5_HWSolution

5_HWSolution - PBM 7.1 To derive Parsevals theorem for the...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
PBM 7.1 To derive Parseval’s theorem for the DFT: 11 22 00 1 () (), NN nk x nX k N −− == = ∑∑ () we start from the R.H.S of (7.1) and proceed as 2 * * () () 12 ( ) ( )exp( ), kk kn Xk XkX k nk x n j π = = () where in (7.2), we use the relationship given by DFT. Interchanging the order of the summation, we can write 111 2 * 000 1 2 0 2 ( ) ( ) ( )exp( ) NNN kkn N k nk j N xn −−− === = = ⎩⎭ = ∑∑∑ () where we have used the IDFT expression. This concludes the proof. PBM 7.4 We start with the DFT expression: 1/ 2 1 / 2 1 /2 0 /2 1 0 2 ()e x p 2 () (1 ) , 2 N nk Nk nk N nn n N k N n jn k N X kx n x n W W W N N x n W = = ⎛⎞ =− = + + ⎜⎟ ⎝⎠ ⎧⎫ + ⎨⎬ () where we have used 2 exp N j W N ± and ( ) exp ( 1) Nk k N Wj k = −= . Considering only the even values of k , i.e., k =2 m for m= 0,1,…,N/2, (7.9) can be written as 0 (2 ) ( ) , for 0,1, , / 2 1 2 N nm N n N Xm x n x nW m N = =+ + = . ( Similarly, for the odd values of k , (7.9) can be represented as 0 (2 1) ( ) 0,1, , / 2 1 2 N m n N x nWW m N = += + = . ()
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Therefore, if we define 01 () a n d 22 n N NN x nx n x n x n W ⎧⎫ ⎛⎞ ++ −+ ⎨⎬ ⎜⎟ ⎝⎠ ⎩⎭ ±± , we observe that X ( k ) can be realized as the N /2 point DFT of 0 x n and 1 x n length- N /2 sequences. Note that, N/2 complex multiplications are required to obtain 1 x n . The Flow-graph of this reduced complexity DFT implementation is shown in the Figure 7.1
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/02/2009 for the course EEE DSP taught by Professor Ap during the Fall '09 term at ASU.

Page1 / 5

5_HWSolution - PBM 7.1 To derive Parsevals theorem for the...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online