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Last time we discussed about, the relationship
between the two types of frequency transforms;
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the one in the continuous domain, the other
one in the discrete domain of time.
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So, we saw that if we know the analogue domain
frequency response, analogue domain frequency
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response say; something like this if it is
band limited then if we can maintain a sampling
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frequency which is much higher than the maximum
limit of the frequency in the band, then we
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can have the frequency domain response of
the system for discrete time systems as repetition
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of the same characteristics with a periodicity
of 2 p i, where 2 p i corresponds to omega
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s the sampling frequency, okay.
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So, it is in terms of radian frequency, this
is in terms of radian per second. You can
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always convert this to radian per second,
this is analogue frequency. And hence; from
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a given characteristic like this given frequency
domain response in the discrete domain of
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time, we can always get back to the analogue
domain response by filtering this in the base
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band and then taking inverse transform. So,
Nike’s frequency criteria suggest, that
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the sampling frequency should be at least
2 times, the maximum frequency that is present
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in the signal.
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Now, for example for music system 44.1 KHz
is the frequency of sampling that we use for
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CD, compact disc. Then what do you want to
really capture is a signal up to 20 KHz or
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so 20 KHz or so. Whereas, for speech it is
8 KHz sampling frequency, that means we want
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to capture within 4 kilo hertz; the frequencies
of speech that will be, within 4 KHz, this
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is for telephonic purpose, all right, for
telephonic we require 8 kilo hertz of sampling.
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If you want, normally the frequency band is
approximately up to 3.4 KHz, that you want
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to capture. So, it is little above twice this
frequency, so this is good enough. If you
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want to identify a person, if you want to
identify a person very accurately then we
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go a little above this. Quite often you must
have observed, you must have experienced when
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you receive a telephone call, you are not
able to distinguish the voices of the speaker
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at the other end all right, unless he introduces
himself quite often, we make mistakes, okay.
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So, if you want to also recognise the voice
then we go for22.05 KHz, that means if you
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want to go for a little higher band; as I
told you yesterday actually it is not suddenly
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coming to a drastic end at a particular frequency,
we are truncating it by pre filtering, others
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it will be going tapering of like this to
a larger value.
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So, where you are cutting? Cutting this off
will be ensuring the quality of the speech,
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that you recovering, all right. So, if you
want to identify a person's voice; that means
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there are certain high frequency components,
if you miss out you will not be able to identify.
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That is why for speech recognition, speaker
recognition we go for a little higher frequency
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of sampling, okay. So, this will be less than
little less than 11 KHz.
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Now, today we shall be taking up Discrete
Fourier transform. So, in discrete Fourier
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transform, what exactly we want to find out?
When a sequence is a finite length, x n is
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a finite length then the DTFT, which we defined
as summation x n e to the power minus j omega
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n. And from this DTFT, we can recover the
signal x n; which means what should we write,
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1 by 2 p i, all right e to the power plus
j omega n d omega, so we have to perform this
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integration.
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You have to know this in the continuous domain
of frequency, all right. So, as I had mentioned
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this x may be of this type, okay. So, you
have to take the inverse transform of this
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to go from frequency domain to time domain.
Since it is periodic, so in the time domain,
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it will be discrete lines which will be x
n, okay. Now, instead of computing this from
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this integral, can we can we realise this
from some discrete values of this frequency
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domain representation?
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That is we shall not take a continuous set
of values but we can take some discrete values
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of this. Can we compute x n from these discrete
lines, if x n is a finite sequence? Yes, then
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it is possible. If, x n is finite then x n
can be recovered from only specific values
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of omega K, okay, we can realise x n. If x
n is having a length of N, say 0, 2 N minus
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1; x 0 x 1 up to x n minus 1 then we can take
those many values, that is n number of values
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K varying from 0 to N minus 1.
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We can take only N number of frequencies and
evaluate the function at those N numbers of
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frequencies, all right. And that is sufficient
to define the function x n. You can always
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retrieve x n from, not from the continuous
function but only from those n number of frequency
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domain representation, so this we defined.
