**2. GASP: Brief description**

GASP is based on the assumption that the frequency-time region *Z* of the noise exists where a signal may be present; for example, there is an observed stochastic sample from this region, relative to which it is necessary to make the decision a "yes" signal (the hypothesis *H*<sup>1</sup> ) or a "no" signal (the hypothesis *H*<sup>0</sup> ). We now proceed to modify the initial premises of the classical and modern signal processing theories. Let us suppose there are two independent frequency-time regions *Z* and *Z* belonging to the space *A* . Noise from these regions obeys the same pdf with the same statistical parameters (for simplicity of considerations). Generally, these parameters are differed. A "yes" signal is possible in the noise region *Z* as before. *It is known a priori that a "no" signal is obtained in the noise region Z* . It is necessary to make the decision a "yes" signal (the hypothesis *H*<sup>1</sup> ) or a "no" signal (the hypothesis *H*<sup>0</sup> ) in the observed stochastic sample from the region *Z* , by comparing statistical parameters of pdf of this

les from two independent frequency-time regions – a "yes" signal is possible in the first region and it is known a priori that a "no" signal is obtained in the second region. The proposed GASP allows us to formulate a decision-making rule based on the determination of *the jointly sufficient statistics of the mean and variance* of the likelihood function (or functional). Classical and modern signal processing theories allow us to define *only the mean* of the likelihood function (or functional). Additional information about the statistical characteristics of the likelihood function (or functional) leads us to better quality signal detection and definition of signal parameters in compared with the optimal signal processing algorithms of classi-

Thus, for any wireless communication systems, we have to consider two problems – analysis and synthesis. The first problem (analysis) – the problem to study a stimulus of the additive and multiplicative noise on the main principles and performance under the use of GASP – is an analysis of impact of the additive and multiplicative noise on the main characteristics of wireless communication systems, the receivers in which are constructed on the basis of GASP. This problem is very important in practice. This analysis allows us to define limitations on the use of wireless communication systems and to quantify the additive and multiplicative noise impact relative to other sources of interference present in these systems. If we are able to conclude that the presence of the additive and multiplicative noise is the main factor or one of the main factors limiting the performance of any wireless communication systems, then the second problem – the definition of structure and main parameters and characteristics of the generalized detector or receiver (GD or GR) under a dual stimulus of

GASP allows us to extend the well-known boundaries of the potential noise immunity set by classical and modern signal processing theories. Employment of wireless communication systems, the receivers of which are constructed on the basis of GASP, allows us to obtain high detection of signals and high accuracy of signal parameter definition with noise components present compared with that systems, the receivers of which are constructed on the basis of classical and modern signal processing theories. The optimal and asymptotic optimal signal processing algorithms of classical and modern theories, for signals with amplitudefrequency-phase structure characteristics that can be known and unknown a priori, are con-

stituents of the signal processing algorithms that are designed on the basis of GASP.

GASP is based on the assumption that the frequency-time region *Z* of the noise exists where a signal may be present; for example, there is an observed stochastic sample from this region, relative to which it is necessary to make the decision a "yes" signal (the hypothesis *H*<sup>1</sup> ) or a "no" signal (the hypothesis *H*<sup>0</sup> ). We now proceed to modify the initial premises of the classical and modern signal processing theories. Let us suppose there are two independent frequency-time regions *Z* and *Z* belonging to the space *A* . Noise from these regions obeys the same pdf with the same statistical parameters (for simplicity of considerations). Generally, these parameters are differed. A "yes" signal is possible in the noise region *Z* as before. *It is known a priori that a "no" signal is obtained in the noise region Z* . It is necessary to make the decision a "yes" signal (the hypothesis *H*<sup>1</sup> ) or a "no" signal (the hypothesis *H*<sup>0</sup> ) in the observed stochastic sample from the region *Z* , by comparing statistical parameters of pdf of this

the additive and multiplicative noise – the problem of synthesis – arises.

cal or modern theories.

**2. GASP: Brief description** 

sample with those of the sample from the reference region *Z* . Thus, there is a need to accumulate and compare statistical data defining the statistical parameters of pdf of the observed input stochastic samples from two independent frequency-time regions *Z* and *Z* . If statistical parameters for two samples are equal or agree with each other within the limits of a given before accuracy, then the decision of a "no" signal in the observed input stochastic process <sup>1</sup> , , *X X <sup>N</sup>* is made – the hypothesis *H*<sup>0</sup> . If the statistical parameters of pdf of the observed input stochastic sample from the region *Z* differ from those of the reference sample from the region *Z* by a value that exceeds the prescribed error limit, then the decision of a "yes" signal in the region *Z* is made – the hypothesis *H*<sup>1</sup> .

Fig. 1. Definition of sufficient statistics under GASP.

The simple model of GD in form of block diagram is represented in Fig.2. In this model, we use the following notations: MSG is the model signal generator (the local oscillator), the AF is the additional filter (the linear system) and the PF is the preliminary filter (the linear system) A detailed discussion of the AF and PF can be found in (Tuzlukov, 2001 and Tuzlukov, 2002).

Fig. 2. Principal flowchart of GD.

Consider briefly the main statements regarding the AF and PF. There are two linear systems at the GD front end that can be presented, for example, as bandpass filters, namely, the PF with the impulse response ( ) *PF h τ* and the AF with the impulse response ( ) *AF h τ* . For simplicity of analysis, we think that these filters have the same amplitude-frequency responses and bandwidths. Moreover, a resonant frequency of the AF is detuned relative to a resonant frequency of PF on such a value that signal cannot pass through the AF (on a value that is higher the signal bandwidth). Thus, the signal and noise can be appeared at the PF output and *the only noise* is appeared at the AF output. It is well known, if a value of detuning between the AF and PF resonant frequencies is more than 4 5 *<sup>a</sup> f* , where *<sup>a</sup> f* is the signal bandwidth, the processes forming at the AF and PF outputs can be considered as independent and uncorrelated processes (in practice, the coefficient of correlation is not more than 0.05). In the case of signal absence in the input process, the statistical parameters at the AF and PF outputs will be the same, because the same noise is coming in at the AF and PF inputs, and we may think that the AF and PF do not change the statistical parameters of input process, since they are the linear GD front end systems.

By this reason, the AF can be considered as a generator of reference sample with *a priori* information *a "no" signal is obtained in the additional reference noise* forming at the AF output. There is a need to make some comments regarding the noise forming at the PF and AF outputs. If the Gaussian noise *n t*( ) comes in at the AF and PF inputs (the GD linear system front end), the noise forming at the AF and PF outputs is Gaussian, too, because the AF and PF are the linear systems and, in a general case, take the following form:

$$n\_{\rm PF}(t) = \int\_{-\infty}^{\infty} h\_{\rm PF}(\tau) n(t - \tau) d\tau \quad \text{and} \quad n\_{\rm AF}(t) = \int\_{-\infty}^{\infty} h\_{\rm AF}(\tau) n(t - \tau) d\tau \,. \tag{1}$$

If, for sake of simplicity, the additive white Gaussian noise (AWGN) with zero mean and two-sided power spectral density <sup>0</sup> 2*N* is coming in at the AF and PF inputs (the GD linear system front end), then the noise forming at the AF and PF outputs is Gaussian with zero mean and variance given by <sup>2</sup> <sup>2</sup> 0 0 <sup>2</sup> 8 *<sup>F</sup> <sup>n</sup> <sup>ω</sup> <sup>σ</sup> <sup>N</sup>* (Tuzlukov, 2002) where in the case if AF (or PF) is the RLC oscillatory circuit, the AF (or PF) bandwidth *<sup>F</sup>* and resonance frequency *ω*<sup>0</sup> are defined in the following manner , *<sup>F</sup>* 0 1 <sup>2</sup> , *<sup>R</sup> LC L* .The main functioning condition of GD is an equality over the whole range of parameters between the model signal *u t*( ) at the GD MSG output and the transmitted signal *u t*( ) forming at the GD input liner system (the PF) output, i.e. ( ) ( ) *ut u t* . How we can satisfy this condition in practice is discussed in detail in (Tuzlukov, 2002; Tuzlukov, 2012). More detailed discussion about a choice of PF and AF and their impulse responses is given in (Tuzlukov, 1998).

#### **3. Diversity problems in wireless communication systems with fading**

In the design of wireless communication systems, two main disturbance factors are to be properly accounted for, i.e. fading and additive noise. As to the former, it is usually taken into account by modeling the propagation channel as a linear-time-varying filter with random impulse response (Bello, 1963 & Proakis, 2007). Indeed, such a model is general enough to encompass the most relevant instances of fading usually encountered in practice, i.e.

