Title: BIEN425
1BIEN425 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
2Ideal filters
- Selectively scale the frequency contents of a
signal
A(f)
Passband
1
Stopband
f
0
fs/2
Fp cut off frequency
3Non-ideal filters
Passband
Stopband
Transition
4Linear 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?
5Lets find the frequency response H(f)
What is the magnitude response A(f)
61-dp
ds
Fp
Fs
7Logarithm design
- Represent the responses in dB scale
Because we want to zoom-in on the dynamics of the
transitions and stopbands
8Ap
As
Fp
Fs
9Linear 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
11Nonlinear 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.
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14Minimum 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)
16Example
17- Minimum phase filter if and only if all of its
zeros lie inside or on the unit cycle.
18Allpass filter
- Amplitude response A(f)1 for all f up to fs/2
- Provides phase compensation
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20Direct form
Direct II requires half the storage of Direct I
21Parallel form
22Cascade form
- Cascade form is less sensitive to coefficient
quantization. - Cascade form needs less number of fan-outs