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ESE 531: Digital Signal Processing
Lec 13: February 23st, 2017 Frequency Response of LTI Systems
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Lecture Outline
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! Frequency Response of LTI Systems " Magnitude Response
" Simple Filters
" Phase Response " Group Delay
" Example: Zero on Real Axis
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Frequency Response of LTI System
! LTI Systems are uniquely determined by their impulse response
! We can write the input-output relation also in the z-domain
! Or we can define an LTI system with its frequency response ! H(ejω) defines magnitude and phase change at each frequency
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y n⎡⎣ ⎤⎦= x k⎡⎣ ⎤⎦ h n− k⎡⎣ ⎤⎦k=−∞
∞
∑ = x k⎡⎣ ⎤⎦∗h k⎡⎣ ⎤⎦
Y z( ) = H z( ) X z( )
Y e jω( ) = H e jω( ) X e jω( )
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Frequency Response of LTI System
! We can define a magnitude response
! And a phase response
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Y e jω( ) = H e jω( ) X e jω( )
Y e jω( ) = H e jω( ) X e jω( )
∠Y e jω( ) =∠H e jω( )+∠X e jω( )
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Phase Response
! Limit the range of the phase response
5 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Phase Response
! Limit the range of the phase response
6 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
2
Group Delay
! General phase response at a given frequency can be characterized with group delay, which is related to phase
! More later…
7 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Linear Difference Equations
8 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Magnitude Response
9 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Magnitude Response
10 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Magnitude Response
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e jω
ω
dk
v1
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Magnitude Response Example
12 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
3
Magnitude Response Example
13 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Magnitude Response Example
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e jω
ωv1
v2
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Magnitude Response Example
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e jω
ωv1
v2
π 2πω
H (e jω )
1
(dB - Log scale)
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Low Pass Filter
16 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Low Pass Filter
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πω
H (e jω )
1
(dB - Log scale)
ωc
1 2
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple High Pass Filter
18 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
4
Simple High Pass Filter
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e jω
ωv1v2
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple High Pass Filter
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πω
H (e jω )
1
(dB - Log scale)
ωc
1 2
e jω
ωv1v2
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Stop (Notch) Filter
21 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Stop (Notch) Filter
22
ω0
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Stop (Notch) Filter
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ω0
πω
H (e jω )
1
(dB - Log scale)
ω0Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Stop (Notch) Filter
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ω0
πω
H (e jω )
1
(dB - Log scale)
ω0Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
5
Simple Band-Stop (Notch) Filter
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ω0
πω
H (e jω )
1
(dB - Log scale)
ω0Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Pass Filter
26 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Pass Filter
27 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Pass Filter
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πω
H (e jω )
1
(dB - Log scale)
ω0Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Simple Band-Pass Filter
29
πω
H (e jω )
1
(dB - Log scale)
ω0
Larger α reduces pass band
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Phase Response
! Limit the range of the phase response
30 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
6
Phase Response Example
31
ωπ
−π
ARG
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Group Delay
! General phase response at a given frequency can be characterized with group delay, which is related to phase
32
ωω1 ω2
- slope Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Phase Response Example
33
ωπ
−π
ARG
For linear phase system, group delay is nd Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Group Delay
! General phase response at a given frequency can be characterized with group delay, which is related to phase
34
ωω1 ω2
- slope Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Group Delay
35
ωω1 ω2
- slope
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Group Delay
36
ωω1 ω2
- slope
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
7
Group Delay Math
37
H (z) =b0a0
(1− ck z−1)
k=1
M
∏
(1− dk z−1)
k=1
N
∏H (e jω ) =
b0a0
(1− cke− jω )
k=1
M
∏
(1− dke− jω )
k=1
N
∏
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Group Delay Math
38
H (z) =b0a0
(1− ck z−1)
k=1
M
∏
(1− dk z−1)
k=1
N
∏H (e jω ) =
b0a0
(1− cke− jω )
k=1
M
∏
(1− dke− jω )
k=1
N
∏
arg[H (e jω )]= arg[1− cke− jω ]
k=1
M
∑ − arg[1− dke− jω ]
k=1
N
∑
grd[H (e jω )]= grd[1− cke− jω ]
k=1
M
∑ − grd[1− dke− jω ]
k=1
N
∑
arg of products is sum of args
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Group Delay Math
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arg[1− re jθe− jω ]= tan−1 rsin(ω −θ )1− rcos(ω −θ )⎛
⎝⎜
⎞
⎠⎟
grd[H (e jω )]= grd[1− cke− jω ]
k=1
M
∑ − grd[1− dke− jω ]
k=1
N
∑
! Look at each factor:
grd[1− re jθe− jω ]= r2 − rcos(ω −θ )
1− re jθe− jω2
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
40
arg[1− re− jω ]
r
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
41
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
r
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
42
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
ωr
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
8
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
43 Penn ESE 531 Spring 2017 - Khanna
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
ωϕ
r
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
44 Penn ESE 531 Spring 2017 - Khanna
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
ωϕ
r
ϕ −ω
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
45 Penn ESE 531 Spring 2017 - Khanna
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
ωϕ
r
ϕ −ω
ω
arg
π
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
46
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
r ω
arg
π
ω = 0
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
47
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
r ω
arg
π
ω = π
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
48
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
ωϕ
r
ϕ −ω
ω
arg
π
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
9
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
49
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
r
ϕ −ω
ω
arg
π
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! Geometric Interpretation for (θ=0)
50
arg[1− re− jω ]= arg[(e jω − r)e− jω ]= arg[e jω − r]− arg[e jω ]
ωϕ
ω
arg
π
grd
πω
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Group Delay Math
51
arg[1− re jθe− jω ]= tan−1 rsin(ω −θ )1− rcos(ω −θ )⎛
⎝⎜
⎞
⎠⎟
grd[H (e jω )]= grd[1− cke− jω ]
k=1
M
∑ − grd[1− dke− jω ]
k=1
N
∑
! Look at each factor:
grd[1− re jθe− jω ]= r2 − rcos(ω −θ )
1− re jθe− jω2
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
52
! For θ≠0
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! Magnitude Response
53 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! For θ=π, how does zero location effect magnitude, phase and group delay?
54 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
10
Example: Zero on Real Axis
! For θ=π, how does zero location effect magnitude, phase and group delay?
55 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Example: Zero on Real Axis
! For θ=π, how does zero location effect magnitude, phase and group delay?
56 Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
2nd Order IIR with Complex Poles
57 Penn ESE 531 Spring 2017 - Khanna
magnitude
2nd Order IIR with Complex Poles
58 Penn ESE 531 Spring 2017 - Khanna
magnitude
phase
group delay
Big Ideas
59
! Frequency Response of LTI Systems " Magnitude Response
" Simple Filters
" Phase Response " Group Delay
" Example: Zero on Real Axis
Penn ESE 531 Spring 2017 – Khanna Adapted from M. Lustig, EECS Berkeley
Admin
! HW 5 " Due Friday 3/3
! Homework solutions to be posted soon
60 Penn ESE 531 Spring 2017 - Khanna
