12 **2. Main functioning principles under employment of GASP in DS-**13 **CDMA wireless communication systems**

#### 14 **2.1. GR flowchart**

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80 Contemporary Issues in Wireless Communications

16 theories.

40 GASP.

30 the problem of synthesis – arises.

1 The main idea is to use the **generalized approach to signal processing** (GASP) in noise in 2 wireless communication systems [1-3]. The generalized approach is based on a seemingly 3 abstract idea: the introduction of an additional noise source that does not carry any 4 information about the signal and signal parameters in order to improve the qualitative 5 performance of wireless communication systems. In other words, we compare statistical 6 data defining the statistical parameters of the probability distribution densities (pdfs) of 7 the observed input stochastic samples from two independent frequency time regions – a 8 "yes" signal is possible in the first region and it is known a priori that a "no" signal is 9 obtained in the second region. The proposed GASP allows us to formulate a decision-ma-10 king rule based on the determination of **the jointly sufficient statistics of the mean and**  11 **variance** of the likelihood function (or functional). Classical and modern signal processing 12 theories allow us to define **only the mean** of the likelihood function (or functional). 13 Additional information about the statistical characteristics of the likelihood function (or 14 functional) leads us to better quality signal detection and definition of signal parameters 15 in compared with the optimal signal processing algorithms of classical or modern

17 Thus, for any wireless communication systems, we have to consider two problems – 18 analysis and synthesis [8]. The first problem (analysis) – the problem to study a stimulus 19 of the additive and multiplicative noise on the main principles and performance under the 20 use of GASP – is an analysis of impact of the additive and multiplicative noise on the 21 main characteristics of wireless communication systems, the receivers in which are 22 constructed on the basis of GASP. This problem is very important in practice. This 23 analysis allows us to define limitations on the use of wireless communication systems and 24 to quantify the additive and multiplicative noise impact relative to other sources of 25 interference present in these systems. If we are able to conclude that the presence of the 26 additive and multiplicative noise is the main factor or one of the main factors limiting the 27 performance of any wireless communication systems, then the second problem – the 28 definition of structure and main parameters and characteristics of the generalized detector 29 or receiver (GD or GR) under a dual stimulus of the additive and multiplicative noise –

31 GASP allows us to extend the well-known boundaries of the potential noise immunity set by 32 classical and modern signal processing theories. Employment of wireless communication 33 systems, the receivers of which are constructed on the basis of GASP, allows us to obtain 34 high detection of signals and high accuracy of signal parameter definition with noise 35 components present compared with that systems, the receivers of which are constructed on 36 the basis of classical and modern signal processing theories. The optimal and asymptotic 37 optimal signal processing algorithms of classical and modern theories, for signals with 38 amplitude-frequency-phase structure characteristics that can be known and unknown a 39 priori, are components of the signal processing algorithms that are designed on the basis of 15 The receiver in wireless communication system has an antenna array with the number of 16 elements equal to *M* and each antenna array element receives *N* samples during the 17 sensing time. The signal detection problem can be modeled as the conventional binary 18 hypothesis test:

$$\begin{cases} \mathfrak{H}\_0 \Rightarrow \boldsymbol{z}\_i[k] = \boldsymbol{w}\_i[k], & i = 1, \dots, M; \ k = 0, \dots, N - 1, \\\mathfrak{H}\_1 \Rightarrow \boldsymbol{z}\_i[k] = h\_i[k] \; \boldsymbol{s}[k] + \boldsymbol{w}\_i[k], & i = 1, \dots, M; \ k = 0, \dots, N - 1, \end{cases} \tag{1}$$

