A Low Complexity Piecewise Suboptimal Detectorfor Signals in Alpha-Stable Interference

Tarik S. Shehata, Ian Marsland, and Mohamed El-Tanany

Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada

Email:{tshehata, ianm, tanany}@sce.carleton.ca

Abstract—The design of near-optimal detectors for binary sig-nals in α-stable interference using two receive antennas is consid-ered. The optimal detector requires more complex computation,whereas the linear (Gaussian) detector is practically simple butsuffers from performance degradation with α-stable noise. Inthis paper, based on the optimal decision regions analysis, wepropose to approximate the optimal decision boundaries by usinglinear segments which results in a novel piecewise detector. Theproposed detector has much less complexity over the optimaldetector. Also, the analytical evaluation shows near-optimal per-formance of the piecewise detector. Moreover, simulation resultsshow that the proposed detector has robust performance fordifferent values of α.

I. INTRODUCTION

The symmetric alpha stable (SαS ) distribution has been

widely used to model the aggregate interference in modern

wireless networks. The SαS distribution is used to model the

interference in the wireless ad hoc networks and wireless

packet networks [1], [2], and is proposed to model the inter-

ference in cognitive radio networks, wireless packet networks

[3], and multiple access interference in ultrawide band systems

[4].

We consider the detection of signals using independent

samples from two antennas at the receiver1. The optimal

maximum likelihood (ML) detector for signals in SαS noise

requires complex computation of numerical integration or

DFT operations due to the lack of a closed form expression

for the probability density function [5]. The conventional

linear (Gaussian) detector, which is optimal when α = 2,

is much simpler, however, it performs poorly when α is

smaller as demonstrated and justified in [6]. In this paper,

we propose a new suboptimal detector that has a near optimal

performance yet has much less complexity compared to the

optimal detector. The design of the proposed detector depends

on the analysis of the optimal decision regions proposed in

[6], which gives insights on the optimal detector behavior

in processing the received samples. Based on the derived

optimal decision boundaries, we propose a simple piecewise

suboptimal detector that approximates the decision boundaries

of the optimal detector using linear segments. The performance

of the proposed piecewise detector approaches the optimal one

for different values of α with much less complexity compared

to the optimal ML detector.

1For the case when one antenna is used, the ML detector reduces to thelinear (Gaussian) detector.

The rest of the paper is organized as follows. In Section

II, the system model is described, and a background on the

analysis of the optimum decision regions is presented. In

Section III, the piecewise detector is proposed. In Section

IV, the performance of the proposed detector is evaluated

analytically and by simulation.

II. SYSTEM MODEL AND BACKGROUND

A. System Model

Consider a wireless communication system where the re-

ceiver uses two antennas. The two channels are assumed to

have equal gain as in the case of line-of-sight reception. The

two antennas are assumed to be far enough apart to have

independent interference samples. The received signal vector

is r = [r1r2], where

ri = s + ni i = 1, 2 (1)

where s ∈ ±A is a BPSK symbol, and n1 and n2 are indepen-

dent SαS interference samples. The α-stable distribution has

no closed form expression for the probability density function

except for two special cases when α = 1 (Cauchy distribution)

and α = 2 (Gaussian distribution), so, the characteristic

function is used to best describe the α-stable distribution:

φX(w) = e−γ|w|α (2)

where γ is the noise dispersion, which measures the dis-

tribution’s spread around its center and it is analogous to

the variance in case of Gaussian distribution, and α is the

characteristic exponent, which measures the thickness of the

distribution’s tails; a small value of α implies more impulses in

the noise samples. The probability density function, fα(x), can

be calculated numerically using the inverse Fourier transform

of (2).

The optimal maximum likelihood (ML) detector computes

the following test statistics:

log{

fα[r1 − A]fα[r2 − A]fα[r1 + A]fα[r2 + A]

} s1�<s0

0 (3)

where s0 is the hypothesis that s = −A was transmitted, and

s1 is the hypothesis that s = +A was transmitted. It can be

seen that the optimal detector requires complex computations

to calculate the likelihood function.

978-1-4244-2519-8/10/$26.00 ©2010 IEEE

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

s1

s0

r1

r2

Fig. 1. Optimal decision regions with α = 1, SNR = 0 dB. Shaded area:decide s1, white area: decide so.