Consider briefly the main statements regarding the AF and PF. There are two linear systems at the GD front end that can be presented, for example, as bandpass filters, namely, the PF with the impulse response ( ) *PF h τ* and the AF with the impulse response ( ) *AF h τ* . For simplicity of analysis, we think that these filters have the same amplitude-frequency responses and bandwidths. Moreover, a resonant frequency of the AF is detuned relative to a resonant frequency of PF on such a value that signal cannot pass through the AF (on a value that is higher the signal bandwidth). Thus, the signal and noise can be appeared at the PF output and *the only noise* is appeared at the AF output. It is well known, if a value of detuning between the AF and PF resonant frequencies is more than 4 5 *<sup>a</sup> f* , where *<sup>a</sup> f* is the signal bandwidth, the processes forming at the AF and PF outputs can be considered as independent and uncorrelated processes (in practice, the coefficient of correlation is not more than 0.05). In the case of signal absence in the input process, the statistical parameters at the AF and PF outputs will be the same, because the same noise is coming in at the AF and PF inputs, and we may think that the AF and PF do not change the statistical parameters of input process,

By this reason, the AF can be considered as a generator of reference sample with *a priori* information *a "no" signal is obtained in the additional reference noise* forming at the AF output. There is a need to make some comments regarding the noise forming at the PF and AF outputs. If the Gaussian noise *n t*( ) comes in at the AF and PF inputs (the GD linear system front end), the noise forming at the AF and PF outputs is Gaussian, too, because the AF and PF

and () ()( ) *nt h AF AF <sup>τ</sup> n t <sup>τ</sup> <sup>d</sup><sup>τ</sup>*

If, for sake of simplicity, the additive white Gaussian noise (AWGN) with zero mean and two-sided power spectral density <sup>0</sup> 2*N* is coming in at the AF and PF inputs (the GD linear system front end), then the noise forming at the AF and PF outputs is Gaussian with zero

RLC oscillatory circuit, the AF (or PF) bandwidth *<sup>F</sup>* and resonance frequency *ω*<sup>0</sup> are de-

GD MSG output and the transmitted signal *u t*( ) forming at the GD input liner system (the PF) output, i.e. ( ) ( ) *ut u t* . How we can satisfy this condition in practice is discussed in detail in (Tuzlukov, 2002; Tuzlukov, 2012). More detailed discussion about a choice of PF

In the design of wireless communication systems, two main disturbance factors are to be properly accounted for, i.e. fading and additive noise. As to the former, it is usually taken into account by modeling the propagation channel as a linear-time-varying filter with random impulse response (Bello, 1963 & Proakis, 2007). Indeed, such a model is general enough to encompass the most relevant instances of fading usually encountered in practice, i.e.

1

<sup>2</sup> , *<sup>R</sup> LC L*

  *<sup>ω</sup> <sup>σ</sup> <sup>N</sup>* (Tuzlukov, 2002) where in the case if AF (or PF) is the

.The main functioning condition of

. (1)

at the

are the linear systems and, in a general case, take the following form:

8 *<sup>F</sup> <sup>n</sup>*

and AF and their impulse responses is given in (Tuzlukov, 1998).

0

**3. Diversity problems in wireless communication systems with fading** 

GD is an equality over the whole range of parameters between the model signal *u t*( )

() ()( ) *nt h PF PF τ n t τ dτ* 

mean and variance given by <sup>2</sup> <sup>2</sup> 0 0 <sup>2</sup>

fined in the following manner , *<sup>F</sup>*

since they are the linear GD front end systems.

frequency- and/or time-selective fading, and flat-flat fading. As to the additive noise, such a disturbance has been classically modeled as a possibly correlated Gaussian random process.

However, the number of studies in the past few decades has shown, through both theoretical considerations and experimental results, that Gaussian random processes, even though they represent a faithful model for the thermal noise, are largely inadequate to model the effect of real-life noise processes, such as atmospheric and man-made noise (Kassam, 1988 & Webster, 1993) arising, for example, in outdoor mobile communication systems. It has also been shown that non-Gaussian disturbances are commonly encountered in indoor environments, for example, offices, hospitals, and factories (Blankenship & Rappaport, 1993), as well as in underwater communications applications (Middleton, 1999). These disturbances have an impulsive nature, i.e. they are characterized by a significant probability of observing large interference levels.

Since conventional receivers exhibit dramatic performance degradations in the presence of non-Gaussian impulsive noise, a great attention has been directed toward the development of non-Gaussian noise models and the design of optimized detection structures that are able to operate in such hostile environments. Among the most popular non-Gaussian noise models considered thus far, we cite the alpha-stable model (Tsihrintzis & Nikias, 1995), the Middleton Class-A and Class-B noise (Middleton, 1999), the Gaussian-mixture model (Garth & Poor, 1992) which, in turn, is a truncated version, at the first order, of the Middleton Class-A noise, and the compound Gaussian model (Conte et al., 1995). In particular, in the recent past, the latter model, subsuming, as special cases, many marginal probability density functions (pdfs) that have been found appropriate for modeling the impulsive noise, like, for instance, the Middleton Class-A noise, the Gaussian-mixture noise (Conte, 1995), and the symmetric alpha-stable noise (Kuruoglu, E. et al., 1998). They can be deemed as the product of a Gaussian, possibly complex random process times a real non-negative one.

Physically, the former component, which is usually referred to as speckle, accounts for the conditional validity of the central limit theorem, whereas the latter, the so-called texture process, rules the gross characteristics of the noise source. A very interesting property of compound-Gaussian processes is that, when observed on time intervals whose duration is significantly shorter than the average decorrelation time of the texture component, they reduce to spherically invariant random processes (SIRPs) (Yao, 1973), which have been widely adopted to model the impulsive noise in wireless communications (Gini, F et al., 1998), multiple access interference in direct-sequence spread spectrum cellular networks (Sousa, 1990), and clutter echoes in radar applications (Sangston & Gerlach, 1994).

We consider the problem of detecting one of *M* signals transmitted upon a zero-mean fading dispersive channel and embedded in SIRP noise by GD based on the GASP in noise. The similar problem has been previously addressed. In (Conte, 1995), the optimum receiver for flat-flat Rayleigh fading channels has been derived, whereas in (Buzzi et al., 1999), the case of Rayleigh-distributed, dispersive fading has been considered. It has been shown therein that the receiver structure consists of an estimator of the short-term conditional, i.e. given the texture component, noise power and of a bank of *M* estimators-correlators keyed to the estimated value of the noise power. Since such a structure is not realizable, a suboptimum detection structure has been introduced and analyzed in (Buzzi et al., 1997).

We design the GD extending conditions of (Buzzi et al., 1997) and (Buzzi et al., 1999) to the case that a diversity technique is employed. It is well known that the adoption of diversity techniques is effective in mitigating the negative effects of the fading, and since conventional diversity techniques can incur heavy performance loss in the presence of impulsive disturbance (Kassam & Poor, 1985), it is of interest to envisage the GD for optimized diversity reception in non-Gaussian noise. We show that the optimum GD is independent of the joint pdf of the texture components on each diversity branch. We also derive a suboptimum GD, which is amenable to a practice. We focus on the relevant case of binary frequency-shift-keying (BFSK) signaling and provide the error probability of both the optimum GD and the suboptimum GD. We assess the channel diversity order impact and noise spikiness on the performance.

#### **3.1 Problem statement**

The problem is to derive the GD aimed at detecting one out of *M* signals propagating through single-input multiple-output channel affected by dispersive fading and introducing the additive non-Gaussian noise. In other words, we have to deal with the following *M*-ary hypothesis test:

$$\mathcal{H}\_i \Rightarrow \begin{cases} \mathbf{x}\_1(t) = \mathbf{s}\_{1,i}(t) + n\_1(t) \\ \dots = \mathbf{s}\_{1,\dots,\mathbf{m}\_i} = \dots = \mathbf{s}\_{1,\dots,\mathbf{m}\_i} \\ \mathbf{x}\_P(t) = \mathbf{s}\_{P,i}(t) + n\_P(t) \end{cases} \quad i = 1, \dots, M \quad t \in [0, T] \,\prime \end{cases} \tag{2}$$

where *P* is the channel diversity order and[0, ] *T* is the observation interval; the waveforms <sup>1</sup> { } ( ) *<sup>P</sup> p p x t* are the complex envelopes of the *P* distinct channel outputs; , 1 { } ( ) , 1, *<sup>P</sup> pi p st i* ,*M* represent the baseband equivalents of the useful signal received on the *P* diversity branches under the *i*th hypothesis. Since the channel is affected by dispersive fading, we may assume (Proakis, 2007) that these waveforms are related to the corresponding transmitted signals ( ) *u t <sup>i</sup>*

$$s\_{p,i}(t) = \bigcap\_{-\alpha}^{\alpha} h\_p(t,\tau)u\_i(t-\tau)d\tau \quad , \qquad t \in [0, T] \tag{3}$$

where ( , ), 1, , *<sup>p</sup> h t p P* is the random impulse response of the channel *p*th diversity branch and is modeled as a Gaussian random process with respect to the variable *t*. In keeping with the uncorrelated-scattering model, we assume that the random processes ( , ), 1, , *<sup>p</sup> h t p P* are all statistically independent; as a consequence, the waveforms , 1 { } ( ) *<sup>P</sup> <sup>p</sup> <sup>i</sup> <sup>p</sup> s t* are themselves independent complex Gaussian random processes that we assume to be zero-mean and with the covariance function

$$\text{Cov}(t,\tau) = \text{E}[\mathbf{s}\_{p,i}(t)\mathbf{s}\_{p,i}^{\star}(\tau)] \quad , \quad i = 1,\ldots,M \quad t,\tau \in [0,T] \tag{4}$$

independent of *p* (the channel correlation properties are identical of each branch) and upper bounded by a finite positive constant. This last assumption poses constraint on the average receive energy in the *i-*th hypothesis <sup>0</sup> (,) *T <sup>E</sup>i i Cov t t dt* . We also assume in keeping with

We design the GD extending conditions of (Buzzi et al., 1997) and (Buzzi et al., 1999) to the case that a diversity technique is employed. It is well known that the adoption of diversity techniques is effective in mitigating the negative effects of the fading, and since conventional diversity techniques can incur heavy performance loss in the presence of impulsive disturbance (Kassam & Poor, 1985), it is of interest to envisage the GD for optimized diversity reception in non-Gaussian noise. We show that the optimum GD is independent of the joint pdf of the texture components on each diversity branch. We also derive a suboptimum GD, which is amenable to a practice. We focus on the relevant case of binary frequency-shift-keying (BFSK) signaling and provide the error probability of both the optimum GD and the suboptimum GD. We assess the channel diversity order impact and noise spikiness on the per-