where *z* [*k*] *<sup>i</sup>* is the discrete-time received signal at the receiver input; *w* [*k*] *<sup>i</sup>* 20 is the discretetime circularly symmetric complex Gaussian noise with zero mean and variance , <sup>2</sup> 21 i.e. [ ] ~ (0, ) <sup>2</sup> *wi k* CN ; *h* [*k*] *<sup>i</sup>* 22 is the discrete-time channel coefficients obeying the circularly symmetric complex Gaussian distribution with zero mean and variance equal to <sup>2</sup> *<sup>h</sup>* 23 , i.e., [ ] ~ (0, ) <sup>2</sup> *<sup>i</sup> <sup>h</sup>* 24 *h k* CN ; and *s*[*k*] is the discrete-time signal, i.e., the signal to be detected. We 25 consider the same initial conditions with respect to *s*[*k* ] as in [4,5]. Throughout this chapter, the signal *s*[*k* ] , the channel coefficients *h* [*k*] *<sup>i</sup>* , and the noise *w* [*k*] *<sup>i</sup>* 26 are independent between 27 each other.

28 We define the *NM* 1 signal vector **Z** that collects all the observed signal samples during 29 the sensing time:

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$$\mathbf{Z} = \begin{bmatrix} z\_1[0], \dots, z\_M[0], \dots, z\_1[N-1], \dots, z\_M[N-1] \end{bmatrix}^T,\tag{2}$$

2 where *T* denotes a transpose. The data distribution in the complex matrix **Z** can be 3 expressed as:

$$\mathbf{Z} \sim \begin{cases} \mathcal{C} \mathcal{N}'(0, \sigma^2 \mathbf{I}) \; , & \Rightarrow \; \mathfrak{H}\_0 \\ \mathcal{C} \mathcal{N}'(0, E\_s \sigma\_h^2 \mathbf{I} + \sigma^2 \mathbf{I}) \; , & \Rightarrow \; \mathfrak{H}\_1 \end{cases} \tag{3}$$

where *Es* 5 is the average signal energy at the receiver input, and **I** is the *MN MN* identity 6 matrix. We consider a situation when the signaling scheme is unknown (the receiver has a 7 total freedom of choosing the signaling strategy) excepting a known power within the limits 8 of the frequency band of interest. Thus, the receiver should be able to detect a presence of 9 any possible signals satisfying the power and bandwidth constraints for robust detection of 10 the signal *s*[*k*] incoming at the receiver input in wireless communication systems.

11 The generalized receiver (GR) has been constructed based on the generalized approach to 12 signal processing (GASP) in noise and discussed in numerous journal and conference papers 13 and some monographs, namely, in [1–21]. GR is considered as a linear combination of the 14 correlation detector that is optimal in the Neyman-Pearson criterion sense under detection 15 of signals with known parameters and the energy detector that is optimal in the Neyman-16 Pearson criterion sense under detection of signals with unknown parameters. The main 17 functioning principle of GR is a matching between the model signal generated by the local 18 oscillator in GR and the information signal incoming at the GR input by whole range of 19 parameters. In this case, the noise component of the GR correlation detector caused by 20 interaction between the model signal generated by the local oscillator in GR and the input 21 noise and the random component of the GR energy detector caused by interaction between 22 the energy of incoming information signal and the input noise are cancelled in the statistical 23 sense. This GR feature allows us to obtain the better detection performance in comparison 24 with other classical receivers.

25 The specific feature of GASP is introduction of additional noise source that does not carry 26 any information about the signal with the purpose to improve a qualitative signal detection 27 performance. This additional noise can be considered as the reference noise without any 28 information about the signal to be detected. The jointly sufficient statistics of the mean and 29 variance of the likelihood function is obtained in the case of GASP implementation, while 30 the classical and modern signal processing theories can deliver only a sufficient statistics of 31 the mean or variance of the likelihood function (no the jointly sufficient statistics of the 32 mean and variance of the likelihood function). Thus, the implementation of GASP allows us 33 to obtain more information about the input process or received information signal. Owing to 34 this fact, an implementation of the receivers constructed based on GASP basis allows us to 1 improve the signal detection performance of wireless communication system in comparison 2 with employment of other conventional receivers.

4 **Figure 1.** GR flowchart.

3

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82 Contemporary Issues in Wireless Communications

3 expressed as:

24 with other classical receivers.

