Difference Threshold Test for M -ary Signaling with Coherent Detection
Young Gil Kim
Sang Wu Kim
University of Seoul
Korea Advanced Institute of Science and Technology
Seoul 130-743, Korea
Taejon 305-701, Korea
ygkim@ieee.org
swkim@san.kaist.ac.kr
We present the difference threshold test (DTT) as a method
for generating a reliability information for M -ary signaling
with coherent detection. The proposed DTT declares an erasure whenever the difference between the shortest Euclidean
distance and the second shortest Euclidean distance from the
received signal is less than a threshold. We show that the DTT
is an approximation of the Bayesian test, and for binary signaling, it is equivalent to the Bayesian test. We also extend the
DTT for L-fold diversity systems. The performance of ReedSolomon (RS) codes with errors-and-erasures decoding is analyzed when the DTT and the conventional SNR threshold test
are applied.
II. S YSTEM M ODEL
We consider M -ary signals in slow frequency-nonselective
fading channels with additive white Gaussian noise (AWGN).
The low-pass equivalent received signal is
I. I NTRODUCTION
where r = (r1 , r2 , ..., rN ), x ∈ {s1 , s2 , ..., sN }, n =
T
(n1 , n2 , ..., nN ), and rk = 0 r(t)fk ∗ (t)dt = a · xk + nk ,
N
M
where {fk (t)}k=1 is an orthonormal basis for {sm (t)}m=1 .
∗
We assume that E[nk ] = 0 for all k, and E[nk nj ] = N0 δkj ,
where δkj is the Kronecker delta function. We assume perfect
estimation of fading at the receiver.
The use of side information permits identification and erasure of symbols that have been impaired by channel effects
such as fading, jamming, background noise, etc. Since more
erasures can be corrected than errors, it is advantageous to determine the reliability of the received symbols and erase unreliable symbols prior to the decoding process. There are a
number of methods for generating side information, and their
performances have been analyzed in [1]-[6]. In [6], difference
threshold test (DTT) was proposed for binary frequency shift
keying (BFSK) with noncoherent detection. In [7], DTT for
M -ary orthogonal signaling with noncoherent detection was
proposed and compared with the Viterbi’s ratio threshold test
(RTT) [8].
In this paper, we present the DTT for M -ary signaling with
coherent detection. The proposed DTT declares an erasure if
the difference between the shortest Euclidean distance and the
second shortest Euclidean distance is less than a threshold. We
show that the DTT is an approximation of the Bayesian test,
and for binary signaling, it is equivalent to the Bayesian test.
Also, we find that the erasure zone, where erasure is declared,
of the Bayesian test is very similar to that of the proposed DTT.
We also extend the DTT for L-fold diversity systems. Finally,
we apply the proposed DTT to Reed-Solomon (RS) codes with
binary phase shift keying (BPSK) signaling, and compare with
the conventional SNR threshold test.
This paper consists of six sections. In Section II, we describe the system model. In Section III, we propose the DTT
for M -ary signaling with coherent detection. In Section IV,
the DTT for L-fold diversity systems is presented. In Section
V, we analyze a RS-coded BPSK signaling with errors-anderasures decoding using the proposed DTT in slow frequencynonselective Rayleigh fading channels. Finally, conclusions
are given in Section VI.
GLOBECOM 2003
r(t) = a · x(t) + n(t),
0≤t≤T
(1)
where a is a complex Gaussian fading, x(t) is the transmitted signal, n(t) is a zero-mean complex AWGN, and T is
