Lecture Digital signal processing: Lecture 5 - Zheng-Hua Tan
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Digital Signal Processing, Fall 2006
Lecture 5: System analysis
Zheng-Hua Tan
Department of Electronic Systems
Aalborg University, Denmark
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Course at a glance
MM1
Discrete-time
signals and systems
MM2
Fourier-domain
representation
Sampling and
reconstruction
System
System
analysis
System
structures
MM6
MM5
Filter
MM4
z-transform
MM3
2
DFT/FFT
Filter structures
MM9,MM10
MM7
Filter design
MM8
Digital Signal Processing, V, Zheng-Hua Tan, 2006
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System analysis
Three domains
Time domain: impulse response, convolution sum
y[n] = x[n] * h[n] =
∞
∑ x[k ]h[n − k ]
k = −∞
Frequency domain: frequency response
Y ( e jω ) = X ( e jω ) H ( e jω )
z-transform: system function
Y ( z) = X ( z)H ( z)
LTI system is completed characterized by …
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Part I: Frequency response
Frequency response
System functions
Relationship between magnitude and phase
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Frequency response
Relationship btw Fourier transforms of input and
output
Y ( e jω ) = X ( e jω ) H ( e jω )
In polar form
Magnitude Æ magnitude response, gain, distortion
| Y (e jω ) |=| X (e jω ) | ⋅ | H (e jω ) |
Phase Æ phase response, phase shift, distortion
∠Y (e jω ) = ∠X (e jω ) + ∠H (e jω )
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Ideal lowpass filter – an example
Frequency response
| ω |< ωc ,
⎧ 1,
H ( e jω ) = ⎨
⎩0, ωc <| ω |< π
Frequency selective filter
Impulse response
hlp [n] =
sin ω c n
,
πn
−∞ < n < ∞
?
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h[n] = 0, n < 0
Noncausal, cannot be implemented!
How to make a noncausal system causal?
Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Make noncausal system causal
Cascading systems
x[n]
Ideal delay
Forward difference
h[n] = δ [ n + 1] − δ [ n]
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One - sameple delay
h[n] = δ [ n − 1]
Backward difference
h[n] = δ [n] − δ [n − 1]
x[n]
h[n] = δ [n − nd ]
y[n]
y[n]
In general, any noncausal FIR system can be made
cause by cascading it with a sufficiently long delay!
But ideal lowpass filter is an IIR system!
Digital Signal Processing, V, Zheng-Hua Tan, 2006
Phase distortion and delay
Ideal delay system
hid [n] = δ [n − nd ]
Delay distortion
H id (e jω ) = e − jωnd
| H id (e jω ) |= 1
∠H id (e jω ) = −ωnd , | ω |< π
Linear phase distortion
Ideal lowpass filter with linear phase
⎧e − jωnd ,
| ω |< ω c ,
H lp (e jω ) = ⎨
0
,
ω
c < ω |< π
⎩
hlp [n] =
8
sin ω c (n − nd )
,
π ( n − nd )
Ideal lowpass filter is
always noncausal!
−∞ < n < ∞
Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Group delay
A measure of the linearity of the phase
Concerning the phase distortion on a narrowband
signal
x[n] = s[n] cos(ω 0 n)
w0
0
For this input with spectrum only around w0, phase
effect can be approximated around w0 as the linear
approximation (though in reality maybe nonlinear)
∠H (e jω ) ≈ −ωnd − φ0
and the output is approximately
y[n] ≈| H (e jω0 ) | s[n − nd ] cos(ω0 (n − nd ) − φ0 )
Group delay
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τ = grd [ H (e jω )] = −
d
{arg[ H (e jω )]}
dω
Digital Signal Processing, V, Zheng-Hua Tan, 2006
An example of group delay
Figure 5.1, 5.2, 5.3
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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An example of group delay
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Part II: System functions
Frequency response
System functions
Relationship between magnitude and phase
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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System function of LCCDE systems
Linear constant-coefficient difference equation
M
N
∑a
k =0
k
y[n − k ] = ∑ bm x[n − m]
m =0
z-transform format
N
M
∑ ak z − k Y ( z ) = ∑ bm z − m X ( z )
k =0
m =0
M
Y ( z)
H ( z) =
=
X ( z)
∑ bm z −m
m =0
N
∑ ak z − k
k =0
M
=(
∏ (1 − cm z −1 )
b0 m =1
)
a0 N
∏ (1 − d k z −1 )
(1 − cm z −1 ) in the numerator
a zero at z = cm a pole at z = 0
(1 − d k z −1 ) in the denominator
a zero at z = 0 a pole at z = d k
k =1
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Stability and causality
Stable
h[n] absolutely summable
H(z) has a ROC including the unit circle
Causal
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h[n] right side sequence
H(z) has a ROC being outside the outermost pole
Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Inverse systems
Many systems have inverses, specially systems with
rational system functions
M
G( z) = H ( z) H i ( z) = 1
1
H ( z)
g[n] = h[n] * hi [n] = δ [n]
H i ( z) =
∏ (1 − cm z −1 )
b
H ( z ) = ( 0 ) mN=1
a0
∏ (1 − d k z −1 )
k =1
N
a
H i ( z) = ( 0 )
b0
∏ (1 − d k z −1 )
k =1
M
∏ (1 − cm z −1 )
m =1
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Poles become zeros and vice versa.
ROC: must have overlap btw the two for the sake of
G(z).