 [0], , [0], , [ 1], , [ 1] , <sup>1</sup> <sup>1</sup> *<sup>T</sup>* <sup>1</sup>**<sup>Z</sup>** *<sup>z</sup> zM <sup>z</sup> <sup>N</sup> zM <sup>N</sup>* (2)

2 2

**I I**

 , (0, ) ,

1

H

H

0

2 where *T* denotes a transpose. The data distribution in the complex matrix **Z** can be

(0, ) ,

2

where *Es* 5 is the average signal energy at the receiver input, and **I** is the *MN MN* identity 6 matrix. We consider a situation when the signaling scheme is unknown (the receiver has a 7 total freedom of choosing the signaling strategy) excepting a known power within the limits 8 of the frequency band of interest. Thus, the receiver should be able to detect a presence of 9 any possible signals satisfying the power and bandwidth constraints for robust detection of

11 The generalized receiver (GR) has been constructed based on the generalized approach to 12 signal processing (GASP) in noise and discussed in numerous journal and conference papers 13 and some monographs, namely, in [1–21]. GR is considered as a linear combination of the 14 correlation detector that is optimal in the Neyman-Pearson criterion sense under detection 15 of signals with known parameters and the energy detector that is optimal in the Neyman-16 Pearson criterion sense under detection of signals with unknown parameters. The main 17 functioning principle of GR is a matching between the model signal generated by the local 18 oscillator in GR and the information signal incoming at the GR input by whole range of 19 parameters. In this case, the noise component of the GR correlation detector caused by 20 interaction between the model signal generated by the local oscillator in GR and the input 21 noise and the random component of the GR energy detector caused by interaction between 22 the energy of incoming information signal and the input noise are cancelled in the statistical 23 sense. This GR feature allows us to obtain the better detection performance in comparison

25 The specific feature of GASP is introduction of additional noise source that does not carry 26 any information about the signal with the purpose to improve a qualitative signal detection 27 performance. This additional noise can be considered as the reference noise without any 28 information about the signal to be detected. The jointly sufficient statistics of the mean and 29 variance of the likelihood function is obtained in the case of GASP implementation, while 30 the classical and modern signal processing theories can deliver only a sufficient statistics of 31 the mean or variance of the likelihood function (no the jointly sufficient statistics of the 32 mean and variance of the likelihood function). Thus, the implementation of GASP allows us 33 to obtain more information about the input process or received information signal. Owing to 34 this fact, an implementation of the receivers constructed based on GASP basis allows us to

*Es <sup>h</sup>* 4 (3)

 

CN

CN

**<sup>I</sup> <sup>Z</sup> <sup>~</sup>**

10 the signal *s*[*k*] incoming at the receiver input in wireless communication systems.

5 The GR flowchart is presented in Fig. 1. As we can see from Fig. 1, the GR consists of three 6 channels:


As follows from Fig. 1, under the hypothesis H<sup>1</sup> 12 (a "yes" signal), the GR correlation channel generates the signal component *s*mod [*k*]*s* [*k*] *<sup>i</sup> <sup>i</sup>* 13 caused by interaction between the model signal 14 (the reference signal at the GR MSG output) and the incoming information signal and the noise component *s*mod [*k*] [*k*] *<sup>i</sup> <sup>i</sup>* caused by interaction between the model signal *s*mod [*k*] *<sup>i</sup>* 15 and the noise [*k*] *<sup>i</sup>* (the PF output). Under the hypothesis H<sup>1</sup> 16 , the GR autocorrelation channel generates the information signal energy [ ] <sup>2</sup> *<sup>s</sup> <sup>k</sup> <sup>i</sup>* and the random component *<sup>s</sup>* [*<sup>k</sup>* ] [*<sup>k</sup>* ] *<sup>i</sup> <sup>i</sup>* 17 caused by interaction between the information signal *s* [*k* ] *<sup>i</sup>* and the noise [*k*] *<sup>i</sup>* 18 . The main purpose of 19 the GR compensation channel is to cancel in the statistical sense the GR correlation channel noise component *s*mod [*k*] [*k*] *<sup>i</sup> <sup>i</sup>* 20 and the GR autocorrelation channel random component *s* [*k* ] [*k*] *<sup>i</sup> <sup>i</sup>* between each other based on the same nature of the noise [*k*] *<sup>i</sup>* 21 .

