variable fractional delay fir filters with sparse coefficients
TRANSCRIPT
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Variable Fractional Delay FIR Filters
with Sparse Coefficients
W.-S. Lu T. Hinamoto
Dept. of Electrical & Computer Engineering Graduate School of Engineering
University of Victoria Hiroshima University Victoria, Canada Hiroshima, Japan
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Outline
• Design Problem and Related Work
• Significance
• Design Method at a Glance
• A Design Example
• Concluding Remarks
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1. Design Problem and Related Work
Design Problem
Find a VFD FIR digital filter of order (N, K)
0 0
( , ) ( ) , ( )N K
n kn n nk
n k
H z p a p z a p a p−
= =
= =∑ ∑
with 0 1p≤ ≤ and sparse A = {aij} ( 1) ( 1)N KR + × +∈ that optimally
approximates a desired frequency response ( )( , ) j D p
dH p e ωω − +=
in a weighted least-squares sense. 1
212
0 0
( ) ( , ) ( , ) ( , )
subject to: sparsity( )
d
z
J W p H p H p dpd
N
π
ω ω ω ω= −
=
∫ ∫A
A
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Related Work
· A. Tarczynski, G. D. Cain, E. Hermanowicz and M. Rojewki, 1997 · W.-S. Lu and T.-B. Deng, 1999. · T.-B. Deng, 2001. · C.-C. Tseng, 2004. · T.-B. Deng and Y. Lian, 2006.
· Linear programming algorithms for sparse FIR filters
(T. Baran, D. Wei, and A.V. Oppenheim, March 2010) · l1-minimization algorithms for sparse 1-D and 2-D
FIR filters (W.-S. Lu and T. Hinamoto, 2010, 2011)
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2. Significance
· Implementing a VFD filter in Farrow model is costly as
each coefficient of a VFD filter is a polynomial rather than
a scalar. Hence VFD digital filters with sparse coefficients
are of interest because the sparsity implies reduced
implementation complexity (cost), hence real-time
application potential.
· VFD filters are not sparse in general.
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3.Design Method at a Glance
Define
1 and 1T Tj jN Ke e p pω ω− −⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦pω
and assume a separable weighting function 1 2( , ) ( ) ( )W p W W pω ω= ,
then up to a constant one can write 1
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0 0
12
( ) ( , ) ( , ) ( , )
tr( ) tr( )
d
T
J W p H p H p dpdπ
ω ω ω ω= −
= −
∫ ∫A
PA A SAΩ
where 1
2 10 0
1 ( )2 10 0
( ) , Re ( )
( ) , Re ( )
T T
T T T j D pp p
W p dp W d
W p dp W e d
π
π ω
ω ω
ω ω+
⎡ ⎤= = ⎢ ⎥⎣ ⎦⎡ ⎤= = ⎢ ⎥⎣ ⎦
∫ ∫
∫ ∫
P pp
S p
ωω
ω ω ω
Ω
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To deal with large condition numbers of P and Ω, Cholesky
decompositions 1 1 1 1,T T= =P P P Ω Ω Ω are used, and up to a
constant J(A) can be written as 21
1 12 2( ) , , TJ −= − = ⊗ =a a y P y sΓ Γ Ω Γ
where a and s are the vectors generated by concatenating the columns of A and S, and ⊗ is the Kronecker product. Design Phase 1
· To identify an index set of the most appropriate locations
in a to be set to zero in order to satisfy a target sparsity.
· This is done subject to maintaining closeness of ( , )jH e pω
to ( , )dH pω .
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· The target sparsity is achieved by introducing sparsity
promoting l1-norm of a into an objective function: 21
21 2minimize μ + −
aa a yΓ (1)
where 1
0
n
ii
a=
= ∑a . Problem (1) is a convex, for which many
fast algorithms are available. E.g. using FISTA, (1) can be
solved with a small number of iterations.
Input: Data Γ and y, parameter μ and iteration number M. Step 1. Compute A0 = Γ−1y, a0 = A0(:). Set b1 = a0, t1 = 1, and m = 1. Step 2. Compute am = Sμ/L{bm − (1 /L)ΓT (Γbm − y)},
where ( ) sgn( ) max{| | ,0}S u u uα α= ⋅ − Step 3. Update tm+1 = 2(1 1 4 ) / 2mt+ + Step 4. Update bm+1 = am + 1(( 1) / )m mt t +− (am − am−1). Step 5. If m < M, set m = m + 1 and repeat from Step 2; otherwise stop and
output am as solution a.
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·Hard thresholding is applied to vector a with an appropriate value of threshold ε ∗ so that the length of the index set ˆ{ , ( ) }I i a i ε∗ ∗= < equals to target sparsity Nz. The index set I ∗ is a key ingredient of design phase 2.
