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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