Digital Signal Processing, V, Zheng-Hua Tan, 2006
Example
1 − 0.5 z −1
, | z |> 0.9
1 − 0.9 z −1
1 − 0.9 z −1
1
0.9 z −1
H i ( z) =
=
−
1 − 0.5 z −1 1 − 0.5 z −1 1 − 0.5 z −1
H ( z) =
So,
| z |> 0.5
hi [n] = (0.5) n u[n] − 0.9(0.5) n −1 u[n − 1]
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Part III: Magnitude and phase
Frequency response
System functions
Relationship between magnitude and phase
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Relationship btw magnitude and phase
In particular, for systems with rational system
functions, there is constraint btw magnitude and
phase
H (e jω ) =| H (e jω ) | e j∠H ( e
jω
)
Consider the square of the magnitude
| H (e jω ) |2 = H (e jω ) H * (e jω ) = H ( z ) H * (1 / z * ) | z = e jω
M
M
H ( z) = (
∏ (1 − cm z −1 )
b0 m =1
)
a0 N
H * (1 / z * ) = (
∏ (1 − d k z −1 )
∏ (1 − cm* z )
b0 m =1
)
a0 N
∏ (1 − d k * z )
k =1
k =1
M
b
C ( z ) = H ( z ) H (1 / z ) = ( 0 ) 2
a0
*
*
∏ (1 − cm z −1 )(1 − cm z )
*
m =1
N
∏ (1 − d k z −1 )(1 − d k * z)
k =1
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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An example
P271, Example 5.11
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
An example
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Part VI: All-pass systems
Frequency response
System functions
Relationship between magnitude and phase
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
All-pass systems
Consider the following stable system function
z −1 − a *
1 − az −1
H ap ( z ) =
e − jω − a *
1 − ae − jω
* jω
− jω 1 − a e
=e
1 − ae − jω
H ap (e jω ) =
| H ap (e jω ) |= 1
all-pass system: for which the
frequency response magnitude is a constant.
General form
z −1 − d k M c ( z −1 − ek )( z −1 − ek )
∏
−1
*
k =1 1 − d z
k =1 (1 − e z −1 )(1 − e z −1 )
k
k
k
Mr
*
H ap ( z ) = A ∏
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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An example
P275 Example 5.13, Firstorder all-pass system
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
An example
Second-order all-pass
system
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Part V: Minimum-phase systems
Frequency response
System functions
Relationship between magnitude and phase
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Minimum-phase systems
Magnitude does not uniquely characterize the
system
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Stable and causal Æ poles inside unit circle, no
restriction on zeros
Zeros are also inside unit circle Æ inverse system is
also stable and causal (in many situations, we need
inverse systems!)
Æ such systems are called minimum-phase systems
(explanation to follow): are stable and causal and
have stable and causal inverses
Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Minimum-phase and all-pass decomposition
Any rational system function can be expressed as:
H ( z ) = H min ( z ) H ap ( z )
Suppose H(z) has one zero outside the unit circle at
z = 1 / c * , | c |< 1
H ( z ) = H 1 ( z )( z −1 − c * )
= H 1 ( z )(1 − cz −1 )
z −1 − c *
1 − cz −1
minimum-phase all-pass
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Frequency response compensation
When the distortion system is not minimum-phase
system:
H d ( z ) = H d min ( z ) H ap ( z )
H c ( z) =
G ( z ) = H d ( z ) H c ( z ) = H ap ( z )
1
H d min ( z )
Frequency response magnitude is compensated
Phase response is the phase of the all-pass
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Properties of minimum-phase systems
From minimum-phase and all-pass decomposition
H ( z ) = H min ( z ) H ap ( z )
arg[ H (e jω )] = arg[ H min (e jω )] + arg[ H ap (e jω )]
From previous figures, the continuous-phase curve
of an all-pass system is negative for 0 ≤ ω ≤ π
So change from minimum-phase to nonminimumphase (+all-pass phase) always decreases the
continuous phase or increases the negative of the
phase (called the phase-lag function). Minimumphase is more precisely called minimum phase-lag
system
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Part VI: Linear-phase systems
Frequency response
System functions
Relationship between magnitude and phase
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Design a system with non-zero phase
System design sometimes desires
Constant frequency response magnitude
Zero phase, when not possible
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accept phase distortion, in particular linear phase since
it only introduce time shift
Nonlinear phase will change the shape of the input
signal though having constant magnitude response
Digital Signal Processing, V, Zheng-Hua Tan, 2006
Ideal delay
H id (e jω ) = e − jωα , | ω |< π
| H id (e jω ) |= 1
∠H id (e jω ) = −ωα , | ω |< π
grd [ H id (e jω )] = α
hid [n] =
when α = nd
sin π (n − α )
π (n − α )
hid [n] = δ [n − nd ]
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Ideal lowpass with linear phase
hlp [n] =
sin ω c (n − nd )
π ( n − nd )
Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Generalized linear phase
Linear phase filters
H (e jω ) =| H (e jω ) | e − jωα
Generalized linear phase filters
H (e jω ) = A(e jω )e − jωα + jβ
A(e jω ) is a real function of ω ,
α and β are real constants
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
Summary
Frequency response
System functions
Relationship between magnitude and phase
All-pass systems
Minimum-phase systems
Linear systems with generalized linear phase
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Digital Signal Processing, V, Zheng-Hua Tan, 2006
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Course at a glance
MM1
Discrete-time
signals and systems
MM2
Fourier-domain
representation
Sampling and
reconstruction
System
System
analysis
System
structure
MM6
MM5
Filter
MM4
z-transform
MM3
35
DFT/FFT
Filter structures
MM9,MM10
MM7
Filter design
MM8
Digital Signal Processing, V, Zheng-Hua Tan, 2006
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