Chapter 5: Frequency Domain Analysis of LTI Systems
Discrete-Time Signals and Systems
Frequency Domain Analysis of LTI Systems
Reference:
Sections 5.1, 5.2 - 5.5 of
Dr. Deepa Kundur
John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
Principles, Algorithms, and Applications, 4th edition, 2007.
University of Toronto
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Chapter 5: Frequency Domain Analysis of LTI Systems
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5.1 Frequency-Domain Characteristics of LTI Systems
Chapter 5: Frequency Domain Analysis of LTI Systems
The Frequency Response Function
I
I
Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
y (n) =
5.1 Frequency-Domain Characteristics of LTI Systems
The Frequency Response Function
∴ y (n) =
Recall for an LTI system: y (n) = h(n) ∗ x(n).
Suppose we inject a complex exponential into the LTI system:
∞
X
=
∞
X
k=−∞
∞
X
h(k)Ae jω(n−k)
h(k)Ae jωn · e −jωk
k=−∞
h(k)x(n − k)
"
k=−∞
jωn
= Ae jωn ·
x(n) = Ae
#
h(k)e −jωk
I
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{z
≡H(ω)=DTFT {h(n)}
}
= Ae jωn H(ω)
Note: we consider x(n) to be comprised of a pure frequency of
ω rad/s
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∞
X
k=−∞
|
I
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Thus, y (n) = H(ω)x(n) when x(n) is a pure frequency.
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.1 Frequency-Domain Characteristics of LTI Systems
Chapter 5: Frequency Domain Analysis of LTI Systems
The Frequency Response Function
LTI System Eigenfunction
I
Eigenfunction of a system:
I
y (n) = H(ω)x(n)
output = scaled input
A·v = λ·v
I
I
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Chapter 5: Frequency Domain Analysis of LTI Systems
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5.1 Frequency-Domain Characteristics of LTI Systems
an input signal that produces an output that differs from the
input by a constant multiplicative factor
multiplicative factor is called the eigenvalue
Therefore, a signal of the form Ae jωn is an eigenfunction of an
LTI system.
Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
Chapter 5: Frequency Domain Analysis of LTI Systems
LTI System Eigenfunction
I
5.1 Frequency-Domain Characteristics of LTI Systems
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5.1 Frequency-Domain Characteristics of LTI Systems
Magnitude and Phase of H(ω)
Implications:
I
I
An LTI system can only change the amplitude and phase of a
sinusoidal signal.
H(ω) = |H(ω)|e jΘ(ω)
An LTI system with inputs comprised of frequencies from set Ω0
cannot produce an output signal with frequencies in the set Ωc0
(i.e., the complement set of Ω0 )
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|H(ω)| ≡ system gain for freq ω
∠H(ω) = Θ(ω) ≡ phase shift for freq ω
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.1 Frequency-Domain Characteristics of LTI Systems
Chapter 5: Frequency Domain Analysis of LTI Systems
Example:
5.1 Frequency-Domain Characteristics of LTI Systems
Example:
Determine the magnitude and phase of H(ω) for the three-point
moving average (MA) system
What is the phase of H(ω) = 13 (1 + 2 cos(ω))?
1
y (n) = [x(n + 1) + x(n) + x(n − 1)]
3
1
|1 + 2 cos(ω)|
3
0 0 ≤ ω ≤ 2π
3
Θ(ω) =
π 2π
≤ω<π
3
|H(ω)| =
By inspection, h(n) = 13 δ(n + 1) + 13 δ(n) + 13 δ(n − 1). Therefore,
H(ω) =
∞
X
h(n)e
−jωn
n=−∞
1
X
1 −jωn
=
e
3
n=−1
See
Figure 5.1.1 of text
.