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This is also very interesting, this will be
the same expression; x n e to the power minus
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j omega n, this we are sorry omega into n,
this this we are evaluating, earlier. We shall
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be taking only some specific values of K,
omega equal to omega K. We take this as, e
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to the power minus j 2 p i by N, we subdivide
the entire angle of 2 p i into n number of
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equi-space points, all right. We call this
basic operation of an angular shift as W N;
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this capital W stands for a unit, basic unit
of the angular shift. See, e to the power
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minus j 2 p i by N is, if this is 2 p i you
divide into n equal parts, okay like this.
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So, each of these will be corresponding to
an angular shift of WN. So, if I have 8 points
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then WN will be; WN 1, that is one unit will
be e to the power minus j 2 p i by N, that
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is multiplying any quantity if I, say multiply
x K by W N 1, it will mean x K multiplied
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by e to the power minus j, this N is now 8,
2 p i by 8. How much is it, 45 degree shift.
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So, it is x K into e to the power minus j
p i by 4; which means x K multiplied by sorry,
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0.7 o 7 minus j 0.7 o 7, 1 by root 2. That
means the entire circle has been divided into
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8 parts, all right. So, this is 45, 90, 135,
180 and so on. So, cosine of minus 45 degrees
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and sin of minus 45 degrees, so that give
me this; so it is basically a complex quantity
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that you are multiplying with the real term
x K, all right.
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So, X suppose you are given some values x
n; so let us take a very simple set of values
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1 minus 1, 2, 3, 0, 1, 2, one two three four
five six seven; let us have one more value
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1, as for simplicity I have taken very simple
values; 1, 0, 2. So, what will be capital
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X 0? We define, okay before we go through
that; X we are evaluating at e to the power
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j omega K, at K equal to 0. We call this as
capital X K at K equal to 0.
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So, that is capital X 0, okay. Hence forth,
in the transform domain we shall be denoting
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this because there are only 8 for an 8 point
sequence; there are only 8 frequency domain
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representations, so I will call them as, capital
X 0, X 1, X 2, all right. They are very similar
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to your Fourier harmonics, in the continuous
domain you talk about harmonic terms all right;
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they are also exactly similar to harmonics.
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So, X 0, how much will be that, X 0? It will
be x n, WN; see this was e to the power minus
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j 2 p i by N into K into n, is it not, if
you look at it we were multiplying by e to
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the power minus j omega n into x n, is it
not summation like this? And what is it, x
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n e to the power minus j omega K into n? And
what is omega K? Which means, e to the power
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minus j 2 p i by capital N into K, is it not?
K equal to 1 corresponds to this, K equal
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to 2, K equal to 0, K equal to 3, 4, 5, 6,
7, so there are eight values. So, you are
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taking different values of K between 0 and
7, okay there there are eight points.
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So, if you substitute here, so W N into n
K which means; e to the power minus j 2 p
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i by N into K into N. So, basically this reduces
to W K into n and W K is nothing but, 2 p
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i by N into e to the power minus j 2 p i by
N into K, okay this small w. So, this is x
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n, W N, n K is that okay? And this is to be
evaluated at K equal to 0, to get these value,
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okay.
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So, what is it? If I put K equal to 0, this
will be always 1; e to the power minus j 2
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p i by N multiplied by 0, is always 1, so
it will be just summation of x n, sum of these
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terms all right. So, it will be 1 minus 1,
0, 2 plus 3, 5 plus 1, 6, 7, 9 is that okay?
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So, let us compute one or two more then we
I will go to the details of this. What will
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be X 1 for the same sequence, what will be
X 1?
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It will be, x 0 plus x 1 into e to the power
minus j 2 p i by 8 into 1 into 1 n into K,
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all right. n is this, K is this. So when I
am going to compute for K, I multiply by this
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plus x 2 into e to the power minus j 2 p i
by 8 into 2 into 1 plus x 3 into e to the
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power minus j 2 p i by 8 into 3 into 1, is
that all right, and so on plus x 7 into e
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to the power minus j 2 p i by 8, 7 into 1,
okay.
