BIEN425 - PowerPoint PPT Presentation

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BIEN425

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Design linear phase FIR filters Compute the amplitude response of linear phase FIR filters ... Represent FIR and IIR filters in direct, parallel and cascade structures – PowerPoint PPT presentation

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Title: BIEN425


1
BIEN425 Lecture 9
  • By the end of the lecture, you should be able to
  • Describe the properties of ideal filters
  • Describe the linear / logarithm design
    specifications for non-ideal filters
  • Design linear phase FIR filters
  • Compute the amplitude response of linear phase
    FIR filters
  • Decompose any filters with rational transfer
    function into product of minimum-phase and
    allpass filters
  • Represent FIR and IIR filters in direct, parallel
    and cascade structures

2
Ideal filters
  • Selectively scale the frequency contents of a
    signal

A(f)
Passband
1
Stopband
f
0
fs/2
Fp cut off frequency
3
Non-ideal filters
Passband
Stopband
Transition
4
Linear design example
Consider
(There is an error in the text book)
Zeros at z -1 Poles at z c
To make sure that the filter is stable, we choose
c lt 1
Now assume c 0.5, and lets evalulate the filter
at 2 ends f 0 f fs/2
So what type of filter is it?
5
Lets find the frequency response H(f)
What is the magnitude response A(f)
6
1-dp
ds
Fp
Fs
7
Logarithm design
  • Represent the responses in dB scale

Because we want to zoom-in on the dynamics of the
transitions and stopbands
8
Ap
As
Fp
Fs
9
Linear phase filters
  • Remember a signal y(k) can be expressed by the
    following equation where A(f) and f(f) are the
    amplitude and phase characteristics of a filter
    respectively.

From equation 2.93
10
  • To preserve the shape integrity of a given input
    signal, the delay term will have to be
    independent of frequency
  • In this case, the phase characteristics is as
    follows

Delay
This is called linear phase
11
Nonlinear phase filters
  • The group delay D(f) is defined by
  • For nonlinear phase filters, D(f) will be a
    function of frequency (f) and distortion is
    unavoidable.

In this case, F(f) can be a-2ptf
12
Linear-phase filter using FIR
  • There is a simple symmetry condition on the
    coefficients that guarantees a linear phase
    filter.

13
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14
Minimum phase filter
  • Every digital filter with rational transfer
    function can be expressed as a product of two
    specialized filters
  • Minimum-phase filter
  • Allpass filter
  • Magnitude response by itself does not provide
    enough information to completely specify a
    filter.
  • For an IIR filter with m-zeros, there are 2m
    number of distinct filters having the same
    magnitude response.

15
  • Define square of magnitude response A2(f)
  • Therefore, we can take zero at the reciprocal of
    b(z)

16
Example
17
  • Minimum phase filter if and only if all of its
    zeros lie inside or on the unit cycle.

18
Allpass filter
  • Amplitude response A(f)1 for all f up to fs/2
  • Provides phase compensation

19
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20
Direct form
Direct II requires half the storage of Direct I
21
Parallel form
22
Cascade form
  • Cascade form is less sensitive to coefficient
    quantization.
  • Cascade form needs less number of fan-outs
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