Frequency Response of
Discrete-Time Systems
EE 327
Signals and Systems 1
© David W. Graham 2006
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
1
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
2
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
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
Frequency response plot
can be taken from the
contour of the pole-zero
plot around the unit circle
(from –π to π)
4
First-Order System (a=0.9)
Impulse Response (h[n])
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
0.8
0.6
0.4
0.2
0
1
0
5
10
Sample Value
(0.9)n u[n]←→
Magnitude Frequency Response
Imaginary Part
1
15
20
z
z − 0 .9
10
8
6
4
2
0
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
5
First-Order System (a=0.5)
Impulse Response (h[n])
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
Faster Decay
0.8
0.6
0.4
0.2
0
1
0
5
10
Sample Value
(0.5)n u[n]←→
Magnitude Frequency Response
Imaginary Part
1
15
20
z
z − 0.5
2
Wider
Bandwidth
1.5
1
0.5
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
6
First-Order System (a=0.1)
Impulse Response (h[n])
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
Even Faster Decay
0.8
0.6
0.4
0.2
0
1
0
5
10
Sample Value
(0.1)n u[n]←→
Magnitude Frequency Response
Imaginary Part
1
15
20
z
z − 0.1
1.15
Even Wider
Bandwidth
1.1
1.05
1
0.95
0.9
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
7
First-Order Systems – Varying Pole Position (a > 0)
Frequency-Domain Response
Ti me-Domain Response
1
0.9
1
a=0.1
0.8
0.7
0.6
Impulse Response (h[n])
Normalized Magnitude Frequency Response
0.8
a=0.5
0.5
0.4
0.6
0.4
a=0.9
0.3
a=0.5
0.2
a=0.1
0.2
a=0.1
0.1
0
-3
-2
-1
0
Frequency (rad/sec)
1
2
• Lowpass filter (from 0 to π)
• Increasing the pole decreases
the corner frequency
3
0
0
2
4
6
8
10
Sample Value
12
14
16
18
20
• Lowpass filter
• The smaller |a| is, the faster the
decay (small time constant =
high corner frequency)
8
First-Order System (a=-0.1)
Impulse Response (h[n])
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
0.5
0
1
0
5
10
Sample Value
(− 0.1)n u[n]←→
Magnitude Frequency Response
Imaginary Part
1
15
20
z
z + 0.1
1.15
1.1
1.05
1
0.95
0.9
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
9
First-Order System (a=-0.5)
Impulse Response (h[n])
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
Slower Decay
0.5
0
-0.5
-1
1
0
5
10
Sample Value
(− 0.5)n u[n]←→
Magnitude Frequency Response
Imaginary Part
1
15
20
z
z + 0.5
2
Narrower
Bandwidth
1.5
1
0.5
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
10
First-Order System (a=-0.9)
Impulse Response (h[n])
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
Even Slower Decay
0.5
0
-0.5
-1
1
0
5
10
Sample Value
(− 0.9)n u[n]←→
Magnitude Frequency Response
Imaginary Part
1
15
20
z
z + 0.9
15
Even Narrower
Bandwidth
10
5
0
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
11
First-Order Systems – Varying Pole Position (a < 0)
Frequency-Domain Response
Ti me-Domain Response
1
1
0.9
a=-0.1
0.8
0.6
0.7
a=-0.9
0.4
0.6
Impulse Response (h[n])
Normalized Magnitude Frequency Response
0.8
a=-0.5
0.5
0.4
a=-0.5
0.2
0
-0.2 a=-0.1
-0.4
0.3
-0.6
0.2
-0.8
0.1
a=-0.1
-1
0
-3
-2
-1
0
Frequency (rad/sec)
1
2
• Highpass filter (from 0 to π)
• Increasing the pole decreases
the corner frequency
3
0
2
4
6
8
10
Sample Value
12
14
16
18
• Highpass filter
• The smaller |a| is, the faster the
decay (small time constant =
high corner frequency)
• Oscillation from a first-order
system
20
12
0.5
2
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
8
6
4
2
0
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
z
z
n
n
←
→ k1 (0.3) u[n] + k 2 (0.8) u[n]
z − 0.3 z − 0.8
Normalized
Normalized
MagnitudeMagnitude
Frequency Response
Frequency Response
Imaginary Part
1
Magnitude Frequency Response
Second-Order System (0.3, 0.8)
1
0.8
0.6
0.4
Single Pole (0.8)
0.2
0
Two Poles (0.3, 0.8)
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
Pole with the slower response dominates
13
Imaginary Part
1
MagnitudeMagnitude
Frequency Response
Frequency Response
Second-Order System (-0.8, 0.8)
3
2.5
0.5
2
0
2
1.5
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
1
0.5
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
z
z
n
n
←
→ k1 (0.8) u[n] + k 2 (− 0.8) u[n]
z − 0 .8 z + 0 .8
14
Magnitude Frequency Response
Complex Poles
Imaginary Part
1
0.5
2
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
z
z
z − p z + p∗
4
3
2
1
0
-3
-2
-1
0
1
Frequency (rad/sec)
p, p ∗ = 0.566 ± j 0.566
2
3
p = 0.8
arg( p ) = π 4
15
Complex Poles – Varying the Magnitude
Magnitude Frequency Response
Previous Position
0.5
2
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
2
1.5
1
0.5
z
z
z − p z + p∗
-3
-2
-1
0
1
Frequency (rad/sec)
p, p ∗ = 0.353 ± j 0.353
Magnitude Frequency Response
Imaginary Part
1
2
3
p = 0.5
arg( p ) = π 4
4
|p|= 0.8
Real Part = 0.8
3
2
1
|p| =Real
0.5 Part = 0.5
0
-3
-2
-1
0
1
(rad/sec)
only Frequency
the magnitude
2
3
•Alters
•Does not change the corner frequency 16
Imaginary Part
1
0.5
2
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
z
z
z − p z + p∗
MagnitudeResponse
NormalizedNormalized
Magnitude Frequency
Frequency Response
Magnitude Frequency Response
Complex Poles – Varying the Angle
8
6
4
2
0
-3
-2
-1
0
1
Frequency (rad/sec)
p, p ∗ = 0.693 ± j 0.4
2
3
p = 0.8
arg( p ) = π 6
1
0.8
0.6
0.4
0.2
0
-3
-2
Alters
-1
0
1
2
3
Frequency
(rad/sec)
only
the corner
frequency
17
Imaginary Part
1
0.5
5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
Magnitude Frequency Response
Higher-Order Frequency Responses
1
0.8
0.6
0.4
0.2
0
-3
-2
-1
0
1
Frequency (rad/sec)
2
3
18
Discrete-Time Frequency Responses in MATLAB
Use the “freqz” function
2.2
2
[H] = freqz(num,den,ww);
1.8
Magnitude Frequency Response
num = [1 0];
den = [1 –0.5];
ww = -pi:0.01:pi;
1.6
1.4
1.2
1
figure;
plot(ww,abs(H));
0.8
0.6
-3
-2
-1
0
Frequency (rad/sec)
1
2
3
19