1 jω
1
=
[e + 1 + e −jω ] = (1 + 2 cos(ω))
3
3
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Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
5.2 Frequency Response of LTI Systems
Chapter 5: Frequency Domain Analysis of LTI Systems
Frequency Response of LTI Systems
5.2 Frequency Response of LTI Systems
Frequency Response of LTI Systems
I
z-Domain
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ω-Domain
z=e jω
If H(z) converges on the unit circle, then we can obtain the
frequency response by letting z = e jω :
H(ω) = H(z)|z=e jωn =
H(z) =⇒ H(ω)
z=e jω
∞
X
h(n)e −jωn
n=−∞
system function =⇒ frequency response
PM
=
z=e jω
Y (z) = X (z)H(z) =⇒ Y (ω) = X (ω)H(ω)
1+
k=0
P
N
bk e
k=1
−jωk
ak e −jωk
for rational system functions.
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.4 LTI Systems as Frequency Selective Filters
Chapter 5: Frequency Domain Analysis of LTI Systems
LTI Systems as Frequency-Selective Filters
I
Ideal Filters
Filter: device that discriminates, according to some attribute of
the input, what passes through it
I
Classification:
I
I
I
For LTI systems, given Y (ω) = X (ω)H(ω)
I
5.4 LTI Systems as Frequency Selective Filters
I
H(ω) acts as a kind of weighting function or spectral shaping
function of the different frequency components of the signal
I
I
lowpass
highpass
bandpass
bandstop
allpass
LTI system ⇐⇒ Filter
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Chapter 5: Frequency Domain Analysis of LTI Systems
See
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Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
5.4 LTI Systems as Frequency Selective Filters
Chapter 5: Frequency Domain Analysis of LTI Systems
Ideal Filters
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5.4 LTI Systems as Frequency Selective Filters
Suppose H(ω) = Ce −jωn0 for all ω:
F
Common characteristics:
I
.
Ideal Filters
I
I
Figure 5.4.1 of text
δ(n) ←→ 1
unity (flat) frequency response magnitude in passband and zero
frequency response in stopband
F
δ(n − n0 ) ←→ 1 · e −jωn0
F
C · δ(n − n0 ) ←→ C · 1 · e −jωn0 = Ce −jωn0
I
linear phase; for constants C and n0
C e −jωn0 ω1 < |ω| < ω2
H(ω) =
0
otherwise
I
Therefore, h(n) = C δ(n − n0 ) and:
y (n) = x(n) ∗ h(n) = x(n) ∗ C δ(n − n0 ) = C x(n − n0 )
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.4 LTI Systems as Frequency Selective Filters
Chapter 5: Frequency Domain Analysis of LTI Systems
Ideal Filters
I
Phase versus Magnitude
Therefore for ideal linear phase filters:
C e −jωn0 ω1 < |ω| < ω2
H(ω) =
0
otherwise
I
I
5.4 LTI Systems as Frequency Selective Filters
What’s more important?
signal components in stopband are annihilated
signal components in passband are shifted (and scaled by
passband gain which is unity)
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5.4 LTI Systems as Frequency Selective Filters
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Chapter 5: Frequency Domain Analysis of LTI Systems
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5.4 LTI Systems as Frequency Selective Filters
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.4 LTI Systems as Frequency Selective Filters
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Chapter 5: Frequency Domain Analysis of LTI Systems
Chapter 5: Frequency Domain Analysis of LTI Systems
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5.4 LTI Systems as Frequency Selective Filters
Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
5.4 LTI Systems as Frequency Selective Filters
Chapter 5: Frequency Domain Analysis of LTI Systems
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5.5 Inverse Systems and Deconvolution
Why Invert?
I
There is a fundamental necessity in engineering applications to
undo the unwanted processing of a signal.
