On Simple and Tight Upper Bounds on the ML Decoding Error ...

On Simple and Tight Upper Bounds on the ML Decoding Error Probability for Block Codes over Interleaved Fading Channels I. Sason and S. Shamai Department of Electrical Engineering Technion, Haifa 32000, Israel E-mails: [email protected], [email protected]

D. Divsalar Jet Propulsion Laboratory Pasadena, CA 91109, USA E-mail: [email protected]

Abstract

assessing the performance of ensembles of codes. The motivation for introducing and applying such

We derive here simple and tight bounds on the ML decoding error probability for block codes operating over fully interleaved Rician fading channels. It is assumed that the fading samples are statistically independent and also that perfect estimates of these samples are provided to the decoder. These upper bounds on the bit and block error probabilities are based on certain variations of Gallager’s bounding techniques. These bounds do not require integration in their final version and they are reasonably tight in a certain portion of the rate region, exceeding the cutoff rate. The proposed upper bound is demonstrated to be a generalization of some reported bounds for the AWGN and the fading channels.

bounds has increased with the recent introduction of the turbo codes and the re-introduction of low density parity check codes [1]. Clearly, the sought for bounds must not be subject to the union bound limitation, as for long block length these families of codes perform reliably at rates above the cutoff rate of the channel. Although, the maximum likelihood (ML) decoding is mostly prohibitively complex for long enough codes, the derivation of upper bounds on the ML decoding error probability is of interest, providing an ultimate indication of the system performance. Further, the fine structure of efficient

1.

Introduction

coding families is usually not available, necessitating efficient bounds to rely on basic features, such as the distance spectrum of the code, which can usually be

Since the error performance of efficiently coded

found by some analytical methods.

communication systems rarely admits exact expressions, tight analytical bounds emerge as a useful

In this work, we derive simple and tight bounds

theoretical and engineering tool for assessing per-

on the ML decoding error probability for block codes

formance and for gaining insight into the effect of

operating over fully interleaved Rician fading chan-

the main system parameters. Since specific good

nels. It is assumed here that the fading samples

codes are hard to identify, one resorts to accurately

1

are statistically independent and also that a per-

Consider a binary and linear (n, k) block code k n

fect estimate of these samples is provided to the

C with rate R =

decoder. These bounds are based on certain vari-

spectrum {Sd }nd=1 is calculable analytically. Sup-

ations of Gallager’s bounding techniques, and we

and suppose that its distance

pose the code is BPSK modulated and then trans-

obtain here upper bounds on the block and bit er-

mitted over a fully interleaved Rician fading channel,

ror probabilities. In parallel to our study here, im-

with a perfect channel side information (CSI) avail-

proved upper bounds for coded communications over

able to the receiver. Here Es = REb , where Es , Eb

fully interleaved fading channels were recently de-

are the energies per coded symbol and information

rived by Divsalar and Biglieri [2], and also by Sason

bit respectively. The following equation holds for

and Shamai [5]. For long enough block codes, the

the conditional probability density functions (pdf)

tightness of these bounds is especially pronounced

with the binary inputs: p0 (y, a) = p1 (−y, a), where

in the rate region exceeding the cutoff rate. It is

y = ax+ν is the received signal corresponding to an√ √ tipodal signaling x ∈ {− Es , + Es }, i.i.d fading

demonstrated in [5] that The upper bound derived in [5] based on generalizing the Duman and Salehi

(a) and additive white Gaussian noise (ν). We de-

framework results in the tightest reported bound.

note the pdf of the non-negative fading a by p(a) (the

However, the bounds derived in this work are simple,

effect of the phases of the fading measurements is

since they do not require any numerical integration

eliminated at the receiver and the fading coefficients

in their final version (as opposed to other reported

during each symbol are treated as non-negative ran-

bounds in [2],[5],[7]). They are also tight in a certain

dom variables).

portion of the rate region exceeding the cutoff rate.

which is ideally given to the receiver, is interpreted

Certain interconnections between these bounds are

as part of the measurements and it is also indepen-

demonstrated and we also show that they are gen-

dent of the transmitted signal. It follows that:

eralized versions of some reported bounds for the AWGN channel [4],[8].

