Digital Signal Processing Fall 1992 Optimal FIR Filter Design Hossein Sameti Department of Computer Engineering Sharif University of Technology Definition of generalized linearphase GLP ID: 207216
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Slide1
CE 40763Digital Signal ProcessingFall 1992Optimal FIR Filter Design
Hossein SametiDepartment of Computer Engineering Sharif University of TechnologySlide2
Definition of generalized linear-phase (GLP):Let’s focus on Type I FIR filter:Optimal FIR filter design
2
It can be shown that
(L+1)
unknown parameters
a(n)
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide3
Problem statement for optimal FIR filter design3
Given
determine coefficients of
G(
ω
)
(i.e.
a(n)
)
such that
L
is
minimized (minimum length of the filter).
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide4
G(ω) is a continuous function of ω and is as many times differentiable as we want.How many local extrema (min/max) does G(ω) have in the range ?In order to answer the above question, we have to write cos(ωn) as a sum of powers of cos(
ω).Observations on G(ω)
4
:
sum
of powers of
cos
(
ω
)
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide5
5Observations on G(ω
)
Find extrema
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide6
6
Observations on G(ω)
Polynomial of degree
L-1
Maximum of
L-1
real
zeros
Max. total number of real zeros:
L+1
Conclusion
: The maximum number of real zeros for
(derivative of the frequency
response of type I FIR filter) is
L+1
,
where
(
N
is the number of taps).
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide7
Problem Statement for optimal FIR filter design7
Given
determine coefficients of
G(
ω
)
(i.e.
a(n)
)
such that
L
is
minimized (minimum length of the filter).
Problem A
Problem B
Problem C
Problem A
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide8
Problem B8
Given
determine coefficients of
G(
ω
)
(i.e.
a(n)
) such that
is
minimized
.
Compute
Guess
L
Algorithm
B
Increase
L
by 1
Decrease
L
by 1
Yes
Stop!Slide9
9Problem C
Define
F
as a union of closed intervals in
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide10
10Problem C
where
W
is a positive weighting function
Desired frequency response
Find
a(n)
to minimize
(same assumption as Problem B)Slide11
We start by showing thatProblem C= Problem B?11
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide12
Problem C= Problem B?12
By definition:
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide13
Problem C= Problem B?13
By definition:
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide14
Problem C= Problem B?14
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide15
Problem C= Problem B?15
in Problem C
in Problem BSlide16
Conclusion:Problem C= Problem B?16
Find
a(n)
such that is
minimized
.
Problem B:
Find
a(n)
such that is
minimized
.
Problem C:
Problem B= Problem C
Problem A= Problem C
We now try to solve Problem C.
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide17
Assumptions:F: union of closed intervalsG(x) to be a polynomial of order L:D = Desired function that is continuous in F.W= positive function
Alternation Theorem
17Slide18
The necessary and sufficient conditions for G(x) to be unique Lth order polynomial that minimizes is that E(x) exhibits at least L+2 alternations, i.e., there are at least L+2 values of x such that
18
Alternation Theorem
for a polynomial of degree 4
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide19
Number of alternations in the optimal case19
Recall
G(
ω
)
can have at most
L+1
local extrema.
According to the alternation theorem
, G(
ω
)
should have at least L+2 alternations(local extrema) in F.
Contradiction!?Slide20
Number of alternations in the optimal case20
can also be alternation frequencies, although they are not local extrema.
G(
ω
)
can have at most
L+3
local extrema in
F
.
Ex: Polynomial of degree 7
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide21
Number of alternations in the optimal case21According to the alternation theorem, we have at least
L+2 alternations.According to our current argument, we have at most L+3 local extrema.Conclusion: we have either L+2 or L+3 alternations in F for the
optimal case.
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide22
22Example: polynomial of degree 7
Extra-ripple case
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide23
23Example: polynomial of degree 7
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide24
For Type I low-pass filters, alternations always occur atIf not, we potentially lose two alternations. Optimal Type I Lowpass Filters
24
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide25
Optimal Type I Lowpass Filters
25
Equi-ripple except possibly at
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide26
Summary of observations26
For
optimal type I low-pass
filters, alternations always occur at
If not, two alternations are lost and the filter is no longer optimal.
Filter will be equi-ripple except possibly at
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide27
Parks-McClellan Algorithm (solving Problem C)27
Given
determine coefficients of
G(
ω
)
(i.e.
a(n)
)
such that is
minimized
.
At alternation frequencies, we have:
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide28
28Parks-McClellan Algorithm
Equating Eq.1 and Eq.2
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide29
29Parks-McClellan AlgorithmSlide30
30Parks-McClellan Algorithm
L
+2 linear equations and
L
+2 unknowns
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide31
31Parks-McClellan Algorithm
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide32
Remez Exchange Algorithm32
It can be shown that if
's
are known, then can be derived using the following formulae:
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide33
Remez Exchange Algorithm33
is an
Lth
-order trigonometric polynomial.
We can interpolate a trigonometric polynomial through L+1 of the L+2 known values of
or G
.
Using Lagrange interpolation formulae we can find the frequency response as:
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide34
Now is available at any desired frequency, without the need to solve the set of equations for the coefficients of . If for all in the passband and
stopband, then the optimum approximation has been found. Otherwise, we must find a new set of extremal frequencies. Remez Exchange Algorithm
34
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide35
Flowchart of P&M Algorithm
35Slide36
Example of type I LP filter before the optimum is found36
Original alternation frequency
Next alternation frequency
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide37
App. estimate of L:App. Length of Kaiser filter:Comparison with the Kaiser window
37
Example:
Optimal filter:
Kaiser filter:
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide38
Demonstration
38
Does it meet the specs?
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology Slide39
Demonstration39
Increase
the
length of the filter by 1.
Does it meet the specs?
Hossein
Sameti, Dept. of Computer Eng., Sharif University of Technology