formula.
We do a change of variables so that the

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00:04:57,567 --> 00:05:01,004
integral is between minus omega n and
omega n.

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00:05:01,004 --> 00:05:05,682
This change is the frequency variable
inside of xc.

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00:05:05,682 --> 00:05:11,358
Then we also note that adding multiples of
2k omega n to the frequency doesn't change

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00:05:11,358 --> 00:05:15,670
the frequency of that c.
[unknown] effect, we can interchange

65
00:05:15,670 --> 00:05:21,367
summation and integral to end up this one
or two pi's, the integral, the summation.

66
00:05:21,367 --> 00:05:27,261
Of the spectrum xc and its shifts by
multiples of 2k omega m n.

67
00:05:27,261 --> 00:05:34,857
And then at the right end we still have
the integration variable for the inverse

68
00:05:34,857 --> 00:05:39,923
fully transform.
Lets define a periodized spectrum, so

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00:05:39,923 --> 00:05:46,927
capital X tilde c is the summation of Xc
and its shifted versions by even multiples

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00:05:46,927 --> 00:05:50,778
of omega n.
Therefore, the sample xn, is integral

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00:05:50,778 --> 00:05:56,988
between minus omega n and omega n, of the
periodized spectrum x tilde multiplied by

72
00:05:56,988 --> 00:06:01,822
e to the j omega nts.
Let's use a falling change of variable.

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00:06:01,822 --> 00:06:05,961
Small omega is equal to capital omega
times Ts.

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00:06:05,961 --> 00:06:12,367
This will change the integration
boundaries from minus omega, omega n, into

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00:06:12,367 --> 00:06:19,167
minus pi to pi, it changes the frequencies
inside the x tilled, and it changes it's

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00:06:19,167 --> 00:06:23,306
exponent to j o Omega N, which is more
convenient.

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00:06:23,306 --> 00:06:30,267
So, we recognize now that xn is the
inverse discrete time Fourier Transform of

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00:06:30,267 --> 00:06:35,015
1 over ts[INAUDIBLE] spectrum X til c
Rescaled by TS.

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00:06:35,015 --> 00:06:41,609
An important point to note is that
this[UNKNOWN] spectrum, rescaled, is

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00:06:41,609 --> 00:06:45,734
actually 2 pi periodic.
So it's a valid DTFT.

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00:06:45,734 --> 00:06:52,630
Since this is the IDTFT, it means that the
DTFT of the sequence, capital X of e to

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00:06:52,630 --> 00:06:57,361
the j omega, is simply up to a scale
factor of 1 over ts.

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00:06:57,361 --> 00:07:02,891
The summation of xe, and these shifts by
multiples of 2 pi over ts.

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00:07:02,891 --> 00:07:09,908
A picture is worth 1,000 equations, maybe.
So let us look at this spectrum and how it

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00:07:09,908 --> 00:07:13,051
evolves.
So we start with capital XC.

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00:07:13,051 --> 00:07:18,555
It's a spectrum, which is inside the
boundary minus omega n to omega n.

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00:07:18,555 --> 00:07:24,352
It doesn't cover the entire interval.
Then the periodized version puts

88
00:07:24,352 --> 00:07:28,860
repetition spectras of 2 omega n, 4 omega
n, et cetera.

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00:07:28,860 --> 00:07:34,976
This is now a periodic spectrum of period
2 capital omega n, so spectrum of x of e

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00:07:34,976 --> 00:07:38,270
to the j omega is A scaled version of
this.

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00:07:38,270 --> 00:07:42,409
It is 2 pi periodic.
And it is shown here at the bottom in

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00:07:42,409 --> 00:07:45,871
blue.
Let's take a band limited signal that is

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00:07:45,871 --> 00:07:49,250
band limited to omega zero, equal to omega
n.

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00:07:49,250 --> 00:07:53,196
This is the limit of the Nyquist sampling
frequency.

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00:07:53,196 --> 00:07:58,434
It will be periodic of period 2 omega n,
which is equal to 2 omega zero.

