1

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

2

Outline

• Design Problem and Related Work

• Significance

• Design Method at a Glance

• A Design Example

• Concluding Remarks

3

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

4

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)

5

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.

6

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

212

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

7

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 .

8

· 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.

9

·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

10

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

11

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.

12

· 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.

13

· 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

)

14

· 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

)

15

======== 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).

16

· 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

)

17

======== 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

18

· 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

19

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.