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2
Outline
- Measures for Performance Robustness
- Design Formulations
- Properties of Error Functions
- Design of Robust Minimax FIR Filters
- An Example
Robust Digital Filters Part 1: Minimax FIR Filters Wu-Sheng Lu - - PowerPoint PPT Presentation
Robust Digital Filters Part 1: Minimax FIR Filters Wu-Sheng Lu - - PowerPoint PPT Presentation
Robust Digital Filters Part 1: Minimax FIR Filters Wu-Sheng Lu Takao Hinamoto University of Victoria Hiroshima University Victoria, Canada Higashi-Hiroshima, Japan May 2019 1 Outline Measures for Performance Robustness Design
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- 1. Measures for Performance Robustness
- Motivation
Typical error measures
2
L :
1/2 2
( ) | ( , ) ( ) |
d
W H H d
x
L: max
( ) | ( , ) ( ) |
d
W H H
x
Let x∗ be a minimizer under a certain measure. The optimal performance of the filter, i.e.
( , ) H
x
, would be achieved only if its implementation is perfectly accurate. In practice, however, neither hardware nor software utilized in a practical implementation are of infinite precision, thus only an approximate version of
( , ) H
x is actually realized. This approximation may be modeled as a frequency
response
( , ) H
x
for some variation due to various reasons ranging from
power-of-two constraints on filter coefficients to rounding errors in multiplications using fixed-point arithmetic. Since the perturbed design represented by
x is no longer a minimizer, the
performance degradation at
x is inevitable even for small .
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- We seek to develop design methods for digital filters that achieve performance
- ptimality subject to variations of filter coefficients.
To this end we introduce new
2
L and L error measures for filters with robust
performance against coefficient variations:
1/2 2 2( )
max ( ) | ( , ) ( ) |
r
d B
e W H H d
x x
and
( ) max max ( ) | ( , ) ( ) |
r
d B
e W H H
x x
where Br is a region for parameter variation . Choices of Br include · Bounding box
{ :| | , 0,1,..., }
r i i
B r i K
· Ball
2
{ :|| || }
r
B r
· Ellipsoid
2
{ :|| || 1}
r
B D
with 1
diag{ , ,..., }
K
d d d D
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- 2. Design Formulations
- Given filter’s order and type (FIR or IIR) and a region of permitted parameter
variations Br, robust lease-square and minimax designs are obtained by solving
2
minimize ( ) e x
and
minimize ( ) e x
- Constraints on stability of H(z) need to be imposed for IIR designs.
- This paper addresses linear-phase FIR filters only, and we consider FIR filters of
- dd length N with transfer function and frequency response
1
( ) , ( , ) ( )
N i jK T i i
H z h z H e
x c x with K = (N − 1)/2.
- Let
( ) ( )
jK d d
H e A
be the desired frequency response. The two error functions become
1/2 2 2( )
max ( ) | ( ) ( ) ( ) |
r
T d B
e W A d
x c x ( ) max max ( ) | ( ) ( ) ( ) |
r
T d B
e W A
x c x
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- 3. Properties of Error Functions
- Property 1: Functions
2( )
e x and ( ) e x are convex. Hence the two design
problems can be addressed as convex optimization problems.
- Property 2: The sub-differential of
( ) e x
with respect to x is given by [ ( ) ( ) ( )]
( ) ( ) ( ) ( )( | |)
T d
y A
e W y
c x
g x x c
(1a)
where
,
( , ) arg (max max ( ) | ( ) ( ) ( ) |)
r
T d B
W A
c x
, and
1 if | | 1 if [ 1,1] if y y y y
(1b)
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- 4. Design of Robust Minimax FIR Filters
This paper is focused on the design of robust minimax linear-phase FIR filters:
minimize max max ( ) | ( ) ( ) ( ) |
r
T d B
W A
x
c x
- Available options include a variation of the gradient descent (GD) method,
known as the heavy ball method (Polyak, 1987), and GD with momentum. The heavy ball algorithm updates iterate
k
x to
1 1
( )
k k k k k k k
x x g x x
(2a) which starts with k = 0 with point
1
x
set to
x . We see the last term of the above
update uses past iterates to provide momentum that pushes the current iterate like a heavy ball to move down hill faster. The step size
k
is calculated using
( ) best 2 2
( ) || ||
k k k k k
e e
x g
(2b) where
( ) best 1,...,
min ( )
k i i k
e e
x keeps track of the best performance achieved so far,
k
> 0
is a sequence satisfying
k k
and 2 k k
, and
k
satisfies
k k
and
2 k k
. Here
and
k k
were simply set to be proportional to 1/(k + 1).
