Transcript Document
ECE 8443 – Pattern Recognition LECTURE 30: SYSTEM ANALYSIS USING THE TRANSFER FUNCTION
•
Objectives:
Stability Response to a Sinusoid Filtering White Noise Autocorrelation Power Spectral Density
•
Resources:
JOS: Transfer Function Analysis RAKL: z-Transform Analysis RWang: Analysis of LTI Systems SST: Stock Market Trading URL: Audio:
Transfer Functions and Stability
•
Consider a causal, linear time-invariant system described by:
•
H
(
z
)
Y
(
z
)
X
(
z
)
i M
1
b i z
i
1
i N
1
a i z
i
b
0 1
a
0
b
1
z
1
a
1
z
1
b
2
z a
2 2
z
2 ...
...
b M
z a N
M z
N
We have previously described how to factor this into a product of its poles:
H
(
z
)
b
0
z
M z
b
1
p
1
M z
z
1
p
2
b
2
z
...
M
z
2 ...
p N
b M
z A
1
p
1
z A
2
p
2 ...
z A N
p N
We note that both p
k
and A
k
can, in general, be complex.
• • Applying the inverse z-Transform:
h
(
t
)
A
1
1
n u
[
n
]
A
2
2
n u
[
n
] ...
A N
n N u
[
n
] • • This implies the poles all lie within a circle of radius 1 in the z-plane.
n
0
bounded, the output will also be bounded. We refer to this as Bounded-Input Bounded-Output (BIBO) stability.
ECE 3163: Lecture 30, Slide 1
Response to a Sinusoid
•
Consider the signal: Z
x
[
n
]
C X
( cos 0
u
[
z
)
z C
2
z
n
] 2 2 (cos cos 0 0
z
)
z
1
z C
z e
2
j
0 (cos
z
e
0 )
j
0
z
•
Recall that a sinewave is a signal that is marginally stable, meaning its poles like on the unit circle.
e j
0 2
f
0
f s
• •
We can compute the output of an LTI system modeled
e
by a rational transform:
Y
(
z
)
H
(
z
)
X
(
z
)
B
(
z A
(
z
) )
z
z
2
e
j
0 (cos
z
e
)
z j
0
We can express the z-transform in terms of these poles using a PFE:
j
0
Y
(
z
)
z
(
z
)
A
(
z
)
z c
e j
0
z
c e
j
0
c
e j
0
Y
(
z
)
z
z
e j
0
CB
(
A
(
z z
) )(
z z
cos
e
j
0 0 )
z
e j
0 ( after simplifica tion )
C H
2
e j
ECE 3163: Lecture 30, Slide 2
Response to a Sinusoid (Cont.)
• We can solve for Y (
z
)
:
Y
(
z
)
z
A
( (
z z
) )
z
C z
/ 2
H j
0
e
z
C z
/ 2
H e
j
0
j
0 •
The first term of this expression is a transient response that decays to zero if the system is stable.
•
The second two terms can be rewritten as a phase-shifted cosine:
y ss
C H
cos
o n
H
u
[
n
] •
Once again we see that the output of a linear system to a sinewave is an amplitude and phase-shifted version of the input (reminiscent of phasor analysis in circuits).
•
This result can be generalized to all periodic signals because a periodic signal can be represented as a sum of sinewaves via the Fourier series.
•
We also see the importance of poles of the system as resonant frequencies. What happens if the input signal is at the same frequency as a pole of the transfer function?
•
What implications does this have for physical structures such as bridges or skyscrapers? circuits? ECE 3163: Lecture 30, Slide 3
Applications: Noise Reduction
•
Consider a signal corrupted by noise:
x
[
n
]
s
[
n
]
w
[
n
] •
Suppose we have reason to believe the underlying
signal, s [
n
]
, is smooth, and that the noise corrupts our measurements.
•
How can we recover the smooth signal?
•
One option is to use the averager we explored earlier:
y
[
n
]
x
[
n
1 ]
x
[
n
]
x
[
n
1 ] •
Another type of filter we might use is a recursive, or infinite impulse response (IIR) filter:
h
[
n
] ( 1
b
)
b n u
[
n
]
H
(
z
) 1 1
bz b
1
y
[
n
]
by
[
n
1 ] ( 1
b
)
x
[
n
]
ECE 3163: Lecture 30, Slide 4
Correlation-Based Processing (Advanced / Statistics)
•
Consider operating on the correlation of the signal:
x
[
n
]
s
[
n
]
w
[
n
]
R xx
x
[
n
]
x
[
n
k
]
n
s
[
n
]
s
[
n
k
]
n
x
[
n
]
x
[
n n
s
[
n
]
w
[
n
k
]
k
]
n
s
[
n
]
w
[
n
]
s
[
n
k
]
w
[
n
k
]
n
s
[
n
k
]
w
[
n
]
n
w
[
n
]
w
[
n
k
] • • •
Assume the noise is statistically uncorrelated with the signal:
E
s
[
n
]
w
[
n
k
]
n
s
[
n
]
w
[
n
k
] 0 and
E
s
[
n
k
]
w
[
n
]
n
s
[
n
k
]
w
[
n
] 0
Also, assume the noise is “white,” meaning it has a flat frequency spectrum:
E
w
[
n
]
w
[
n
k
]
n
w
[
n
]
w
[
n
k
] 0
w
2
Our correlation operator simplifies to:
k k
0 0
R xx
s
[
n
R ss
n
]
s
[
n
2
w
k
]
n
w
[
n
]
w
[
n
k
]
R ss
R ww
•
Hence, the effects of additive white noise can be removed through the use of a correlation function. What is the significance of this result?
ECE 3163: Lecture 30, Slide 5
Correlation and the Power Spectrum
•
Consider the discrete Fourier transform of the correlation function:
R j
k
R xx
e
j
n n
x
[
n
]
k
x
[
n X j
j
k
n
k
]
e
j
n
x
[
n
]
x
[
n
k
]
e
j
n
n
k
x
[
n
]
x
[
n n
x
[
n
]
m
x
[
m
]
e
j
(
m
k
)
n
x
[
n
]
e
k
]
e
j
n j
k m
x
[
m
]
e j
m
X
2 •
This relationship is known as the Wiener-Khintchine Theorem:
R xx
xx
X
2 •
Hence, the spectrum of the correlation function is just the power spectral density. Two important observations:
The autocorrelation function is “blind” to phase.
The spectrum of the autocorrelation function of our noisy signal is:
R xx
F
1
N
n s
[
n
]
s
[
n
k
] 1
N
n w
[
n
]
w
[
n
k
]
S
(
e j
) 2
F
1
N
n w
[
n
]
w
[
n
k
]
S
(
e j
) 2
F
2
w
[
n
]
S
(
e j
) 2 2
w
ECE 3163: Lecture 30, Slide 6
Summary
•
Discussed stability of a linear, time-invariant system: poles must be inside the unit circle.
•
Demonstrated that the response to a sinusoid is a amplitude and phase shifted sinusoid at steady state.
•
Introduced an application of filtering to reduce the effects of additive noise.
•
Introduce the concept of an autocorrelation function and demonstrated its relationship to the power spectrum.
ECE 3163: Lecture 30, Slide 7