B. Optimal decision regions

Analyzing the decision regions of the optimal detector is a

new tool to understand the behavior of the optimum detector

of signals in SαS noise [6]. By assuming that the channel

is constant and has equal gain for the two antennas at the

receiver, the optimal decision boundaries can be derived for

the case when α = 1 to be:

r2 = −r1 (4a)

r2 = −γ2 + A2

r1(4b)

Fig. 1 shows the optimal decision regions for α = 1. By

comparing these optimal decision regions with those of the

linear detector2, it can be seen that the linear detector has poor

performance compared to the optimum detector. Also, Fig. 2

shows the optimal decision boundaries (found numerically)

in the fourth quadrant only evaluated at different values of

SNR. As shown in the figure, the optimal decision boundaries

for α �= 1 shares the linear boundary in (4a) and the curved

boundaries are getting close to the case when α = 1 as the

SNR increases. So, at higher SNR, the curved boundaries

for different values of α can be approximated by (4b). In

the following section, we will show how to use the optimal

decision boundaries to design a near optimal low-complexity

detector.

III. PIECEWISE SUBOPTIMAL DETECTORS

Based on the analysis of the optimal decision regions, we

propose new suboptimal detectors that linearly approximate

the optimal decision boundaries shown in Fig. 1.

1) Inverse-linear Detector:As shown in Fig. 1, the optimal decision boundaries in the

2The decision boundary for the linear detector is r2 = −r1.

0 10 20 30 40−40

−30

−20

−10

0 0 dB10 dB

15 dB

20 dB

r1

r2

α = 0.7α = 1α = 1.5α = 1.9

Fig. 2. Optimal curved decision boundaries.

second and fourth quadrants can be approximated such

that the decision regions are{s = +A if r2 � −r1

s = −A if r2 > −r1

(5)

This detector looks like the linear detector but with

reverse decision regions; however, as shown in the next

section, it has the same performance as the linear detector.

2) Piecewise Detector:In this detector, the curved boundaries in (4b) are approx-

imated by using piecewise symmetric linear segments

passing by the boundaries cross point (±a0,∓ao) as

shown in Fig. 3. Due to symmetry, the fourth quadrant is

considered where the linear segment is

r2 = mr1 − (m + 1)ao (6)

where ao =√

γ2 + A2 and m is the slope of the

segment. The linear segment crosses the boundary (4b)

in two points (a0,−ao) and ( ao

m ,−aom). When m = 1,

the line in (6) is a tangent to the boundary at the point

(ao,−ao), while at m > 1 and m < 1, the line forms

the two segments shown in Fig. 3. As can be seen,

the accuracy of the linear approximation in (6) depends

on the slope m. We propose to choose m such that

the excess probability of error, β, is minimized. The

excess probability of error is defined as the difference

between the probability of bit error of the piecewise and

optimal detectors. By comparing the decision regions in

which both the optimum and the piecewise detector have

incorrect decisions, the excess probability of error, β, is

reduced to:

β = 2×⎡⎣∫∫

R1

Λα(r1, r2) dr2 dr1 +∫∫R2

Λα(r1, r2) dr2 dr1

⎤⎦

(7)

−10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

s1

s0 (ao,−ao)

(ao

m,−aom)

r1

r2

Fig. 3. Decision boundaries of the piecewise detector.

1 1.5 2 2.5 3 3.5 410−5

10−4

10−3

10−2

40 dB

0 dB

m

β

Fig. 4. The excess probability of error (β) evaluated at SNR = 0, 1, ..., 40dB.

where Λα(r1, r2) = fα(r1, r2|s = +A) − fα(r1, r2|s =−A) and R1 : {r2 = −r1 → −−a2

o

r1, r1 = ao

m → ao} and

R2 : {r2 = −−a2o

r1→ −r1, r1 = 0 → ao

m }.

As can be seen, it is difficult to minimize β analytically,

instead, it is evaluated numerically. Fig. 4 shows β at

different values of the SNR. The optimal slope, mopt,

can be calculated numerically for different values of γand SNR and stored in memory, and also, it can be fitted

linearly with SNR. Moreover, Fig. 4 suggests m = 1 as a

simple choice of the slope value. As shown in the figure,

the difference between the excess probability of error, β,

when m = 1 and m = mopt is small enough to use

m = 1 as a good approximation of the optimal slope.