the symbol duration. We assume that each of M signals,
s1 (t), s2 (t), ..., sM (t), is transmitted with equal probability.
The received signal can be expressed by a vector with N
components:
r=a·x+n ,
(2)
III. DTT FOR COHERENT DETECTION
Derivation of DTT
In this section, we derive the DTT from the Bayesian test.
The Bayesian test declares an erasure if the a posteriori probability of correct decision is less than a threshold [1]:
Pr(x = x̂|r, a) < φ
(3)
where x is the transmitted signal, φ is a given threshold,
and x̂ is the maximum a posteriori (MAP) detection among
{s1 , s2 , ..., sM }:
x̂ = arg
max
sm ∈{s1 ,s2 ,...,sM }
Pr(x = sm |r, a) .
(4)
We define LLR Λm for m = 1, 2, ..., M as
Λm = ln
Pr(x = x̂|r, a)
.
Pr(x = sm |r, a)
(5)
Then,
- 1999 -
e−Λm Pr(x = x̂|r, a)
M
e−Λm Pr(x = x̂|r, a)
=
=
m=1
Pr(x = x̂|r, a)
Pr(x = sm |r, a)
M
(6)
Pr(x = sm |r, a) (7)
m=1
M
e−Λm
=
1.
(8)
m=1
0-7803-7974-8/03/$17.00 © 2003 IEEE
Thus, the Bayesian test declares an erasure if
1
≤φ .
Pr(x = x̂|r, a) = M
−Λm
m=1 e
Erasure Zone
(9)
Now, we derive LLR Λm . Assuming that each symbol is transmitted with an equal probability, the LLR Λm in (5) can be
expressed as
Λm = ln
Pr(r|x = x̂, a)
.
Pr(r|x = sm , a)
(10)
Since nk ’s are zero-mean Gaussian and E[nk nj ∗ ] = N0 δkj ,
nk ’s are independent and so are rk ’s. Therefore,
Pr(r|x = sm , a) =
N
Pr(rk |smk , a) ,
(11)
k=1
where
2
mk |
1 − |rk −as
N0
.
e
πN0
Substituting (11) in (10), yields
Pr(rk |smk , a) =
Λm
=
(12)
N
|rk − asmk |2 − |rk − ax̂k |2 /N0 (13)
k=1
(14)
||r − asm ||2 − ||r − ax̂||2 /N0 ,
N
where, for x = (x1 , x2 , ..., xN ), ||x||2 = k=1 |xk |2 .
Now, we approximate the a posteriori probability of correct
decision by taking the largest two terms in the denominator
of (9), yielding
1
Pr(x = x̂|r, a) = M
(15)
−Λm
m=1 e
1
(16)
≈
1 + e−Λm∗
where
Erasure zone is the region where we declare an erasure
based on (9) or (19). For the proposed DTT, we can simplify (19) as
ˆ
Re{r • (ax̂ − ax̂)}/N
0 <γ ,
N
where x • y = i=1 xi yi∗ and γ is a threshold that depends
ˆ Therefore, the boundary of erasure zone with
on ax̂ and ax̂.
ˆ
the DTT is a plane that is orthogonal to the vector from ax̂
to ax̂. In Figure 1(a) and (b), erasure zones for BPSK and
QPSK signals are plotted, respectively. Shaded area represents the erasure zone. Since the proposed DTT is equivalent
to the Bayesian test for binary signals, the erasure zones for
the proposed DTT and the Bayesian test are the same. In Figure 1(b), the erasure zone of the proposed DTT is darker than
the Bayesian test. Note that the boundary of erasure zone for
the proposed DTT is always a line for two-dimensional signals. The erasure zone of the Bayesian test looks similar to
that of the proposed DTT, but includes the area near the origin.
=
Λ m∗
=
=
(22)
IV. DTT IN L- FOLD D IVERSITY S YSTEM
In this section, we consider extending the DTT with L-fold
diversity. We consider a multichannel which consists of L independent frequency nonselective, slow fading channels. The
received vector rl , l = 1, 2, ..., L, corresponding to the lth
diversity branch is
rl = al x + nl
(23)
where al and nl are the complex fading and the noise vector,
respectively, at the lth diversity branch. We assume that {rl }
are independent and so are {al } and {nl }. If we let
min Λm
(17)
min ||r − asm ||2 − ||r − ax̂||2 /N0 (18)
sm =x̂
x̂L = arg max Pr(x = sm |r1 , r2 , ..., rL , a1 , a2 , ..., aL )
sm
(24)
sm =x̂
is the difference between the shortest Euclidean distance and
the second shortest Euclidean distance from r, normalized
