ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU Applications of Distributed Arithmetic to Digital Signal Processing: A Tutorial Review

ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU

Applications of Distributed Arithmetic to Digital Signal Processing:

A Tutorial Review

VLSI Signal Processing台灣大學電機系吳安宇

Ref: Stanley A. White, “Applications of Distributed Arithmetic to Digital Signal Processing: A Tutorial

Review,” IEEE ASSP Magazine, July, 1989


Outline

Distributed Arithmetic (DA) Introduction

Technical overview of DA

Increasing the speed of DA multiplication

Application of DA to a biquadratic digital filter

Conclusions


Distributed Arithmetic (DA, 1974) Introduction

The most-often encountered form of computation in DSP:

Sum of product

Dot-product

Inner-product

Executed most efficiently by DA

By careful design one may reduce gate count in a signal processing arithmetic unit by a number seldom smaller then 50% and often as large as 80%


Technical overview of DA

Advantage of DA: efficiency of mechanization

A frequently argued:

Slowness because of its inherent bit-serial nature (not true)

Some modifications to increase the speed by employing techniques:

Plus more arithmetic operations

Expense of exponentially increased memory


Technique overview -continued I

The sum of product:

where xk is a 2’s-complement binary number scaled such that | xk |<1 Ak is fixed coefficients

xk : {bk0, bk1, bk2……, bk(N-1) } , word length=N

where bk0 is the sign bit

Express each xk as:

(2) 代入 (1) =>

K

kkk xAy

1(1)

1

10 2

N

n

nknkk bbx (2)

K

k

N

n

nknkk bbAy

1

1

10 2 (3)


Technique overview -continued II

Critical step

where K=No. of taps (inputs), N: Wordlength of Data

K

k

N

n

nknkk bbAy

1

1

10 2 (3)

1

1 10

1

)(2N

n

K

kkk

nK

kknk bAbAy (4)


Technique overview -continued III

Consider the equation (4)

has only 2K possible values

has only 2K possible values

We can store it in a lookup-table(ROM): size=2*2K

1

1 10

1

)(2N

n

K

kkk

nK

kknk bAbAy

K

kknkbA

1

K

kkk bA

10 )(


Technique overview -continued IV

ExampleLet no. of taps K=4

and the fixed coefficients are A1=0.72, A2=-0.3, A3=0.95, A4=0.11

We need 22K = 32-word ROM (k=4)

1

1 10

1

)(2N

n

K

kkk

nK

kknk bAbAy


ExampleUnfolding

nnnnk

knk bAbAbAbAbA 44332211

4

1

)()()()(

)(

0,440,330,220,11

4

10

bAbAbAbA

bAk

kk


Example -continued I

Hardware architecture


Example -continued II

Shorten the table

eq. (4)

nnnnk

knk bAbAbAbAbA 44332211

4

1

4

10

4

10 )()(

kkk

kkk bAbA

1

1 10

1

)(2N

n

K

kkk

nK

kknk bAbAy (5)



Only 16 words of ROM are required,now.

Example -continued II


Technique overview -advanced

Input

1

1

)1(0 22

N

n

Nnknkk bbx

1

10 2

N

n

nknkk bbx

2‘s-complement

1

1

)1(00 22

2

1 N

n

Nnknknkkk bbbbx (7)

(6)},......,{)]([2

1)1(10 nkkkkkk bbbxxx


Technique overview -advanced I

1

1

)1(00 22

2

1 N

n

Nnknknkkk bbbbx

1

0

)1(222

1 N

n

Nnknk cx

K

kkk xAy

1代入

1

0

)1(

1

)1(1

0

022222

1 N

n

Nnn

K

k

Nnkn

N

nk QbQcAy

Where and

K

kkn

kn c

AbQ

1 2)(

}1,1{0,

0,

)(

knknkn

knknkn cwhere

n

n

bb

bbc

K

k

kAQ

1 2)0(


Technique overview -advanced II



Increase the speed of DA multiplication

The way I:Plus more arithmetic operations

1

0

)1( 022N

n

Nnn QbQy Initial

condition

1

0

)1(1

)2(2

11

00 22....222

N

n

NN

NN

nn bQbQbQbQbQ

Even part Odd part


InitialCondition1/2*Q(0)

1

0

)1( 022N

n

Nnn QbQy

Odd part(sign}

Even part

Increase the speed of DA multiplicationHardware architecture of way I -at the expense of linearly increased memory & arithmetic operation


Increase the speed of DA multiplicationThe way II: - at the expense of exponentially increased memory

ROM : 2*7 words => 1*128 words




Transfer function of a typical biquadratic digital filter:

The time-domain description:

22

11

22

110

1)(

)(

zBzB

zAzAA

zX

zY

Tnnnnn yyxxxBBAAAzY ]][[)( 212121210






The vector matrix equation

The relationships between two equations

1

1

1

210

222

111

n

n

n

n

n

n

v

u

x

dda

bca

cba

y

v

u

2121

211

2111212221212102

21022111

00

)()()(

)(

ccbbB

bbB

dbdcadbdcaccbbaA

bbadadaA

aA



Normal form biquadratic filter

b1

z-1

c1

a1

d1

z-1

d2

a0

c2

z-1

b2

a2

xn yn

vn

un



DA realization



Increase speed

1*3 BAAT DA structure 4*3 BAAT DA structure



Increase speed II2048 word/ROM

require 4 ROM

4 operations



Increase speed III32 word/ROM

require 8 ROM

8 operations


ConclusionsDA is a very efficient means to mechanize computations that are dominated by inner products

If performance/cost ratio is critical, DA should be seriously considered as a contender

When a many computing methods are compared, DA should be considered. It is not always (but often) best, and never poorly.

Documents

ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU Applications of Distributed Arithmetic to Digital Signal Processing: A Tutorial Review