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1 For description of the GR flowchart we consider the discrete-time processes without loss of 2 any generality. Evidently, the cancelation between the GR correlation channel noise component *s*mod [*k*] [*k*] *<sup>i</sup> <sup>i</sup>* and the GR autocorrelation channel random component *s* [*k* ] [*k* ] *<sup>i</sup> <sup>i</sup>* 3 is possible only based on the same nature of the noise [*k*] *<sup>i</sup>* 4 satisfying the condition of equality between the signal model *s*mod [*k*] *<sup>i</sup>* and incoming signal *<sup>s</sup>* [*<sup>k</sup>* ] *<sup>i</sup>* 5 over the whole range of parameters. The condition *s*mod [*k* ] *s* [*k* ] *<sup>i</sup> <sup>i</sup>* 6 is the **GR main functioning condition**. To 7 satisfy this condition, we are able to define the incoming signal parameters. Naturally, in 8 practice, signal parameters are random. How we can satisfy the GR main functioning 9 condition and define the signal parameters in practice if there is no a priori information 10 about the signal and there is an uncertainty in signal parameters, i.e. information signal 11 parameters are stohastic, is discussed in detail in [1,3,21].

Under the hypothesis H <sup>0</sup> 12 (a "no" information signal), satisfying the GR main functioning condition *s*mod [*k*] *<sup>s</sup>* [*k*] *<sup>i</sup> <sup>i</sup>* , we obtain only the background noise [ ] [ ] <sup>2</sup> <sup>2</sup> *<sup>k</sup> <sup>k</sup> <sup>i</sup> <sup>i</sup>* <sup>13</sup> at the GR 14 output. Additionally, the practical implementation of the GR decision statistics requires an estimation of the noise variance <sup>2</sup> *<sup>w</sup>* using the reference noise [*k*] *<sup>i</sup>* 15 at the AF output. AF is 16 the reference noise source and the PF bandwidth is matched with the bandwidth of the information signal *s* [*k* ] *<sup>i</sup>* 17 to be detected. The threshold apparatus (THRA) device defines the 18 GR threshold. PF and AF can be considered as the linear systems, for example, as the bandpass filters, with the impulse responses *h* [*m*] *PF* and *h* [*m*] *AF* 19 , respectively. For 20 simplicity of analysis, we assume that these filters have the same amplitude-frequency 21 characteristics or impulse responses by shape. Moreover, the AF central frequency is 22 detuned with respect to the PF central frequency on such a value that the information signal 23 cannot pass through the AF. Thus, the information signal and noise can be appeared at the 24 PF output and the only noise is appeared at the AF output. If a value of detuning between the AF and PF central frequencies is more than *<sup>s</sup>* 4 or 5*f* , where *<sup>s</sup>* 25 *f* is the signal bandwidth, 26 the processes at the AF and PF outputs can be considered as the uncorrelated and 27 independent processes and, in practice, under this condition, the coefficient of correlation 28 between PF and AF output processes is not more than 0.05 that was confirmed 29 experimentally in [22,23].

30 The processes at the AF and PF outputs present the input stochastic samples from two 31 independent frequency-time regions. If the noise *w*[*k*] at the PF and AF inputs is Gaussian, 32 the noise at the PF and AF outputs is Gaussian, too, owing to the fact that PF and AF are the 33 linear systems and we believe that these linear systems do not change the statistical 34 parameters of the input process. Thus, the AF can be considered as a reference noise 35 generator with a priori knowledge a "no" signal (the reference noise sample). A detailed 36 discussion of the AF and PF can be found in [6,7]. The noise at the PF and AF outputs can be 37 presented in the following form:

