Circular Folding and Circular Convolution: Concepts in Signal Processing, Slides of Digital Signal Processing

Download Circular Folding and Circular Convolution: Concepts in Signal Processing and more Slides Digital Signal Processing in PDF only on Docsity!

Circular folding of a sequence is given as'( )^ (^ ,mod

x^ n^ x^ n^ '( )^ ((^ ))^ N Circular folding is equivalent toplotting^ ( ) in clockwise fashion!

N

^  x n x^ n^  x n

docsity.com

^ ^

^  4

Let^ ( )^ 1,2,^ 2,

Then^ ((^ ))^ 1,1,

x n^

x^ n   ^ ^ ^  (0) (^1) x   (1)^2 x  (2) (^2) x   (3)^1 x 

'(1)^1 x  '(2) (^2) x  ^ '(0)^1 x^ ^  '(3)^2 x 

( )x n^

((^ ))x n^4 docsity.com

^ We know if corresponding spectra of two signalsare multiplied in frequency domain, the twosignals undergo^ convolution

in time domain. ^ If spectra consist of Discrete Fourier transform(DFT) their multiplication corresponds to newtype of convolution in time domain called

Circular convolution.

docsity.com

^ FindX(k) DFT of x(n) andY(k) DFT of y(n) ^ Then findZ(k) = X(k)Y(k) ^ Then takeIDFT Z(k)which is equivalent to circular convolution of x(n) andy(n)

docsity.com

3 21 3 3 0 21 3 1 2 0 2

2 2 1 1

1 3 1

2 0 0 0 21 1 (^

) 3 1

2 0 0 0

The IDFT of^ ( )^1 (^ )^ ( )^1 (^ )^ ( )

kN j mN k kN j^ mN ( )k k^1 ( ) ( ) ( ) 1 ( ) ( ) ( )

k^ k N^ N^

Nj n j^ l^ j^

m N^ N^

N k^ n^

l k N^ N^ N^

j^ m^ n^ l N

X^ k x m X^ k eN x m X^ k X^ k eN x m x^ n e^ n l^ k

x^ l e^ e

N x m x^ n^ x^ l^

e

 N

 

      ^ ^

  ^ ^

 ^ ^ ^

 

  ^   ^ ^ 

^ ^

 ^   ^ 

   ^ 

 (^1)   

docsity.com

21 (^ (^ ) 0 21 ( ) 0 21 ( ) 0

)^ 0,^ ,2^ ,...

Now^

0 (^ )^

or^0

or^ ((^ ))

kN j m^ n^ l N k kN j m n^ l N k kN j m n^ l^ or 0

N

N^ m N k

n^ l^ N^ Ne otherwiseN m n^ l^ pN where p

e^ otherwiseN^ l^

m^ n^ pN^ l^ m

n

          e o

^ ^    ^ ^    ^ ^ ^

   So we substitute^ ((^1 13 1 20

(^ )^ ( )^ ((^

therwisel m n^ pN^ m^ n^ N N N ))N n l

x^ m^ x^ n^ x^

   ^ ^ ^ ^    m^ n   

docsity.com

N-point DFT

N-point DFT Let ( )^

( ) Then ((^ ))^

kj p N( ) x n^ X k x^ n^ p^ N

 X k e  ^ 

 docsity.com

^ DFT is computationally very exhaustive! ^ Number of complex Multiplications requiredare^ for^ N -point DFT.^2 N ^ Just Imagine if N = 4096, how manymultiplications are needed?

docsity.com

^ Assume that we are calculating

N^ point DFT where N is a power of 2.  We divide our N-point signal into two signalsas follows:( )^ (2 )^1 ( )^ (2^ 1),^2

0,1,2,...,^12 f^ n^ x^ n and

N  f n x^ n^ n^ ^ ^ 

docsity.com

1 0 odd/ 2^1

/ 2^1 ( 2 ) ( 2^

0

0 2 / 2 / 2^1

/ 2^11 / 2 2 / 2 0

0 (^ )^ (^ )^

0,1, 2...,^1

(^ )^ (^ )^

(^ )

(^ )^ (2^ )^

(2^ 1)

But^ T herefore(^ )^ (^ )^

(^ )

N^ k N n kn^

knN N n even^ n N^

Nk m k^

m N^

N m^

m N^ N N^

Nkm k k N N N m^

m X^ k^ x n W^

k^ N X^ k^ x n W^

x n W X^ k^ x^ m W

x^ m^

W

W^ W X k f^ m W^   ^ ^ W^ f^ m W

 ^

 ^

 ^

 ^

^ 

^  ^

^  

 ^  ^  

 ^

 (^ )^ (^ )^ (^ )^1

m kX k F k W F^ k N

docsity.com