Lecture 5 Frequency Response of FIR Systems (III)

Transcription

1 EE3054 Signal and Sytem Lecture 5 Frequency Repone of FIR Sytem (III Yao Wang Polytechnic Univerity Mot of the lide included are extracted from lecture preentation prepared by McClellan and Schafer Licene Info for SPFirt Slide Thi work releaed under a Creative Common Licene with the following term: Attribution The licenor permit other to copy, ditribute, diplay, and perform the work. In return, licenee mut give the original author credit. Non-Commercial The licenor permit other to copy, ditribute, diplay, and perform the work. In return, licenee may not ue the work for commercial purpoe unle they get the licenor' permiion. Share Alike The licenor permit other to ditribute derivative work only under a licene identical to the one that govern the licenor' work. Full Text of the Licene Thi (hidden page hould be kept with the preentation 2/29/ , JH McClellan & RW Schafer 2 1

2 Previou Lecture Review LTI: Linear & Time-Invariant COMPLETELY CHARACTERIZED by: FREQUENCY RESPONSE, or IMPULSE RESPONSE h[n] Sinuoid IN -----> Sinuoid OUT At the SAME Frequency j M = h[ ne ] n= 0 j n 2/29/ , JH McClellan & RW Schafer 3 SINUSOID thru FIR IF Multiply the Magnitude Add the Phae x[ n] = H Aco( n + φ y[ n] = A * jωˆ j ( e = 1 j co( n + φ + H ( e 1 ˆ1 1 jω 2/29/ , JH McClellan & RW Schafer 4 2

3 Propertie of Frequency Repone Periodicity H(e^jw=H(e^j(w+2 \pi Only need to evaluate over (-pi,pi Conjugate ymmetry (true when h[n] i real H(e^jw = H* (e^-jw H(e^jw = H(e^-jw, <H(e^jw = -<H(e^-jw Uing MATLAB freqz( ww = -pi:(pi/100:pi; HH = freqz(bb,1,ww; Subplot(2,1,1,plot(ww,ab(HH; Subplot(2,1,2,plot(ww,angle(HH; angle( give the principle phae (-pi,pi freqz(bb,1,ww will give the plot directly, with the amplitude repone plotted in log cale (10 log10(a, phae in degree, in normalized frequency (-1,1 with 1 correponding to pi. 3

4 Typical FIR filter All pa: delay ytem Low pa: averaging or weighted averaging H(0 \=0, H(pi=0 High pa: difference ytem H(0=0, H(pi \=0 Application of Filter Low pa: remove noie High pa: detect tranient change, edge detection in image Demo uing noiy peech ignal Try different averaging length, weighted averaging, play ound before and after, plot ignal before and after Uing image demo (nrfiltdemo, edgedemo 4

5 CASCADE SYSTEMS Doe the order of S 1 & S 2 matter? NO, LTI SYSTEMS can be rearranged!!! WHAT ARE THE FILTER COEFFS? {b k } WHAT i the overall FREQUENCY RESPONSE? [n] h 1n [ ] h h 2 [ n] h n] h [ ] δ S 1 1n [ ] S 2 1[ 2 n 2/29/ , JH McClellan & RW Schafer 9 CASCADE EQUIVALENT MULTIPLY the Frequency Repone x[n] H 1 ( e jωˆ H 2 ( e jωˆ y[n] x[n] j ˆω y[n] EQUIVALENT SYSTEM j = H 1 ( e j H 2 ( e j 2/29/ , JH McClellan & RW Schafer 10 5

6 Convolution Theorem h1(n*h2(n <-> H1(jw H2(jw! Convolution in time domain <-> Multiplication in freq. domain Proof! (Two way Example h1[n]={1,2,1}/4 (averaging h2[n]={1,-1} (difference h=h1*h2=? H=? H1=? H2=? 6

7 How to determine impule repone from freq repone? START with x[n] j ˆω and findh[ n] or h[ n = y[n] h [n] ]? b k x[n] ωˆ j ˆω y[n] ωˆ jωˆ j2 = 7e co( 2/29/ , JH McClellan & RW Schafer 13 FREQ DOMAIN --> TIME jωˆ j2 = 7e = 7e = (3.5e co( ˆ ω j2 j j (0.5e + 3.5e + 0.5e jωˆ j3 EULER Formula h[ n] = 3.5δ [ n 1] + 3.5δ [ n 3] b k = { 0, 3.5, 0,3.5} 2/29/ , JH McClellan & RW Schafer 14 7

