Applications of Distributed Arithmetic to Digital Signal Processing: A Tutorial Review

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Applications of Distributed

Arithmetic to Digital Signal

Processing:

A Tutorial Review

Ref: Stanley A. White, “Applications of

Distributed Arithmetic to Digital Signal

Processing: A Tutorial Review,” IEEE ASSP Magazine, July, 1989

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Outline:

• Distributed Arithmetic (DA) Introduction • Technical overview of DA

• Increasing the speed of DA multiplication

• Application of DA to a biquadratic digital filter

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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

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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

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Technique overview

-continued I-continued I

• The sum of product:

– where xk is a 2’s-complement binary number scale

d 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 k k kx A y 1 (1)



      1 1 0 2 N n n kn k k b b x ₍₂₎         _ _  K N n kn k k b b A y ₀ 1 2 (3)

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Technique overview

-continued II-continued II

• Critical step

– where K=No. of taps (inputs), N: Wordlength of Data            _ _  K k N n n kn k k b b A y 1 1 1 0 2 (3)                 1 1 1 0 1 ) ( 2 N n K k k k n K k kn kb A b A y ₍₄₎

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Technique overview

-continued III-continued III

• Consider the equation (4)

– has only 2K possible values

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

                1 1 1 0 1 ) ( 2 N n K k k k n K k kn kb A b A y       K k kn kb A 1    K k k k b A 1 0) (

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Technique overview

-continued IV-continued IV

• Example

– Let no. of taps K=4

– and the fixed coefficients are A₁=0.72, A₂=-0.3,

A3=0.95, A4=0.11

– We need 22K = 32-word ROM (k=4)

                1 1 1 0 1 ) ( 2 N n K k k k n K k kn kb A b A y

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Example

• Unfolding n n n n k kn kb Ab A b A b A b A ₁ ₁ ₂ ₂ ₃ ₃ ₄ ₄ 4 1             ) ( ) ( ) ( ) ( ) ( 0 , 4 4 0 , 3 3 0 , 2 2 0 , 1 1 4 1 0 b A b A b A b A b A k k k           

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Example

-continued I-continued I

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Example

• Shorten the table

– eq. (4) n n n n k kn kb Ab A b A b A b A ₁ ₁ ₂ ₂ ₃ ₃ ₄ ₄ 4 1                          4 1 0 4 1 0) ( ) ( k k k k k k b A b A                1 1 1 0 1 ) ( 2 N n K k k k n K k kn kb A b A y ₍₅₎

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Example

Hardware architecture

Only 16 words of ROM are required,now.

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Technique overview

-advanced-advanced

• Input      _     1 1 ) 1 ( 0 2 2 N n N n kn k k b b x



      1 1 0 2 N n n kn k k b b x 2‘s-complement









_  _ _ _ _ _  1  ( 1) 0 0 2 2 1 N N n kn kn k k k b b b b x ₍₇₎ (6) } ,... , { )] ( [ 2 1 ) 1 ( 1 0      _k _k _k _k _k _n k x x b b b x

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Technique overview

-advanced I-advanced I









_     _ _ _ _ _       1 1 ) 1 ( 0 0 2 2 2 1 N n N n kn kn k k k b b b b x     _       1 0 ) 1 ( 2 2 2 1 N n N n kn k c x    K k k kx A y 1 代入                           _  1 0 ) 1 ( 1 ) 1 ( 1 0 0 2 2 2 2 2 1 N n N n n K k N n kn N n k c Q b Q A y Where   and K k _c A b Q( )  { 1,1} 0 , 0 , ) (         _kn kn kn kn kn kn where c n n b b b b c    K Ak Q(0)

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Technique overview

-advanced II-advanced II

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multiplication

• The way I:

– Plus more arithmetic operations

         _  1 0 ) 1 ( ₀ 2 2 N n N n n Q b Q y _Initial condition                      _ _ _ _ _ 1 0 ) 1 ( 1 ) 2 ( 2 1 1 0 0 2 2 .... 2 2 2 N n N N N N n n Q b Q b Q b Q b b Q

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multiplication

• Hardware architecture of way I

-at the expense of linearly

increased memory & arithmetic operation

Initial Condition 1/2*Q(0)       _    N 1_Q _b ₂ n ₂ (N 1)_Q ₀ y Odd part(sign} Even part

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multiplication

• The way II:

- at the expense of exponentially

increased memory

• ROM : 2*7 words => 1*128 words

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digital filter

• Transfer function of a typical biquadratic digital filter:

• The time-domain description:

2 2 1 1 2 2 1 1 0 1 ) ( ) (          z B z B z A z A A z X z Y T n n n n nx x y y x B B A A A z Y( )  [ ₀ ₁ ₂  ₁  ₂][ _₁ _₂ _₁ _₂]

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digital filter

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digital filter

• The vector matrix equation

• The relationships between two equations                                   1 1 1 2 1 0 2 2 2 1 1 1 n n n n n n v u x d d a b c a c b a y v u 2 1 1 2 1 1 1 2 1 2 2 2 1 2 1 2 1 0 2 2 1 0 2 2 1 1 1 0 0 ) ( ) ( ) ( ) ( b b B d b d c a d b d c a c c b b a A b b a d a d a A a A             

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digital filter

• Normal form biquadratic filter

b 1 z- 1 c 1 a1  d 1 z- 1 d 2 a 0  c 2 z- 1 a2  x_n y_n v_n u_n

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digital filter

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digital filter

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digital filter

• Increase speed II – 2048 word/ROM – require 4 ROM – 4 operations

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digital filter

• Increase speed III

– 32 word/ROM – require 8 ROM – 8 operations

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Conclusions

• DA 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.