Clearly, the fading realization a

# " p (y − a 2Es /N0 )2 1 p0 (y, a) = p1 (−y, a) = √ p(a) , exp − 2 2π (1)

2.

where −∞ < y < +∞ and a ≥ 0.

Analysis and Discussion

Let the block code C be partitioned to a set of

2.1.

subcodes {Cd }nd=1 , where each subcode Cd includes

Derivation of a closed form upper bound

all the codewords possessing a constant Hamming weight of d (d = 1, 2 . . . n) and also the all-zero

In this section we derive a closed form upper

codeword. Let Pe (d) denote the conditional block

bound on the maximum likelihood (ML) decoding

error probability associated with ML decoding and

error probability of binary and linear block codes

the subcode Cd , where we assume that the all-zero

operating over fully interleaved Rician fading chan-

codeword is transmitted (incurring no loss of gener-

nels.

ality, as the binary block code C is linear and the

2

binary input channel is symmetric). Based on the

where −∞ < y < +∞, a ≥ 0, α is an arbitrary

union bound, we get an upper bound on the average

non-negative parameter and u, v are arbitrary real

ML decoding error probability of the binary, linear

valued parameters.

block code C:

We assume here that the fading (a) during each Pe ≤

dX max

Pe (d) ,

symbol is Rician distributed, and p(a) admits the

(2)

form

d=dmin

where dmin and dmax denote the minimal and max-

p(a) = 2(1 + K)a e−(1+K)a

2

−K

imal Hamming weights of the nonzero codewords of the code C, respectively. Based on (1), the general-

³ p ´ I0 2a K(K + 1) ,

where a ≥ 0 and the Rician parameter K stands for

ized second version of Duman & Salehi bounds [4],[7]

the power ratio of the direct to the diffused received

which relies on Gallager’s 1965 bounding technique

paths.

yields the following upper bound on Pe (d): Pe (d) ≤ (Sd ) µZ µZ

+∞ −∞ +∞ −∞

Z Z

By the substitution of (1), (4) into (3), we obtain

ρ

after some algebra an upper bound associated with

∞

ψ(y, a)

1 1− ρ

1 ρ

p0 (y, a) da dy

0 ∞

1

¶(1−δ)ρn

1

the ML decoding:

ρ −

n(1−ρ) 2

where d = dmin , . . . dmax is the Hamming weight of the subcode Cd , δ

4 = nd

α−1 α− ρ

¶− nρ 2

¶δρn Pe (d) ≤(Sd ) α , ¶n(1−ρ) ¶ µ µ n(1 − ρ) Kt 1+K · · exp − (3) 1+K +t 1+K +t

ψ(y, a)1− ρ p0 (y, a) ρ −λ p1 (y, a)λ da dy 0

µ

is the normalized Hamming

weight (0 ≤ δ ≤ 1), 0 ≤ ρ ≤ 1, λ > 0 and ψ(y, a) is an arbitrary pdf of the measurements y, a.

·

µ

1+K 1+K +ε

¶dρ

·

µ

1+K 1+K +ν

¶(n−d)ρ

µ

Kε · dρ · exp − 1+K +ε

¶

µ

Kν(n − d)ρ · exp − 1+K +ν

¶

,

(5)

In contrast to [5], where the optimal function ψ where:

(which can be viewed as a tilting with respect to the two measurements y and a) was pursued, here for

t=

the sake of closed form results, only an exponential tilting is examined. Let ψ be the following tilted



1  ε = α(u2 + v 2 ) 1 − ρ

pdf:

ψ(y, a) =

p

α 2π

µ

αv 2 Es , N0

¶

+

1 − ρ

³

αu−1 − ρ α − α−1 ρ

αu −

2λ

´2 

 Es ,  N0

q h ´2 i ³ αv 2 a2 Es s − exp − α2 y − au 2E · p(a) N0 N0 , Z ∞  ³ ´2  ³ αv 2 a2 E ´ s αu−1 ¶ µ αu − da p(a) · exp − ρ 1 1  Es  N0 0 , + − ν = α(u2 + v 2 ) 1 −  ρ ρ N0 α − α−1 ρ (4) 3

where 0 < α
0), 0 ≤ ρ ≤ 1, λ > 0 and u, v

weight and information Hamming weight of d and w, respectively.

are real numbers. In order to get the tightest upper bound within this family, these parameters should be optimized.