8 Steady-State and Tranient Repone x[n]=e^{jwn} i infinite in duration and not realizable in practice In practice, a ignal tart at a certain time, call it t=0 What will be the repone to uch ignal? Tranient repone: initial output after t=0 Steady-tate: after ome initial time (depending on filter length, ame a if x[n] tart from infinity! Example x[n]=co(0.1 pi n u[n] H[n]={1,2,1}, H(e^jw=(1+co w e^(-jw y[n]=? 8

9 What if the ignal i finite duration? x[n]=e^{jwn}, n=0,1,.,n-1 y[n]=? Dicrete Time v. Continuou Time Frequency x[n]=co(0.5 p n w_d=0.5 p, f_d=w_d/2p=0.25, what doe thi mean? T_d (period=1/f_d=4 ample, f_d: how much of a cycle by one ample (cycle/ample. f_d<1 (need more than 1 ample to complete a cycle, w_d<2pi Think of thi ignal a the reult of ampling a continuou time ignal at interval T_ (ec., or uing ampling frequency f_=1/t_ x_c(t=co (0.5 p t/t_, x[n]=x_c(t=nt_ w=0.5 pi/t_, f=0.25/t_(cycle/ec, T=1/f=4T_ (ec. E.g. T_=0.1 ec., f=2.5 cycle/ec., T=0.4 ec. 9

10 General Relation ωt = ; fˆ = ω = f f ˆ ω = ωf = ; f = fˆ f = T f ; 2π For a given continuou time ignal, the dicrete freq. of the ampled ignal decreae with the ampling freq. f ; Range of frequencie : ( π, π fˆ ( 1 2,1 2 ω ( πf, πf f ( f 2, f 2 Above i true only if the original continuou ignal maximum freq. < f/2 (no aliaing EFFECTIVE Freq. Repone Aume NO Aliaing, then ANALOG FREQ <--> DIGITAL FREQ?ω =ωt = ω f DIGITAL FILTER So, we can plot: Scaled Freq. Axi jω /f v. ω ANALOG FREQUENCY 2/29/ , JH McClellan & RW Schafer 20 10

11 DIGITAL FILTER j ˆω EFFECTIVE RESPONSE LOW-PASS FILTER 2/29/ , JH McClellan & RW Schafer pt AVERAGER Example x(t x[n] A-to-D j ˆω y[n] y(t D-to-A 10 ω?ω?ω y[ n] = 11 1 x[ n k ] k= Hz 11? j in( ˆ 2 ω j5 25 Hz = e x( t = co(2π (25 t + co(2π (250 t 1 π 2 2/29/ , JH McClellan & RW Schafer 22 11in( 1 2 ω 11

12 TRACK the FREQUENCIES x(t A-to-D x[n] j ˆω y[n] D-to-A y(t ω 250 Hz 0.5π?ω?ω ω j0.5π 0.5π 250 Hz j0.05π 25 Hz.05π.05π 25 Hz F = 1000 Hz NO new freq 2/29/ , JH McClellan & RW Schafer pt AVERAGER NULLS or ZEROS = 0. 05π = 0. 5π 2/29/ , JH McClellan & RW Schafer 24 12

13 EVALUATE Freq. Repone f = 1000 j2π (25 /1000 j2π (250 /1000 MAG SCALE PHASE CHANGE 2/29/ , JH McClellan & RW Schafer 25 Summary Type of filter and their application Low pa, high pa, bandpa Convolution Theorem: h1[n]*h2[n] <-> H1(e^jw H2(e^jw Tranient and teady tate repone Dicrete-time v. continuou-time frequency How to filter continuou time ignal with dicrete filter? 13

14 READING ASSIGNMENTS Thi Lecture: Chapter 6, Section 6-6, 6-7 & 6-8 Next Lecture: Chapter 7, Z-tranform 14