Frequency Response of Discrete-Time Systems

Transcription

1 Frequency Response of Discrete-Time Systems EE 37 Signals and Systems David W. Graham 6

2 Relationship of Pole-Zero Plot to Frequency Response Zeros Roots of the numerator Pin the system to a value of ero Poles Roots of the denominator Cause the system to shoot to infinity

3 3D Visualiation of the Pole-Zero Plot Visualie The real-imaginary plane is a stretchy material Every ero pins this material down to a value of ero Every pole can be imagined as an infinitely tall pole/stick that pushes the stretchy material up to infinity The system is then defined by the contour of this material

4 Frequency Response Determination Frequency Response Ignores the transients (magnitude of the poles) Only looks at the steady-state response (frequency is given by the angle of the poles) = re jω Let r = on the unit circle e jω gives the angle 3

5 Frequency Response Determination Frequency Response Ignores the transients (magnitude of the poles) Only looks at the steady-state response (frequency is given by the angle of the poles) = re jω Let r = on the unit circle e jω gives the angle Frequency response plot can be taken from the contour of the pole-ero plot around the unit circle (from π to π) 4

6 Impulse Response (h[n]) First-Order System (a=.9) n (.9 ) u [ n ]

7 First-Order System (a=.5) Impulse Response (h[n]) Faster Decay 5 5 (.5) n u[ n].5.5 Wider Bandwidth

8 First-Order System (a=.) Impulse Response (h[n]) Even Faster Decay 5 5 (.) n u[ n] Even Wider Bandwidth

9 First-Order Systems Varying Pole Position (a > ) Frequency-Domain Response Ti me-domain Response.9 a=..8 Normalied a=. a=.5 Impulse Response (h[n]) a=. a=.5 a= Frequency (rad/sec) Lowpass filter (from to π) Increasing the pole decreases the corner frequency Sample Value Lowpass filter The smaller a is, the faster the decay (small time constant = high corner frequency) 8

10 Impulse Response (h[n]) First-Order System (a=-.) n (. ) u [ n ]

11 First-Order System (a=-.5) Impulse Response (h[n]) Slower Decay (.5) n u[ n] +.5 Narrower Bandwidth

12 First-Order System (a=-.9) Impulse Response (h[n]) Even Slower Decay (.9) n u[ n] +.9 Even Narrower Bandwidth

13 First-Order Systems Varying Pole Position (a < ) Frequency-Domain Response Ti me-domain Response.9 a=-..8 Normalied a=-.5 Impulse Response (h[n]) a=-. a=-.5 a=-.9. a= Frequency (rad/sec) Sample Value Highpass filter (from to π) Increasing the pole decreases the corner frequency Highpass filter The smaller a is, the faster the decay (small time constant = high corner frequency) Oscillation from a first-order system

14 Second-Order System (.3,.8) n k Two Poles (.3,.8) n (.3 ) u [ n ] k (.8 ) u [ n ] Single Pole (.8) malied Normalied Magnitude Magnitude Frequency Res Frequency Response Pole with the slower response dominates 3

15 Second-Order System (-.8,.8) Magnitude Frequency Response n k n (.8 ) u [ n ] + k (.8 ) u [ n ]

16 Complex Poles p + p p =. 8 p, p =.566 ± j.566 arg ( p) = π 4 5

17 Complex Poles Varying the Magnitude Previous Position p + p 4 3 p =. 5 p, p =.353 ± j.353 arg p = Real.5 Part =.5 ( p) = π 4 p =.8 Real Part = Alters only the magnitude Does not change the corner frequency 6

18 Complex Poles Varying the Angle malied Normalied Magnitude Frequency Magnitude Res Frequency Response p + p p =. 8 p, p =.693 ± j.4 ( ) 6 arg p = π Alters only the corner frequency

19 Higher-Order Frequency Responses

20 Discrete-Time Frequency Responses in MATLAB Use the freq function. num = [ ]; den = [.5]; ww = -pi:.:pi; [H] = freq(num,den,ww); figure; plot(ww,abs(h)); Frequency (rad/sec) 9