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So, 1 comes every time because I am evaluating
it, for K equal to 1 that is corresponding
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to K. And this 7, 3, 2 etcetera, they will
be coming along with the term x n, all right
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because that n is varying. So, it is summated
over n. So, how does it look like? 1 plus
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first one is 1 then minus 1 next value is
minus 1 into e to the power minus j; now,
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there is a common multiplier that you can
see, e to the power minus j 2 p i by 8 into
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1. So long as this is 1, e to the power minus
j 2 p i by 8 into 1, 2 p i by 8 into 1, so
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e to the power minus j 2 p i by 8 is a basic
element of rotation minus 45 degrees.
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And it is to be multiplied by 1, 2, 3 depending
on the sequence element, x 2 x 3 and so on.
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So, the first element, it will be e to the
power minus j p i by 4 okay then 2 into e
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to the power minus j, 2 times this plus 3
into e to the power minus j three times this,
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this is the basic angle plus 0 plus 1 into
e to the power minus j 5 p i by 4 plus 2 into
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e to the power minus j 6 p i by 4 plus 1 into
e to the power minus j 7 p i by 4, is that
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all right. Whatever be that result, sum A
plus j B, you can write all right.
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If I go for say, I want to compute X 3, what
should I do? I am arbitrarily taking some
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value other than one, to make my point clear.
x 0 plus; what should be the multiplication
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of x 1, now the basic angle will be three
times p i by four that is minus forty-five,
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minus ninety and then minus 135 degrees, okay.
It is minus 135 degrees from your side, minus
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135 degree should be the basic angle and multiples
of minus 135 degrees, is that all right.
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So, x 1 into, if you permit me to write in
terms of degrees which you are probably all
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convenient with; minus 135 degrees I will
write like this, only for is of understanding
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then x 2 into e to the power minus j, 270
degree all right plus x 3 into e to the power
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minus j 405 degrees and so on, is that okay.
And you compute the values for minus 45 degrees,
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it is plus 0.7 o 7, minus j point 7 o 7. For
90 degrees, it will be just j. 135 degrees
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it is in the third quadrant, both will be
negative. All the time, it will be 1 by root
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2.
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Then if I take x 4, what will be the basic
angel, four times 45 degrees. So 180 on; so
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it will start with minus 1, 2 times 180 plus
1, 3 times 180 minus 1 again four times 180
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plus 1 and so on. When we compute X 4; X 4
which is half of capital N that is 8 point
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sequence, if you take half of it, the basic
angle will be 180 degrees all right. So, alternately
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then science will be changing okay.
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So, if I ask you to compute X 4; blindly you
can write 1 minus of this, which is plus 1
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minus then plus of 2, minus of three plus
of 0, minus of one plus of 2, minus of 1 that
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will be the net result. So, whenever you are
going to compute X at N by 2, X at N by 2
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the sequence will be just alternately appearing
with the negative sign
and so on, is that okay. The reverse relation,
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that is to get x n from x K is also very simple,
it is 1 by N, X K, W N minus n K.
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See, the forward sequence was X K sigma x
small x n, W N n K; mind you W has associated
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with it negative power here, all right minus
j 2 p i by N. So this means, plus j 2 p i
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by N, okay. If you are given this set of X
K; 8 discrete Fourier transforms, if I give
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you, you can always get back x n. Earlier,
you for the continuous domain you are integrating
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from minus p i to plus p i, here you do not
have to do that; you just take summation of
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those eight points and you have to multiply
by just complex conjugate.
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You see, earlier it was minus p i by 4, now
if I put a power of plus one, it will be plus
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p i by 4, all right. So, it will be just complex
conjugate. So, W N, n K is nothing but W N
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minus N K conjugate, all right. It is a just
a complex operator. Now, how do you get these?
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Where is the proof? So, let us go for the
proof. So, X K is n varies from 0 to N minus
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1. I can write x n, W N, n K, n vary from
0 to N minus 1. Now, this x n we have written
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as, 1 by N X say, p you can write X p W N
minus p K, okay; p varying from this is n
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equal to 0 to N minus 1, I should write p
N W minus n K okay, plus n K, is that all
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right.