96
00:07:58,434 --> 00:08:03,847
So the repetitions will just.
Touch each other, it is therefore the

97
00:08:03,847 --> 00:08:10,026
usual periodic spectrum and now we look at
the spectrum of x of e to the j omega.

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00:08:10,026 --> 00:08:13,480
It fills the entire interval minus pi to
pi.

99
00:08:13,480 --> 00:08:17,315
What happens when omega 0 is bigger than
omega N.

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00:08:17,315 --> 00:08:22,624
Well we know trouble is going to come
because aliasing will happen.

101
00:08:22,624 --> 00:08:29,116
So, we see that the spectrum is beyond the
central interval of minus omega N to omega

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00:08:29,116 --> 00:08:31,936
N.
So the repetition will overlap with each

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00:08:31,936 --> 00:08:34,619
other.
The result is still 2 omega N periodic

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00:08:34,619 --> 00:08:39,587
spectrum, we see it has a funny shape now,
it's not the original shape because of the

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00:08:39,587 --> 00:08:43,447
overlaps and when we look at the spectrum
of x of e to the j omega.

106
00:08:43,447 --> 00:08:48,210
So blue spectrum, it indeed does not
resembles the original spectrum because an

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00:08:48,210 --> 00:08:51,175
entire interval has been messed up by
aliasing.

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00:08:51,175 --> 00:08:54,767
Let us look at the non-band limited
signal.

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00:08:54,767 --> 00:09:01,307
So this is a goshan type of single, which
has decayed but goes well beyond the

110
00:09:01,307 --> 00:09:07,382
central interval minus omega m, 2 omega m.
It is after sampling the periodized

111
00:09:07,382 --> 00:09:12,926
version is to omega n periodic of course,
but there are overlaps all over, and it

112
00:09:12,926 --> 00:09:18,890
has in the center of the interval shapes
that resemble the original spectrum, but

113
00:09:18,890 --> 00:09:24,182
then it has overlap And so the blue
spectrum, which is between minus pi and pi

114
00:09:24,182 --> 00:09:28,024
resembles the original spectrum around the
origin 0.

115
00:09:28,024 --> 00:09:32,442
But is different from the original at the
boundary of the Intro.

116
00:09:32,442 --> 00:09:38,070
Since it leads us to sampling strategies
for different signal.

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00:09:38,070 --> 00:09:45,312
So if you are given a sampling period Ts
and the nyquist frequency omega n is equal

118
00:09:45,312 --> 00:09:50,154
to pi over Ts, if the signal is band
limited to omega n.

119
00:09:50,154 --> 00:09:53,628
Then row sampling is fine.
As I pull in to sink sampling up to a

120
00:09:53,628 --> 00:09:56,937
scaling factor.
If the signal is not band limited, we have

121
00:09:56,937 --> 00:09:59,884
two choices.
Either we band limit it first with a low

122
00:09:59,884 --> 00:10:03,569
pass filter that will cut off all
frequencies beyond omega n in the

123
00:10:03,569 --> 00:10:06,296
continuous time domain before we do
sampling.

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00:10:06,296 --> 00:10:10,992
Then, everything will work just fine as we
have seen or we roll sample the signal,

125
00:10:10,992 --> 00:10:14,726
and we have aliasing.
Now let's face it, aliasing usually sounds

126
00:10:14,726 --> 00:10:17,510
horrible, so usually it's not the first
choice.

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00:10:17,510 --> 00:10:23,024
So let's investigate a little bit how we
sample signals that are not strictly band

128
00:10:23,024 --> 00:10:26,717
limited to begin with.
So how do we do sync sampling and

129
00:10:26,717 --> 00:10:30,646
interpolation.
So sample x of n is simply the inner

130
00:10:30,646 --> 00:10:36,982
product of sync function shifting to the
location n times Ts was x(t) so in put

131
00:10:36,982 --> 00:10:43,813
single, this can be written simply as the
combination of the sync with x at location

132
00:10:43,813 --> 00:10:47,787
nTs.
Then the interpolation, x hat of t, is the

133
00:10:47,787 --> 00:10:54,776
sum of the samples multiplied by the same
functions shifted to the location n t s.

