Frequency Shift and Windowing-Digital Signal Processing-Lecture Slides, Slides of Digital Signal Processing

Download Frequency Shift and Windowing-Digital Signal Processing-Lecture Slides and more Slides Digital Signal Processing in PDF only on Docsity!

Frequency Shift and Windowing

Theorem

• If the spectra of two signals are convoluted

in frequency domain, the signals aremultiplied in time domain. Thus

1

1

2

2

1

2

1

2

(^

(^

(^

)^

(^

F F

F

x^

n

X

x^

n

X

Thenx

n x

n

X

X

^ 

Solution



(^1

X

(^

X

W

c

c

W^

W

^

C^

C







(^2

X

Solution /

1

1

1

/2 /

1

2

2

/

1

1

2

1

2

sin(

)

1

2

(^

)^

( )

2 1

1

(^

)^

( )

[^

(^

)^

(^

)]

cos(

)

2

2

sin(

)

1

2

(^

)^

(^

)^

( )

( )

cos(

)

2

W

j^ n

F

W W

j^ n

F

c^

c^

c

W

F

c

W

n

X^

x^

n^

e^

d^

n

X^

x^

n^

e^

d^

n

W

n

X^

X^

x^

n x

n^

n

n















 



 

















^

















^







^

^

^

^



^







 

docsity.com

Parseval’s Theorem

2

2

(^

n n

x n

X

 







The amount of energy contained in theAperiodic signal is same whether calculatedin time or calculated in frequency domain.

Example

-^

How much is the energy available in followingsignal:

sin(

)

( )

nc

x n

 n 



• We know from Fourier Transform Pairs

that the Fourier transform of

x

(n) is

1

(^

)^

0

c^

c

for

X

elsewhere













  



Sampling And Reconstruction

• Let us turn our attention to continuous

time signals and Spectra for a while.

• Suppose we have a continuous time

signal band Limited signal of B Hz.

• Let us multiply x(n) with impulse train as

follows

'^

( )

( )

( )

(^

)^

( )

(^

2

)^

...

x^

x t

t^

x t

t^

T

x t

t^

T



















Steps of Sampling and

Reconstrcution

You are given a continuous time signal, then

following are steps involved in samplingand reconstruction.

-^

Sample the signal x(t)

-^

Apply Low-pass filter and reconstruct x(t)

Example

( )x t

……

t

(^

X w

w

(^

)s

x^ nT

……

t

'(^

X^

w

w

w^ c

2 wc

wc 

2 wc ^

B 2 

B  2 

docsity.com

Sampling..contd

• Thus we conclude that when a signal is

sampled at frequency ‘Ts’ then thespectrum contains replicas of X(w)appearing at f

or

s^

s

w

f

T

T

• This is called Sampling Frequency

Interpretation of Sampling

• The Continuous time Fourier transform /

Spectrum of continuous time signalincludes frequencies from –Infinity to+Infinity.

• Theoretically analog frequency can have

value from 0 Hz to Infinity Hz.

Reconstruction

• Now we apply Low-Pass Filter to sampled

signal. The bandwidth of Low-pass filter issame as that of signal that is B Hz.

w

X(w)

B 2 

B  2 

2 (Band width)

'(^

X^

w

w

w^ c

2 wc

wc 

2 wc ^

B 2 

B  2 



w

B 2 

B  2 

c w

docsity.com