Abstract
Abstract-The performance of weak signal detectors is addressed in additive noise described by the first order moving average (FOMA) of an impulsive process. The decision regions of the maximum likelihood (ML) and suboptimum ML (S-ML) detectors are derived in the FOMA model. The ML and S-ML detectors are employed in the antipodal signaling system, and compared in terms of the bit-error-rate performance in impulsive environment. Numerical results show that the S-ML detector, despite its reduced complexity and simpler structure, exhibits practically the same performance as the ML detector. It is also shown that the performance gap between detectors for FOMA and independent and identically distributed noise becomes larger as the degree of noise impulsiveness increases.
I. INTRODUCTION
It is usually assumed that the sampled noise components are statistically independent in signal detection problems. The independent noise assumption, however, is frequently violated in discrete-time signal detection applications. As a consequence, a detector optimized for the independent noise is often not guaranteed to be optimum in many practical signal detection systems, which becomes more critical as the sampling rate gets higher.
A number of dependent observation models, taking into account the dependence among noise components, have been proposed and investigated to address such a situation [1] - [5] . Typical examples of dependent noise models include the mdependent [1] , φ-mixing [2] , and transformation noise [4] , [5] models.
Among the models employed for the description of dependent environment, the transformation noise designates a linear transformation of an independent random process, and has been shown to be useful in deriving simpler structures of the locally optimum detectors in dependent noise environment. When the dependence of noise is weak, we can model the dependent noise components as a simple first order moving average (FOMA) [1] . The FOMA model has proved to be a good approximation to the weakly-dependent noise environment [1] , [5] .
In the meantime, the problem of designing detectors for weak signals in the FOMA noise has been considered in several studies, in which the asymptotic and finite samplesize performance of the LO detector with or without memory units is investigated. It is noteworthy that the LO detectors have been extensively investigated, for example, in [6] - [11] , because of the simple detector structure in non-Gaussian noise environment and of the almost optimum performance even at large signal strength in many cases, In this paper, we consider detection of weak signals in weakly-dependent impulsive noise described by the FOMA model. It is worthwhile to consider signal detection in impulsive noise environment since many noise sources involved in practical communication systems are clearly non-Gaussian [12] , [13] .
Specifically, we obtain the decision regions of the maximum likelihood (ML) and suboptimum ML (S-ML) detectors in the FOMA noise model, and compare the performance of the ML and S-ML detectors. In the antipodal signaling system, the bit-error-rate (BER) performance of the S-ML detector is investigated and compared to that of the ML detector in the FOMA noise environment.
The organization of this paper is as follows. Details of the signal detection problem and observation model are described in Section 2. The decision regions of the ML and S-ML detectors are obtained for the FOMA noise model in Section 3. An application of the ML and S-ML detectors in the antipodal signaling systems is discussed in Section 4, where the BER performance of the ML and S-ML detectors is obtained and compared via Monte Carlo simulations. 
where w(t) is a sample function of the additive noise process and T s is the signaling interval. When we extract samples from the received signal x(t), let the sampling period (the time between samples) t ∆ satisfy
Then, n observations can be obtained over every signaling interval T s , with which we construct the discrete-time observation model given by
where {X i } are the discrete-time observations (data), {s k,i } are the transmitted signal components, and {W i } are the random noise components. We assume that the transmitted signal vector
in (3) can be expressed as
In (5), θ is the common factor of the signal strength, k is the non-negative proportionality constant for the strength of signal vector s k , ands
is the unit energy version of s k , where · 2 is the Euclidean norm.
The problem of detecting M -ary signals can be modelled as an M -ary hypothesis testing problem in statistical inference, in which the hypotheses {H k } M k=1 are described by
where
is the observation vector and
is the random noise vector.
Assuming that the degree of dependence among noise components is weak, the dependent noise components {W i } in this paper are modelled by the FOMA of independent and identically distributed (i.i.d.) random variables.
Specifically, let
where {Λ i } form a zero-mean i.i.d. random process with Λ 0 = 0 and univariate probability density function (pdf) f Λ (·), and ρ d is called the dependence parameter. Clearly, the noise components {W i } are independent when ρ d = 0. The dependence parameter ρ d allows a simple single-parameter characterization of the degree of dependence among noise components. For the observation model (7) with the noise components {W i } described by (10) , the joint pdf of X is
where
Defining the transformed observations
and noting that the Jacobian of the transformation from X to
is 1, the joint pdf of Y is
where y = (y 1 , y 2 , · · · , y n ) and
is the transformed signal component. When ρ d = 0, the model (10) is simply W i = Λ i and consequently
is equal to the joint pdf φ X (x|H k , θ) of X. In essence, the original observation space described by
can be moved into the transformed observation space described by
where
and
III. DETECTION IN IMPULSIVE FOMA NOISE