The problem is to derive the GD aimed at detecting one out of *M* signals propagating through single-input multiple-output channel affected by dispersive fading and introducing the additive non-Gaussian noise. In other words, we have to deal with the following *M*-ary hy-

............................... 1, , [0, ]

where *P* is the channel diversity order and[0, ] *T* is the observation interval; the waveforms

,*M* represent the baseband equivalents of the useful signal received on the *P* diversity branches under the *i*th hypothesis. Since the channel is affected by dispersive fading, we may assume (Proakis, 2007) that these waveforms are related to the corresponding transmitted si-

> 

and is modeled as a Gaussian random process with respect to the variable *t*. In keeping with the uncorrelated-scattering model, we assume that the random processes ( , ), 1, , *<sup>p</sup> h t*

independent complex Gaussian random processes that we assume to be zero-mean and with

, , ( , ) ( ) ( ) , 1, , , [0, ] [ ] *Cov t E s t s i M t T pi pi*

independent of *p* (the channel correlation properties are identical of each branch) and upper bounded by a finite positive constant. This last assumption poses constraint on the average

*T*

*p P* is the random impulse response of the channel *p*th diversity branch

*p p x t* are the complex envelopes of the *P* distinct channel outputs; , 1 { } ( ) , 1, *<sup>P</sup>*

*i Mt T*

(3)

 (4)

*<sup>E</sup>i i Cov t t dt* . We also assume in keeping with

*pi p st i*

*<sup>p</sup> <sup>i</sup> <sup>p</sup> s t* are themselves

*p P*

*H* , (2)

1 1, 1

, ( ) ( , ) ( ) , [0, ] *p i p i s t h t ut d t T*

receive energy in the *i-*th hypothesis <sup>0</sup> (,)

are all statistically independent; as a consequence, the waveforms , 1 { } ( ) *<sup>P</sup>*

() () ()

*xt s t nt*

*i*

,

*P Pi P*

() () ()

*xt s t nt*

formance.

pothesis test:

<sup>1</sup> { } ( ) *<sup>P</sup>*

gnals ( ) *u t <sup>i</sup>*

where ( , ), 1, , *<sup>p</sup> h t* 

the covariance function

**3.1 Problem statement** 

*i*

the model (Van Trees, 2001) that , , [ ( ) ( )] 0 *Es ts pi pi* . This is not a true limitation in most practical instances, and it is necessarily satisfied if the channel is wide sense stationary. Finally, as to the additive non-Gaussian disturbances <sup>1</sup> { } ( ) *<sup>P</sup> p p n t* , we resort to the widely adopted compound model, i.e. we deem the waveform ( ) *n t <sup>p</sup>* as the product of two independent processes:

$$m\_p(t) = \upsilon\_p(t)\varrho\_p(t) \quad , \qquad p = 1, \ldots, P \tag{5}$$

where ( ) *<sup>p</sup> v t* is a real non-negative random process with marginal pdf ( ) *<sup>p</sup> f* and ( ) *<sup>p</sup> g t* is a zeromean complex Gaussian process. If the average decorrelation time of ( ) *<sup>p</sup> v t* is much larger than the observation interval[0, ] *T* , then the disturbance process degenerates into SIPR (Yao, 1973)

$$m\_p(t) = \nu\_p \mathbf{g}\_p(t) \quad , \qquad p = 1, \ldots, P \; . \tag{6}$$

From now on, we assume that such a condition is fulfilled, and we refer to (Conte, 1995) for further details on the noise model, as well as for a list of all of the marginal pdfs that are compatible with (5). Additionally, we assume <sup>2</sup> [ ] 1 *E <sup>p</sup>* and that the correlation function of the random process ( ) *<sup>p</sup> g t* is either known or has been perfectly estimated based on (5). While previous papers had assumed that the noise realization 1( ), , ( ) *nt n t <sup>P</sup>* were statistically independent, in this paper, this hypothesis is relaxed. To be more definite, we assume that the Gaussian components 1( ), , ( ) *<sup>P</sup> gt g t* are uncorrelated (independent), whereas the random variables <sup>1</sup> , , *<sup>P</sup>* are arbitrary correlated. We thus denote by <sup>1</sup> ,, 1 (,, ) *<sup>P</sup> <sup>P</sup> f* their joint pdf. It is worth pointing out that the above model subsumes the special case that the random variables <sup>1</sup> , , *<sup>P</sup>* are either statistically independent or fully correlated, i.e. <sup>1</sup> *<sup>P</sup>* . Additionally, it permits modeling a much wider class of situations that may occur in practice. For instance, if one assumes that the *P* diversity observations are due to a temporal diversity, it is apparent that if the temporal distance between consecutive observations is comparable with the average decorrelation time of the process ( )*t* , then the random variables <sup>1</sup> , , *<sup>P</sup>* can be assumed to be neither independent nor fully correlated. Such a model also turns out to be useful in clutter modeling in that if the diversity observations are due to the returns from neighboring cells, the corresponding texture components may be correlated (Barnard & Weiner, 1996). For sake of simplicity, consider the white noise case, i.e. ( ) *n t <sup>p</sup>* possesses an impulsive covariance *p*

$$\text{Cov}\_n(t, \tau) = \mathcal{Z}\mathcal{N}\_0 E[\nu\_p^2] \delta(t - \tau) = \mathcal{Z}\mathcal{N}\_0 \delta(t - \tau) \,. \tag{7}$$

where <sup>0</sup> 2*N* is the power spectral density (PSD) of the Gaussian component of the noise processes 1( ), , ( ) *<sup>P</sup> gt g t* . Notice that this last assumption does not imply any loss of generality should the noise possess a non-impulsive correlation Then, due to the closure of SIRP with respect to linear transformations, the classification problem could be reduced to the above form by simply preprocessing the observables through a linear whitening filter. In such a situation, the , ( ) *p i s t* represent the useful signals at the output of the cascade of the channel and of the whitening filter. Due to the linearity of such systems, they are still Gaussian processes with known covariance functions. Finally, we highlight here that the assumption that the useful signals and noise covariance functions (3) and (6) are independent of the index *p*  has been made to simplify notation.

#### **3.2 Synthesis and design**

#### **3.2.1 Optimum GD structure design**

Given the *M*-ary hypothesis test (1), the synthesis of the optimum GD structure in the sense of attaining the minimum probability of error *PE* requires evaluating the likelihood functionals under any hypothesis and adopting a maximum likelihood decision-making rule. Formally, we have

$$
\hat{\boldsymbol{\Theta}} = \boldsymbol{\mathsf{H}}\_{i} \Longrightarrow \Lambda \Big[ \mathbf{x}(t); \boldsymbol{\mathsf{H}}\_{i} \Big] > \max\_{k \neq i} \Lambda \Big[ \mathbf{x}(t); \boldsymbol{\mathsf{H}}\_{k} \Big] \tag{8}
$$

with <sup>1</sup> ( ) ( ), , ( ) [ ]*<sup>T</sup> <sup>P</sup>* **x** *t xt x t* . The above functionals are usually evaluated through a limiting procedure. We evaluate the likelihood <sup>|</sup> ( ) *f***<sup>x</sup>***Q i <sup>H</sup>* **x***<sup>Q</sup>* of the *Q*-dimensional random vector **x** <sup>1</sup> [,, ]*<sup>T</sup> x x <sup>Q</sup>* whose entries are the projections of the received signal along the first *Q* elements of suitable basis *B<sup>i</sup>* . Therefore, the likelihood functional corresponding to *H<sup>i</sup>* is

$$\Lambda[\mathbf{x}(t); \mathcal{H}\_i] = \lim\_{Q \to \ast} \frac{f\_{\mathbf{x}\_Q|\mathcal{H}\_i}(\mathbf{x}\_Q)}{f\_{\mathbf{n}\_{A\tilde{Q}\_Q}}(\mathbf{n}\_{A\tilde{F}\_Q})},\tag{9}$$

where ( ) *AFQ AFQ <sup>f</sup>***<sup>n</sup> <sup>n</sup>** is the likelihood corresponding to the reference sample with *a priori* information a "no" signal is obtained in the additional reference noise forming at the AF output, i.e. no useful signal is observed at the *P* channel outputs. In order to evaluate the limit (9), we resort to a different basis for each hypothesis. We choose for the *i-*th hypothesis the Karhunen-Loeve basis *B<sup>i</sup>* determined by the covariance function of the useful received signal under the hypothesis *H<sup>i</sup>* . Projecting the waveform received on the *p-*th diversity branch along the first *N* axes of the *i-*th basis yields the following *N*-dimensional vector:

$$\mathbf{x}\_{\text{N},p}^{i} = \mathbf{s}\_{\text{N},p}^{i} + \nu\_p \mathbf{g}\_{\text{N},p}^{i} \quad , \qquad p = 1, \ldots, P \tag{10}$$

where , *i <sup>N</sup> <sup>p</sup>* **s** and , *<sup>i</sup>* **<sup>g</sup>***<sup>N</sup> <sup>p</sup>* are the corresponding projections of the waveforms , ( ) *p i s t* and ( ) *<sup>p</sup> g t* . Since *B<sup>i</sup>* is the Karhunen-Loeve basis for the random processes 1, , ( ), , ( ) *i i <sup>P</sup> st s t* , the entries of , *i <sup>N</sup> <sup>p</sup>* **s** are a sequence of uncorrelated complex Gaussian random variables with the variances 1, , 2 2 ( ) , , *s s i i <sup>N</sup>* which are the first *N* eigenvalues of the covariance function (, ) *Cov t u <sup>i</sup>* , whereas the entries of **g***N*,*<sup>p</sup>* are a sequence of uncorrelated Gaussian variables with variance <sup>0</sup> 2*N* . Here we adopt the common approach of assuming that any complete orthonormal system is an orthonormal basis for white processes (Conte, 1995 and Poor, 1988). Upon defining the following *NP*-dimensional vector