by N0 . Since the a posteriori probability of correct decision
in (16) increases monotonically with increasing Λm∗ , we propose to declare an erasure if
ˆ 2 − ||r − ax̂||2 /N0 < γ ,
(19)
||r − ax̂||
for some constant γ and call it the DTT, where
ˆ = arg min ||r − asm ||2 .
x̂
sm =x̂
(20)
then
Pr(x = x̂L |r1 , r2 , ..., rL , a1 , a2 , ..., aL ) = M
m=1
sm
Λm,L = ln
Pr(r1 , r2 , ..., rL |x = x̂L , a1 , a2 , ..., aL )
. (26)
Pr(r1 , r2 , ..., rL |x = sm , a1 , a2 , ..., aL )
Since r1 , r2 , ..., rL are independent,
Pr(r1 , r2 , ..., rL |x = x̂L , a1 , a2 , ..., aL )
(21)
Thus, the reliability of the ML decision (or MAP decision for
equally probable source) can be measured approximately by
the difference between the shortest Euclidean distance and the
second shortest Euclidean distance from the received vector r.
For binary signals such as BPSK, the proposed DTT is equivalent to the Bayesian test because M = 2 for binary signals.
GLOBECOM 2003
e−Λm,L
(25)
where
Note that for equally probable source,
x̂ = arg min ||r − asm ||2 .
1
L
Pr(rl |x = x̂L , al )
(27)
||rl − al sm ||2 − ||rl − al x̂L ||2 /N0 .
(28)
=
l=1
and thus
- 2000 -
Λm,L =
L
l=1
0-7803-7974-8/03/$17.00 © 2003 IEEE
Note that for equally probable source,
L
2
x̂L = arg min
||rl − al sm ||
.
sm
Probability of Incorrect Decoding
(29)
l=1
Following the procedure toward (19) yields that the DTT declares an erasure if
L ˆ L ||2 − ||rl − al x̂L ||2 /N0 < γL
||rl − al x̂
(30)
l=1
for some constant γL , where
L
2
ˆ
x̂L = arg min
.
||rl − al sm ||
sm =x̂L
(31)
pe t (1 − pe )
l=1
r =a·x+n
2
+
+1
N
Per (i),
where Per (i) is the probability of i code symbol erasures taking place in a codeword, i.e.
Per (i) =
(33)
= k · log2 Q · Eb .
(34)
Thus,
n
.
(35)
k
We assume that E{a2 } = 1 for convenience of analysis. Thus,
the average received energy per information bit Ēb is the same
as the transmitted energy per information bit Eb :
Eb = Es,b
(36)
n i
p (1 − per )n−i ,
i er
per = 1 − (1 − per,b )log2 Q ,
(41)
where per,b is the BPSK symbol erasure probability. In (39),
pe denotes the probability of code symbol error when the code
symbol is not erased, and is given by
pe = 1 − (1 − pe,b )log2 Q ,
(42)
where pe,b is the BPSK symbol error probability when not
erased. In the next subsections, we derive per,b and pe,b for
SNR threshold test, and the proposed DTT.
SNR Threshold Test: For SNR threshold test, we erase a
E
BPSK symbol if the received SNR a2 Ns,b
is less than a thresh0
old, i.e.
(43)
if a2 < γ0 ,
for some constant γ0 . Since a2 is exponentially distributed,
the BPSK symbol erasure probability per,b is given by
per,b
=
0
=
γ0
e−x dx
1 − e−γ0 .
(44)
(45)
The BPSK symbol error probability when not erased is given
by
DTT for BPSK
2Es,b
From (19), the proposed DTT declares an erasure if
γ0
−γ0
e Q
pe,b = e
γ0
N0
2
2
(r + a Es,b ) − (r − a Es,b ) /N0 = 4 Es,b ·|ar|/N0 < γ .
Es,b
Es,b + N0
(37)
−
Q 2
,
γ0
Es,b + N0
N0
Thus, the DTT, which is equivalent to the Bayesian test for
binary signaling, declares an erasure if |ar| is less than a given
threshold.
A detailed derivation of (46) is given in Appendix A.
GLOBECOM 2003
(40)
where per is the code symbol erasure probability given by
(32)
= log2 Q · Es,b
Ēb = E{a2 } · Eb = Eb .
n−i−t
i=n−k+1
where a is the fading amplitude, x is Es,b or − Es,b
with equal probability 1/2, and n is the AWGN with variance
N0 /2. Es,b is the energy per BPSK subsymbol, Eb is the energy per information bit, and Es,c is the energy per RS code
symbol. They are related by
nEs,c
Thus, the probability of incorrect decoding PE is given by [9]