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 7 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 85

$$\begin{cases} \boldsymbol{w}\_{PF}[k] = \boldsymbol{\xi}[k] = \sum\_{m=-\eta}^{n} h\_{PF}[m] \boldsymbol{\upkappa}[k-m] \ ,\\ \boldsymbol{w}\_{AF}[k] = \boldsymbol{\eta}[k] = \sum\_{m=-\eta}^{n} h\_{AF}[m] \boldsymbol{\upkappa}[k-m] \ . \end{cases} \tag{4}$$

Under the hypothesis , H<sup>1</sup> the signal at the PF output can be defined as *x* [*k*] *s* [*k*] [*k*] *<sup>i</sup> <sup>i</sup> <sup>i</sup>* 2 (see Fig. 1), where [*k*] *<sup>i</sup>* is the observed noise at the PF output and *s* [*k*] *h* [*k* ] *s*[*k*] *<sup>i</sup> <sup>i</sup>* ; *h* [*k* ] *<sup>i</sup>* 3 are the channel coefficients indicated here only in a general case. Under the hypothesis H<sup>0</sup> 4 and for all *i* and *k*, the process *x* [*k*] [*k*] *<sup>i</sup> <sup>i</sup>* 5 at the PF output is subjected to the complex 6 Gaussian distribution and can be considered as the independent and identically distributed (i.i.d.) process. The process at the AF output is the reference noise [*k*] *<sup>i</sup>* 7 with the same statistical parameters as the noise [*k*] *<sup>i</sup>* 8 in the ideal case (we make this assumption for simplicity). In practice, the statistical parameters of the noise [*k*] *<sup>i</sup>* and [*k*] *<sup>i</sup>* 9 are different.

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component *s*mod [*k*] [*k*] *<sup>i</sup> <sup>i</sup>*

Under the hypothesis

29 experimentally in [22,23].

37 presented in the following form:

estimation of the noise variance <sup>2</sup>

84 Contemporary Issues in Wireless Communications

11 parameters are stohastic, is discussed in detail in [1,3,21].

H

1 For description of the GR flowchart we consider the discrete-time processes without loss of 2 any generality. Evidently, the cancelation between the GR correlation channel noise

 *<sup>i</sup>* 4 satisfying the condition of equality between the signal model *s*mod [*k*] *<sup>i</sup>* and incoming signal *<sup>s</sup>* [*<sup>k</sup>* ] *<sup>i</sup>* 5 over the whole range of parameters. The condition *s*mod [*k* ] *s* [*k* ] *<sup>i</sup> <sup>i</sup>* 6 is the **GR main functioning condition**. To 7 satisfy this condition, we are able to define the incoming signal parameters. Naturally, in 8 practice, signal parameters are random. How we can satisfy the GR main functioning 9 condition and define the signal parameters in practice if there is no a priori information 10 about the signal and there is an uncertainty in signal parameters, i.e. information signal

<sup>0</sup> 12 (a "no" information signal), satisfying the GR main functioning

14 output. Additionally, the practical implementation of the GR decision statistics requires an

*<sup>i</sup>* 15 at the AF output. AF is 16 the reference noise source and the PF bandwidth is matched with the bandwidth of the information signal *s* [*k* ] *<sup>i</sup>* 17 to be detected. The threshold apparatus (THRA) device defines the 18 GR threshold. PF and AF can be considered as the linear systems, for example, as the bandpass filters, with the impulse responses *h* [*m*] *PF* and *h* [*m*] *AF* 19 , respectively. For 20 simplicity of analysis, we assume that these filters have the same amplitude-frequency 21 characteristics or impulse responses by shape. Moreover, the AF central frequency is 22 detuned with respect to the PF central frequency on such a value that the information signal 23 cannot pass through the AF. Thus, the information signal and noise can be appeared at the 24 PF output and the only noise is appeared at the AF output. If a value of detuning between the AF and PF central frequencies is more than *<sup>s</sup>* 4 or 5*f* , where *<sup>s</sup>* 25 *f* is the signal bandwidth, 26 the processes at the AF and PF outputs can be considered as the uncorrelated and 27 independent processes and, in practice, under this condition, the coefficient of correlation 28 between PF and AF output processes is not more than 0.05 that was confirmed

