EE123 Digital Signal Processing

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1 EE123 Digital Signal Processing Lecture 2A D.T Systems D. T. Fourier Transform

2 A couple of things Read Ch It s OK to use 2nd edition My office hours: posted on-line W 4-5pm Cory 506 ham radio lectures. Wednesday 6:30-8:30pm Cory 521

3 Intro Last time Convinced you why you should take this class DSP is, just F$#!ing awesome D.T signals vs C.T Today: D.T systems DTFT

4 Discrete Time Systems x[n] T { } y[n] What properties? Causality: y[n0] depends only on x[n] for - n n0

5 Properties of D.T. Systems Cont. Memoryless: y[n] depends only on x[n] Linearity: Example: y[n] = x[n] 2 Superposition: T {x 1 [n]+x 2 [n]} = T {x 1 [n]} + T {x 2 [n]} = y 1 [n]+y 2 [n] Homogeneity: T {ax 1 [n]} = at {x[n]} = ay[n]

6 Properties of D.T. Systems Cont. Time Invariance: If: Then: y[n] = T {x[n]} y[n n 0 ] = T {x[n n 0 ]} BIBO Stability If: Then: x[n] apple B x < 1 y[n] apple B y < 1 8 n 8 n

7 Example: Time Shift y[n] =x[n n d ] Causal L TI memoryless if nd>=0 Y Y if nd=0 Y BIBO stable Accumulator y[n] = nx k= 1 Compressor y[n] =x[mn] M>1 x[k] Y Y Y N N N Y N N Y

8 Examples Why the compressor is NOT Time Invariant? Suppose M=2, x[n] = cos[ /2 n] y[n] n n x[n 1] n = y[n 1] n

9 Examples * j0 cl] Non-Linear system: Median Filter /MIL y[n]a_= NtJ»-l.f).)fN?_ S'(<;r M.!.Mf/2!/VI ffltf-f( MED{x[n k],, x[n + k]} =- m o[ r[jj-k.'],...1 :X{n+-tJ1, - 0 Q 1 tft, 6J '}CxJ :> -i - -1 s?- e ee }'lhl

10 Example: Removing Shot Noise From NPR This American Life, ep.203 The Greatest Phone Message in the World em.. (giggle) There comes a time in life, when.. eh... when we hear the greatest phone mail message of all times and... well here it is... eh.. you have to hear it to believe it.. corrupted message

11 Spectrum of Speech Speech Corrupted Speech

12 Low Pass Filtering LP-Filter Spectrum

13 Low-Pass Filtering of Shot Noise Corrupted LP-filtered

14 Low-Pass Filtering of Shot Noise Corrupted Med-filter

15 The Greatest Message of All Times... I thought you ll get a kick out of a message from my mother... Hi Fred, you and the little mermaid can go blip yourself. I told you to stay near the phone... I can t find those books.. you have other books here.. it must be in Le hoya.. call me back... I m not going to stay up all night for you Bye Bye...

16 Discrete-Time LTI Systems The impulse response h[n] completely characterizes an LTI system DNA of LTI [n] LTI h[n] discrete convolution x[n] LTI y[n] =h[n] x[n] y[n] = 1X N= 1 h[m]x[n m] Sum of weighted, delayed impulse responses!

17 BIBO Stability of LTI Systems An LTI system is BIBO stable iff h[n] is absolutely summable 1X k= 1 h[k] < 1

18 BIBO Stability of LTI Systems Proof: if y[n] = apple 1X h[k]x[n k] k= 1 1X k= 1 h[k] x[n k] apple B x apple B x 1 X k= 1 h[k] < 1

19 BIBO Stability of LTI Systems Proof: only if P 1 k= 1 h[k] = 1 suppose show that there exists bounded x[n] that gives unbounded y[n] Let:

20 Discrete-Time Fourier Transform (DTFT) X(e j! )= x[n] = 1 Alternative 2 1X k= 1 Z X(f) = x[n] = x[k]e j!k X(e j! )e j!n d! 1X k= 1 Z x[k]e j2 fk X(f)e j2 fn df Why one is sum and the other integral? Why use one over the other? M. Lustig, EE123 UCB

21 Example 1: w[n] window -N N DTFT: W (e j! ) = NX e j!k k= N = e j!n 1+e j! + + e j!2n Recall: 1+p + p p M = 1 pm+1 1 p p = e j! M = 2N M. Lustig, EE123 UCB

22 Example 1 cont. DTFT: M. Lustig, EE123 UCB

23 Example 1 cont. DTFT: W (e j! ) = e j!n 1+e j! + + e j!2n M. Lustig, EE123 UCB = e jwn 1 ei!(2n+1) 1+e j! = e j!n e j!n e j! 1 e j! = e j!(n+ 1 2 ) e j!(n+ 1 2 ) e j! 2 e j! 2 = sin[(n )!] sin(! 2 ) j - periodic sinc e j! 2 e j! 2

24 Example 1 cont. W (e j! )= sin[(n )!] sin(! 2 ) also, Σx[n]! (2N + 1) as!! 0 from l Hôpital 2N +1 N =1, why? M. Lustig, EE123 UCB

25 Properties of the DTFT Periodicity: X(e j(!+2 ) )=X(e j! ) Conjugate Symmetry: X (e j! )=X(e j! ) if x[n] is real Re X(e j! ) = Re X(e j! ) Im X(e j! ) = Im X(e j! ) Big deal for: MRI, Communications, more... M. Lustig, EE123 UCB

26 Half Fourier Imaging in MR k-space (Raw Data) Image Complete based on conjugate symmetry Half the Scan time! Discrete Fourier transform M. Lustig, EE123 UCB

27 SSB Modulation Real Baseband signal has conjugate symmetric spectrum m[n] cos(! 0 n) AM modulation (DSB-SC)!! Single sideband (USB) half bandwidth! M. Lustig, EE123 UCB

28 SSB Amateur radio on shortwaves often use SSB modulation Example: Websdr M. Lustig, EE123 UCB

29 Properties of the DTFT cont. Time-Reversal x[n] $ X(e i! ) x[ n] $ X(e i! ) = X (e j! ) if x[n] 2Real If x[n] = x[-n] and x[n] is real, then: X(e j! ) = X (e j! )! X(e j! ) 2Real M. Lustig, EE123 UCB

30 Q: Suppose: x[n] $ X(e j! )? $ Re X(e j! ) A: Decompose x[n] to even and odd functions x[n] =x e [n]+x o [n] x e [n] := 1 2 x o [n] := 1 2 (x[n]+x[ n]) (x[n] x[ n]) x e [n]+x o [n]!re X(e j! ) + jim X(e j! ) M. Lustig, EE123 UCB

31 M. Lustig, EE123 UCB Oops!

32 Properties of the DTFT cont. Time-Freq Shifting/modulation: x[n] $ X(e j! ) Good for MRI! Why x[n n d ] $ e j!n d X(e j! ) e j! 0n x[n] $ X(e j(!! o) ) M. Lustig, EE123 UCB

33 M. Lustig, EE123 UCB Thursday, January 26, 12

34 Example 2 What is the DTFT of: 5 High Pass Filter 2N +1= e j n M. Lustig, EE123 UCB See 2.9 for more properties

EE123 Digital Signal Processing

EE123 Digital Signal Processing Lecture 2 Discrete Time Systems Today Last time: Administration Overview Announcement: HW1 will be out today Lab 0 out webcast out Today: Ch. 2 - Discrete-Time Signals and