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Prof. Gunaratna Setty delivered this lecture at Gujarat Ayurved University for Digital Signal Processing course. It includes: Operation, Principles, Digital, Signal, Processing, Aperiodic, Spectrum, Continuous
Typology: Slides
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Time Domain
Frequency
Domain
N o w w e can say d ivid e th e ran g e - in to in fin ite p erio d s each w ith len g th
, g o t th at!
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(
)
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(
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L et u s su b stitu te
k
k
lN
j
n
j
n
s
n
l
n
lN
k
lN
j
n
s
l
n
lN
n
N
X
k
x n e
x n e
X
k
x n e
n
, th en
(
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W e k n o w rearran g e th e o rd er o f su m m atio n.
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(1 )
k
j
n
lN
s
l
n
k
j
n
s
l
n
k
j
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s
n
l
n
lN
X
k
x n
lN
e
X
k
x n
lN
e
X
k
x n
lN
e
Now
(
) is a periodic signal
(
) with
period
.
According to fourier series analysis equation
1
(
)
and it can be compared with eq(1)
(
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(
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l
p
l
k
j
n
k
p
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s
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l
x n
lN
x
n
N
c
x
n e
N
X
k
x n
lN
(1)
(
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and according to fourier series synthesis equation
1
(
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(
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k
j
n
k
s
k
k
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p
k
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k
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c
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N
( )
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, ,
,....
N
kn
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n j
N
N
X k
x n W
k
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where W
e
1
0
2
0 1 2
1
1
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4
2(
3
6
3(
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2(
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1
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(0)
1
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(1)
(1)
1
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(2)
(2)
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N
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N
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X
x
W
W
W
X
x
W
W
W
X
x
W
W
W
W
W
W
X N
x N
( )
p
x
n
(
p
x
n
(
p
x
n
(
p
x
n
Circular shift of a sequence is given as
'( )
(
, mod
)
'( )
((
))
Circular shift is same as performing rotationon a signal.
N
x
n
x n
k
N
x
n
x
n
k