Lecture 4 Frequency Response of FIR Systems (II)

Transcription

1 EE3054 Signals and Systems Lecture 4 Frequency Response of FIR Systems (II Yao Wang Polytechnic University Most of the slides included are extracted from lecture presentations prepared by McClellan and Schafer License Info for SPFirst Slides This wor released under a Creative Commons License with the following terms: Attribution The licensor permits others to copy, distribute, display, and perform the wor. In return, licensees must give the original authors credit. Non-Commercial The licensor permits others to copy, distribute, display, and perform the wor. In return, licensees may not use the wor for commercial purposes unless they get the licensor's permission. Share Alie The licensor permits others to distribute derivative wors only under a license identical to the one that governs the licensor's wor. Full Text of the License This (hidden page should be ept with the presentation 2/29/ , JH McClellan & RW Schafer 2 1

2 PREVIOUS LECTURE REVIEW SINUSOIDAL INPUT SIGNAL OUTPUT has SAME FREQUENCY DIFFERENT Amplitude and Phase FREQUENCY RESPONSE of FIR MAGNITUDE vs. Frequency PHASE vs. Freq PLOTTING MAG e PHASE j H ( e 2/29/ , JH McClellan & RW Schafer 3 LTI SYSTEMS LTI: Linear & Time-Invariant COMPLETELY CHARACTERIZED by: FREQUENCY RESPONSE, or IMPULSE RESPONSE h[n] Sinusoid IN -----> Sinusoid OUT At the SAME Frequency M h[ ne ] n 0 j ˆ ωn 2/29/ , JH McClellan & RW Schafer 4 2

3 3 2/29/ , JH McClellan & RW Schafer 5 TIME & FREQUENCY FIR DIFFERENCE EQUATION is the TIME-DOMAIN M M n x h n x b n y 0 0 ] [ ] [ ] [ ] [ M j j e h e H 0 ˆ ˆ ] [ ( ω ω ω ω ω ω ˆ 3 ˆ 2 ˆ ˆ [3] [2] [1] [0] ( j j j j e h e h e h h e H Properties of Frequency Response Periodicity H(e^jwH(e^j(w+2 \pi Only need to evaluate over (-pi,pi Conjugate symmetry (true when h[n] is real H(e^jw H* (e^-jw H(e^jw H(e^-jw, <H(e^jw -<H(e^-jw Proof

4 FREQ. RESPONSE PLOTS DENSE GRID (ww from -π to +π ww -pi:(pi/100:pi; HH freqz(bb,1,ww VECTOR bb contains Filter Coefficients ww -pi:(pi/100:pi; Without specifying ww, the frequency axis is -1 to 1 1 corresponds to fs/2 in continuous freq. Go through freqz( in class PLOT of FREQ RESPONSE b } { {1,2,1} j ˆ ω (2 + 2cos ˆ ω e RESPONSE at π/3 ωˆ π ˆω (radians 2/29/ , JH McClellan & RW Schafer 8 π 4

5 Response to complex exponential input x[n]ae^j(w0 n+phi y[n]? Response to sinusoidal input x[n]a cos(w0 n+phi y[n]? 5

6 SINUSOID thru FIR IF Multiply the Magnitudes Add the Phases x[ n] H Acos( ˆ ωn + φ y[ n] A * jωˆ j ˆ ω ( e 1 cos( ˆ ωn + φ + H ( e 1 ˆ1 1 jω 2/29/ , JH McClellan & RW Schafer 11 Periodic signal thru FIR MULTIPLY MAGS ADD PHASES 2/29/ , JH McClellan & RW Schafer 12 6

7 What does H(e^jw tell you? Sinusoid-IN Ae^j(wt+phi or A cos (wt+phi gives Sinusoid-OUT Be^j(wt+psi or B cos (wt+psi The amplitude response H(e^jw tells how does the system change the magnitude of the input: BA H(e^jw The phase response <H(e^jw tells how does the system change the phase of the input \psi \phi+<h(e^jw When input is not an arbitrary signal x[n], it can be decomposed into many sinusoids with different frequencies and each will be affected by H(e^jw at corresponding freq. LTI Demo with Sinusoids (dltidemo x[n] FILTER y[n] 2/29/ , JH McClellan & RW Schafer 14 7

8 BLOCK DIAGRAMS Equivalent Representations x[n] h[n] y[n] x[n] j ˆω y[n]?ω ωˆ 2/29/ , JH McClellan & RW Schafer 15 UNIT-DELAY SYSTEM Findh[ n] andh ( e j ˆω x[n] δ [ n 1] x[n] jωˆ ωˆ e j for y[ n] x[ n 1] y[n] { b } y[n]?ω?ω { 0, 1} 2/29/ , JH McClellan & RW Schafer 16 8

9 GENERAL DELAY PROPERTY Find h[ n] and for y[ n] x[ n n d ] h[ n] δ [ n nd ] M δ [ 0 n d ] e j ˆ ω e j ˆ ω n d ONLY ONE non-zero TERM for at n d 2/29/ , JH McClellan & RW Schafer 17 Delay system is all pass! H[n]d(n-n d <-> H(e^jwe^{-jn d w} H(e^jw 1: All Pass! <H(e^jwn d w: Linear Phase! Slope indicates the delay Show plots using freqz 9

10 Averaging System y[n](x[n}+2x[n-1]+x[n-2]/4 H(e^jw? H? <H? Plot Averaging System is Low Pass! H(e^jw(1/2(1+cosw e^-jw H(01 H(pi0 x[n]3+4 cos(0.5π n-π/2 y[n]? 10

11 First Difference System y[n]x[n]-x[n-1] H(e^jw? H? <H? Plot! First Difference System is High Pass! H(e^jw 2 sin(w/2 H 0 at w0 Removes DC component (0 frequency A high-pass system x[n]3+4 cos(0.5π n-π/2 y[n]? 11

12 FILTER TYPES LOW-PASS FILTER (LPF BLURRING ATTENUATES HIGH FREQUENCIES HIGH-PASS FILTER (HPF SHARPENING for IMAGES BOOSTS THE HIGHS REMOVES DC BAND-PASS FILTER (BPF 2/29/ , JH McClellan & RW Schafer 23 B & W IMAGE 2/29/ , JH McClellan & RW Schafer 24 12

13 B&W IMAGE with COSINE (noise FILTERED: 11-pt AVG 2/29/ , JH McClellan & RW Schafer 25 FILTERED B&W IMAGE LPF: BLUR 2/29/ , JH McClellan & RW Schafer 26 13

14 Summary How to compute frequency response h[n] -> H(e^jw Physical meaning of freq resp., amplitude resp, and phase resp. Input with multiple sinusoids -> output? Type of filters Low pass, high pass, bandpass Readings Textboo, Sec