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See, x n X K, I started of this X K is equal
to x n W N n K. And x n, I can write in terms
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of this. I am not using the same variable
K, I am taking I can have any running variable
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K. So, I have taken p okay; X p W N p K, it
should be p n, all right and W n K okay, equal
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to 0 two N minus 1 without loss of generality.
P is the running variable, so summated from
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this range 0 to N minus 1, okay. K is not
a running variable, K is at a particular value
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K we are evaluating and over this again; I
am having a running variable of n, I can interchange
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p and n summation n equal to 0 to N minus
1, okay. I can put X p outside W N n into
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K minus p okay, n is varying.
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Now, most of you have studied in three phase
power systems, balanced and unbalanced voltages,
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all right. Any unbalanced system, we can always
represent in terms of sequence components
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of three sets of balanced voltages, all right.
So, if you remember the matrix of transformation
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that we used, was having an angular shift
of 120 degrees, okay.
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We are writing that as an operator a, okay.
So, we had a to the power 0, a to the power
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1 and a to the power 2; where a cube was making
1, all right. So a was basically, e to the
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power j 2 p i by 3, 2 p i by 3, hundred and
twenty degree shift, we are multiplying in
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the positive reaction. So, a 0 means 1, this
was a, this was a square, all right.
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Now, you take any one of them, a to the power
any of them okay and if you sum them together;
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that is a, suppose I take a basic element
of a 1 then a 1 to the power 0 plus a 1 to
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the power 1 plus a 1 to the power 2, will
be equal to 0, a 2 to the power 0 plus a 2
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to the power 2 plus a 2 to the power, sorry
1 plus a 2 to the power 2; so by the extent,
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that same logic here, we will find W N is
a basic element all right.
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If you take any difference K minus p other
than 0, now if I take a 0 to the power 0 plus
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a 0 to the power 1 plus a 0 to the power 2
then I will get 3, all right. So, it is basically
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the same thing that is how, we get the 0 sequence
component, is it not. W N is e to the power
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j 2 p i by N, okay. Now, if I multiply by
n and summit over n of a quantity other than
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00:30:10,600 --> 00:30:17,600
0; if K minus p is other than 0, 1, 2, 3,
4, whatever you take if you summit, it will
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be always equal to 0.
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But at K equal to p, it will be equal to n;
it has come to 3, is it not. So, it will be
183
00:30:28,620 --> 00:30:35,570
equal to n. So, this n will get cancelled
with these and this will stay only for K equal
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00:30:35,570 --> 00:30:42,570
to p, is that all right. So, it becomes only
X p, after summation it becomes only 1, 1
185
00:30:44,120 --> 00:30:51,120
into N, N gets cancelled. So, we get X K is
equal to X p where p is equal to K; this this
186
00:30:59,159 --> 00:31:06,159
is valid only for p equal to K, okay.
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00:31:07,610 --> 00:31:14,610
So, basically 1 plus W plus W square plus
W cube and so on, W N minus 1 is always 0;
188
00:31:31,149 --> 00:31:38,149
sum of these will be always 0. If we take
1 plus W square plus W square squared W squared
189
00:31:40,529 --> 00:31:47,529
cubed W square to the power N minus 1 will
be also equal to 0. You take any power of
190
00:31:49,779 --> 00:31:54,789
W, as a basic element and then you add up
these terms, it will be always equal to 0,
191
00:31:54,789 --> 00:32:01,789
except for W equal to W to the power 0, okay.
Now, there are certain interesting functions,
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00:32:17,929 --> 00:32:22,450
that we shall discuss now, delta n all right.
193
00:32:22,450 --> 00:32:29,450
What will be its DFT, what will be its DFT?
Could someone suggest, what will be the DFT
194
00:32:39,269 --> 00:32:46,269
of delta n, n? Delta n is basically, a sequence
like suppose we have an eight point sequence;
195
00:32:51,259 --> 00:32:58,259
we take only eight points n 1, 0, 0, there
is seven 0, all right. So, if we have an eight
196
00:33:00,990 --> 00:33:07,990
point DFT, what will be this? Okay, what is
X 0? 1. X 1? 1. X 2? All will be 1, okay.