n−k
n−i

n−i
PE =
Per (i) ·
·
(39)

t
n−k−i
i=0
t=
V. E RRORS - AND -E RASURES D ECODING FOR RS- CODED
BPSK S IGNALS
System Model
We apply the proposed DTT to a RS-coded BPSK signals
in slow frequency-nonselective Rayleigh fading channels. We
use (n,k) Q-ary RS code, where n is the number of code symbols in a codeword and k is the number of information symbols
in a codeword. Each Q-ary RS code symbol is composed of
log2 Q BPSK symbols. We assume that each of log2 Q BPSK
symbols in a code symbol experiences an independent fading.
At receiver, we declare a code symbol erasure if the code symbol contains at least one BPSK symbol erasure.
The low-pass equivalent received signal for BPSK signaling
after phase compensation is
Es,c
We derive the probability of incorrect decoding with errorsand-erasures decoding for the SNR threshold test [9] and the
proposed DTT. With (n, k) RS code, we can correct t errors
and e erasures if
t + 2e ≤ n − k.
(38)
- 2001 -
(46)
0-7803-7974-8/03/$17.00 © 2003 IEEE
DTT: For the proposed DTT, we erase the BPSK symbol if
|ar| < γ0
(47)
Then, the BPSK symbol erasure probability per,b is given by
1
1
1
per,b = 1 − e−αγ0 +
1 − e−βγ0 ,
β
Es,b + N0 α
(48)
where
2 Es,b + N0 + Es,b
α=
(49)
N0
and
2 β=
Es,b + N0 − Es,b .
(50)
N0
The BPSK symbol error probability when not erased, pe,b , is
given by
pe,b
=
Pr(ar < 0|ar < −γ0 or ar > −γ0 ) (51)
Pr(ar < −γ0 )
=
,
(52)
1 − per,b
where
N0
Pr(ar < −γ0 ) = 2 Es,b + N0
√
√
2(
Es,b +N0 +
Es,b )γ0
1
−
N
0
·
e
(53)
.
Es,b + N0 + Es,b
See Appendix A for the derivation of (48) and (53).
Numerical Results and Discussions
Figure 2 is a plot of the probability of incorrect decoding
PE versus threshold γ0 . We find that the probability of incorrect decoding PE of each system is minimized at different
thresholds, and the proposed DTT provides a lower PE . Figure 3 is a plot of the optimum code symbol erasure probability
p∗er versus Ēb /N0 that minimizes PE . We find that the proposed DTT erases more code symbols than the SNR threshold
test. Figure 4 is a plot of the probability of incorrect decoding
PE versus Ēb /N0 when the threshold γ0 is optimally chosen
to minimize PE . We also plotted PE of the errors-only decoding, which is a special case of γ0 = 0. We find that the
proposed DTT provides a power gain of 1 dB over the SNR
threshold test for PE of 10−5 and the power gain increases as
the required probability of incorrect decoding decreases.
VI. C ONCLUSION
We presented the DTT as a method for generating a reliability information for M -ary signaling with coherent detection. The proposed DTT declares an erasure if the difference
between the shortest Euclidean distance and the second shortest Euclidean distance from the received signal is less than a
threshold. It is shown that the DTT is an approximation of
the Bayesian test, and for binary signaling, the DTT is equivalent to the Bayesian test. The proposed DTT is applied to a
RS-coded BPSK signaling with errors-and-erasures decoding.
The proposed DTT provides a power gain of 1 dB over the
SNR threshold test for the probability of incorrect decoding of
10−5 . We also derived the DTT for L-fold diversity.
GLOBECOM 2003
R EFERENCES
[1] C. W. Baum and M. B. Pursley, “Bayesian methods for erasure
insertion in frequency-hop communication systems with partialband interference,” IEEE Trans. Commun., vol. 40, No. 7, pp.
1231-1238, Jul. 1992.
[2] C. M. Keller and M. B. Pursley, “Diversity combining for channels with fading and partial-band interference,” IEEE J. Select.
Areas Commun., vol. SAC-5, No. 2, pp. 248-260, Feb. 1987.
[3] L. Yang and L. Hanzo, “Performance analysis of coded M ary orthogonal signaling errors-and-erasures decoding over
frequency-selective fading channels,” IEEE J. Select. Areas
Commun., vol. 19, No. 2, pp. 211-221, Feb. 2001.
[4] L. Yang, K. Yen and L. Hanzo, “A Reed-Solomon coded DSCDMA system using noncoherent M -ary orthogonal modulation over multipath fading channels,” IEEE J. Select. Areas
Commun., vol. 18, No. 11, pp. 2240-2251, Nov. 2000.
[5] L. Yang and L. Hanzo, “Low-complexity erasure insertion in
frequency-hopping spread-spectrum communications subject to
fading and partial-band interference,” IEEE Globecom, Nov. 2529, 2001, San Antonio, Texas, USA, pp. 796-800.
[6] I. Su and J. Wu, “Difference threshold test for asynchronous
BFSK frequency-hopped multiple access systems over Rician
channels,” Electronic Letters, vol. 35, No. 18, pp. 1512-1513,
Sept. 1999.
[7] Y. G. Kim and S. W. Kim, “Difference threshold test for M-ary
orthogonal FSK signaling in Rayleigh fading channels,” Proc.
of IEEE ICC, Anchorage, USA, May 2003.
[8] A. J. Viterbi, “A robust ratio-threshold technique to mitigate
tone and partial band jamming in coded MFSK systems,” IEEE
MILCOM, pp. 22.4.1-22.4.5, Oct. 1982.
[9] J. Hagenauer and E. Lutz, “Forward error correction coding for
fading compensation in mobile satelite channels,” IEEE J. Select. Areas Commun., vol. SAC-5, pp. 215-225, Feb. 1987.
[10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series
and Products. Academic Press, corrected and enlarged edition,
p. 307, 1980.
A PPENDIX A
In this Appendix, we derive (46), (48), and (53).
Derivation of (46)
The BPSK symbol error probability when not erased, pe,b
in (46), is
∞ 2Es,b
Q
pe,b =
x e−(x−γ0 ) dx
(54)
N0
γ0
∞ ∞
t2
1
= √ · eγ0
e− 2 e−x dtdx
(55)
2Es,b
2π
γ0
x
N
1
=√
2π
∞
· eγ0 2Es,b
N0
0
γ0
N0
2
2Es,b t
γ0
t2
e− 2 e−x dxdt
(56)
∞
N0
t2
1
−
t2
= √ · eγ0 2E
e− 2 e−γ0 − e 2Es,b
dt
s,b
2π
N0 γ0