$$\mathbf{x}\_{N}^{i} = \begin{bmatrix} \mathbf{x}\_{N,1}^{iT}, \mathbf{x}\_{N,2}^{iT}, \dots, \mathbf{x}\_{N,P}^{iT} \end{bmatrix}^{T} \tag{11}$$

the likelihood functional taking into consideration subsection 3.1 and (Tuzlukov, 2001) can be written in the following form

$$\Delta[\mathbf{x}\_{N}^{i}, \boldsymbol{\mu}\_{i}] = \frac{f\_{\mathbf{x}\_{N}^{i}|\boldsymbol{\mu}\_{i}}(\mathbf{x}\_{N}^{i})}{f\_{\mathbf{n}\_{A\mathcal{E}\_{N}}^{i}|\boldsymbol{\mu}\_{0}}(\mathbf{n}\_{A\mathcal{E}\_{N}}^{i})} = \frac{\int \prod\_{p=1}^{p} \prod\_{j=1}^{N} \frac{1}{\sigma\_{s\_{j,j}}^{2} + 4\sigma\_{n}^{4}y\_{p}^{2}} \exp\left[-\frac{\left|\boldsymbol{\boldsymbol{x}}\_{j,p}^{i}\right|^{2}}{\sigma\_{s\_{j,i}}^{2} + 4\sigma\_{n}^{4}y\_{p}^{2}}\right] f\_{\mathbf{v}}(\mathbf{y}) d\mathbf{y}}{\int \prod\_{p=1}^{p} \frac{1}{\left(4\sigma\_{n}^{4}y\_{p}^{2}\right)^{N}} \exp\left[-\frac{\left|\boldsymbol{\boldsymbol{\boldsymbol{n}}}\_{AF\_{\boldsymbol{f},p}}^{i}\right|^{2}}{4\sigma\_{n}^{4}y\_{p}^{2}}\right] f\_{\mathbf{v}}(\mathbf{y}) d\mathbf{y}}.\tag{12}$$

where , *i j p x* is the *j*-th entry of the vector , *i <sup>N</sup> <sup>p</sup>* **x** , the integrals in (12) are over the set [0, ) , *<sup>P</sup>* **ν** 1 1 [ , , ], [ , , ] *P P* **<sup>y</sup>** *y y* , and <sup>1</sup> *P <sup>i</sup> <sup>i</sup> d dy* **<sup>y</sup>** . The convergence in measure of (12) for increasing *N* to the likelihood functional [ ( ); ]*<sup>i</sup>* **x** *t H* is ensured by the Grenander theorem (Poor, 1988). In order to evaluate the above functional, we introduce the substitution

$$y\_p = \frac{\left\| \mathbf{x}\_{N,p}^i \right\|}{\sqrt{4\sigma\_n^4 z\_p}} \quad , \qquad p = 1, 2, \dots, P \tag{13}$$

where denotes the Euclidean norm. Applying the same limiting procedure as in (Buzzi, 1999), we come up with the following asymptotical expression:

$$\Lambda[\mathbf{x}(t); \mathcal{H}\_i] = \lim\_{N \to \infty} \prod\_{p=1}^{P} \Lambda\_{\mathbb{S}^N}^p \left[ \mathbf{x}\_{N, p}^i, \left\| \frac{\mathbf{x}\_{N, p}^i}{4\sigma\_n^4 N} \right\|^2; \mathcal{H}\_i \right], \tag{14}$$

where

312 Wireless Communications and Networks – Recent Advances

the useful signals and noise covariance functions (3) and (6) are independent of the index *p* 

Given the *M*-ary hypothesis test (1), the synthesis of the optimum GD structure in the sense of attaining the minimum probability of error *PE* requires evaluating the likelihood functionals under any hypothesis and adopting a maximum likelihood decision-making rule. For-

<sup>ˆ</sup> [] [] ( ); max ( ); *<sup>k</sup> k i i i t t*

<sup>|</sup> ( ) ( ); lim ( ) [ ] *<sup>i</sup>*

where ( ) *AFQ AFQ <sup>f</sup>***<sup>n</sup> <sup>n</sup>** is the likelihood corresponding to the reference sample with *a priori* information a "no" signal is obtained in the additional reference noise forming at the AF output, i.e. no useful signal is observed at the *P* channel outputs. In order to evaluate the limit (9), we resort to a different basis for each hypothesis. We choose for the *i-*th hypothesis the Karhunen-Loeve basis *B<sup>i</sup>* determined by the covariance function of the useful received signal under the hypothesis *H<sup>i</sup>* . Projecting the waveform received on the *p-*th diversity branch

> ,, , , 1, , *ii i Np Np p Np* **xs g**

nce *B<sup>i</sup>* is the Karhunen-Loeve basis for the random processes 1, , ( ), , ( ) *i i <sup>P</sup> st s t* , the entries of

*<sup>N</sup> <sup>p</sup>* **s** are a sequence of uncorrelated complex Gaussian random variables with the variances

 which are the first *N* eigenvalues of the covariance function (, ) *Cov t u <sup>i</sup>* , whereas the entries of **g***N*,*<sup>p</sup>* are a sequence of uncorrelated Gaussian variables with variance <sup>0</sup> 2*N* . Here we adopt the common approach of assuming that any complete orthonormal system is an orthonormal basis for white processes (Conte, 1995 and Poor, 1988). Upon defining the

> ,1 ,2 , [ ] , ,, *iT iT iT N N N NP*

the likelihood functional taking into consideration subsection 3.1 and (Tuzlukov, 2001) can

*<sup>t</sup> <sup>f</sup>* **<sup>x</sup>**

along the first *N* axes of the *i-*th basis yields the following *N*-dimensional vector:

**x**

*AF Q*

*<sup>i</sup> <sup>Q</sup> AF f*

**n**

*Q Q*

**n**

*<sup>i</sup>* **<sup>g</sup>***<sup>N</sup> <sup>p</sup>* are the corresponding projections of the waveforms , ( ) *p i s t* and ( ) *<sup>p</sup> g t* . Si-

*Q*

*<sup>H</sup> <sup>H</sup>* , (9)

*p P* (10)

*i T* **x xx x** (11)

**x**

*<sup>P</sup>* **x** *t xt x t* . The above functionals are usually evaluated through a limiting procedure. We evaluate the likelihood <sup>|</sup> ( ) *f***<sup>x</sup>***Q i <sup>H</sup>* **x***<sup>Q</sup>* of the *Q*-dimensional random vector **x** <sup>1</sup> [,, ]*<sup>T</sup> x x <sup>Q</sup>* whose entries are the projections of the received signal along the first *Q* elements of suitable basis *B<sup>i</sup>* . Therefore, the likelihood functional corresponding to *H<sup>i</sup>* is

*HH H* **x x** *H* (8)

has been made to simplify notation.

**3.2.1 Optimum GD structure design** 

**3.2 Synthesis and design** 

with <sup>1</sup> ( ) ( ), , ( ) [ ]*<sup>T</sup>*

mally, we have

where , *i*

1, , 2 2 ( ) , , *s s i i <sup>N</sup>*

 

, *i*

*<sup>N</sup> <sup>p</sup>* **s** and ,

following *NP*-dimensional vector

be written in the following form

$$\boldsymbol{\Lambda}\_{\rm gN}^{p}\left(\mathbf{x}\_{N,p}^{i},\boldsymbol{y}\_{p}^{2};\boldsymbol{\mathcal{H}}\_{i}\right) = \exp\left\{\sum\_{j=1}^{N} \left[\frac{\sigma\_{s\_{j,i}}^{2}\left\|\mathbf{x}\_{j,p}^{i}\right\|^{2}}{4\sigma\_{n}^{4}\boldsymbol{y}\_{p}^{2}\left(\sigma\_{s\_{j,i}}^{2} + 4\sigma\_{n}^{4}\boldsymbol{y}\_{p}^{2}\right)} - \ln\left(1 + \frac{\sigma\_{s\_{j,i}}^{2}}{4\sigma\_{n}^{4}\boldsymbol{y}\_{p}^{2}}\right)\right]\right\}\tag{15}$$

represents the ratio between the conditional likelihoods for *H<sup>i</sup>* and *H*<sup>0</sup> based on the observation of the signal received on the *p-*th channel output only. Equation (14) also requires evaluating

$$Z\_p = \lim\_{N \to \infty} \frac{\left\| \mathbf{x}\_{N,p}^i \right\|^2}{N} \,, \tag{16}$$

that, following in (Buzzi, 1999), can be shown to converge in the mean square sense to the random variable 4 2 4 *n <sup>p</sup>* for any of the Karhunen-Loeve basis , 1, , *<sup>i</sup> B i M* . Due to the fact that the considered noise is white, this result also holds for the large signal-to-noise ratios even though, in this case, a large number of summands is to be considered in order to achieve a given target estimation accuracy. Notice also that 4 2 4 *n <sup>p</sup>* can be interpreted as a shortterm noise power spectral density (PSD), namely, the PSD that would be measured on sufficiently short time intervals on the *p-*th channel output. Thus, the classification problem under study admits the sufficient statistics

$$\ln \Lambda[\mathbf{x}(t); \mathsf{H}\_i] = \sum\_{p=1}^{P} \sum\_{j=1}^{\alpha} \left[ \frac{\sigma\_{s\_{j,i}}^2 \left| \boldsymbol{\omega}\_{j,p}^i \right|^2}{Z\_p \left( \sigma\_{s\_{j,i}}^2 + Z\_p \right)} - \ln \left( 1 + \frac{\sigma\_{s\_{j,i}}^2}{Z\_p} \right) \right]. \tag{17}$$

