113_1_Week8_print

113_1_Week8_print - EE 113: Digital Signal Processing Week...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
EE 113: Digital Signal Processing Week 8 Fourier-Domain - continued 1. Discrete Fourier Transform (DFT) 2. Relation to DTFT 3. Use DFT to compute the linear convolution 4. Circular Convolution
Background image of page 1

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

View Full DocumentRight Arrow Icon
Discrete FT (DFT) ± A finite or periodic sequence has only N unique values, x [ n ] for 0 n < N ± Spectrum is completely defined by N distinct frequency samples ± Divide 0..2 π into N equal steps, { ω k } = 2 π k / N Discrete finite/pdc X [ k ] Discrete finite/pdc x [ n ] Discrete FT (DFT)
Background image of page 2
± Uniform sampling of DTFT spectrum: ± DFT: where i.e. 1/ N th of a revolution DFT and IDFT X [ k ] = X ( e j ω ) ω= 2 π k N = x [ n ] e j 2 k N n n = 0 N 1 X [ k ] = x [ n ] W N kn n = 0 N 1 W N = e j 2 N
Background image of page 3

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

View Full DocumentRight Arrow Icon
IDFT ± Inverse DFT IDFT ± Check: Sum of complete set of rotated vectors = 0 if l n ; = N if l = n re im W N W N 2 x [ n ] = 1 N X [ k ] W N nk k = 0 N 1 x [ n ] = 1 N x [ l ] W N kl l ( ) W N nk k = 1 N x [ l ] W N k ( l n ) k = 0 N 1 l = 0 N 1 = x [ n ]
Background image of page 4
DFT examples ± Finite impulse ± Periodic sinusoid: ( r Z) x [ n ] = 1 n = 0 0 n = 1.. N 1 X [ k ] = x [ n ] W N kn n = 0 N 1 = W N 0 = 1 k x [ n ] = cos 2 π rn N = 1 2 W N rn + W N rn ( ) X [ k ] = 1 2 W N rn + W N rn ( ) n = 0 N 1 W N kn = N /2 k = r , k = N r 0 o . w .
Background image of page 5

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

View Full DocumentRight Arrow Icon
DFT and DTFT ± DFT ‘samples’ DTFT at discrete freqs: DTFT DFT • continuous freq ω • infinite x [ n ] , - <n< • discrete freq k=N ω/2π • finite x [ n ] , 0 n<N X ( e j ) X [ k ] k =1. ..
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 26

113_1_Week8_print - EE 113: Digital Signal Processing Week...

This preview shows document pages 1 - 7. Sign up to view the full document.

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