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Frequency Response
A. Introduction
The fundamental model of an LTI system is a
LCCDE of the form;
+ + + ⋯ = + + ⋯
Solution to such equations are known but are
not convenient or insightful.
Instead we characterize these systems in two
different but related way.
Impulse Response
x(t) = δ(t)
LTI
System
y(t) = h(t)
Frequency Response
(1) In the time domain via a time function
description of the system called the impulse
response.
(2) In the frequency domain via a frequency
function description of the system called the
frequency response (also called the transfer
function).
Impulse Response
Thus we characterize the system by how it
responds to the most basic of time signals the
impulse function.
h(t)
δ(t)
The impulse response h(t) of a system is the
response of the system to a unit impulse or
dirac delta function.
h(t)
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Physical Interpretation of Convolution
Physical Interpretation of Convolution
If we envision the arbitrary input x(t) as consisting
of many closely spaced impulses each of which
has the amplitude equal to the value of x(t).
Then the output is simply the sum (superposition)
of the responses to all the weighted impulses.
Thus convolution is simply a sum of the weighted
and shifted impulse responses.
Frequency Response
The frequency domain description is characterized
by it’s response to a frequency domain impulse
(a time domain complex exponential signal).
; = ∗ ℎ ℎ; { } = { ∗ ℎ }
Frequency Response
Using the convolution property
Y(f) = X(f) H(f)
H(f) =
()
()
So the output spectrum Y(f) is simply the input
spectrum X(f) times the system frequency
response H(f).
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Physical Meaning of H(f)
X(f) = δ(f-fo)
LTI
System
Y(F) = H(fo)
= !"#$ ↔ & ' = ((' − '*)
+ = , ' ∗ !"#$ ↔ - ' = , ' ((' − '*)
- ' = , '* ((' − '*)
Physical Meaning of H(f)
The response of an LTI system to a single
frequency input is a single frequency output.
The output is the same frequency as the input,
but is multiplied by the value H(fo). H(fo) is the
value of the system frequency response function
at frequency fo.
Converting the output back to the time domain,
Physical Meaning of H(f)
Converting the output back to the time domain,
! - '
Plotting the Frequency Response H(f)
We can express H(f) in polar/ exponential form
as
= ! , ' . ' − '
= , ' ! . ' − '
= , ' !"#/$
= , ' ()
, ' = , ' "0()
Where
|H(f)| is the magnitude/amplitude of the
frequency response and
Θ(f) is the phase of the frequency response.
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Plotting the Frequency Response H(f)
Plotting the Frequency Response H(f)
Ex… Plotting H(f)
Given: an LTI system with an impulse response
of
ℎ = !1$ Plot the frequency response H(f)
Solution:
, ' = ℎ =
1
10 + 426'
Plotting the Frequency Response H(f)
Amplitude Response
Symmetry of Frequency Response H(f)
If h(t) is real valued,
the frequency response exhibits the following symmetry
|H(f)| = |H(-f)| {even symmetry}
Θ(f) = -θ(-f) {odd symmetry}
Phase Response
Many times we only show H(f) for f > 0, since the
above symmetry is applied.
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LTI System Response Summary
LTI System Response Summary
Time Domain Response
x(t)
h(t)
y(t)
Frequency Domain Response
X(f)
:
= ∗ ℎ() = 7 8 ∗ ℎ − 8 98
!:
LTI System Response Summary
H(f)
Y(f)
- ' = , ' &(')
Response of LTI systems to Sinusoids
Response to sinusoid
"#$
H(f)
, '* "#$
,('*) = |, '* | "0(<)
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