IV. PERFORMANCE EVALUATION AND SIMULATION

RESULTS

In this section, the performance of the optimal, the linear

and the piecewise detectors will be evaluated analytically in

the case when α = 1.

A. Optimum Detector

By using the decision regions in Fig. 1, the conditional

probability of correct decision can be evaluated as follows:

Pc|−A =∫∫Z

fα(r1, r2|s = −A) dr1 dr2 (8)

= 2 ×ao∫

r1=0

−r1∫r2=− a2

or1

fα(r1, r2|s = −A) dr1 dr2+

2 ×∞∫

r1=ao

− a2o

r1∫r2=−r1

fα(r1, r2|s = −A) dr1 dr2+

0∫r1=−∞

0∫r2=−∞

fα(r1, r2|s = −A) dr1 dr2

where ao =√

γ2 + A2 and fα(r1, r2|s = −A) is the

conditional joint distribution of the received samples. For the

case when α = 1,

fα(r1, r2|s = −A) =γ2

π2

2∏i=1

1γ2 + (ri + A)2

. (9)

Assuming that the transmitted symbols are equally likely, the

average probability of error is:

Pe = 1 − (Pc|−A) (10)

The double integral in (8) can be reduced to a single integral,

which can be simply evaluated numerically, by applying the

following identity:

b∫r1=a

f2(r1)∫r2=f1(r1)

fα(r1, r2|s = −A) dr1 dr2

=γ2

π2

b∫r1=a

tan−1[

A+f2(r1)γ

]− tan−1

[A+f1(r1)

γ

]γ [γ2 + (r1 + A)2]

dr1 (11)

B. Linear Detector

The probability of correct decision of the linear detector can

be calculated analytically by changing the decision boundaries

in (8):

P c|−A =∫∫Z

fα(r1, r2|s = −A) dr1 dr2

= 2 ×∞∫

r1=0

−r1∫r2=−∞

fα(r1, r2|s = −A) dr1 dr2+

0∫r1=−∞

0∫r2=−∞

fα(r1, r2|s = −A) dr1 dr2 (12)

Then, the probability of error can be calculated using (10).

For the case of the Inverse-linear detector, by using the

corresponding decision regions as illustrated by (5), it can be

shown that the resultant probability of error is the same as that

of the linear detector.

C. Piecewise Detector

To evaluate the performance of the piecewise detector, the

decision regions in Fig. 3 are used to construct the integration

limits of the joint probability density function in (8). So the

conditional probability of correct decision of the piecewise

detector is:

P c|−A =

= 2 ×ao∫

r1=0

∫ −r1

r2=mr1−(m+1)ao

fα(r1, r2|s = −A) dr1 dr2+

2 ×(1+ 1

m )ao∫r1=ao

mr1−(m+1)ao∫r2=−r1

fα(r1, r2|s = −A) dr1 dr2+

2 ×∞∫

r1=(1+ 1m )ao

0∫r2=−r1

fα(r1, r2|s = −A) dr1 dr2+

0∫r1=−∞

0∫r2=−∞

fα(r1, r2|s = −A) dr1 dr2 (13)

The performance of the piecewise detector is compared to

the optimal and linear detectors and validated by simulation

results in Fig. 5. It can be seen that the performance of the

piecewise detector approaches the optimal one (within 0.2 dB).

D. Simulation Results

To evaluate the performance of the proposed detectors, the

probability of error of the optimal and suboptimal piecewise

detector is evaluated by simulation for the case when α = 0.7,

1, 1.5 and 1.9. Because the noise variance is not defined for

values of α < 2, the noise dispersion, γ, is used instead to

define the SNR as Es4γ , where Es = 2A2 is the received energy

per symbol3. γ is set to 1 in all simulations. The receiver uses

two antennas to get two independent interference samples. To

calculate an accurate Pe, up to 108 frames with 250 bits each

3Note that when α = 2, the noise variance is defined as 2γ, whichcorresponds to No

2for an AWGN channel.

are transmitted, and the simulation is stopped when the number

of bit errors reaches 104. The simulation results validate the

analytic performance of the piecewise detector in case when

α = 1, and also, the performance is evaluated when α �= 1.

