Ideal Filters -Frequency-Domain Analysis

7/30/2019 Ideal Filters -Frequency-Domain Analysis

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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!!!

Ideal Filters -Frequency-Domain Analysis

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