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Soft decision syndrome decoding

Citation for published version (APA):

Schalkwijk, J. P. M., & Vinck, A. J. (1976). Soft decision syndrome decoding. (EUT report. E, Fac. of Electrical Engineering; Vol. 76-E-63). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1976

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SOFT DECISION SYNDROME DECODING

by

J. P. M. Schalkwijk and

(3)

AFDELING DER ELE~TROTECHNIE~ VA~GROEP TELECOMMUNUCATIE

DEPARTMENT OF ELECTRICAL ENGINEERING GROUP TELECOMMUNICATIONS

SOFT DECISION SYNDROME DECODING

by J.P.M. Schalkwijk and A.J. Vinck TH-Report 76-E-63 March 1976 ISBN 90 6144 063 7

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SOFT DECISION SYNDROME DECODING

Abstract

A method is given for soft decision syndrome decoding.

Then the influence of soft decision on the hardware complexity

(5)

C,o (,' )

1. INTRODUCTION

The principle of "Viterbi like" syndrome decoding for binary con-volutional codes will be explained using the R=~ binary code gene-rated by the encoder of Fig. 1.

n, (" ) C20 C22

+

C12 z(,,) ~

c

2(")n,(,,) +C,(" )n2(<>j encoder Cn

+

n2(acj C,o C'2

channel syndrome former

Fig. 1. Encoding and synd"f'Ome forming ci!'cui

t

fa!' a R='> code

The encoder has connection polynomials C

1 (a) and C2(a). Hence, the

encoder outputs are C

l (a)x(a) and C2(a)x(a). The syndrome z(a) only depends on n

l (a) and n2(a), for z(a) = c

2(a)[C1 (a)x(a) + nl (a)] + Cl (a) [C2(a)x(a) + n2(a)]

= C

2(a)n1 (a) + Cl (a)n2(a) For a non catastrophic code, C

l (a) and C2(a) are relatively prime.

Hence, there exist polynomialS Dl(a) and D

2(a) such that Dl (a)C

1 (a) + D2(a)C2(a) = 1. The estimate x(a) of the data sequence x(a) can be written as

(6)

2

where

and

According to [1], we can draw the state diagram, see

Fig.

2, for the syndrome forming circuit of

Fig.

1. Solid transitions

-

--

--

,.- "-

...

"

/

"

"-,

"

...

"

/

"

"-/ 01:1 / \00;1 '\.

I

I \ \ I I

-

00;1 ... \

,

I 11;1 \ , 11;0 10;1 10:0

\

\ /

,

....

\ 01;0 I \ I

"

01;!') /00;0 / / "-

"

/

"-

"-

. / "-

...

...

/

,.-,.-

. /

...

-

-

.--

-Fig.

2.

State diagram of the syndrome former

in

Fig.

2 correspond to Zk = 0, and dashed transitions to Zk = 1,

k

= ...

,-1,0,1,.· •• Next to each transition one finds the values of

fi

k1, fik2; Wk' k = ••• ,-1,0,1,·.·. With each state in

Fig.

2, we

associate a metric register, and a path register. The metric register contains the logarithm of the path likelihood of the most likeli path

v-1

entering a particular state s, (k), j = 0,1, ••• ,2 at the particular

]

time k, k = ••• ,-1,0,1, ••.• The logarithm of the likelihood is taken,

for then the new metric at time k+1 is the sum of the old metric at

time k and the branch metric. The coefficients

W~_D+1' W~_D+2'

, "'k

. j

(2)

(3)

(7)

k = ••• ,-1,0,1,···, associated with the path [fi

1 (a),fi2(a)]j of maximum log likelihood are stored in the path register for the

j-th state. The oldest bit

w

j on the path with largest metric k-D+l

is added to

[Dl (a)Yl(a) + D2 (a)Y2(a)]k_D+1

to give the estimate x(a) of (2).

(8)

4

2. QUANTIZATION OF THE RECEIVED CODE SYMBOLS

AS pointed out in [21, 180· binary phase shift keying (BPSK) in combination with coding is an efficient way of communication over Gaussian channels. Quantization of the demodulated received code symbols, is to facilitate digital processing at the decoder. When 8-level quantization is used, about 0.25 dB in received signal to noise ratio is lost, compared with infinitely fine quantization. Hence, further quantization is questionable. With 2-level (binary) quantization the loss in SNR is roughly 2 dB.

Fig.

3 shows the quantization schemes for 2, 4 and 8 levels, where +1

1+1+ 0 I -1 +1 3 2 1 0 ~1 I +1 7 6 5 4 3 2 1 0 ... ---+-----.-+----+----~- I -1 +1

Fig.

3.

Quantization scheme for

2, 4

and

8

levels

corresponds with a code symbol 1, and -1 with a code symbol O.

The spacings in the above schemes can be shown to be almost optimum. The Gaussian channel with modulator and demodulator is then equi-valent to a discrete channel with two inputs, and 2, 4 or 8 outputs, respectively. The channel transition probabilities are equal to the

"00""":;7: ""' •

"I::, ..

