Digital Signal Processing
Frequency-Domain Analysis of LTI Systems
Moslem Amiri, V´aclav Pˇrenosil
Embedded Systems Laboratory
Faculty of Informatics, Masaryk University
Brno, Czech Republic
amiri@mail.muni.cz
prenosil@fi.muni.cz
April, 2012
Frequency-Domain Characteristics of LTI Systems
We develop characterization of LTI systems in frequency domain
Basic excitation signals are complex exponentials and sinusoidal
functions
Characteristics of system are described by a function of ω called
frequency response, which is Fourier transform of impulse response h(n)
of system
Frequency response function completely characterizes an LTI system in
frequency domain
This allows us to determine steady-state response of system to any
arbitrary weighted linear combination of sinusoids or complex
exponentials
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The Frequency Response Function
We know that response of any relaxed LTI system to an arbitrary input
signal x(n) is given by convolution sum formula
y(n) =
∞
k=−∞
h(k)x(n − k)
System is characterized in time domain by its unit sample response h(n)
To develop a frequency-domain characterization of system, we excite
system with complex exponential
x(n) = Aejωn, −∞ < n < ∞
A = amplitude
ω = any arbitrary frequency confined to [−π, π]
y(n) =
∞
k=−∞
h(k)[Aejω(n−k)
] = A
∞
k=−∞
h(k)e−jωk
ejωn
(1)
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The Frequency Response Function
The term in brackets in (1) is Fourier transform of unit sample
response h(k) of system
H(ω) =
∞
k=−∞
h(k)e−jωk
Hence response of system to x(n) = Aejωn is
y(n) = AH(ω)ejωn
Response is also in form of a complex exponential with the same
frequency as input, but altered by multiplicative factor H(ω)
As a result of this characteristic behavior, x(n) = Aejωn
is called an
eigenfunction of system
An eigenfunction of a system is an input signal that produces an output
that differs from input by a constant multiplicative factor
Multiplicative factor (in this case H(ω)) is called an eigenvalue of
system
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The Frequency Response Function
Example
Determine output sequence of system with impulse response
h(n) = (1
2)nu(n)
when input is
x(n) = Aejπn/2, −∞ < n < ∞
Fourier transform of h(n)
H(ω) =
∞
n=−∞
h(n)e−jωn
=
1
1 − 1
2 e−jω
ω=π/2
−−−−→ H(
π
2
) =
1
1 + j 1
2
=
2
√
5
e−j26.6◦
y(n) = AH(ω)ejωn
= A
2
√
5
e−j26.6◦
ejπn/2
=
2
√
5
Aej(πn/2−26.6◦
)
, −∞ < n < ∞
The only effect of system on input signal is to scale amplitude by 2/
√
5 and
shift phase by −26.6◦
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The Frequency Response Function
Example (continued)
If input sequence is
x(n) = Aejπn, −∞ < n < ∞
at ω = π
H(π) =
1
1 − 1
2e−jπ
=
1
3
2
=
2
3
and output of system is
y(n) = 2
3Aejπn, −∞ < n < ∞
If we alter frequency of input signal, effect of system on input also
changes and hence output changes
H(π) is purely real
Phase associated with H(ω) is zero at ω = π
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The Frequency Response Function
In general, H(ω) is a complex-valued function of ω
H(ω) = |H(ω)|ejΘ(ω)
|H(ω)| = magnitude of H(ω)
Θ(ω) = H(ω), which is phase shift imparted on input signal by system
at frequency ω
Since H(ω) is Fourier transform of {h(k)}, H(ω) is a periodic function
with period 2π
H(ω) =
∞
k=−∞
h(k)e−jωk
Unit impulse h(k) is related to H(ω) through integral expression
h(k) =
1
2π
π
−π
H(ω)ejωk
dω
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The Frequency Response Function
For an LTI system with a real-valued impulse response, magnitude and
phase functions possess symmetry properties
H(ω) =
∞
k=−∞
h(k)e−jωk
=
∞
k=−∞
h(k) cos ωk − j
∞
k=−∞
h(k) sin ωk
= HR(ω) + jHI (ω) = |H(ω)|ejΘ(ω)
= H2
R(ω) + H2
I (ω)ej tan−1[HI (ω)/HR (ω)]
where
HR(ω) =
∞
k=−∞
h(k) cos ωk and HI (ω) = −
∞
k=−∞
h(k) sin ωk
HR (ω): even, HI (ω): odd −→ |H(ω)|: even, Θ(ω): odd
If we know |H(ω)| and Θ(ω) for 0 ≤ ω ≤ π, we also know these
functions for −π ≤ ω ≤ 0
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The Frequency Response Function
Example
Determine magnitude and phase of H(ω) for three-point moving
average (MA) system
y(n) = 1
3[x(n + 1) + x(n) + x(n − 1)]
and plot these two functions for 0 ≤ ω ≤ π
We have
h(n) = {
1
3
,
1
3
↑
,
1
3
}
H(ω) =
∞
k=−∞
h(k)e−jωk
=
1
3
(ejω
+ 1 + e−jω
) =
1
3
(1 + 2 cos ω)
|H(ω)| =
1
3
|1 + 2 cos ω| and Θ(ω) =
0, 0 ≤ ω ≤ 2π/3
π, 2π/3 ≤ ω < π
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The Frequency Response Function
Figure 1: Magnitude and phase responses for the MA system in Example.
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References
John G. Proakis, Dimitris G. Manolakis, Digital Signal
Processing: Principles, Algorithms, and Applications, Prentice
Hall, 2006.
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