A. Decision criteria
When the signal strength approaches zero, the probability P e (θ) of error is minimized under the S-ML criterion [14] if the decision region D S k for the hypothesis H k is determined as
where
is called the locally optimum nonlinearity [5] , and
denotes the difference between the signal components of the k-th and m-th transformed observations with the dependence taken into account. Employing the ML criterion, on the other hand, we obtain the decision region
for the ML detector optimized for the dependent noise W i described by (10) . By replacing y i , b k,i , and b m,i with x i ,s k,i , ands m,i , respectively, in (21) and (24), it is straightforward to get the decision regions of the S-ML and ML detectors optimized for the i.i.d. noise.
B. Decision regions
In practical communication systems Many signal and noise sources are obviously non-Gaussian. The statistical characteristics of the non-Gaussian (impulsive) sources may be described by heavy-tailed distributions including the symmetric α-stable (SαS), t, and Cauchy [5] , [12] , [15] distributions.
Particularly, the SαS distribution has been tested with a variety of real data and found to match the real data with high fidelity. The t-distribution is also an important class in the theory of statistics as it arises naturally in sampling from a Gaussian distributed population [5] .
1) Decision regions in FOMA SαS noise: Assume that the common pdf of {Λ i } is the SαS pdf
where the dispersion parameter γ > 0 is related to the spread of the SαS distribution and the characteristic exponent α (0 < α ≤ 2) controls the heaviness of tails. When α = 2, the SαS pdf is a Gaussian pdf, and we get
and
In the i.i.d. Gaussian noise, the decision regions of the S-ML and ML detectors can be obtained from (26) and (27), respectively, with
and y i = x i as mentioned before. Clearly,
when θ → 0. When α = 1, the SαS pdf is the Cauchy pdf, and the decision regions are
and
for the S-ML and ML detectors, respectively, since
The decision regions of the S-ML and ML detectors for i.i.d. Cauchy noise can be obtained easily from (31) and (32), respectively.
We would like to mention that the lack of a closed-form expression of the SαS pdf (25) except for the cases α = 1 and 2 prohibits the evaluation of the general decision regions. Thus, the Cauchy-optimized detector (32) has been used as a useful detector under general impulsive circumstances [12] .
2) Decision regions in FOMA t-noise: Assume that
where the degree of freedom ν is related to the heaviness of the tail of the t-pdf (34), with a smaller value representing a more impulsive pdf. The t-pdf (34) becomes a Cauchy pdf when ν = 1, and its limit as ν → ∞ is a Gaussian pdf since
as ν → ∞. It is straightforward to obtain the decision regions
and
for the S-ML and ML detectors, respectively, since
The decision regions (36) and (37) are essentially the same as (31) and (32) obtained for the Cauchy noise, respectively, because the Cauchy noise is a special case of the t-noise. The decision regions of the S-ML and ML detectors for i.i.d. tnoise can be obtained easily from (36) and (37), respectively.
IV. PERFORMANCE IN ANTIPODAL SIGNALING

A. Decision structure
Detection of antipodal signals in the transformed observation space described by Y in (18) can be expressed as a binary testing problem of the hypotheses
Assuming that
ands
without loss of generality, we have 
Then, the decision rules are
and
for the S-ML and ML detectors, respectively. To fully exploit the dependence among n noise components in the FOMA model, a detector is to use n − 1 memory units. The S-ML detector (44), on the other hand, can be implemented with one-memory unit as shown in Fig. 1 , by noting that
with
B. Numerical results and discussions
In order to assess the performance of detection schemes in SαS environment, we employ the geometric SNR (G-SNR) [16] . The G-SNR for the transformed observation space described by Y with SαS noise can be defined as
where
is the power of the transformed signal θ k b k and
is the exponential of the Euler constant. Note that the G-SNR represents the standard SNR when α = 2. For convenience, we have named various detectors using F and I to represent the FOMA and i.i.d. noise, respectively, and G, C, and T to denote the Gaussian, Cauchy, and t distributions, respectively. For example, the FG ML detector denotes the ML detector optimized for FOMA Gaussian noise and the IC S-ML detector denotes the S-ML detector optimized for i.i.d. Cauchy noise. Monte Carlo simulations are performed with 10 6 runs at each BER assuming γ = 1 without loss of generality.
As clearly shown in Figure 2 , the FG and IG detectors become practically useless when the impulsiveness of noise gets severer, while the performance of the FC and IC detectors gets better. In addition, the BER performance gaps between the FC S-ML and FC ML detectors and between the IC S-ML and IC ML detectors become negligible as the degree of noise impulsiveness increases. The performance of the IC S-ML and IC ML detectors is better than that of the FG and IG detectors: yet, it deviates farther from the optimum performance of the FC detectors as the value of α decreases.
V. CONCLUSION
Detection of weak signals is addressed in the FOMA model. We have derived the decision regions of the S-ML and ML detectors in FOMA noise: specific decision regions of the S-ML and ML detectors in SαS and t distributions are obtained for performance analysis. The BER performance of the S-ML and ML detectors in antipodal signaling systems is then investigated in impulsive FOMA noise environment via Monte Carlo simulations.
Through numerical results, we have shown that, despite the S-ML detector generally has a simpler structure than the ML detector, the performance difference between the ML and S-ML detectors is negligibly small in FOMA environment. It is observed that a detector designed for FOMA noise offers better performance than that for i.i.d. noise when there exists dependence between noise components. It is also observed that the performance gap between detectors for FOMA and i.i.d. noise becomes more significant as the impulsiveness of the FOMA noise gets severer.

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