197
00:33:17,330 --> 00:33:24,330
So, X K equal to 1, this is what we have observed
also in DTFT, will you remember?
198
00:33:27,409 --> 00:33:34,409
What will be delta; say n minus 3, what will
be delta n minus 3? Okay, delta n minus 3
199
00:33:45,360 --> 00:33:52,360
is a sequence, like 0, 0, 0, 1 then again
all zeros, okay. So, what will be capital
200
00:33:56,860 --> 00:34:03,860
X 0? 1. X 1? W N 3, do you all agree? What
will be X 2? What is W N 3? Minus 1 by root
201
00:34:20,310 --> 00:34:27,310
2, minus j 1 by root 2; third quadrant 135
degrees, is it not? If n is equal to eight,
202
00:34:28,429 --> 00:34:32,500
you are considering an eight point sequence
then it will be 135 degrees.
203
00:34:32,500 --> 00:34:39,500
And what will be X 2? Two times 135 degrees,
so 270 degrees, so minus 270 degrees, remind
204
00:34:46,800 --> 00:34:53,800
you. Minus 270 is plus 90 degree, so that
will be plus j, okay. I will write W N 6,
205
00:34:55,720 --> 00:35:02,720
that is nothing but plus j, we have taken
n is equal to; you should put 8 then only
206
00:35:04,200 --> 00:35:11,200
then will be clear. Yes, we are considering
only eight point sequence for ease of understanding,
207
00:35:16,240 --> 00:35:21,130
you change the value of N; obviously these
values will be changing, okay.
208
00:35:21,130 --> 00:35:28,130
So, this is how you can obtain the DFT’s.
There are now, can you compute DTFT from DFT’s
209
00:35:45,920 --> 00:35:52,920
that is what we are trying to achieve is;
if you are given say the DFT’s of course
210
00:35:56,109 --> 00:35:59,670
their complex, there is an angle associated
with that.
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00:35:59,670 --> 00:36:06,670
I will show only the magnitudes on this side
and so on. Suppose this this is a DFT, what
212
00:36:13,690 --> 00:36:20,690
we are trying to find out is; what is that
continuous function whose discrete values
213
00:36:24,300 --> 00:36:29,880
are these, that means you are trying to reconstruct,
you are trying to reconstruct the original
214
00:36:29,880 --> 00:36:36,300
DTFT whose sampled values are these DFT’s.
215
00:36:36,300 --> 00:36:43,300
So from the sample values, we are trying to
reconstruct the original DTFT, okay. So, we
216
00:36:45,450 --> 00:36:52,450
go by this, X e to the power j omega is; sigma
x n e to the power minus j n omega, and what
217
00:37:05,250 --> 00:37:12,250
is x n? You can write in terms of the DTFT’s,
the DTFT; we can write in terms of DFT, it
218
00:37:19,940 --> 00:37:26,940
will be
X K W N minus K n, all right. Then e to the
power minus j n omega, this is for x n from
219
00:37:47,980 --> 00:37:54,579
the discrete domain, is it not.
220
00:37:54,579 --> 00:38:01,579
I can take 1 by N outside, this is summated
over K, I can interchange the summation operation
221
00:38:09,930 --> 00:38:16,930
and then X K summation. What is W N minus
K n? We can write in terms of e to the power
222
00:38:23,770 --> 00:38:30,770
minus j 2 p i by N into K, is this all right.
This is minus that means; here it will be
223
00:38:38,140 --> 00:38:41,060
plus minus n omega.
224
00:38:41,060 --> 00:38:48,060
So, minus omega whole thing into n is that
all right, is this okay? n varying from, so
225
00:39:01,530 --> 00:39:08,530
what you are adding up is; capital N number
of summation of this exponential terms, all
226
00:39:12,500 --> 00:39:18,160
right for a particular value of omega. We
are now trying to find out all continuous,
227
00:39:18,160 --> 00:39:25,040
the continuous domain representations of that
frequency function.