,'ono. 'ou.,"" , •• 'om •• ,"", ••

"0

variance ---20 and mean + 1 lies in the intervals indicated in

Fig. 3.

Es

-The problem we are now faced with is the adjustment of the syndrome decoder. Take, for example, a 4-level quantizer as indicated in

Fig. 4.

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1 +

- ... --~3"--+--;;-2 -~'l,--"Ii---no- - - - ----+

Fig. 4.

Probability density function of the received

signal

Let a received signal lie in interval 2. The syndrome forming circuit only accepts the symbols 0 and 1. Hence, a binary quantizer is used to set the received signal equal to O. Now there are two possabilities, the relevant noise bit could either be zero or one with probability Pr(O)

=

Q

1 and Pr(l)

=

Q2' respectively. The same can be said about a received signal lying in interval 1. For the intervals

a

and 3, Pr(O)

=

~ and Pr(l)

=

Q3' In fact, we only need the absolute value of the received signal to determine Pr(O) and Pr(l) and thus the branch metric. From simulations, it follows that the decoder is quite insensitive to branch metric quantization. Hence, use of integers instead of exact log likelihoods gives a very small performance degradation.

Fig.

5 shows a possible set of metrics for the case of 4-level quantization.

Hard quantized

noise

o

1

Received quantized

level

o

o

3 1 1 2 2 1 2

o

3

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6

In [1] is shown that in general, the number of different state metrics

"

for a R = ~ "Viterbi like" syndrome decoder with 2 states is equal 3 "

to - • 2

4 For a special class of codes, this can even be further

reduced to (/3)". The reduction was mainly based on the fact that the

branch metric contribution of the noise pairs [0,1] and [1,0] are equal. From

Fig.

5 it follows that in the soft decision decoder, the above mentioned contributions are not always equal. For example, let a received signal pair be in the intervals

a

and 1. Then, the branch metric for a [O,l]-branch is 2, while the contribution for a [1,0]-branch is 4. A direct consequence is that the R

=

~ syndrome decoder

"

has 2 different state metrics. Each state having 4 entries, which leads to a more complicated path register reshuffling and metric calculation scheme as compared to the Viterbi decoder.

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CONCLUSIONS

A method is given for soft decision "Viterbi like" syndrome decoding. This method is not only valid for R

=

'>.

but for all R

=

kin "Viterbi

like" syndrome decoders. The number of different state metrics in the R

=

'>

case cannot be reduced. It seems that the advantages of syndrome decoding for long constraint length codes disappear.

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8

Implementation of a R = ~ convolutional decoder

The three major functions to be perfomed by the encoder are 1/ determination of the branch metrics,

2/ determination of the new state metrics and 3/ generation of the survivor sequence.

The hardware complexity of the Viterbi and of the syndrome decoder strongly depends upon the constraint length ~ of the encoder, the code rate, and also if hard (Q

=

2) or shoft (Q

=

4,8) decision is used on the received data stream. First a general comparison will be made for R

=

~ and Q 2. Then the influence of soft decision will be discussed. For Viterbi decoding the new metric of any state (S) is calculated according to

{M _v(k) + d(~'E)}l

as+a

where d(~,~) denotes the hamming distance between the transition from

-1

state (a S) to state (S), and the received bit pair ~. Note that d(c,r) = 2 - d(~,~), whenever the outermost stages of the encoder are connected with the mod-2 adders. The hardware translation of (6) is given in

Fig.

6. A constraint length v code, requires a Viterbi decoder with basicly 2v of these sections.

oompare

and seleot

Fig.

6.

Add and oompare logio

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In the state table of the syndrome decoder [1], the transitions oCcur

b

in groups. One group is given in Fig. 7, where 52 is

state transition pair

00 01 11 10 5 a. -1 5 a. -1 5+0. -1 a. -1 5+s 2 b - 1 +0. a. -1 5+s 2 b b -1 5+( s 2 +0. ) a. -1 5+5 b - 1 2 +0. -1 a. S+s 2 b a. -1 5 a. -1 5+0. -1 5+0. -v a -1 5 a -1 5+a -1 a -1 5+s 2 b - 1 +0. a -1 5+S 2 b -v b - 1 5+a +(s2 +0. )0. a. -1 5+5 b - 1 2 +0. -1 'b-1 a. 5+5 2 +0. -1 a. S a -1 S+a. -1

Fig. 7. Group of transitions

the base state of the syndrome former. In general, the syndrome decoder

has 2v/4 of these groups. Assume that, without loss of generality, state (S)

b -1

and state (s+(s2 +0. )0.) cause the same syndrome outputs. Otherwise

-v -1 -1 b

interchange (S) and (S+a ). The metrics for state (a.S+a ) and (a. s+s2)

are calculating according

M -1 -1 (k+1) = M -1 b (k+l) M b -1 (k)+1]+ S+ (s2 +0. ) a. a. S+a. a S+s2 + ~ min[M (k)+l, M k -v s+a. b -1 _v(k)+l] s+(s2 +a )0.+0. (7 ) -1