I
I
I
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reverse intersymbol interference in data symbols in
telecommunications applications to improve error rate; called
equalization
correct blurring effects in biomedical imaging applications for
more accurate diagnosis; called restoration/enhancement
increase signal resolution in reflection seismology for improved
geologic interpretation; called deconvolution
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Chapter 5: Frequency Domain Analysis of LTI Systems
Invertibility of Systems
I
I
5.5 Inverse Systems and Deconvolution
Invertibility of LTI Systems
Identity System
Invertible system: there is a one-to-one correspondence between
its input and output signals
the one-to-one nature allows the process of reversing the
transformation between input and output; suppose
Direct
System
Inverse
System
y (n) = T [x(n)] where T is one-to-one
w (n) = T −1 [y (n)] = T −1 {T [x(n)]} = x(n)
Impluse response =
Identity System
Direct
System
Direct
System
Inverse
System
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Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Chapter 5: Frequency Domain Analysis of LTI Systems
Invertibility of LTI Systems
I
Inverse
System
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5.5 Inverse Systems and Deconvolution
Invertibility of Rational LTI Systems
Therefore,
I
Suppose, H(z) is rational:
h(n) ∗ hI (n) = δ(n)
A(z)
B(z)
B(z)
HI (z) =
A(z)
poles of H(z) = zeros of HI (z)
zeros of H(z) = poles of HI (z)
H(z) =
I
I
For a given h(n), how do we find hI (n)?
Consider the z−domain
H(z)HI (z) = 1
HI (z) =
1
H(z)
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Chapter 5: Frequency Domain Analysis of LTI Systems
Example
5.5 Inverse Systems and Deconvolution
Common Transform Pairs
Determine the inverse system of the system with impulse response
h(n) = ( 12 )n u(n).
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5.5 Inverse Systems and Deconvolution
1
2
3
4
5
6
Signal, x(n)
δ(n)
u(n)
an u(n)
nan u(n)
−an u(−n − 1)
−nan u(−n − 1)
7
(cos(ω0 n))u(n)
8
(sin(ω0 n))u(n)
9
(an
cos(ω0 n)u(n)
10
(an
sin(ω0 n)u(n)
z-Transform, X (z)
1
1
1−z −1
1
1−az −1
az −1
(1−az −1 )2
1
1−az −1
az −1
(1−az −1 )2
1−z −1 cos ω0
1−2z −1 cos ω0 +z −2
z −1 sin ω0
1−2z −1 cos ω0 +z −2
1−az −1 cos ω0
1−2az−1 cos ω0 +a2 z −2
1−az −1 sin ω0
1−2az−1 cos ω0 +a2 z −2
ROC
All z
|z| > 1
|z| > |a|
|z| > |a|
|z| < |a|
|z| < |a|
|z| > 1
|z| > 1
|z| > |a|
|z| > |a|
Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
Chapter 5: Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Example
Another Example
Determine the inverse system of the system with impulse response
h(n) = ( 12 )n u(n).
Determine the inverse system of the system with impulse response
h(n) = δ(n) − 12 δ(n − 1).
I
I
I
H(z) = 1− 11z −1 , ROC: |z| > 12 ; note: direct system is causal +
2
stable.
Therefore,
1
1
HI (z) =
= 1 − z −1
H(z)
2
By inspection,
1
hI (n) = δ(n) − δ(n − 1)
2
I
H(z) =
∞
X
h(n)z −n =
n=−∞
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∞
X
1
[δ(n) − δ(n − 1)]z −n
2
n=−∞
1
= 1 − z −1
2
1
HI (z) =
1 − 12 z −1
Is the inverse system stable? causal?