228
00:39:25,040 --> 00:39:32,040
So, here if I take e to the power j 2 p i
by N K minus omega as a basic element x all
229
00:39:40,760 --> 00:39:47,760
right; so what is this? 1 plus x plus x square,
n is equal to 0, 1, 2, 3 means up to x to
230
00:39:50,690 --> 00:39:57,690
the power N minus 1, okay. I can write this
as, multiplied by 1 minus x divide by 1 minus
231
00:39:58,810 --> 00:40:05,810
x; so 1 minus x to the power N by 1 minus
x, is it not?
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00:40:06,089 --> 00:40:13,089
So, you can write this as equal to 1 by N
summation X K, then 1 minus e to the power
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00:40:26,819 --> 00:40:33,819
minus j omega. I can always take omega on
this side, so I put a minus; minus 2 p i by
234
00:40:37,970 --> 00:40:44,970
N K, okay whole to the power N divided by
1 minus e to the power minus j omega minus
235
00:40:53,230 --> 00:41:00,230
2 p i by N into K, correct me if I am wrong,
K equal to 0 to N minus 1, is that okay. I
236
00:41:09,520 --> 00:41:16,440
can write this, this divided by 2; if I take
out then it will be sin function.
237
00:41:16,440 --> 00:41:23,440
Similarly 1 minus e to the power minus j theta;
I can take minus j theta by two common, so
238
00:41:24,040 --> 00:41:30,270
e to the power minus j theta by 2 into e to
the power j theta by 2 minus e to the power
239
00:41:30,270 --> 00:41:37,270
minus j theta by 2, okay; which will give
me, e to the power minus j theta by 2 into
240
00:41:38,079 --> 00:41:45,079
twice sin of theta by 2 into j, okay. Now,
both of them will generate j, so they I will
241
00:41:46,740 --> 00:41:51,109
get cancelled. Both of them will generate
2, so they will also get cancel. So, 2 j will
242
00:41:51,109 --> 00:41:52,750
get cancelled from both.
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00:41:52,750 --> 00:41:59,750
So, I will take out 1 by N sigma X K e to
the power minus j omega minus 2 p i by N K
244
00:42:11,099 --> 00:42:18,099
into N by 2, divided by e to the power minus
j omega minus 2 p i by N K divided by 2. And
245
00:42:26,890 --> 00:42:33,890
this side, I will have sin of omega minus;
okay if I call this as theta okay, 2 p i by
246
00:42:38,530 --> 00:42:45,530
N K into N by 2 divided by sin of omega minus
2 p i by N K divided by 2 okay, this angle
247
00:42:57,480 --> 00:43:02,290
divided by 2.
So, that gives you 1 by N, in a compact form;
248
00:43:02,290 --> 00:43:09,290
if I write X K e to the power minus j, this
is this by 2 if I take as theta omega minus
249
00:43:15,619 --> 00:43:22,619
2 p i by N K by 2 as theta. Then it is N theta
and this is theta, so N minus 1 into theta,
250
00:43:28,599 --> 00:43:35,599
e to the power j theta 1 e to the power j
theta 2, so this will be theta 1 minus theta
251
00:43:37,349 --> 00:43:44,349
2; okay into sin of N theta by 2 by sin of
theta by 2, where or this I have taken as
252
00:43:53,230 --> 00:43:58,400
theta. So, sin N theta by sin theta okay.
253
00:43:58,400 --> 00:44:05,400
So, given the values of X K, you can always
reconstruct capital X e to
the power j omega. The other alternative is;
254
00:44:19,839 --> 00:44:26,839
this will be giving you an analytical expression,
all right, so that anybody can put in any
255
00:44:27,510 --> 00:44:33,339
value of omega we can evaluate this, corresponding
to omega you have to evaluate theta and then
256
00:44:33,339 --> 00:44:39,880
we can make a program. So, normally what we
do, we do not make an analogue plot. We take
257
00:44:39,880 --> 00:44:46,880
large number of points, you know. Say, if
I ask you to sketch X versus omega capital
258
00:44:48,059 --> 00:44:54,970
X versus omega, like this, what do you do?