30 The processes at the AF and PF outputs present the input stochastic samples from two 31 independent frequency-time regions. If the noise *w*[*k*] at the PF and AF inputs is Gaussian, 32 the noise at the PF and AF outputs is Gaussian, too, owing to the fact that PF and AF are the 33 linear systems and we believe that these linear systems do not change the statistical 34 parameters of the input process. Thus, the AF can be considered as a reference noise 35 generator with a priori knowledge a "no" signal (the reference noise sample). A detailed 36 discussion of the AF and PF can be found in [6,7]. The noise at the PF and AF outputs can be

*<sup>w</sup>* using the reference noise [*k*]

condition *s*mod [*k*] *<sup>s</sup>* [*k*] *<sup>i</sup> <sup>i</sup>* , we obtain only the background noise [ ] [ ] <sup>2</sup> <sup>2</sup> *<sup>k</sup> <sup>k</sup> <sup>i</sup> <sup>i</sup>* <sup>13</sup>

*<sup>i</sup>* 3

is possible only based on the same nature of the noise [*k*]

and the GR autocorrelation channel random component *s* [*k* ] [*k* ] *<sup>i</sup>*

 

at the GR

10 The decision statistics at the GR output presented in [1,3] is extended to the case of antenna 11 array employment when an adoption of multiple antennas and antenna arrays is effective to 12 mitigate the negative attenuation and fading effects [4,5]. The GR decision statistics can be 13 presented in the following form:

$$T\_{\rm GR}(\mathbf{X}) = \sum\_{k=0}^{N-1} \sum\_{i=1}^{M} 2\mathbf{x}\_i[k]\mathbf{s}\_{\rm mod\_i}[k] - \sum\_{k=0}^{N-1} \sum\_{i=1}^{M} \mathbf{x}\_i^2[k] + \sum\_{k=0}^{N-1} \sum\_{i=1}^{M} \eta\_i^2[k] \gtrsim\_{\mathfrak{H}\_0}^{\mathfrak{H}\_1} \text{THR}\_{\mathbf{GR}} \tag{5}$$

where *<sup>T</sup>* [*x* [0], , *xM* [0], , *x* [*N* 1], , *xM* [*N* 1] 15 **X** <sup>1</sup> <sup>1</sup> is the vector of random process at the PF output, and *THRGR* 16 is the GR detection threshold. We can rewrite (5) using the vector 17 form:

$$T\_{\rm GR}(\mathbf{X}) = 2\mathbf{S}\_{\rm mod}\mathbf{X} - \mathbf{X}^2 + \boldsymbol{\eta}^2 \underset{\boldsymbol{\mathsf{H}}}{\gtrless} \frac{\mu\_{\boldsymbol{\mathsf{H}}}}{\mu\_{\boldsymbol{\mathsf{H}}}} \text{THR}\_{\rm GR} \tag{6}$$

19 where **X** [**x**(0),..., **x**(*N* 1)] is the *M* 1 vector of the random process at the PF output with elements defined as *<sup>T</sup> k x k x k <sup>M</sup>* [ ] [ [ ], , [ ]] <sup>20</sup>**x** <sup>1</sup> ; **<sup>S</sup>**mod [**s**mod (0),,**s**mod (*<sup>N</sup>* 1)] is the *<sup>M</sup>* <sup>1</sup> 21 vector of the process at the MSG output with the elements defined as **s**mod[*k*] *<sup>T</sup> <sup>s</sup> <sup>k</sup> <sup>s</sup> <sup>k</sup> <sup>M</sup>* [ mod [ ], , mod [ ]] <sup>22</sup><sup>1</sup> ; **<sup>η</sup>** [**η**(0),, **<sup>η</sup>**(*<sup>N</sup>* 1)] is the *<sup>M</sup>* 1 vector of the random process at the AF output with the elements defined as *<sup>T</sup>* [*k*] [ [*k*], , *<sup>M</sup>* [*k*]] **η** <sup>1</sup> and *THR GR* 23 is the GR 24 detection threshold.

