Frequency Response of Discrete-Time Systems EE 37 Signals and Systems David W. Graham 6
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
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
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
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
.8.6.4. 5 5 5 Impulse Response (h[n]).5 -.5 - First-Order System (a=.9) - -.5.5 n (.9 ) u [ n ].9 8 6 4-3 - - 3
First-Order System (a=.5).5 -.5 - - -.5.5 Impulse Response (h[n]).8.6.4. Faster Decay 5 5 (.5) n u[ n].5.5 Wider Bandwidth.5-3 - - 3 6
First-Order System (a=.).5 -.5 - - -.5.5 Impulse Response (h[n]).8.6.4. Even Faster Decay 5 5 (.) n u[ n]..5..5.95 Even Wider Bandwidth.9-3 - - 3 7
First-Order Systems Varying Pole Position (a > ) Frequency-Domain Response Ti me-domain Response.9 a=..8 Normalied.7.6.5.4.3.. a=. a=.5 Impulse Response (h[n]).8.6.4. a=. a=.5 a=.9-3 - - 3 Frequency (rad/sec) Lowpass filter (from to π) Increasing the pole decreases the corner frequency 4 6 8 4 6 8 Sample Value Lowpass filter The smaller a is, the faster the decay (small time constant = high corner frequency) 8
.5 5 5 9 Impulse Response (h[n]).5 -.5 - First-Order System (a=-.) - -.5.5.5..5.95.9 n (. ) u [ n ]. -3 - - 3 +
First-Order System (a=-.5).5 -.5 - - -.5.5 Impulse Response (h[n]).5 -.5 - Slower Decay 5 5.5 (.5) n u[ n] +.5 Narrower Bandwidth.5-3 - - 3
First-Order System (a=-.9).5 -.5 - - -.5.5 Impulse Response (h[n]).5 -.5 - Even Slower Decay 5 5 5 5 (.9) n u[ n] +.9 Even Narrower Bandwidth -3 - - 3
First-Order Systems Varying Pole Position (a < ) Frequency-Domain Response Ti me-domain Response.9 a=-..8 Normalied.8.7.6.5.4.3. a=-.5 Impulse Response (h[n]).6.4. -. -.4 -.6 a=-. a=-.5 a=-.9. a=-. -.8 - -3 - - 3 Frequency (rad/sec) 4 6 8 4 6 8 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
Second-Order System (.3,.8) 8 6 4-3 - - 3.5 -.5 - - -.5.5.8.6.4. n k +.3.8 Two Poles (.3,.8) n (.3 ) u [ n ] k (.8 ) u [ n ] Single Pole (.8) -3 - - 3 malied Normalied Magnitude Magnitude Frequency Res Frequency Response Pole with the slower response dominates 3
Second-Order System (-.8,.8).5 -.5 - - -.5.5 3.5.5.5-3 - - 3 4 Magnitude Frequency Response n k.8 +.8 n (.8 ) u [ n ] + k (.8 ) u [ n ]
Complex Poles.5 -.5 - - -.5.5 4 3-3 - - 3 p + p p =. 8 p, p =.566 ± j.566 arg ( p) = π 4 5
Complex Poles Varying the Magnitude.5 -.5 - - -.5.5 Previous Position.5.5-3 - - 3 p + p 4 3 p =. 5 p, p =.353 ± j.353 arg p = Real.5 Part =.5 ( p) = π 4 p =.8 Real Part =.8-3 - - 3 Alters only the magnitude Does not change the corner frequency 6
Complex Poles Varying the Angle 8 6 4-3 - - 3.8.6.4. -3 - - 3 7 malied Normalied Magnitude Frequency Magnitude Res Frequency Response.5 -.5 - - -.5.5 p + p p =. 8 p, p =.693 ± j.4 ( ) 6 arg p = π Alters only the corner frequency
Higher-Order Frequency Responses.8.6.4. -3 - - 3 8.5 5 -.5 - - -.5.5
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));.8.6.4..8.6-3 - - 3 Frequency (rad/sec) 9