259
00:44:54,970 --> 00:45:00,650
You compute for omega equal to 0.001 radian,
0.002 radian and so on. Large number of points
260
00:45:00,650 --> 00:45:07,650
you compute and then you sketch it, is it
not. So, if you have to do it for getting
261
00:45:09,309 --> 00:45:14,800
a continuous domain function, if you have
to do it in a discrete way then why not do
262
00:45:14,800 --> 00:45:21,800
it with the help of DFT, so what do you do?
Suppose, there are eight points given to you,
263
00:45:22,569 --> 00:45:29,410
so eight points will give you this kind of
a characteristic; out of which you are trying
264
00:45:29,410 --> 00:45:36,410
to identify, with those eight points, if you
take direct DFT, you get only eight points
265
00:45:36,829 --> 00:45:43,829
here, okay. Instead of eight points, I could
have selected sixteen points, all right; sixteen
266
00:45:45,849 --> 00:45:52,849
points and I could have got sixteen such values.
So, x 0, x 1 up to x 7, there are seven points
267
00:45:57,359 --> 00:45:58,770
given to you.
268
00:45:58,770 --> 00:46:05,480
There is no harm, if I put some more zeros,
eight more zeros. So that becomes a sixteen
269
00:46:05,480 --> 00:46:12,480
point sequence, okay. Now, my multiplier will
be e to the power minus j 2 p i by 16, basic
270
00:46:13,559 --> 00:46:19,790
element will be 2 p i by 16, 22.5 degrees.
I will keep on doing that and then I will
271
00:46:19,790 --> 00:46:26,790
get sixteen components, all right. If I appended
with say, large number of zeros to make it
272
00:46:28,329 --> 00:46:35,329
a length of 256, 256 then I will get two-fifty
six such points, all right. So, basically
273
00:46:37,710 --> 00:46:44,710
you append this by zeros to make it of a length
M, which is much much greater than the given
274
00:46:46,000 --> 00:46:48,990
sequence length N.
275
00:46:48,990 --> 00:46:55,829
Well, I am getting large number of points;
I can get a very smooth curve. So, my idea
276
00:46:55,829 --> 00:47:02,829
is to get DTFT from DFT, one is using this
relationship all right. But then this will
277
00:47:05,240 --> 00:47:12,240
remain in any case this will remain only on
paper, as a as an analytic function, as an
278
00:47:12,430 --> 00:47:19,430
algebraic expression but when you want to
compute it, for making a plot you take discrete
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00:47:19,890 --> 00:47:23,950
points and compute, is it not. Compute these
values and then you make a plot.
280
00:47:23,950 --> 00:47:30,950
So, when you are going to take discrete points;
that means what values of theta will you take?
281
00:47:31,740 --> 00:47:38,740
What values of omega will you take? Say, 0.1,
0.2, 0.3 so many radians. So, I suggest you
282
00:47:38,780 --> 00:47:44,349
take thousand such point, thousand twenty-four
points, 2 p i divided by thousand twenty-four
283
00:47:44,349 --> 00:47:51,349
that will also give you of that order, 0.1,
0.2, is it not. So, if I can take 1024 points
284
00:47:52,270 --> 00:47:58,170
that means these eight points I append with
zeros; to make it a length of 10, 24 then
285
00:47:58,170 --> 00:48:03,490
I will get those ten, twenty-four points by
direct competition of DFT.
286
00:48:03,490 --> 00:48:10,490
Now, if a faster algorithm exists for computation
of DFT, for large number of points where number
287
00:48:12,020 --> 00:48:19,020
of points is 2 to the power N then the problem
is very easy, simple, all right. So, that
288
00:48:19,490 --> 00:48:26,490
algorithm is known as FFT, so fast Fourier
transform. Basically, the discrete Fourier
289
00:48:26,760 --> 00:48:32,040
transform is computed via very fast algorithm.
So, fast Fourier transform is not something
290
00:48:32,040 --> 00:48:39,040
different, it is only an algorithm. So, whenever
you are asked to compute the DFTs of a sequence,
291
00:48:40,910 --> 00:48:47,369
we normally pad it with zeros all right; we
get large number of points and we try to take
292
00:48:47,369 --> 00:48:54,119
the advantage of FFT algorithm that is very
simple.