∞

2
1
− t2
γ0
−γ0 = √ ·e
e
e
dt
2Es,b

2π
N0 γ0

∞
2
N

− t2 1+ E 0
s,b
− 2E
e
(57)
dt .
s,b

γ0
N0
(46) can be derived from (57) by using the definition of Qfunction.
- 2002 -
0-7803-7974-8/03/$17.00 © 2003 IEEE
X
as 2
Symbol erasure probability using optimum threshold
0.2025
X
as 1
O
(a)
X
X
as3
as2
0.135
0.09
0.06
0.04
10
DTT
12
-
14
16
18
20
Eb/N0
Fig. 3. Optimum symbol erasure probability vs. E¯b /N0 for (31,16) 32-ary
RS code.
Bayesian
X
as 1
O
DTT
SNR
1
Errors-only
SNR
DTT
0.1
as4 X
0.01
PE
(b)
Fig. 1. Erasure zones of the Bayesian test and the proposed DTT for (a)
BPSK signals and (b) QPSK signals
0.001
0.0001
0.24
1e-05
10
11
12
14
-
15
16
17
18
19
Fig. 4. Probability of incorrect decoding PE vs. E¯b /N0 for (31,16) 32-ary
RS code.
SNR
DTT
PE
13
Eb/N0
0.12
0.06
where we use the equality [10]
∞
q
1 π
√
exp(−px2 − 2 )dx =
exp(−2 pq) .
x
2 p
0
0.03
(59)
The BPSK symbol erasure probability per,b in (48) is
0.015
Threshold γ0
Fig. 2. Probability of incorrect decoding PE vs. threshold γ0 for E¯b /N0 =
12 dB and (31,16) 32-ary RS code.
Derivation of (48) and (53)
0
For the derivation of the BPSK symbol erasure probability (48) for the proposed DTT, the pdf of ar should be derived
first. The pdf of ar, far (y), is
∞
1
−(y−a2
√
Es,b )2
2
e
2ae−a da1
2πa2 N0 /2
0
√
√
1
=
e2y Es,b /N0 e−2|y| N0 +Es,b /N0 (, 58)
N0 + Es,b
far (y) =
GLOBECOM 2003
per,b = Pr(−γ0 < ar < γ0 ) =
√
√
2y
N0 +Es,b
Es,b
2y
0
1
N
N0
0
e
dy
e
N0 + Es,b
−γ0
γ0 2y√Es,b 2y√N0 +Es,b N0
e N0 e−
+
dy
a2 N0
(60)
(61)
1
1
,
(1 − e−αγ0 ) + (1 − e−βγ0 )
β
α
Es,b ) and β =
where α = N20 ( N0 + Es,b +
2
E
).
Also,
(53)
can
be derived from
N
+
E
−
(
s,b
0
s,b
N0
the following integration:
=
1
N0 + Es,b
(62)
Pr(ar < −γ0 ) =
−γ0
√
√
1
e2y Es,b /N0 e2y N0 +Es,b /N0 dy.
N0 + Es,b
−∞
- 2003 -
0-7803-7974-8/03/$17.00 © 2003 IEEE