2.2.

Generalization of the Viterbi and Viterbi bound [8] for fully interleaved Rician fad-

We wish to reduce the number of parameters

ing channels

which have to be numerically optimized in the upper bound (5). Nullifying the partial derivative with respect to λ, yields that the optimal value for the pa-

The generalized Viterbi and Viterbi bound for

rameter λ is determined by the equality λ=

1 2

µ

αu −

αu − 1 ρ

where we assume that αu
0).

Pe (d) ≤

By optimizing the upper bound (5) with respect to 4

the parameter r = αv 2 , it was shown in [6] that the

(Sd )ρ

number of parameters which should be optimized numerically is reduced to three, namely (α, u, ρ).

·

Further details are presented in [6].

µ

µ

1+K 1 + K + β3

1+K 1 + K + β1

µ

1+K · 1 + K + β2

Based on the geometrical region which is associated with the proposed upper bound here (see the

¶dρ

¶n(1−ρ)

µ · exp −

¶(n−d)ρ

¶ µ Kβ3 n(1 − ρ) · exp − 1 + K + β3

Kβ1 dρ 1 + K + β1

¶

¶ µ Kβ2 (n − d)ρ · exp − , 1 + K + β2 (7)

discussion in [7]), it was demonstrated in [6] that the proposed upper bound here is as tight as the Divsalar & Biglieri bound [2] and its advantage over

where β1 =

the later bound stems from its closed form expres-

β3 =

sion.

Es N0

Es N0 ,

β2 =

Es N0

· [1 − (1 − ξρ)2 ].

h

³ ´2 i 1 − 1 + ξ(1 − ρ) and

By comparing the upper bounds (5),(7), we want A similar upper bound on the bit error probability

to determine the parameters in (5), such that the

is derived in [5], based on the input-output weight

upper bound (7) results. To this end, we set α = 1

enumeration function of the ensemble of binary and

in (5) and we also wish to choose the remaining pa-

systematic linear (n, k) codes. In its final form the

rameters in (5), such that the following equations

n

distance spectrum {Sd }d=1 which appears in the up-

hold: β1 = ε , β2 = ν , β3 = t.

per bound on the block error probability is replaced

Some straightforward algebra shows that the bound

for the upper bound on the bit error probability by Pk ³ w ´ Aw,d , where Aw,d designates the Sd0 = w=1 k

(5) specializes to the generalized Viterbi and Viterbi bound (9), by the following setting of the parame-

4

ters:

0

10

v=

p

u = 1 − ξρ ,

1 − (1 − ξρ)2 ,

−1

10

λ=

1 + ξ(1 − ρ) . 2

(8) −2

10

Bit error probability

α=1,

An alternative generalization of the Viterbi & Viterbi bound for fully interleaved Rician fading channels was derived in [6]. It was also demonstrated in [6] that the Duman and

5 −3

10

40 iterations −4

10

−5

10

Salehi first version bound, as derived in [4] for the

1

2

3

4

−6

10

particular case of an AWGN channel is also a particular case of (5).

−7

10

1

1.5

2

2.5

3

3.5

Eb/No [dB]

3.