293
00:48:54,119 --> 00:49:01,119
So, we go for appending, also known as padding.
Appending the data of sequence x n by zeros.
294
00:49:18,280 --> 00:49:25,280
Sequence length was N and M is a appended
sequence length, okay. So, M is much much
295
00:49:39,470 --> 00:49:46,470
greater than N. So, this is the practical
way of computing DFT from DTFT. Now, there
296
00:49:54,809 --> 00:50:01,809
are certain important properties of DFT that
we can just discuss. The properties are most
297
00:50:09,700 --> 00:50:16,700
of the properties are same as DTFT okay, most
of the properties are similar to DTFT. There
298
00:50:22,540 --> 00:50:29,540
are some interesting features of DFT; circular
time shifting.
299
00:50:38,470 --> 00:50:45,470
Now while computing DFT, when we talk about
DFTs, we introduce this terms circular sequence.
300
00:50:48,609 --> 00:50:55,609
Suppose, we have okay we have a sequence say
zero, one, two, three, four like this; I mark
301
00:51:02,000 --> 00:51:09,000
it 0, 1, 2, 3, 4, 5, then. Suppose, zero,
one, two, three, four, five this five point
302
00:51:25,089 --> 00:51:31,619
sequence; I have not shown you the values
for which a particular variable, say x n exists,
303
00:51:31,619 --> 00:51:38,619
that is x 0 is a plus 3 okay, x 1 may be 4,
x 2 may be 1 and so on.
304
00:51:38,970 --> 00:51:44,890
There are some values at these points and
after that, the function becomes zero, this
305
00:51:44,890 --> 00:51:51,890
is a normal linear sequence, okay. If we roll
it that is next point is made 0; that is this
306
00:51:57,450 --> 00:52:04,450
zero is brought here, next to 5 then what
will you see? If I keep on shifting it, it
307
00:52:11,380 --> 00:52:16,599
will be periodically appearing, is it not?
If I keep on rotating it, we I will get zero,
308
00:52:16,599 --> 00:52:20,609
one, two, three, four, five, zero, one, two,
three, four, four, five and so on.
309
00:52:20,609 --> 00:52:27,609
So such a periodic sequence, can we generated
from any finite length sequence, by just putting
310
00:52:29,470 --> 00:52:35,119
it like this, rolling it all right, winding
it and then you get a sequence like this.
311
00:52:35,119 --> 00:52:41,530
Now, if I just move it, I will get the same
sequence with different starting point, all
312
00:52:41,530 --> 00:52:48,530
right. So, this is known as a circular shift;
I mean this kind of a sequence we shall be
313
00:52:48,859 --> 00:52:55,859
using, in circular shift property. So, circular
time shifting property is, suppose there is
314
00:52:56,339 --> 00:53:01,950
a function f module N.
315
00:53:01,950 --> 00:53:08,950
Module N means; basically one, two, three,
four, five, six, this capital N is six. So,
316
00:53:14,380 --> 00:53:21,380
after every six points, it is repeated, okay.
So, if I take n is equal to 3, n is equal
317
00:53:24,869 --> 00:53:31,869
to 3 and n 0 is any particular shift; so 3
minus a n 0 is 1, so 3 minus 1. And then I
318
00:53:34,270 --> 00:53:41,270
keep on 4 minus 1, 7 minus 1, 8 minus 1; if
I want to compute 8 minus 1, if I take module,
319
00:53:43,280 --> 00:53:50,280
module means you subtract keep on subtracting
six, all right. If it is in the negative region,
320
00:53:53,140 --> 00:53:59,030
then you keep on adding six.
So, x of minus two is same as, if I add six
321
00:53:59,030 --> 00:54:06,030
instead of four all right, same as x of ten.
So, if if I evaluate this, the corresponding
322
00:54:12,559 --> 00:54:19,559
DFT will be W N K times n 0, F F K. So, if
I want to compute the K th element of that
323
00:54:29,730 --> 00:54:36,730
shifted sequence then it will be the original
sequence value of F K multiplied by this W
324
00:54:38,030 --> 00:54:49,490
N K n 0, okay. We I will stop here for today,
we will take it up in the next class.