The above equations demonstrate that the optimum GD structure for the problem given in (1) is completely canonical in that for any <sup>1</sup> ,, 1 (,, ) *<sup>P</sup> <sup>P</sup> f* and, for any noise model in the class of compound-Gaussian processes and for any correlation of the random variables 1 , , *<sup>P</sup>* , the likelihood functional is one and the same. Equation (17) can be interpreted as a bank of *P* estimator-GDs (Van Trees, 2003) plus a bias term depending on the eigenvalues of the signal correlation under the hypothesis *H<sup>i</sup>* . The optimum test based on GASP can be written in the following form:

$$\begin{split} \hat{H} = \mathsf{H}\_{i} &\Longrightarrow \sum\_{p=1}^{p} \frac{1}{Z\_{p}} \Bigg\{ \Big\Vert \int\_{0}^{T} [\mathbf{2x}\_{p}(t)\hat{\mathbf{s}}\_{p,i}^{\star}(t) - \mathbf{x}\_{p}(t)\mathbf{x}\_{p}(t-\tau)]dt + \int\_{0}^{T} \boldsymbol{n}\_{AF\_{p}}^{2}(t)dt \bigg\} - b\_{p,i} \\ &> \sum\_{p=1}^{p} \frac{1}{Z\_{p}} \Bigg\llbracket \Big\Vert \int\_{0}^{T} [\mathbf{2x}\_{p}(t)\hat{\mathbf{s}}\_{p,k}^{\star}(t) - \mathbf{x}\_{p}(t)\mathbf{x}\_{p}(t-\tau)]dt + \int\_{0}^{T} \boldsymbol{n}\_{AF\_{p}}^{2}(t)dt \bigg\} - b\_{p,k} \quad \forall k \neq i \end{split} \tag{18}$$

where , ˆ ( ) *p i s t* is the linear minimum mean square estimator of , ( ) *p i s t* embedded in white noise with PSD *Zp* , namely,

$$\hat{\mathbf{s}}\_{p,i}(t) = \int\_{0}^{T} h\_{p,i}(t,\mu) \mathbf{x}\_{p}(\mu) d\mu \,\,\,\,\tag{19}$$

where , (, ) *p i h tu* is the solution to the Wiener-Hopf equation

$$\int\_{0}^{T} \text{Cov}\_{i}(t, z)h\_{p,i}(z, \tau)dz + Z\_{p}h\_{p,i}(t, \tau) = \text{Cov}\_{i}(t, \tau) \,. \tag{20}$$

As to the bias terms *<sup>p</sup>*,*<sup>i</sup> b* , they are given by

$$b\_{p,i} = \sum\_{j=1}^{\alpha} \ln \left[ 1 + \frac{\sigma\_{s\_{j,i}}^2}{Z\_p} \right] \quad \text{i} = 1, \dots, M \quad \text{and} \quad p = 1, \dots, P \; . \tag{21}$$

The block diagram of the corresponding GD is outlined in Fig.3. The received signals 1 *x t*( ), , () *<sup>P</sup> x t* are fed to *P* estimators of the noise short-term PSD, which are subsequently used for synthesizing the bank of *MP* minimum mean square error filters , ( , ), 1, , , *p i h p P i* 1, , *M* to implement the test (18). The newly proposed GD structure is a generalization, to the case of multiple observations, of that proposed in (Buzzi, 1999), to which it reduces to *P* 1 .

,

*j*

 *<sup>P</sup>* , the likelihood functional is one and the same. Equation (17) can be interpreted as a bank of *P* estimator-GDs (Van Trees, 2003) plus a bias term depending on the eigenvalues of the signal correlation under the hypothesis *H<sup>i</sup>* . The optimum test based on GASP can be wri-

*i p p i p p AF p i*

, ,

*x t s t x t x t dt n t dt b k i*

(18)

*H H*

*p pk p p AF p k*

where , ˆ ( ) *p i s t* is the linear minimum mean square estimator of , ( ) *p i s t* embedded in white noi-

The above equations demonstrate that the optimum GD structure for the problem given in

*i*

*x*

1 1

*i*

*t*

(1) is completely canonical in that for any <sup>1</sup> ,, 1 (,, ) *<sup>P</sup> <sup>P</sup> f*

*p p*

*Z*

where , (, ) *p i h tu* is the solution to the Wiener-Hopf equation

0

As to the bias terms *<sup>p</sup>*,*<sup>i</sup> b* , they are given by

,

*p i*

1

*j p*

*T*

1 0 0

, , 0 ˆ () (, ) ( ) *T*

> , 2

*Z*

 

, ,

*Cov t z h z dz Z h t Cov t i pi p pi <sup>i</sup>* 

(, ) (, ) (, ) (, )

ln 1 1, , and 1, , *<sup>j</sup> <sup>i</sup> <sup>s</sup>*

*b i M p P*

The block diagram of the corresponding GD is outlined in Fig.3. The received signals 1 *x t*( ), , () *<sup>P</sup> x t* are fed to *P* estimators of the noise short-term PSD, which are subsequently used for synthesizing the bank of *MP* minimum mean square error filters , ( , ), 1, , , *p i h p P i* 1, , *M* to implement the test (18). The newly proposed GD structure is a generalization, to the case of multiple observations, of that proposed in (Buzzi, 1999), to which it reduces

. (20)

. (21)

*P T T*

, 

tten in the following form:

*p p*

se with PSD *Zp* , namely,

to *P* 1 .

*Z*

ln [ ( ); ] ln 1 | |

 

1 0 0 <sup>1</sup> <sup>ˆ</sup> [2 ( ) ( ) ( ) ( )] ( ) <sup>ˆ</sup> *<sup>p</sup> P T T*

<sup>1</sup> [2 ( ) ( ) ( ) ( )] ( ) , ˆ *<sup>p</sup>*

class of compound-Gaussian processes and for any correlation of the random variables 1

, 2

*<sup>i</sup> <sup>P</sup> s jp <sup>s</sup>*

*p j ps p p*

( )

, ,

*j j*

*Z Z Z*

*i i*

**<sup>x</sup>** *<sup>H</sup>* . (17)

 

2

*pi pi p s t h t u x u du* , (19)

 

, ,

*x t s t x t x t dt n t dt b*

2

and, for any noise model in the

,

22 2

Fig. 3. Flowchart of optimum GD in compound Gaussian noise.

#### **3.2.2 Suboptimal GD: Low energy coherence approach**

Practical implementation of the decision rule (18) requires an estimation of the short-term noise PSDs on each diversity branch and evaluation of the test statistic. This problem requires a real-time design of *MP* estimator-GDs that are keyed to the estimated values of the short-term PSDs. This would require a formidable computational effort, which seems to prevent any practical implementation of the new receiving structure. Accordingly, we develop an alternative suboptimal GD structure with lower complexity. Assume that the signals , { } ( ) : 1, , , 1, , *p i s t <sup>p</sup> Pi M* possess a low degree of coherence, namely, that their energy content is spread over a large number of orthogonal directions. Since

$$\overline{\mathbf{E}}\_i = \sum\_{j=1}^{+\infty} \sigma\_{s\_{j,i}}^2 \; , \tag{22}$$

The low degree of coherence assumption implies that the covariance functions (, ) *Cov t <sup>i</sup>* have a large number of nonzero eigenvalues and do not have any dominant eigenvalue. Under these circumstances, it is plausible to assume that the following low energy coherence condition is met:

$$
\sigma\_{s\_{j,i}}^2 << 2\mathbf{N}\_0 \qquad i = 1, \dots, M \qquad j = 1, 2, \dots \tag{23}
$$

If this is the case, we can approximate the log-likelihood functional (17) with its first-order McLaurin series expansion with starting point , <sup>2</sup> / 0 *j i s p Z* . Following the same steps as in (Buzzi, 1997), we obtain the following suboptimal within the limits GASP decision-making rule:

$$\begin{split} \hat{\boldsymbol{\mathcal{H}}} &= \boldsymbol{\mathcal{H}}\_{i} \Longrightarrow \sum\_{p=1}^{P} \frac{1}{Z\_{p}^{2}} \Bigg\{ \Big\Vert \int\_{0}^{T} \boldsymbol{\mathbf{x}}\_{p}(t) \mathbf{x}\_{p}^{\*}(\tau) \mathbf{Cov}\_{i}(t,\tau) dt d\tau + \int\_{0}^{T} \boldsymbol{n}\_{\mathrm{AF}\_{p}}^{2}(t) dt \Big\Vert - \frac{\overline{\mathbf{E}}\_{i}}{Z\_{p}} \\ &> \sum\_{p=1}^{P} \frac{1}{Z\_{p}^{2}} \Bigg\Vert \int\_{0}^{T} \boldsymbol{\uphat{\mathbf{x}}}\_{p}(t) \mathbf{x}\_{p}^{\*}(\tau) \mathbf{Cov}\_{k}(t,\tau) dt d\tau + \int\_{0}^{T} \boldsymbol{n}\_{\mathrm{AF}\_{p}}^{2}(t) dt \right\Vert - \frac{\overline{\mathbf{E}}\_{k}}{Z\_{p}} \qquad \forall k \neq i. \end{split} \tag{24}$$

The new GD again requires estimating the short-term noise PSDs <sup>1</sup> , , *Z Z <sup>P</sup>* . Unlike the optimum GD (18), in the suboptimum GD (24), the *MP* minimum mean square error filters , ( , ), 1, , , 1, , *p i h p Pi M* whose impulse responses depend on <sup>1</sup> , , *Z Z <sup>P</sup>* through (20) are now replaced with *M* filters whose impulse response (, ) *Cov t <sup>i</sup>* is independent of the short-term noise PSDs realizations, which now affect the decision-making rule as mere proportionality factors. The only difficulty for practical implementation of such a GD scheme is the short-term noise PSD estimation through (16). However, as already mentioned, such a drawback can be easily circumvented by retaining only a limited number of summands.

