1 finite wordlength

1 Professor A G Constantinides

Finite Wordlength EffectsFinite Wordlength Effects

• Finite register lengths and A/D converters cause errors in:-

(i) Input quantisation.

(ii) Coefficient (or multiplier) quantisation

(iii) Products of multiplication truncated or rounded due to machine length



• Quantisation

Q

Output

Input

2)(

2 ,Q

keQ

oi

)(kei

)(keo



• The pdf for e using rounding

• Noise poweror

2Q

2Q

Q1

2

2

222 }{).(Q

QeEdeepe

12

22 Q



• Let input signal be sinusoidal of unity amplitude. Then total signal power

• If b bits used for binary then

so that

• Hence

or dB

21P

bQ 22

32 22 b

bP 22 2.23

b68.1SNR



• Consider a simple example of finite precision on the coefficients a,b of second order system with poles

• where

211

1)(

bzazzH

je

221 ..cos21

1)(

zzzH

2b cos2a



•

bit pattern

000 0 0

001 0.125 0.354

010 0.25 0.5

011 0.375 0.611

100 0.5 0.707

101 0.625 0.791

110 0.75 0.866

111 0.875 0.935

1.0 1.0 1.0

2 ,cos2



• Finite wordlength computations

+

INPUT OUTPUT

+

+


Limit-cycles; "Effective Pole"Limit-cycles; "Effective Pole"Model; Deadband Model; Deadband

• Observe that for

• instability occurs when• i.e. poles are

• (i) either on unit circle when complex• (ii) or one real pole is outside unit

circle.• Instability under the "effective pole" model

is considered as follows

)1(1)( 2

21

1

zbzb

zH

12 b



• In the time domain with

•

• With for instability we have

indistinguishable from

• Where is quantisation

)()()( zXzYzH

)2()1()()( 21 nybnybnxny

12 b

)2(2 nybQ )2( ny

Q



• With rounding, therefore we have

are indistinguishable (for integers)

or

• Hence

• With both positive and negative numbers

5.0)2(2 nyb )2( ny

)2(5.0)2(2 nynyb

215.0

)2(b

ny

215.0

)2(b

ny


Finite Wordlength EffectsFinite Wordlength Effects• The range of integers

constitutes a set of integers that cannot be individually distinguished as separate or from the asymptotic system behaviour.

• The band of integers

is known as the "deadband".• In the second order system, under rounding, the

output assumes a cyclic set of values of the deadband. This is a limit-cycle.

215.0b

22 1

5.0 ,

15.0

bb



• Consider the transfer function

• if poles are complex then impulse response is given by

)1(1)( 2

21

1

zbzb

zG

2211 kkkk ybybxy

kh

)1(sin.

sin kh

k

k



• Where

• If then the response is sinusiodal with frequency

• Thus product quantisation causes instability implying an "effective “ .

2b

2

11

2cos

bb

12 b

2cos1 11 bT

12 b



• Consider infinite precision computations for

21 9.0 kkkk yyxy0 ; 0

100

kx

x

k

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10



• Now the same operation with integer precision

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10



• Notice that with infinite precision the response converges to the origin

• With finite precision the reponse does not converge to the origin but assumes cyclically a set of values –the Limit Cycle



• Assume , ….. are not correlated, random processes etc.

Hence total output noise power

• Where and

)(1 ke )(2 ke

0

2220 )(

kiei kh

12

22 Qe

02

22

22

022

012

0sin

)1(sin.

122

.2k

kb k

bQ 2

0 ;

sin)1(sin

.)()( 21 kk

khkh k



• ie

2cos21

1.

1

16

2242

222

0

b



• For FFTA(n)

B(n)

B(n+1)

B(n+1)-

A(n)

B(n+1)B(n)W(n)B(n)

A(n+1)

)().()()1(

)().()()1(

nBnWnAnB

nBnWnAnA

W(n)



• FFT

• AVERAGE GROWTH: 1/2 BIT/PASS

2)1()1( 22 nBnA

)(2)(

)(2)1( 22

nAnA

nAnA



• FFT

• PEAK GROWTH: 1.21.. BITS/PASS

IMAG

REAL1.0

1.0

-1.0

-1.0

....414.2)()(0.1)(

)1(

)()()()()()1(

)()()()()()1(

nSnCnA

nA

nSnBnCnBnAnA

nSnBnCnBnAnA

x

x

yxxx

yxxx



• Linear modelling of product quantisation

• Modelled as

x(n) )(~ nx

x(n)

q(n)

+ )()()(~ nqnxnx

Q



• For rounding operations q(n) is uniform distributed between , and where Q is the quantisation step (i.e. in a wordlength of bits with sign magnitude representation or mod 2, ).

• A discrete-time system with quantisation at the output of each multiplier may be considered as a multi-input linear system

2Q

2Q

bQ 2



• Then

• where is the impulse response of the system from the output of the multiplier to y(n).

h(n)

)()...()...( 21 nqnqnq p

)(nx )(ny

p

rrrnhrqrnhrxny

1 00)().()().()(

)(nh



• For zero input i.e. we can write

• where is the maximum of which is not more than

• ie

nnx ,0)(

p

rrnhqny

1 0)(.ˆ)(

q̂ rrq , ,)(

2Q

p

nrnh

Qny

1 0)(.

2)(



• However

• And hence

• ie we can estimate the maximum swing at the output from the system parameters and quantisation level

0 0)()(

n nnhnh

0)(.

2)(

nnh

pQny

1 finite wordlength

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