134
00:10:54,776 --> 00:11:00,830
We see this in this block diagram from
left to right we have a continuous time

135
00:11:00,830 --> 00:11:04,870
signal x of t.
It goes through an ideal low pass filter

136
00:11:04,870 --> 00:11:10,412
with the cut off frequency capital omega
N, it is sampled every Ts seconds.

137
00:11:10,412 --> 00:11:16,277
So samples are interpolated by sync
interpolation and the result is x hat of

138
00:11:16,277 --> 00:11:19,673
t.
It is interesting to look at this scheme

139
00:11:19,673 --> 00:11:24,950
in purely diametrical terms.
So, we have an input signal x, it's a

140
00:11:24,950 --> 00:11:30,072
vector in a Hilbert's space.
We have a subspace of band limited

141
00:11:30,072 --> 00:11:33,504
signals, BL.
The first thing we do is we have an

142
00:11:33,504 --> 00:11:38,898
orthonormal basis for BL, it's given by
the sync function and its shifts by

143
00:11:38,898 --> 00:11:44,257
integer multiples of capital Ts.
Based on this orthonormal basis, we write

144
00:11:44,257 --> 00:11:49,833
an orthonormal expansion, which is the
orthonormal project of x onto the space of

145
00:11:49,833 --> 00:11:53,949
band limited signal.
So it's written as a sum of the sync in a

146
00:11:53,949 --> 00:11:59,169
product with x times the sinc functions.
And so we see that the scheme we had on

147
00:11:59,169 --> 00:12:04,462
the previous slide is simply the projector
of x on to the subspace of band limited

148
00:12:04,462 --> 00:12:10,056
signal, that projection of course can be
written in terms of the sampling theorem.

149
00:12:10,056 --> 00:12:14,431
But we can not write x we can only write x
hat, the projection.

150
00:12:14,431 --> 00:12:20,302
Now, let's look at the concrete example,
one that we have encountered before, which

151
00:12:20,302 --> 00:12:24,691
is a non band limited signal.
Here's is its quotient shaped signal.

152
00:12:24,691 --> 00:12:29,572
And we are going to do least squares
approximation on the space of band limited

153
00:12:29,572 --> 00:12:32,661
signals using sync sampling and
interpolation.

154
00:12:32,661 --> 00:12:37,371
So, we start with capital Xc as a
continuous time spectrum, non band

155
00:12:37,371 --> 00:12:40,787
limited.
We are going to band limit it to minus

156
00:12:40,787 --> 00:12:44,995
omega N to omega N.
So this is given by this green ideal

157
00:12:44,995 --> 00:12:49,128
filter.
After filtering, the spectrum is strictly

158
00:12:49,128 --> 00:12:53,624
band limited.
It's zero outside this interval of size 2

159
00:12:53,624 --> 00:12:57,665
omega n.
Then we do the sampling and this reads to

160
00:12:57,665 --> 00:13:03,552
a periodic spectrum X tilde c.
Given in these pink shape here and it

161
00:13:03,552 --> 00:13:10,702
repeats at multiplies of 2 omega N as we
have seen before then we look at the DTFT

162
00:13:10,702 --> 00:13:14,946
of this sequence and the DTFT is 2 pi
periodic.

163
00:13:14,946 --> 00:13:21,602
It's given by this blue function and
finally we re-interpolate a continuous

164
00:13:21,602 --> 00:13:28,674
time spectrum using sync interpolations
that gives us x hat c which now is exactly

165
00:13:28,674 --> 00:13:34,545
the reconstruction of the band limited
version of the initial x c.

166
00:13:34,545 --> 00:13:40,751
This band limited version, of course
easier orthogonal projection of the

167
00:13:40,751 --> 00:13:45,623
initial spectrum onto the space of band
limited signals.