#### **3.3 Special cases**

#### **3.3.1 Channels with flat-flat Rayleigh fading**

Let us consider the situation where the fading is slow and non-selective so that the signal observed on the *p*-th channel output under the hypothesis *H<sup>i</sup>* takes the form

$$s\_{p,i}(t) = A\_p \exp\{j\theta\_p\} u\_i(t) \, , \tag{25}$$

where exp{ } *Ap p j* is a complex zero-mean Gaussian random variable. The signal covariance function takes a form:

$$\text{Cov}\_i(t, \tau) = \overline{\mathbb{E}}\_i u\_i(t) u\_i^\* \left(\tau\right) \,. \tag{26}$$

where the assumption has been made that ( ) *u t <sup>i</sup>* possesses unity norm. Notice that this equation represents the Mercer expansion of the covariance in a basis whose first unit vector is parallel to ( ) *u t <sup>i</sup>* . It should be noted that since the Mercer expansion of the useful signal covariance functions contains just one term, the low energy coherence condition is, in this case, equivalent to a low *SNR* condition. It thus follows that the low energy coherence GD can be now interpreted as a locally optimum GD, thus implying that for large *SNR*s, its performance is expectedly much poorer than that of the optimum GD. The corresponding eigenvalues are

$$
\sigma\_{s\_{1,j}}^2 = \overline{\mathbf{E}}\_i \quad , \qquad \sigma\_{s\_{k\_{\cdot,j}}}^2 = 0 \quad , \qquad \forall k \neq 1 \,. \tag{27}
$$

Accordingly, the minimum mean square error filters to be substituted in (18) have the following impulse responses:

$$h\_{p,i}(t,\tau) = \frac{\overline{\mathbf{E}}\_i}{\overline{\mathbf{E}}\_i + Z\_p} u\_i(t) u\_i^\*(\tau) \; , \tag{28}$$

where the bias term is simply , ln 1{ }*<sup>i</sup> <sup>p</sup> p i <sup>Z</sup> <sup>b</sup> <sup>E</sup>* .We explicitly notice here that such a bias term turns out to depend on the estimated PSD *Zp* . Substituting into (18), we find the optimum test

$$\hat{\boldsymbol{\Theta}} = \boldsymbol{\mathsf{H}}\_{i} \Rightarrow \sum\_{p=1}^{p} \frac{\overline{\mathbf{E}}\_{i}}{Z\_{p}(\overline{\mathbf{E}}\_{i} + \mathbf{Z}\_{p})} \Bigg|\_{0}^{T} [\mathbb{1}\mathbf{x}\_{p}(t)\boldsymbol{\upmu}\_{i}^{\star}(t) - \boldsymbol{x}\_{p}(t)\boldsymbol{x}\_{p}(t-\tau)]dt + \int\_{0}^{T} \boldsymbol{n}\_{\mathrm{AF}\_{p}}^{2}(t)dt \Bigg|^{2} - b\_{p,i}$$

The new GD again requires estimating the short-term noise PSDs <sup>1</sup> , , *Z Z <sup>P</sup>* . Unlike the optimum GD (18), in the suboptimum GD (24), the *MP* minimum mean square error filters , ( , ), 1, , , 1, , *p i h p Pi M* whose impulse responses depend on <sup>1</sup> , , *Z Z <sup>P</sup>* through (20)

short-term noise PSDs realizations, which now affect the decision-making rule as mere proportionality factors. The only difficulty for practical implementation of such a GD scheme is the short-term noise PSD estimation through (16). However, as already mentioned, such a drawback can be easily circumvented by retaining only a limited number of summands.

Let us consider the situation where the fading is slow and non-selective so that the signal

(, ) () ( ) *Cov t u t u i ii i* 

where the assumption has been made that ( ) *u t <sup>i</sup>* possesses unity norm. Notice that this equation represents the Mercer expansion of the covariance in a basis whose first unit vector is parallel to ( ) *u t <sup>i</sup>* . It should be noted that since the Mercer expansion of the useful signal covariance functions contains just one term, the low energy coherence condition is, in this case, equivalent to a low *SNR* condition. It thus follows that the low energy coherence GD can be now interpreted as a locally optimum GD, thus implying that for large *SNR*s, its performance is expectedly much poorer than that of the optimum GD. The corresponding eigenvalues are

2 2 , 0 , 1 *<sup>i</sup> <sup>k</sup> <sup>i</sup> si s*

Accordingly, the minimum mean square error filters to be substituted in (18) have the follo-

, (, ) () ( ) *<sup>i</sup> p i i i i p h t u tu Z*

turns out to depend on the estimated PSD *Zp* . Substituting into (18), we find the optimum

1 0 0 <sup>ˆ</sup> [2 ( ) ( ) ( ) ( )] ( ) ( ) *<sup>p</sup> P T T*

 *E*

*i p i p p AF p i*

*E*

  is a complex zero-mean Gaussian random variable. The signal covariance

*u t* , (25)

*E* , (26)

*E k* . (27)

*<sup>E</sup>* , (28)

*<sup>E</sup>* .We explicitly notice here that such a bias term

2

2

,

*x t u t x t x t dt n t dt b*

is independent of the

are now replaced with *M* filters whose impulse response (, ) *Cov t <sup>i</sup>*

observed on the *p*-th channel output under the hypothesis *H<sup>i</sup>* takes the form

1, ,

*<sup>p</sup> p i <sup>Z</sup> <sup>b</sup>*

*<sup>E</sup> H H*

where the bias term is simply , ln 1{ }*<sup>i</sup>*

*i*

*p pi p*

*Z Z*

**3.3 Special cases** 

where exp{ } *Ap p j*

function takes a form:

wing impulse responses:

test

**3.3.1 Channels with flat-flat Rayleigh fading** 

( ) exp{ } ( ) *p,i p p i st A j*

$$\geq \max\_{k \neq i} \sum\_{p=1}^{p} \frac{\overline{\mathsf{E}}\_{k}}{Z\_{p}(\overline{\mathsf{E}}\_{k} + Z\_{p})} \left| \int\_{0}^{T} [\mathbbm{1}\chi\_{p}(t)\mu\_{k}^{\star}(t) - \mathbbm{x}\_{p}(t)\mathbbm{x}\_{p}(t-\tau)]dt + \int\_{0}^{T} \mu\_{AF\_{p}}^{2}(t)dt \right|^{2} - b\_{p,k\ \prime} \tag{29}$$

whereas its low energy coherence suboptimal approximation can be written in the following form:

$$\begin{split} \hat{H} = \boldsymbol{\mathcal{H}}\_{i} &\Rightarrow \sum\_{p=1}^{P} \frac{1}{Z\_{p}^{2}} \Bigg| \Bigg[ \Bigg[ \textrm{2} \, x\_{p}(t) \boldsymbol{u}\_{i}^{\*}(t) - \boldsymbol{x}\_{p}(t) \boldsymbol{x}\_{p}(t-\tau) \Bigr] dt + \int\_{0}^{T} \boldsymbol{n}\_{AF\_{p}}^{2}(t) dt \bigg]^{2} - \frac{\overline{\mathbf{E}}\_{i}}{Z\_{p}} \\ &> \max\_{k \neq i} \sum\_{p=1}^{P} \frac{1}{Z\_{p}^{2}} \Bigg[ \Bigg[ \textrm{2} \, x\_{p}(t) \boldsymbol{u}\_{i}^{\*}(t) - \boldsymbol{x}\_{p}(t) \boldsymbol{x}\_{p}(t-\tau) \Bigr] dt + \int\_{0}^{T} \boldsymbol{n}\_{AF\_{p}}^{2}(t) dt \bigg]^{2} - \frac{\overline{\mathbf{E}}\_{i}}{Z\_{p}}. \end{split} \tag{30} \end{split} \tag{31}$$

It is worth pointing out that both GDs are akin to the "square-law combiner" (Tuzlukov, 2001) GD that is well known to be the optimum GD in GASP (Tuzlukov, 2005 and Tuzlukov, 2012) viewpoint for array signal detection in Rayleigh flat-flat fading channels and Gaussian noise. The relevant difference is due to the presence of short-term noise PSDs <sup>1</sup> , , *Z Z <sup>P</sup>* which weigh the contribution from each diversity branch. In the special case of equienergy signals, the bias terms in the above decision-making rules end up irrelevant, and the optimum GD test (29) reduces to a generalization of the usual incoherent GD, with the exception that the decision statistic depends on the short-term noise PSD realizations.

