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EECE 301
Signals & Systems
Prof. Mark Fowler
Note Set #20
C-T Systems: Ideal Filters - Frequency-Domain Analysis Reading Assignment: Section 5.3 of Kamen and Heck
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Ch. 1 Intro
C-T Signal Model
Functions on Real Line
D-T Signal Model
Functions on Integers
System Properties
LTICausal
Etc
Ch. 2 Diff EqsC-T System Model
Differential Equations
D-T Signal ModelDifference Equations
Zero-State Response
Zero-Input Response
Characteristic Eq.
Ch. 2 Convolution
C-T System Model
Convolution Integral
D-T Signal Model
Convolution Sum
Ch. 3: CT FourierSignalModels
Fourier Series
Periodic Signals
Fourier Transform (CTFT)
Non-Periodic Signals
New System Model
New Signal
Models
Ch. 5: CT FourierSystem Models
Frequency Response
Based on Fourier Transform
New System Model
Ch. 4: DT Fourier
SignalModels
DTFT
(for Hand Analysis)DFT & FFT
(for Computer Analysis)
New Signal
Model
Powerful
Analysis Tool
Ch. 6 & 8: LaplaceModels for CT
Signals & Systems
Transfer Function
New System Model
Ch. 7: Z Trans.
Models for DT
Signals & Systems
Transfer Function
New System
Model
Ch. 5: DT Fourier
System Models
Freq. Response for DT
Based on DTFT
New System Model
Course Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between
the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).
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5.3 Ideal FiltersOften we have a scenario where we have a good signal, xg(t), corrupted by a
bad signal, xb(t), and we want to use an LTI system to remove (or filter out) thebad signal, leaving only the good signal.
How do we do this? What H() do we want?
h(t)
H()
)()()( txtxtx bg += )()( txty g=
Called a Filter
Desired Output
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)(gX
)(bX
Case #1: is a low-frequency signal)(txg is a high-frequency signal)(txb
)(X
Spectrum of the
Input Signal
In this case, we want
a filter like this:
)(H
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H())(X )()()( HXY =
Then:
)()()()()( bg XHXHY +=
)(gX= 0=)(gX= as desired
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Case #2: is a high-frequency signal
is a low-frequency signal
)(gX
)(bX
We then want:
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)(bX
)(gX
)(H
1
Case #4:
Note that in Cases #3 and #4 the filter cant remove the bad signal without
causing some damage to the desired signal
this is not specific to bandpass and bandstop filters
it can also happen with low-pass and high-pass filters.
In practice this is almost always the case!!
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What about the phase of the filters H()?
Wellwe could tolerate a small delay in the output so
From the time-shift property of the FT then we need:
dtjg eXY
= )()(
h(t)
H()
)(txg )()( dg ttxty =Put in the signal
we want passed
Want to get out the
signal we want
passed but we canaccept a small delay
d
tj
tj
teH
eH
d
d
==
==
)(
1)( For in thepass band
of the filter
Thus we should treat the exponential term here as H(), so we have:
Line of slope tdLinear Phase
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So for an ideal low-pass filter (LPF) we have:
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Example of the effect of a nonlinear phase but an ideal magnitude
Here is the scenario: Imagine we have a a signalx(t) given by
)32cos()22cos(3)2cos(59)( ttttx =
0 H 1 H 2 H 3 H
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)32cos()22cos(3)2cos(59)( ttttx =
H())(tx ?)( =ty
0 Hz 1 Hz 2 Hz 3 Hz
Circles Show
the Frequencies
in Input Signal
Filter does NOTchange amplitudes of
input components
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So, at the filters output we have four sinusoids at the same frequencies and
amplitudes as at the inputBUT, they are not aligned in time in the same way
they were at the input
)10
32cos()22cos(3)4
2cos(59)(
= tttty
Point of this Example
A filter with an ideal magnitude
response but non-ideal phase
response can degrade a signal as
much as a filter with a non-idealmagnitude response!!!