25 According to GASP and GR structure shown in Fig. 1, the GR test statistics takes the following form under the hypotheses H1 and H<sup>0</sup> 26 , respectively:

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$$T\_{GR}(\mathbf{X}) = \begin{cases} \sum\_{k=0}^{N-1} \sum\_{\iota=1}^{M} s\_{\iota}^{2}(k) + \sum\_{k=0}^{N-1} \sum\_{\iota=1}^{M} \eta\_{\iota}^{2}(k) - \sum\_{k=0}^{N-1} \sum\_{\iota=1}^{M} \xi\_{\iota}^{2}(k) & \Rightarrow \mathcal{H}\_{1} \\\\ \sum\_{k=0}^{N-1} \sum\_{\iota=1}^{M} \eta\_{\iota}^{2}(k) - \sum\_{k=0}^{N-1} \sum\_{\iota=1}^{M} \xi\_{\iota}^{2}(k) & \Rightarrow \mathcal{H}\_{0} \end{cases} \tag{7}$$

The term *<sup>s</sup> N k M <sup>i</sup> si <sup>k</sup> <sup>E</sup>* 1 0 1 <sup>2</sup> 2 [ ] is the average signal energy and the term <sup>1</sup> 0 1 2 1 0 1 2[ ] [ ] *N k M <sup>i</sup> <sup>i</sup> N k M <sup>i</sup> <sup>i</sup>* <sup>3</sup> *k k* presents the background noise at the GR output that is a 4 difference between the noise power at the PF and AF outputs. It is important to mention that 5 the main GR functioning condition is the equality between parameters of the model signal *s*mod [*k*] *<sup>i</sup>* and the signal *s* [*k* ] *<sup>i</sup>* , i.e. *s*mod [*k*] *<sup>i</sup> s* [*k*] *<sup>i</sup>* 6 over all range of parameters and, in 7 particular, by amplitude. How we can satisfy this condition in practice is discussed in detail 8 in [1] and [3, Chapter 7, pp 611-621] when there is no a priori information about the signal *s* [*k* ] *<sup>i</sup>* 9 . This condition is essential for complete compensation in the statistical sense between the noise component of the correlation channel 2*s*mod [*k*] [*k* ] *<sup>i</sup> <sup>i</sup>* 10 , the GR correlation channel, caused by interaction between the model signal *s*mod [*k*] *<sup>i</sup>* and noise [*k*] *<sup>i</sup>* 11 , and the random component of the GR autocorrelation channel 2*s* [*k*] [*k*] *<sup>i</sup> <sup>i</sup>* 12 , the GR ED, caused by interaction between the signal *s* [*k* ] *<sup>i</sup>* and noise [*k*] *<sup>i</sup>* 13 [1] and [6, Chapter 3]. The complete matching between the model signal *s*mod [*k*] *<sup>i</sup>* and the incoming signal *s* [*k* ] *<sup>i</sup>* 14 , especially by amplitude, is 15 a very hard problem in practice and only in the ideal case the complete matching is possible. 16 How the GR detection performance can be deteriorated under mismatching between the model signal *s*mod [*k*] *<sup>i</sup>* and the incoming signal *s* [*k* ] *<sup>i</sup>* 17 is discussed in [24]. Additionally, a 18 practical implementation of the GR decision statistics requires an estimation of the noise variance <sup>2</sup> *<sup>w</sup>* using the reference noise [*k* ] *<sup>i</sup>* 19 at the AF output.

The mean *GR m*H0 and the variance *GR Var*H<sup>0</sup> 20 of the test statistics *TGR* (**X**) under the hypothesis H<sup>0</sup> 21 are given in the following form [6, Chapter 3]:

$$\begin{cases} m\_{\mathfrak{H}\_0}^{GR} = \mathbb{E} \left[ T\_{GR}(\mathbf{X}) | \mathfrak{H}\_0 \right] = 0 \end{cases},\tag{8}$$

$$Var\_{\mathfrak{H}\_0}^{GR} = Var \left[ T\_{GR}(\mathbf{X}) | \mathfrak{H}\_0 \right] = 4NM\sigma^4 \,\,\,.$$