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- Computing Sub-gradient
k
g
A major step of the algorithm is to calculate sub-gradient
k
g which is given by
[ ( ) ( ) ( )]
( ) ( ) ( ) ( )( | |)
T d
y A
e W y
c x
g x x c
where
,
( , ) arg(max max ( ) | ( ) ( ) ( ) |)
r
T d B
W A
c x
is not trivial to compute.
- Formulas for calculating (
, )
were derived in Sec. 3B of the paper for the case of Br being a bounding box:
max ( ) | cos( ) | ( ) ( ) |
d
K T i d i
W r i A
c x
(3a)
sgn( ( ) ( ) ){sgn( ( ))}
T d
A
c x c r
(3b) where
1 T K
r r r r
collects permitted upper bounds for individual design variables. ◊ Note that (3a) is a 1-D maximization problem and hence straightforward to perform.
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Algorithm for Robust Minimax FIR Filters inputs: desired amplitude response
( )
d
A , filter length N, weight ( ) W , frequency grids
d
, initial design x0, and number of iterations Nt.
for k = 0, 1, . . . , Nt, Step 1: use (3a) and (3b) to compute and
; use (1) to compute
k
g .
Step 2: use (2a) and (2b) to compute
1 k
x
. end
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- 5. An Example
We consider designing a robust minimax low-pass FIR filter of length N = 21 with normalized pass-band edge
p
= 0.4 and stop-band edge
a
= 0.5 . Assume ( ) 1 W
and a bounding box
{ :| | 0.005, 0,1,...,10}
r i
B i
The set
d
consists of 100 frequency grids that are evenly placed over the union of
the pass-band and stop-band [0, 0.4 ]
[0.6 , ]
. We use a least-squares low-pass filter with the same passband and stopband edges as the initial point of the proposed algorithm. The parameters
k
and
k
were set to 0.16 1
k
k
and
0.08 1
k
k
The algorithm was able to reduce the object function from 0.2837 to 0.0906 in 100 iterations, see below. Fig. 2 depicts the amplitude response of the robust filter
- btained after 104 iterations, at which the objective function was reduced to
( ) 0.0898 e
x
.
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10 20 30 40 50 60 70 80 90 100
iteration
0.05 0.1 0.15 0.2 0.25 0.3
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0.5 1 1.5 2 2.5 3
normalized frequency
- 40
- 35
- 30
- 25
- 20
- 15
- 10
- 5
5
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For comparison, a conventional linear-phase minimax FIR filter of length 21 was designed using the Parks-McClellan (PM) algorithm. Let
(PM)
x
be its parameter vector, the robustness measure of the PM filter was found to be
(PM)
( ) 0.1099 e x
. This is to say, the approximation error of an FIR filter whose coefficients vary from the PM filter
(PM)
x
within the bounding box Br is bounded by 0.1099, while the approximation error of an FIR filter whose coefficients vary from the robust minimax filter
x within the bounding box Br is guaranteed not exceeding 0.0898,
representing a 18.23% reduction. The first 11 coefficients of the robust minimax and PM filters are given in the Table below.
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TABLE I First 11 Coefficients Robust Minimax PM filter 0.037739257611475 0.038776212428159 0.003955891303322 0.002476747389280 −0.030947825075270 −0.030327895328723 −0.017617078016662 −0.018181020733260 0.034656023078736 0.035632966126753 0.040944804608671 0.039290923363811 −0.045087860127533 −0.045102726246706 −0.091116616564451 −0.092430917385436 0.046215874348786 0.047087467600674 0.313562923929549 0.311837559676779 0.449135853464320 0.448729827357856
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