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Chapter 5: Frequency Domain Analysis of LTI Systems
Common Transform Pairs
1
2
3
4
5
6
Signal, x(n)
δ(n)
u(n)
an u(n)
nan u(n)
−an u(−n − 1)
−nan u(−n − 1)
7
(cos(ω0 n))u(n)
8
(sin(ω0 n))u(n)
9
(an
10
(an
cos(ω0 n)u(n)
sin(ω0 n)u(n)
Common Transform Pairs
z-Transform, X (z)
1
1
1−z −1
1
1−az −1
az −1
(1−az −1 )2
1
1−az −1
az −1
(1−az −1 )2
1−z −1 cos ω0
1−2z −1 cos ω0 +z −2
z −1 sin ω0
1−2z −1 cos ω0 +z −2
1−az −1 cos ω0
1−2az−1 cos ω0 +a2 z −2
1−az −1 sin ω0
1−2az−1 cos ω0 +a2 z −2
ROC
All z
|z| > 1
|z| > |a|
|z| > |a|
|z| < |a|
|z| < |a|
1
2
3
4
5
6
Signal, x(n)
δ(n)
u(n)
an u(n)
nan u(n)
−an u(−n − 1)
−nan u(−n − 1)
|z| > 1
7
(cos(ω0 n))u(n)
|z| > 1
8
(sin(ω0 n))u(n)
|z| > |a|
|z| > |a|
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Chapter 5: Frequency Domain Analysis of LTI Systems
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cos(ω0 n)u(n)
10
(an
sin(ω0 n)u(n)
1
1−z −1
1
1−az −1
az −1
(1−az −1 )2
1
1−az −1
az −1
(1−az −1 )2
1−z −1 cos ω0
1−2z −1 cos ω0 +z −2
z −1 sin ω0
1−2z −1 cos ω0 +z −2
1−az −1 cos ω0
1−2az−1 cos ω0 +a2 z −2
1−az −1 sin ω0
1−2az−1 cos ω0 +a2 z −2
Chapter 5: Frequency Domain Analysis of LTI Systems
ROC
All z
|z| > 1
|z| > |a|
|z| > |a|
|z| < |a|
|z| < |a|
|z| > 1
|z| > 1
|z| > |a|
|z| > |a|
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5.5 Inverse Systems and Deconvolution
Homomorphic Deconvolution
There are two possibilities for inverses:
I Causal + stable inverse:
n
1
hI (n) =
u(n)
2
I
The complex cepstrum of a signal x(n) is given by:
cx (n) = Z −1 {ln(Z {x(n)}} = Z −1 {ln(X (z))} = Z −1 {Cx (z)}
Anticausal + unstable inverse:
n
1
hI (n) = −
u(−n − 1)
2
Z
x(n) ←→ X (z)
Z
cepstrum ≡ cx (n) ←→ Cx (z) = ln(X (z))
I
Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
9
(an
z-Transform, X (z)
1
Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Another Example
I
5.5 Inverse Systems and Deconvolution
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We say, cx (n) is produced via a homomorphic transform of x(n).
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Chapter 5: Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Chapter 5: Frequency Domain Analysis of LTI Systems
Homomorphic Deconvolution
Homomorphic Deconvolution
I
Y (z) = X (z)H(z)
Cy (z) = ln Y (z)
= ln X (z) + ln H(z)
= Cx (z) + Ch (z)
Z −1 {Cy (z)} = Z −1 {Cx (z)} + Z −1 {Ch (z)}
cy (n) = cx (n) + ch (n)
where:
1 |n| ≤ N1
0 |n| > N1
0 |n| ≤ N1
1 |n| > N1
ŵlp (n) =
convolution in time-domain ←→ addition in cepstral domain
Dr. Deepa Kundur (University of Toronto) Frequency Domain Analysis of LTI Systems
ŵhp (n) =
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5.5 Inverse Systems and Deconvolution
Homomorphic Deconvolution
I
In many applications, the characteristics of cx (n) and ch (n) are
sufficiently distinct that temporal windows can be used to
separate them:
ĉh (n) = cy (n)ŵlp (n)
ĉx (n) = cy (n)ŵhp (n)
Therefore,
Chapter 5: Frequency Domain Analysis of LTI Systems
5.5 Inverse Systems and Deconvolution
Obtaining the inverse homomorphic transforms of ch (n) and
cx (n) give estimates of h(n) and x(n), respectively.
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