The states with a term (a ) are called odd states. The others are called

-1 -1 b - l

even states. For the even states (a S) and (a S+5

2 +a ), the metrics are calculated according to

M_l (k+l)

=

zkmin[M _v(k), a. S S+a. M -v b -1 (k)+2]+ S+a. +(s2 +0. )et + ~ min[l~ (k), M b -1 (k) +2] k S S+(s2 +a )et (8 ) and

(14)

M -1 b _1(k+l) a S+s2 +a 10 Zk min[M (k)+2, -'J S+(t M -\! b -1 (k)]+ S+a +(s2 +,x )" + zk min[M (k)+2, M b -1 (k)] S s+(s2 +a )a

respectively. The hardware sections for the even and odd state metrics

are given in Fig. 8 and Fig. 9, respectively.

M (k) M (k) M

b -1 (k) i1 b -1 -\! (k)

3 3+a -v 3+(s2 +a )a 3+(s2 +a )a+a

I

1--

zk

I

I

"i--

zk

I

I

/ADD O/IADD 21 ADD

2J

IADD

01

compare

f-

compare

:--select select

I I

Fig. 8. Metric calculation scheme for an even state pair

M (k) M -v (k) M b -1 (k) M b -1 -v (k) 3 3+a 3+(s2 +a )'" 3+(s2 +a )a+a

I'

;<Zk'J

l

;>(Zk

'I

I

compare

l

I

select

I

I

ADD 1

I

M -1 _1(k+1) = M -1 b (k+ 1) a S+a a 3+s 2

Fig. 9 Metric calculation scheme for an odd state pair

(15)

The total amount of nardware, needed to calculate the new state metrics follows from Fig. 6, 8 and 9, and is given in Fig. 10

adders compara"tors mu l tip lexers

Syndrome decoder 2'0-2*2+2'0-2 2'0-2*2+2'0-2 2'0-2*2+2'0-1

Viterbi decoder 2'0-2*8 4.2'0-2

-Fig. 10. Hardware needed to caZculate the new state metrics

If equal complexity for all elements is assumed, the syndrome decoder

'0-2 '0-2

requires 10*2 elements, while the Viterbi decoder does 12*2 .

In fact, tnis difference is even larger, because multiplexing is less

complex as compared to adding. An additional advantage is the number of different state metrics of a syndrome decoder. In [1] is proven that in general this number is equal to

<f)*2

V Hence, less subtractions from each state metric with the most likely state metriC, and less comparisons

for searching this most likely metric need to be made. Next we are discussing the complexity of the path register scheme. The comparator

in the metric calculator, determines which path is the survivor for a certain state. This means that in a Viterbi decoder, each path register has two possible entries, as can be seen in Fig. 11

s e lec ten, ;:t;:;.r.;z.Y __

from comparator L--r~~~~,-__ ~-J

Fig. 11. Four entries of a path register

In general, the path registers of the syndrome decoder, have 4 entries, as can be seen in Fig. 11. The multiplexing for two even states is

(16)

12

mUltiplexer multiplexer

camp camp

multiplexer multiplexer

storage storage

Fig. 12. Path register organization scheme for an even pair

combined, as was done in Fig. 8 for the metric calculation. The same scheme can be drawn for odd pairs of states. If a path register length of 4

is assumed, the hardware needed for the path register organization in the syndrome decoder and in the Viterbi decoder is given in Fig. 13,

respectively. storage muUip~exers syndrome 2*2v- 2*4v +2v- 2• 4v 4 4 4*2 v- 2*4v 4 + 2v-2*4v *3 ~ 4 Viterbi

4

4v *2\J 4\J *2"

4"

Fig. 13. Hardware requirement for the path register

organization scheme

If equal complexity is assumed for mUltiplexers and storage cells, the difference between syndrome and Viterbi path register organization is

v-2

2*v*2 elements in favor of the Viterbi decoder. Together with the metric calculation scheme, we can say that the complexity of the syndrome decoder and Viterbi decoder is about equal.

(17)

But as proven in [1], the number of different states in the syndrome

decoder can be reduced to (/3)V~ Hence, for this special class of codes, the syndrome decoder is a reasonable alternative to the Viterbi decoder, as it achieves an exponentional saving in hardware. Note, that this

is only true for Q=2, i.e. hard decisions. If soft decisions are used,

the number of different states cannot be reduced and the Viterbi decoder is preferable.

(18)

14

REFERENCES

[1] J.P.M. Schalkwijk, "Symmetries of the State Diagram of the Syndrome Former of a Binary Rate-~ Convolutional Code", Proceedings of the C.I.S.M. advanced School on Open Problems in Information Theory, September 1975.

[2] J .A. Heller and I.M. Jacobs, "Viterbi Decoding for Satellite and Space Conununica tion 11, IEEE Trans. on Comm. Techn., Vol. COM-18, October 1971, pp. 835-848.

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