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 <
1
1−ρ
α−1
ρ
and also α −
(since α should be non-negative
number of codewords possessing a total Hamming
> 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 <
1
1−ρ
¶
,
fully interleaved Rician fading channels [6] admits
the form (2), where the upper bound on Pe (d) (d =
(6)
dmin , . . . , dmax ) is:
(such that λ > 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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6