1

BER performance of BPSK transmissions over

multipath channels
Otto Fonseca and Ioannis N. Psaromiligkos

4 P
∞
Abstract— A new closed form expression for the Bit-Error- as z = bk gk , we may write the probability of error
Rate (BER) performance of binary phase shift keying (BPSK) k=−∞, k6=0
transmissions over frequency selective channels is presented. The 4
Pe = P r[b̂0 6= b0 ] using the Gaussian Q(·) function as Pe =
expression is obtained through a novel approximation of the
Gaussian Q(·) function by a fixed series of sinusoids with expo- Ez {Q(g0 − z)}, where E{·} denotes the expectation operator
nentially decreasing amplitudes. Numerical results demonstrate [1]. Since in practice only Nc coefficients are non-zero, Pe
the accuracy of the derived expression. can be calculated by averaging over all 2Nc −1 possible values
of z. When Nc is large this is computationally unfeasible.
A popular approach to Pevaluate Pe is to approximate Q(x)
I. I NTRODUCTION by a series Q̂(x) =
NT −1
ci qi (x), ci ∈ R, of functions
i=0
Accurate BER performance evaluation is crucial for the suc- qi (x) whose expected values are easy to evaluate. In [2]
cessful design and fine-tuning of a digital communications sys- Q̂(x) was obtained by truncating the Fourier Series expansion
tem. In many cases, exact BER expressions either do not exist with period T of Q(x), while in [4] Q̂(x) was formed by
or they are too complex to be of any practical value. In the case truncating the Taylor Series expansion of Q(x). Finally, in [3]
of BPSK transmissions over Inter-Symbol Interference (ISI)- the functions qi (x) were set equal to e−iβx , β ∈ R, or to the
inducing channels, several closed form BER approximations ith Legendre Polynomial Pi (x). However, all these methods
have been proposed [2], [3], [4]. These approximations are rely on tuning parameters (T , β and NT ) that have to be
based on series expansions of the Gaussian Q(·) function. chosen empirically limiting their practical value.
However, they require tuning of their parameters through a
III. P ROPOSED BER APPROXIMATION
trial-and-error process. In this Letter, we propose a novel ap-
proximation of the Q(·) function by a finite series of sinusoids Motivated by the excellent accuracy of the methods in [1] and
with exponentially decreasing amplitudes. The parameters of [2] in the low BER and high BER regions, respectively, we
the series are obtained through numerical minimization of propose in this Letter to approximate Q(x) by the following
a relative approximation error measure. The resulting series series:
NX
"N −1 #
T −1 X
requires much fewer terms than previously proposed series- λi x
T
(λi +jwi )x
based methods to yield a highly accurate closed form BER Q̂(x) = ci e cos(wi x) = Re ci e
i=0 i=0
expression. (1)
where ci , λi and wi are real parameters. In contrast to [2]
II. P ROBLEM DESCRIPTION AND PREVIOUS WORK and [4], we obtain the parameters ci , λi , and wi by explicitly
minimizing the error between Q̂(x) and Q(x). However,
We consider BPSK transmissions over frequency selective
instead of minimizing the Mean-Square-Error (MSE) as in [3]
channels. We denote the real part of the receiver output by
4 P
∞ (which overemphasizes accuracy in the region of high values
y(t) = bk g(t − kTB ) + n(t), where bk ∈ {±1} is of Q(x) at the expense of accuracy in the low value region
k=−∞
the 0-mean transmitted bits sequence, TB is the information [1]), we minimize the quantity [1]
Z ·¯ ¯2 ¯ ¯2 ¸
bit period, and n(t) is white Gaussian noise. Finally, the ¯ ¯ ¯ ¯
impulse response g(t) represents the transmit filter, channel, E= ¯ 1 − Q̂(x)/Q(x) ¯ + ¯ 1 − Q(x)/ Q̂(x) ¯ dx (2)
X
receive filter and equalizer. We focus on the detection of where X is the range over which we wish to approximate
bit b0 . Sampling at t = 0, we form the decision statistic Q(x). The error in (2) was minimized numerically over the
4 P ∞ 4
y= bk gk + n, where gk = g(−kTB ) and n is N (0, 1). set X containing 1,000 points placed uniformly between x = 1
k=−∞
4
(Q(x) = 1.587e−1 ) and x = 7 (Q(x) = 1.2798e−12 ).
Then, the decision on b0 is b̂0 = sign(y). Denoting the ISI As we will demonstrate later, this choice of X results in
approximations of Pe that are potentially accurate down to
This paper is a postprint of a paper submitted to and accepted for publi-
cation in IET Electronics Letters and is subject to Institution of Engineering
a BER of 10−10 . It was found that NT = 7 terms suffice
and Technology Copyright. The copy of record is available at IET Digital for an accurate approximation of Q(x) over X . The obtained
Library. coefficients are shown in Table 1. From (1) we obtain
Authors’ affiliations: Otto Fonseca and Ioannis N. Psaromiligkos (De- "N −1 #
partment of Electrical and Computer Engineering, McGill University, XT
(λi +jwi )g0
3480 University St., Montréal, Québec, H3A 2A7, Canada). E-mail: yan- Pe ' Re ci e Mz (λi + jwi ) (3)
nis@ece.mcgill.ca i=0
 4
where Mz (t) = E{etz } is the moment generating function
(MGF) of z which is equal to
Y∞ Y∞
1 tgk
Mz (t) = (e + e−tgk ) = Cosh(gk t). (4)
k=−∞
2 k=−∞
k6=0 k6=0