Specific Results

Fig. 2: Comparison among bounds on performance. The generalization of the Duman and Salehi

We examine here the ensemble performance of

bound with the optimal tilting measure (curve 1, see

the uniformly interleaved repeat-accumulate codes

[5]) is the tightest bound among the upper bounds

1 [3] of rate– , where the information block length is 4 N = 1024 and every information is repeated q = 4

depicted in Fig. 2. The upper bound proposed here which is based on the generalization of the Duman and Salehi bound with the exponential tilting mea-

times (see Fig. 1). The ensemble of codes is as-

sure in (4) (curve 2, see [6]) is clearly looser than

sumed to be transmitted through a fully interleaved

the former bound which relies on the optimal tilt-

Rayleigh fading channel with perfect CSI at the re-

ing measure, but is tighter than the generalization

ceiver. In Fig. 2, several upper bounds on the bit

of the Viterbi & Viterbi bound (curve 3, see [6]) for

error probability which apply to the optimal ML de-

fully interleaved fading channels. That observation

coding are presented, and are also compared to com-

is expected as it has been established in section 2.2

puter simulation results of the sum-product iterative

that the Viterbi & Viterbi bound is a particular case

decoding algorithm with 40 iterations (see curve 5).

of the bound (5) proposed here (see also [6]). The three bounds depicted in curves 1–3, are evidently

N information bits

a repetition code (q repetitions)

uniform interleaver

differential

of length qN

encoder

tighter than the ubiquitous union bound (curve 4),

qN coded bits

and are also effective at a portion of the rate region exceeding the cutoff rate, which for the rate

Fig. 1: Repeat and accumulate codes.

5

Eb N0

= 2.71 dB. The capacity for rate

Eb N0

= −0.08 dB.

1 4

1 4

equals

is attained at

4.

Summary and Conclusions

[2] D. Divsalar and E. Biglieri, “Upper bounds

We derive here an upper bound on the decod-

to error probabilities of coded systems over

ing error probability associated with ML decoding,

AWGN and fading channels”, Proceedings 2000

which is applicable for a fully interleaved Rician

IEEE Global Telecommunications Conference,

fading channel with perfect channel state informa-

pp. 1605–1610, San Francisco, California, USA,

tion. This upper bound is a particular case of the

November 2000.

generalization of the second version of the Duman

[3] D. Divsalar, H. Jin and R.J. McEliece, “Coding

and Salehi bound [5, 7]. The function ψ in (3) is

theorems for ‘turbo-like’ codes”, Proceedings of

taken here as an exponential type tilting with re-

the 1998 Allerton Conference, Monticello, Illi-

spect to the fading (a) and the received vector (y)

nois, pp. 201–210, September 1998.

(see eq. (4)), rather than the optimal function ‘ψ’ [4] T.M. Duman and M. Salehi, “New performance

which yields the best upper bound within this class

bounds for turbo codes”, IEEE Trans. on Com-

(whose calculation involves numerical integration, as

munications, vol. 46, pp. 717–723, June 1998.

has been demonstrated in [5, 7]). The calculation of the bound derived here does not involve any numer-

[5] I. Sason and S. Shamai, “On Improved bounds

ical integrations, but a numerical optimization with

on the decoding error probability of block codes

respect to three parameters is required. This bound

over interleaved fading channels, with appli-

yields the generalized Viterbi & Viterbi bound as a

cations to turbo-like codes”, to appear in the

particular case (a bound that generalizes the Viterbi

IEEE Trans. on Information Theory, Septem-

& Viterbi bound [8] for the AWGN channel) (see

ber 2001.

also Fig. 2), and it is also a generalization of the

[6] I. Sason, S. Shamai and D. Divsalar, “On simple

first version of Duman and Salehi bound derived for

and tight upper bounds on the ML decoding

the AWGN channel [4] (as was also demonstrated

error probability for block codes operating over

in [6]) . It can also be interpreted as a closed form

interleaved fading channels”, in preparation.

expression of the bound proposed by Divsalar and Biglieri [2]. That observation was demonstrated in

[7] S. Shamai and I. Sason, “Variations on Gal-

[6] by showing that both bounds define the same ge-

lager’s bounding technique with applications”,

ometrical region in terms of Gallager’s 1961 bound-

Proceedings of the Second International Sym-

ing terminology [1], [7]. The current bound admits a

posium on Turbo Codes and Related Topics,

closed form expression (up to three parameters that

pp. 27–34, France, September 2000.

should be optimized numerically) for a general fully

[8] A.M. Viterbi and A.J. Viterbi, “Improved union

interleaved Rician fading channel.

bound on linear codes for the binary-input AWGN channel, with application to turbo de-

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