lab 5 dtfs & dtft
TRANSCRIPT
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(DTFS) applies only for periodic signals
3k k NC C
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5
DTFS
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Solution
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n
5
1
Given: [2,2,3,1,8]
and ( ) ( ) * , k=1,2,3,4,5
Write a Matlab code to get the value of y
n
x
y k x n k
QUIZ
x=[2,2,3,1,8];
for k=1:5
sum=0;
for n=1:5
y=x(n)*k
sum=sum+y
end
y1(k)=sum
end
y1 = 16 32 48 64 80
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% y(1)=(2*1)+(2*1)+(3*1)+(1*1)+(8*1)=16
% y(2)=(2*2)+(2*2)+(3*2)+(1*2)+(8*2)=32
% y(3)=(2*3)+(2*3)+(3*3)+(1*3)+(8*3)=48
% y(4)=(2*4)+(2*4)+(3*4)+(1*4)+(8*4)=64
% y(5)=(2*5)+(2*5)+(3*5)+(1*5)+(8*5)=80
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Solution
N1=3
N2=5
N=LCM(3,5)=15
See P.4.6
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2 2 2 2
3 3 5 51 1 1 1
2 2 2
2 2( ) cos sin
3 5
2
j n j n j n j n
e e e e
n
j
n
j
x n
5
10
3
12
2 2 15
3 15 2
2 2 15 5 15 10
3 15 2
2 2 13
5 15 2
2 2 13 3 15 12
5 15 2
kn n k C
kn n k C
kn n k C
j
kn n k C
j
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2
2
10.5 0.5
2
10.5 0.5
2
j
j
j ej
j ej
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Power spectral density of periodic signal
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( ) ( ) ( ) | ( ) j
j j n
z e n
DTFT X e X z x n e
2
1( ) ( ) ( )
2
j j ninverse DTFT x n X e e d
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DTFT
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: ( ) ( )
: ( ) ( ) ( )( )
( )( )
j
jwn
n
n jwn jw n
n n
n
n
DTFT X x n e
LT X z x n r
Z
e x n re
x
e
n Z
r
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| ( ) | | ( ) | | ( ) |(DTFT)Stable: jwn
n n
X x n e x n
| ( ) | | ( ) | (Z) S table: n
n
X z x n z
If R.O.C includes the unit circle
X(z) is converge at |Z|=1 | ( ) |
Fourier transform of the sequence conver ges.
Otherwise if the R.O.C doesn't include t he unit circle
Fourier transform does
n
x n
not converge absolutely.
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-1 -0.5 0 0.5 1 1.5 2 2.5 3-1
0
1
2
n
h[n
]h[n] = (n) + (n-1) + (n-2)
-4 -3 -2 -1 0 1 2 3 40
2
4
w
|H(w
)|
DTFT of h[n] = (n) + (n-1) + (n-2)
-4 -3 -2 -1 0 1 2 3 4-5
0
5
w
phase
DTFT of h[n] = (n) + (n-1) + (n-2)
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The discrete-time Fourier transform of an impulse response is called the Frequency
Response of an LTI system .
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0.05 2 0.025
25 10.025
1000 40
40
40sample 2
?? sample 0.5377
40 0.53773.42 sample
2
kf
N
N
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LOGO
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