#### **3.3.2 Channels with slow frequency-selective Rayleigh fading**

Now, assume that the channel random impulse response can be written in the following form:

$$\chi\_p(t,\tau) = \chi\_p(\tau) = \sum\_{k=0}^{L-1} A\_{p,k} \exp\{j\theta\_{p,k}\} \delta\{\tau - k\mathcal{W}^{-1}\},\tag{31}$$

where , , exp{ } *A j <sup>p</sup> <sup>k</sup> <sup>p</sup> <sup>k</sup>* is a set of zero-mean, independent complex Gaussian random variables, and *L* is the number of paths. Equation (31) represents the well known taped delay line channel model, which is widely encountered in wireless mobile communications. It is readily shown that in such a case, the received useful signal, upon transmission of ( ) *u t <sup>i</sup>* , has the following covariance function:

$$\text{Cov}\_{i}(t,\tau) = \sum\_{k=0}^{L-1} \overline{A\_{k}^{2}} \mu\_{i}(t - k\mathcal{W}^{-1}) \mu\_{i}^{\*}(\tau - k\mathcal{W}^{-1}) \quad , \qquad i = 1, \ldots, M \tag{32}$$

where <sup>2</sup> *Ak* is the statistical expectation (assumed independent of *p*) of the random variables <sup>2</sup> *Ap*,*<sup>k</sup>* . These correlations admit *L* nonzero eigenvalues, and a procedure for evaluating their eigenvalues and eigenfunctions can be found in (Matthews, 1992). In the special case that the *L* paths are resolvable, i.e. <sup>1</sup> *T W* , the optimum GD (18) assumes the following simplified form:

$$
\hat{H} = \mathcal{H}\_i \Rightarrow \sum\_{p=1}^{p} \sum\_{j=0}^{L-1} \left| \frac{\overline{A\_j^2} \left| \begin{matrix} \overline{A\_j} \\ 0 \end{matrix} \overline{x\_p(t)} u\_i^\* \left( t - j \mathcal{V}^{-1} \right) dt + \overline{\int\_0^T n\_{AF\_p}^2(t) dt} \right|^2}{Z\_p \left( Z\_p + \mathsf{E}\_i \overline{A\_j^2} \right)} - \ln \left\{ 1 + \frac{\mathsf{E}\_i \overline{A\_j^2}}{Z\_p} \right\} \right|^2
$$

$$
> \max\_{k \neq i} \sum\_{p=1}^{p} \sum\_{j=0}^{L-1} \left| \frac{\overline{A\_j^2} \left| \begin{matrix} \overline{A\_j^2} \\ 0 \end{matrix} \overline{x\_p(t)} u\_k^\* \left( t - j \mathcal{V}^{-1} \right) dt + \int\_0^T n\_{AF\_p}^2(t) dt \right|^2}{Z\_p \left( Z\_p + \mathsf{E}\_k \overline{A\_j^2} \right)} - \ln \left\{ 1 + \frac{\mathsf{E}\_k \overline{A\_j^2}}{Z\_p} \right\} \right|, \tag{33}
$$

where *E<sup>i</sup>* is the energy of the signal ( ) *u t <sup>i</sup>* . The low energy coherence suboptimal GD (24) is instead written as

$$\begin{split} \hat{\boldsymbol{\mathcal{H}}} = \boldsymbol{\mathcal{H}}\_{i} &\Longrightarrow \sum\_{p=1}^{P} \sum\_{j=0}^{L-1} \left| \overline{\frac{A\_{j}^{2}}{Z\_{p}^{2}}} \right|\_{0}^{T} \mathbf{x}\_{p}(t) \boldsymbol{\mu}\_{i}^{\*} \left(t - j\mathcal{V}\boldsymbol{V}^{-1}\right) dt + \int\_{0}^{T} \boldsymbol{n}\_{\boldsymbol{A}\_{F}^{\*}}^{2}(t) dt \right|^{2} - \frac{\mathsf{E}\_{i} \overline{\boldsymbol{A}\_{j}^{2}}}{Z\_{p}} \\ &> \max\_{k \neq i} \sum\_{p=1}^{P} \sum\_{j=0}^{L-1} \left| \overline{\frac{A\_{j}^{2}}{Z\_{p}^{2}}} \right| \left| \overline{\boldsymbol{\mathcal{X}}}\_{p}(t) \boldsymbol{\mu}\_{k}^{\*} \left(t - j\mathcal{V}\boldsymbol{V}^{-1}\right) dt + \int\_{0}^{T} \boldsymbol{n}\_{\boldsymbol{A}\_{F}^{\*}}^{2}(t) dt \right|^{2} - \frac{\mathsf{E}\_{k} \overline{\boldsymbol{A}\_{j}^{2}}}{Z\_{p}} \Big|\_{-} \end{split} \tag{34}$$

Optimality of (33) obviously holds for one-short detection, namely, neglecting the intersymbol interference induced by the channel band limitedness.

#### **3.4 Performance assessment**

In this section, we focus on the performance of the proposed GD structures. A general formula to evaluate the probability of error *PE* of any receiver in the presence of spherically invariant disturbance takes the following form:

$$P\_E = \int P\_E(e \mid \mathbf{v}) f\_\mathbf{v}(\mathbf{v}) d\mathbf{v} \,\,\,\,\,\tag{35}$$

where (| ) *P e <sup>E</sup>* **ν** is the receiver probability of error in the presence of Gaussian noise with PSD on the *p*-th diversity branch <sup>2</sup> <sup>0</sup> 2 *<sup>p</sup> N v* . The problem to evaluate *PE* reduces to that of first analyzing the Gaussian case and then carrying out the integration (35). In order to give an insight into the GD performance, we consider a BFSK signaling scheme, i.e. the baseband equivalents of the two transmitted waveforms are related as

$$
\mu\_2(t) = \mu\_1(t) \exp\{j2\pi\Delta ft\},\tag{36}
$$

where <sup>1</sup> *f T* denotes the frequency shift. Even for this simple case study, working out an analytical expression for the probability of error of both the optimum GD and of its low energy coherence approximation is usually unwieldy even for the case of Gaussian noise. With

regard to the optimum GD structure, upper and lower bounds for the performance may be established via Chernoff-bounding techniques. Generalizing to the case of multiple observations, the procedure in (Van Trees, 2003), the conditional probability of error given <sup>1</sup> , , *P* can be bounded as

$$\frac{\exp\{2\mu(0.5\mid\mathbf{v})\}}{2\left(1+\sqrt{0.25\pi\bar{\mu}(0.5\mid\mathbf{v})}\right)} \le P\_E(e\mid\mathbf{v}) \le \frac{\exp\{2\mu(0.5\mid\mathbf{v})\}}{2\left(1+\sqrt{0.25\pi\bar{\mu}(0.5\mid\mathbf{v})}\right)}\tag{37}$$

where (| ) **ν** is the following conditional semi-invariant moment generating the function

$$\begin{split} \mu(\mathbf{x} \mid \mathbf{v}) &= \lim\_{N \to \infty} \ln E \left\{ \exp \left[ \mathbf{x} \sum\_{p=1}^{P} \ln \Lambda\_{\mathcal{S}^N}^p \{ \mathbf{x}\_{N,p}^i; \nu\_p^2 : \mathcal{H}\_1 \} \right] \right\} \Big| \mathcal{H}\_0, \mathbf{v} \right\} \\ &= \sum\_{j=1}^{\infty} \sum\_{p=1}^{P} \left\{ (1-\mathbf{x}) \ln \left[ 1 + \frac{\sigma\_{s\_j}^2}{4 \sigma\_n^4 \nu\_p^2} \right] - \ln \left[ 1 + \frac{\sigma\_{s\_j}^2 (1-\mathbf{x})}{4 \sigma\_n^4 \nu\_p^2} \right] \right\} \end{split} \tag{38}$$

with <sup>2</sup> <sup>1</sup> { }*<sup>j</sup> s j* being the set of common eigenvalues. Substituting this relationship into (37) and averaging with respect to 1 , , *<sup>P</sup>* yields the unconditional bounds on the probability of error for the optimum GD (18).

#### **3.5 Simulation results**

318 Wireless Communications and Networks – Recent Advances

( ) ( )

*<sup>E</sup>*

*jpi AF P L i j*

( )

( ) ( )

 *<sup>E</sup> E*

( )

where *E<sup>i</sup>* is the energy of the signal ( ) *u t <sup>i</sup>* . The low energy coherence suboptimal GD (24) is

*k i p j <sup>p</sup> p p kj*

*E*

*p j p p p ij*

2 1

*A x t u t jW dt n t dt <sup>A</sup>*

<sup>2</sup> 2 2 <sup>1</sup> 1 2

*A A x t u t jW dt n t dt <sup>Z</sup> <sup>Z</sup>*

*p j p p*

<sup>2</sup> 2 2 <sup>1</sup> 1 2

*A A x t u t jW dt n t dt <sup>Z</sup> <sup>Z</sup>*

*j k j*

 *<sup>E</sup> H H*

*p k AF k i p j <sup>p</sup> <sup>p</sup>*

Optimality of (33) obviously holds for one-short detection, namely, neglecting the intersym-

In this section, we focus on the performance of the proposed GD structures. A general formula to evaluate the probability of error *PE* of any receiver in the presence of spherically in-

(| ) ( ) *P Pe E E <sup>f</sup> <sup>d</sup>* **ν ν <sup>ν</sup> ν** , (35)

where (| ) *P e <sup>E</sup>* **ν** is the receiver probability of error in the presence of Gaussian noise with PSD

zing the Gaussian case and then carrying out the integration (35). In order to give an insight into the GD performance, we consider a BFSK signaling scheme, i.e. the baseband equiva-

2 1 *u t u t j ft* ( ) ( )exp{ 2 }

where <sup>1</sup> *f T* denotes the frequency shift. Even for this simple case study, working out an analytical expression for the probability of error of both the optimum GD and of its low energy coherence approximation is usually unwieldy even for the case of Gaussian noise. With

<sup>0</sup> 2 *<sup>p</sup> N v* . The problem to evaluate *PE* reduces to that of first analy-

, (36)

 *<sup>E</sup>* .