Finally, combining (3) with (4) we have

 
TABLE I
NX
T −1 ∞
Y
  C OEFFICIENT VALUES F OR P ROPOSED A PPROXIMATION (NT = 7)
Pe ' Re  ci e(λi +jwi )g0 Cosh[(λi + jwi )gk ] .
i=0 k=−∞
i ci λi wi
k6=0
(5) 0 −2.140052 × 101 −3.5085 1.0654
The proposed BER approximation is obtained by using the 1 −1.392160 × 10−1 −3.6807 0.10374
2 −4.557493 × 10−2 −3.5876 −0.14603
coefficients of Table 1 in (5). 3 6.831091 × 10−2 −2.6201 0.24457
4 3.277368 −3.2368 −0.17281
5 3.606592 −3.2422 −0.16649
IV. N UMERICAL RESULTS AND C ONCLUSIONS 6 3.740890 −3.1919 −0.2413
We consider an impulse response with Nc = 16 coefficients
given by g−5 , ..., g10 = −0.0081, 0.0117, 0.0034, −0.0190,
−0.0129, 1.0000, −0.0580, 0.1199, 0.0321, −0.1356, 0.1002,
0.0592, 0.0476, 0.0974, 0.0271, and 0.0207, respectively. In
Fig. 1 we show the BER approximations as functions of the
signal-to-noise ratio (SNR) |g0 |2 . Based on the recommen-
dations in [2], [3], and [4], we used 16 nonzero terms with
T = 30 for the Fourier Series method, and NT = 13, 15
terms for the Taylor Series and for the Legendre Polynomials
method, respectively. The negative exponentials series [3] was
excluded from our comparisons since use of a constant β for
a range of SNRs yielded poor results. The presented study
shows that the derived expression compares favorably to the
methods in [2], [3], and [4]. Indeed, the derived expression
significantly extends the range of BER values for which
accurate estimates can be obtained and it provides accurate
performance estimates for BER values that are several orders −1
10
of magnitude lower requiring significantly fewer terms in the
−2

finite series approximation of Q(·). 10

−3
10

R EFERENCES −4
10

[1] O. Fonseca and I. N. Psaromiligkos, “Approximation of the Bit-Error- −5

Rate of BPSK Transmissions Over Frequency Selective Channels,” in 10

Proc. IEEE Intern. Conf. on Wirel. and Mob. Comp., Netw. and Commun.
BER

−6
10
(WiMob 2005), August 2005.
[2] N. C. Beaulieu, “The Evaluation of Error Probabilities for Intersymbol −7
10
and Cochannel Interference,” IEEE Trans. Commun., vol. 39, no. 12 ,
Dec. 1991, pp. 1740 - 1749. −8
10
[3] J. Murphy, “Binary Error Rate Caused by Intersymbol Interference and
Gaussian Noise,” IEEE Trans. Commun., vol. 21, no. 9, Sep. 1973, pp. −9
10
Proposed
1039 - 1046. Fourier Series [2]
Taylor Series [4]
[4] E. Y. Ho and Y. S. Yeh, “A new approach for evaluating the error
−10
10
Legendre Polynomials [3]
probability in the presence of intersymbol interference and additive −11
Exact

Gaussian Noise,” Bell Syst. Tech. J., vol. 49, Nov. 1970, pp. 2249-2265. 10
10 12 14 16 18 20 22 24 26
SNR,dB

Fig. 1. BER approximations comparisons (Note: The method in [3] produces

negative BER estimates for SNRs greater than 21dB).