Design Phase 2
· To find a coefficient matrix A that minimizes the WLS error
J(A) subject to the sparsity constraint. This part of the design
is carried out by solving the convex problem 21
2 2minimize ( )
subject to: ( ) 0 for
J
a i i I ∗= −
= ∈a
a a yΓ
· By simply substituting the constraints into the objective
function, the above problem becomes an unconstrained least
square problem whose solution is given by
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1( ) and ( )sI I∗ ∗ − ∗ ∗= = 0a y aΓ
where I ∗ denotes the set of index not in I ∗ and sΓ is
composed of those columns of Γ with indices in set I ∗.
4. A Design Example
The Design Problem
· Design a VFD FIR filter of order (N = 65 and K = 7) with cutoff 0.9cω π= .
· Design performance is evaluated in terms of maximum error
{ }max max ( , ),0 0.9 , 0 1e e p pω ω π= ≤ ≤ ≤ ≤ with
10( , ) 20log ( , ) ( , )de p H p H pω ω ω= − and L2-error
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1/ 20.9 12
20 0
( , ) ( , )de H p H p dpdπ
ω ω ω⎡ ⎤
= −⎢ ⎥⎣ ⎦∫ ∫
· The weighting function was set to 1 2( , ) ( ) ( )W p W W pω ω= with
W2(p) = 1 for p in [0, 1] and
1
1 for [0, 0.88 )( ) 3 for [0.88 , 0.8994 )
0 for [0.88 , ]W
ω πω ω π π
ω π π
∈⎧⎪= ∈⎨⎪ ∈⎩
· The target sparsity was set to Nz = 198 which means a
37.5% of coefficients were set to zero. To achieve this
sparsity, the two key parameters in phase-1 were set to 5 310 , 10μ ε− ∗ −= = . It took 60 FISTA iterations for the
algorithm in phase 1 to converge.
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· Phase 2 of the design then produced an optimal A*
with sparsity(A*) = 198. Below are the numerical
evaluation results:
· maximum error emax = 0.0021.
· L2-error e2 = –75.25 dB.
======== Comparison 1 =======
· Compare with an equivalent nonsparse VFD filter of
order (N = 65, K = 4) with the same specifications:
· maximum error emax = 0.0609.
· L2-error e2 = –45.28 dB.
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· Profile of frequency response error
| ( , ) | over 0 0.9 , 0 1e p pω ω π≤ ≤ ≤ ≤ : the sparse VFD filter:
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
1−200
−150
−100
−50
0
50
Normalized frequency Fractional delay p
|e(
,p)|
(dB
)
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· Profile of frequency response error
| ( , ) | over 0 0.9 , 0 1e p pω ω π≤ ≤ ≤ ≤ : the nonsparse VFD filter:
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
1−200
−150
−100
−50
0
50
Normalized frequency Fractional delay p
|e(
,p)|
(dB
)
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======== Comparison 2 =======
· To justify phase 1 of the design, we compare the above
result with the following: We design a conversional
(nonsparse) VFD filter of order (N = 65, K = 7), then
applied hard thresholding to generate exactly 198
locations that may be considered appropriate to set to
zero. We then went on the carried out phase 2 to yield
a sparse VFD filter. It was found that
· maximum error emax = 0.0025 (vs 0.0021 with phase 1)
· L2-error e2 = –73.41 dB (vs –75.25 dB with phase 1).
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· Profile of frequency response error
| ( , ) | over 0 0.9 , 0 1e p pω ω π≤ ≤ ≤ ≤ : the sparse VFD filter
without phase 1:
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
1−200
−150
−100
−50
0
50
Normalized frequency Fractional delay p
|e(
,p)|
(dB
)
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======== Comparison 3 =======
· Fractional delay over 0 0.9 , 0 1pω π≤ ≤ ≤ ≤ . The sparse filter:
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
Normalized frequency Fractional delay p
Fra
ctio
nal d
elay
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· Fractional delay over 0 0.9 , 0 1pω π≤ ≤ ≤ ≤ . The equivalent
nonsparse filter:
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
10
0.2
0.4
0.6
0.8
1
Normalized frequency Fractional delay p
Fra
ctio
nal d
elay
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5. Concluding Remarks
· A two-phase technique for the WLS design of VFD FIR
filters subject to a target coefficient sparsity constraint
has been proposed.
· The design algorithm is easy to implement abd
computationally efficient because it is based on l1 – l2
convex optimization.
·The performance of the filter appears to be satisfactory
compared with its nonsparse counterpart.