*j i j*

*jpk AF P L k j*

2 1

*A x t u t jW dt n t dt <sup>A</sup>*

 

*p*

*ZZ A Z*

*p*

*ZZ A Z*

2 1 2

*T T*

0 0

ˆ ln 1 ( )

2 1 2

*T T*

0 0

max ln 1 ( )

1 0 0 0 <sup>ˆ</sup> ( ) ( ) ( ) *<sup>p</sup> P L T T*

*i p i AF*

1 0 0 0 max ( ) ( ) ( ) *<sup>p</sup> P L T T*

<sup>2</sup> 1 0

<sup>2</sup> 1 0

2

2

bol interference induced by the channel band limitedness.

variant disturbance takes the following form:

lents of the two transmitted waveforms are related as

**3.4 Performance assessment** 

on the *p*-th diversity branch <sup>2</sup>

*i*

*H H*

instead written as

2

2

(33)

(34)

,

To proceed further in the GD performance there is a need to assign both the marginal pdf, as well as the channel spectral characteristics. We assume hereafter the generalized Laplace noise, i.e. the marginal pdf of the *p*-th noise texture component takes the following form:

$$f\_{\nu\_p}(\mathbf{x}) = \frac{2\nu^{\nu}}{\Gamma(\nu)} \mathbf{x}^{2\nu - 1} \exp\{-\nu \mathbf{x}^2\} \quad , \qquad \mathbf{x} > \mathbf{0} \tag{39}$$

where is a shape parameter, ruling the distribution behavior. In particular, the limiting case implies ( ) ( 1) *<sup>p</sup> fx x* and, eventually, Gaussian noise, where increasingly lower values of account for increasingly spikier noise distribution. Regarding the channel, we consider the case of the frequency-selective, slowly fading channel, i.e. the channel random impulse response is expressed by (31), implying that the useful signal correlation is that given in (32). For simplicity, we also assume that the paths are resolvable. In the following plots the *PE* is evaluated a) through a semianalytic procedure, i.e. by numerically averaging the Chernoff bound (37) with respect to the realizations of the <sup>1</sup> , , *<sup>P</sup>* , and b) by resorting to a Monte Carlo counting procedure. In this later case, the noise samples have been generated by multiplying standard, i.e. with zero-mean and unit-variance, complex Gaussian random variates times the random realizations of <sup>1</sup> , , *<sup>P</sup>* .

The Chernoff bound for the optimum GD versus the averaged received radio-frequency energy contrast that is defined as 2 4 0.5 <sup>0</sup> <sup>1</sup> ( ) <sup>4</sup> *j <sup>L</sup> <sup>P</sup> <sup>j</sup> s n* at *<sup>P</sup>* <sup>2</sup> and for two values of the noise shape parameter is shown in Fig.4. The noise texture components have been assumed to be independent. Inspecting the curves, we see that the Chernoff bound provides a very reliable estimate of the actual *PE* , as the upper and lower bound very tightly follow each other. As expected, the results demonstrate that in the low *PE* region, the spikier the noise, i.e. the lower , the worse the GD performance. Conversely, the opposite behavior is observed for small values of <sup>0</sup> . This fact might appear, at a first look, surprising. It may be analytically justified in light of the local validity of Jensen's inequality (Van Trees, 2003) and is basically the same phenomenon that makes digital modulation schemes operating in Gaussian noise to achieve, for low values of <sup>0</sup> , superior performance in Rayleigh flat-flat fading channels than in no-fading channels. Notice, this phenomenon is in accordance with that observed in (Conte 1995). In order to validate the Chernoff bound, we also show, on the same plots, some points obtained by Monte Carlo simulations. These points obviously lie between the corresponding upper and lower probability of error bounds. Additionally, we compare the GD Chernoff bound with that for the conventional optimum receiver (Buzzi et al., 2001). A superiority of GD structure is evident.

Fig. 4. Chernoff bounds for *PE* of the optimum GD.

estimate of the actual *PE* , as the upper and lower bound very tightly follow each other. As expected, the results demonstrate that in the low *PE* region, the spikier the noise, i.e. the low-

, the worse the GD performance. Conversely, the opposite behavior is observed for

justified in light of the local validity of Jensen's inequality (Van Trees, 2003) and is basically the same phenomenon that makes digital modulation schemes operating in Gaussian noise

than in no-fading channels. Notice, this phenomenon is in accordance with that observed in (Conte 1995). In order to validate the Chernoff bound, we also show, on the same plots, some points obtained by Monte Carlo simulations. These points obviously lie between the corresponding upper and lower probability of error bounds. Additionally, we compare the GD Chernoff bound with that for the conventional optimum receiver (Buzzi et al., 2001). A sup-

. This fact might appear, at a first look, surprising. It may be analytically

, superior performance in Rayleigh flat-flat fading channels

er

small values of <sup>0</sup>

to achieve, for low values of <sup>0</sup>

eriority of GD structure is evident.

Fig. 4. Chernoff bounds for *PE* of the optimum GD.

*λ* = 5 *λ* = 0.1 In Fig. 5, the effect of the channel diversity order is investigated. Indeed, the optimum GD performance versus <sup>0</sup> is represented for several values of *P* and with 1 . The <sup>1</sup> , , *Z Z <sup>P</sup>* have been assumed exponentially correlated with correlation coefficient 0.2 . A procedure for generating these exponentially correlated random variables for integer and semiinteger values of is reported in (Lombardo et al., 1999). As expected, as *P* increases, the GD performance ameliorates, thus confirming that diversity represents a suitable means to restore performance in severely hostile scenarios. Also, we compare the GD performance with that for the conventional optimum receiver (Buzzi et al., 2001) and we see that the GD keeps superiority in this case, too.

Fig. 5. *PE* at several values of *P*.

The optimum GD performance versus <sup>0</sup> for the generalized Laplace noise at 1, 4 *P* and for several values of the correlation coefficient is demonstrated in Fig. 6. It is seen that the probability of error improves for vanishingly small . For small , the GD takes much advantage of the diversity observations. For high values of , the realizations <sup>1</sup> , , *Z Z <sup>P</sup>* are very similar and much less advantage can be gained through the adoption of a diversity strategy. Such GD performance improvement is akin to that observed in signal diversity detection in the presence of flat-flat fading and Gaussian noise. We see that the GD outperforms the conventional optimum receiver (Buzzi et al., 2001) by the probability of error.

In Fig. 7, we compare the optimum GD performance versus that of the low energy coherence GD. We assumed 0.2 and *P* 4 . It is seen that the performance loss incurred by the low energy coherence GD with respect to the optimum GD is kept within a fraction of 1 dB at <sup>4</sup> 10 . *PE* Simulation results that are not presented in the paper show that the crucial factor ruling the GD performance is the noise shape parameter, whereas the particular noise distribution has a rather limited effect on the probability of error.

Fig. 6. *PE* at several values of the correlation coefficient.

Fig. 7. *PE* for the optimum and low energy coherence GDs.

#### **3.6 Discussion**

322 Wireless Communications and Networks – Recent Advances

ry similar and much less advantage can be gained through the adoption of a diversity strategy. Such GD performance improvement is akin to that observed in signal diversity detection in the presence of flat-flat fading and Gaussian noise. We see that the GD outperforms

In Fig. 7, we compare the optimum GD performance versus that of the low energy coheren-

low energy coherence GD with respect to the optimum GD is kept within a fraction of 1 dB

 Simulation results that are not presented in the paper show that the crucial factor ruling the GD performance is the noise shape parameter, whereas the particular noise dis-

for the generalized Laplace noise at

. For small

and *P* 4 . It is seen that the performance loss incurred by the

, the GD takes much ad-

, the realizations <sup>1</sup> , , *Z Z <sup>P</sup>* are ve-

is demonstrated in Fig. 6. It is seen that the

1, 4 *P* and

the conventional optimum receiver (Buzzi et al., 2001) by the probability of error.

The optimum GD performance versus <sup>0</sup>

ce GD. We assumed 0.2

at <sup>4</sup> 10 . *PE*

for several values of the correlation coefficient

probability of error improves for vanishingly small

vantage of the diversity observations. For high values of

tribution has a rather limited effect on the probability of error.

Fig. 6. *PE* at several values of the correlation coefficient.

We have considered the problem of diversity detection of one out of *M* signals transmitted over a fading dispersive channel in the presence of non-Gaussian noise. We have modeled the additive noise on each channel diversity branch through a spherically invariant random process, and the optimum GD has been shown to be independent of the actual joint pdf of the noise texture components present on the channel diversity outputs. The optimum GD is similar to the optimum GD for Gaussian noise, where the only difference is that the noise PSD <sup>0</sup> 2*N* is substituted with a perfect estimate of the short-term PSD realizations of the impulsive additive noise. We also derived a suboptimum GD matched with GASP based on the low energy coherence hypothesis. At the performance analysis stage, we focused on frequency-selective slowly fading channels and on a BFSK signaling scheme and evaluated the GD performance through both a semianalytic bounding technique and computer simulations. Numerical results have shown that the GD performance is affected by the average received energy contrast, by the channel diversity order, and by the noise shape parameter, whereas it is only marginally affected by the actual noise distribution. Additionally, it is seen that in impulsive environments, diversity represents a suitable strategy to improve GD performance.