Fig. 5 shows the performance of the piecewise detector with

m = 1 and m = mopt. It can be seen that the performance of

the proposed piecewise detector approaches the optimal one at

much less complexity compared to the optimal ML detector

which requires numerical integration and/or DFT operations

to implement the test statistics in (3). Also, when m = 1 is

used, the performance of the piecewise detector is only 0.2dB less than the optimal one at a BER of 4 × 10−3. When

m = mopt is used, the performance of the piecewise detector

becomes more close to the optimal performance because it has

the minimum excess probability of error, which is consistent

with the analysis in Section III.

Although the proposed piecewise detector is designed for

the case when α = 1, it shows robust performance for

different values of α. Fig. 6 and Fig. 7 show the near optimal

performance of the proposed detector for α = 0.7 and 1.5,

respectively. Fig. 8 shows the performance when α = 1.9. It

can be seen that the performance of the piecewise detector

approaches the optimal performance at higher SNR, however,

it is almost 1 dB less that the optimal at lower SNR. This

performance is expected because, at higher SNR, the optimal

decision boundaries when α approaches 2 become more close

to the case when α = 1, as shown in Fig. 2, so the optimal

boundaries can be well approximated by (4). However, this

approximation becomes loose at lower values of SNR, which

results in performance degradation of the piecewise detector.

This degradation can be improved by adjusting the piecewise

detector parameters such as the cross point, ao, and the slope

m to give a better approximation of the optimal boundaries

when α �= 1.

V. CONCLUSION

A new piecewise suboptimal detector is proposed to detect

signals in symmetric α−stable interference using two receive

antennas. The proposed detector approximates the optimal

decision boundaries by using linear segments which reduces

the complexity compared to the optimal detector. The analyt-

ical performance evaluation and the simulation results show

that the piecewise detector has a superior performance over

the conventional linear detector and also it has near optimal

performance. Moreover, the piecewise detector shows robust

performance for different values of α.

REFERENCES

[1] E. Sousa, “Interference modeling in a direct-sequence spread-spectrumpacket radio network,” IEEE Transactions on Communications, vol. 38,no. 9, pp. 1475–1482, Sep 1990.

[2] J. Ilow and D. Hatzinakos, “Analytic alpha-stable noise modeling in apoisson field of interferers or scatterers,” IEEE Transactions on SignalProcessing, vol. 46, no. 6, pp. 1601–1611, Jun 1998.

[3] M. Win, P. Pinto, and L. Shepp, “A mathematical theory of networkinterference and its applications,” Proceedings of the IEEE, vol. 97, no. 2,pp. 205–230, Feb. 2009.

0 5 10 15 20 25 3010−4

10−3

10−2

10−1

SNR (dB)

Pe

OptimumPiecewise (m = 1)Piecewise (m = mopt)Linear

Fig. 5. Analytical and simulated probability of error with α = 1. Analytic:solid lines, simulations: markers only.

20 25 30 35 40 45 5010−4

10−3

10−2

10−1

SNR (dB)

BE

R

OptimumPiecewise (m = 1)Piecewise (m = mopt)Linear

Fig. 6. Performance comparison with α = 0.7.

[4] S. Niranjayan and N. C. Beaulieu, “A myriad filter detector for UWBmultiuser communication,” in IEEE International Conference on Com-munications, 2008. ICC ’08., May 2008, pp. 3918–3922.

[5] C. L. Nikias and M. Shao, Signal Processing with Alpha-Stable Distri-butions and Applications. John Wiley & Sons Canada, 1995.

[6] T. S. Shehata, I. Marsland, and M. El-Tanany, “A novel frameworkfor signal detection in alpha-stable interference,” in IEEE VehicularTechnology Conference, 2010. VTC ’10., May 2010.

0 5 10 15 2010−5

10−4

10−3

10−2

10−1

SNR (dB)

BE

R

OptimumPiecewise (m = 1)Piecewise (m = mopt)Linear

Fig. 7. Performance comparison with α = 1.5.

0 5 10 15 2010−5

10−4

10−3

10−2

10−1

SNR (dB)

BE

R

OptimumPiecewise (m = 1)Piecewise (m = mopt)Linear

Fig. 8. Performance comparison with α = 1.9.