Frequency Response of Discrete-Time Systems

EE 327

Signals and Systems 1© David W. Graham 2006

1

Relationship of Pole-Zero Plot to Frequency Response

Zeros • Roots of the numerator• “Pin” the system to a value of zero

Poles• Roots of the denominator• Cause the system to shoot to infinity

2

3D Visualization of the Pole-Zero Plot

Visualize• The real-imaginary plane is a “stretchy material”• Every zero pins this material down to a value of zero• 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

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)

z = rejω

•Let r = 1 � on the unit circle•ejω� gives the angle

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)

z = rejω

•Let r = 1 � on the unit circle•ejω� gives the angle

Frequency response plot can be taken from the contour of the pole-zero plot aroundthe unit circle (from –π to π)

5

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Sample Value

Impu

lse

Res

pons

e (h

[n])

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

First-Order System (a=0.9)

-3 -2 -1 0 1 2 30

2

4

6

8

10

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

( ) [ ]9.0

9.0−

→←z

znun

6

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

First-Order System (a=0.5)

( ) [ ]5.0

5.0−

→←z

znun

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Sample Value

Impu

lse

Res

pons

e (h

[n])

Faster Decay

-3 -2 -1 0 1 2 30.5

1

1.5

2

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

Wider Bandwidth

7

First-Order System (a=0.1)

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

( ) [ ]1.0

1.0−

→←z

znun

-3 -2 -1 0 1 2 30.9

0.95

1

1.05

1.1

1.15

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

Even Wider Bandwidth

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Sample Value

Impu

lse

Res

pons

e (h

[n])

Even Faster Decay

8

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (rad/sec)

Nor

mal

ized

Mag

nitu

de F

requ

ency

Res

pons

e

a=0.1

a=0.5

a=0.1

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

a=0.9

a=0.5

a=0.1

Sample Value

Impu

lse

Res

pons

e (h

[n])

First-Order Systems – Varying Pole Position (a > 0)

Frequency-Domain Response Ti me-Domain Response

•Lowpass filter (from 0 to π)• Increasing the pole decreases the corner frequency

•Lowpass filter•The smaller |a| is, the faster the decay (small time constant = high corner frequency)

9

0 5 10 15 20

0

0.5

1

Sample Value

Impu

lse

Res

pons

e (h

[n])

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

-3 -2 -1 0 1 2 30.9

0.95

1

1.05

1.1

1.15

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

First-Order System (a=-0.1)

( ) [ ]1.0

1.0+

→←−z

znun

10

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

First-Order System (a=-0.5)

( ) [ ]5.0

5.0+

→←−z

znun

0 5 10 15 20

-1

-0.5

0

0.5

1

Sample Value

Impu

lse

Res

pons

e (h

[n])

Slower Decay

-3 -2 -1 0 1 2 30.5

1

1.5

2

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

Narrower Bandwidth

11

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

First-Order System (a=-0.9)

( ) [ ]9.0

9.0+

→←−z

znun

-3 -2 -1 0 1 2 30

5

10

15

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

Even Narrower Bandwidth

0 5 10 15 20

-1

-0.5

0

0.5

1

Sample Value

Impu

lse

Res

pons

e (h

[n])

Even Slower Decay

12

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (rad/sec)

Nor

mal

ized

Mag

nitu

de F

requ

ency

Res

pons

e

a=-0.1

a=-0.5

a=-0.1

0 2 4 6 8 10 12 14 16 18 20

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

a=-0.9

a=-0.5

a=-0.1

Sample Value

Impu

lse

Res

pons

e (h

[n])

First-Order Systems – Varying Pole Position (a < 0)

Frequency-Domain Response Ti me-Domain Response

•Highpass filter (from 0 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

13

-3 -2 -1 0 1 2 30

2

4

6

8

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

Frequency (rad/sec)

Nor

mal

ized

Mag

nitu

de F

requ

ency

Res

pons

e

Single Pole (0.8)

Two Poles (0.3, 0.8)

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

2

Real Part

Imag

inar

y P

art

Second-Order System (0.3, 0.8)

Nor

mal

ized

Mag

nitu

de

Fre

quen

cy R

espo

nse

Pole with the slower response dominates

( ) [ ] ( ) [ ]nuknukz

z

z

z nn 8.03.08.03.0 21 +→←

−−

14

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

2

Real Part

Imag

inar

y P

art

Second-Order System (-0.8, 0.8)

-3 -2 -1 0 1 2 30.5

1

1.5

2

2.5

3

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

eM

agni

tude

F

requ

ency

Res

pons

e

( ) [ ] ( ) [ ]nuknukz

z

z

z nn 8.08.08.08.0 21 −+→←

+−

15

-3 -2 -1 0 1 2 30

1

2

3

4

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

2

Real Part

Imag

inar

y P

art

Complex Poles

∗+− pz

z

pz

z566.0566.0, jpp ±=∗ 8.0=p

( ) 4arg π=p

16

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

2

-3 -2 -1 0 1 2 30.5

1

1.5

2

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

Complex Poles – Varying the MagnitudePrevious Position

∗+− pz

z

pz

z353.0353.0, jpp ±=∗ 5.0=p

( ) 4arg π=p

-3 -2 -1 0 1 2 30

1

2

3

4

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

Real Part = 0.8

Real Part = 0.5

|p|= 0.8

|p| = 0.5

•Alters only the magnitude•Does not change the corner frequency

17

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

Frequency (rad/sec)

Nor

mal

ized

Mag

nitu

de F

requ

ency

Res

pons

e

-3 -2 -1 0 1 2 30

2

4

6

8

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

2

Complex Poles – Varying the Angle

Nor

mal

ized

Mag

nitu

de

Fre

quen

cy R

espo

nse

∗+− pz

z

pz

z4.0693.0, jpp ±=∗ 8.0=p

( ) 6arg π=p

Alters only the corner frequency

18

Higher-Order Frequency Responses

-3 -2 -1 0 1 2 30

0.2

0.4

0.6

0.8

1

Frequency (rad/sec)

Mag

nitu

de F

requ

ency

Res

pons

e

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

5

19

Discrete-Time Frequency Responses in MATLAB

Use the “freqz” function

num = [1 0];den = [1 –0.5];ww = -pi:0.01:pi;

[H] = freqz(num,den,ww);

figure;plot(ww,abs(H));

-3 -2 -1 0 1 2 30.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Frequency (rad/sec)M

agni

tude

Fre

quen

cy R

espo

nse