**2.2. MGF of the GR partial test statistics** ( ) *TGR* **X***<sup>k</sup>* 23

24 To complete consideration of the GR main functioning principles the moment generating function (MGF) definition of the GR partial test statistics ( ) 25 *TGR* **X***<sup>k</sup>* under the main GR functioning condition *s*mod [*k*] *<sup>s</sup>* [*k*] *<sup>i</sup> <sup>i</sup>* and the hypothesis H<sup>1</sup> 26 given by

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 9 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 87

$$T\_{GR}(\mathbf{X}\_k) = \sum\_{l=1}^{M} s\_l^2 \{k\} + \sum\_{l=1}^{M} \eta\_l^2 \{k\} - \sum\_{l=1}^{M} \xi\_l^2 \{k\} \tag{9}$$

2 is required. We say that the random variable *x* has a chi-square distribution with degree 3 of freedom if its probability density function (pdf) is determined as

$$p(\mathbf{x}) = c\mathbf{x}^{0.5\nu - 1} \exp(-0.5\mathbf{x})\,,\tag{10}$$

5 where *c* is a constant given by [25]

8 BookTitle

 

*i i*

*M*

1 2

*M*

0 1

*k k*

*i i*

2

*s k k k*

*N*

1

*k*

*N*

*k*

<sup>2</sup> 2 [ ] is the average signal energy and the term

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

*<sup>i</sup>* 11 , and the random

*<sup>i</sup>* 12 , the GR ED, caused by interaction

0

1 **(X** (7)

1

[ ] [ ] .

0

, the GR correlation channel,

H

1

H

*k*

[ ] [ ] [ ] ,

*N*

1

0 1

*i i*

*M*

2

*k* presents the background noise at the GR output that is a

 

*N*

1

*k*

0

*N*

1

*k*

0

1 2

*i i*

1 2

*i i*

*M*

*M*

)

*M <sup>i</sup> si <sup>k</sup> <sup>E</sup>* 1 0 1

2[ ] [ ] *N k*

0 1

*M <sup>i</sup> <sup>i</sup>*

the noise component of the correlation channel 2*s*mod [*k*] [*k* ] *<sup>i</sup> <sup>i</sup>* 10

component of the GR autocorrelation channel 2*s* [*k*] [*k*] *<sup>i</sup>*

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

*<sup>i</sup>* 19 at the AF output.

0 and the variance *GR Var*

 

functioning condition *s*mod [*k*] *<sup>s</sup>* [*k*] *<sup>i</sup> <sup>i</sup>* and the hypothesis

<sup>1</sup> 26 given by

0

H

*GR*

0

H

*GR*

*m T*

 

<sup>0</sup> 21 are given in the following form [6, Chapter 3]:

**2.2. MGF of the GR partial test statistics** ( ) *TGR* **X***<sup>k</sup>* 23

between the signal *s* [*k* ] *<sup>i</sup>* and noise [*k*]

caused by interaction between the model signal *s*mod [*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

H<sup>0</sup> 20 of the test statistics *TGR* (**X**) under the hypothesis

*Var Var T NM*

*GR*

24 To complete consideration of the GR main functioning principles the moment generating function (MGF) definition of the GR partial test statistics ( ) 25 *TGR* **X***<sup>k</sup>* under the main GR

[ ]

**X**

E ( ) 0 ,

0

H

( ) 4 .

0

H

4

H

*GR*

[ ]

**X** 22 (8)

*GR*

*T*

86 Contemporary Issues in Wireless Communications

The term *<sup>s</sup> N k*

2 1

*k*

<sup>1</sup>

variance <sup>2</sup> 

H

The mean *GR m*

H

*N k*

*<sup>i</sup> <sup>i</sup>* <sup>3</sup>

0 1

*M*

$$c = \frac{1}{2^{0.5\nu} \Gamma(0.5\nu)}\,,\tag{11}$$

7 () is the gamma function. The MGF general form for the chi-square distributed random 8 variable *x* is given by [25]

$$\mathcal{M}\_{\mathbf{x}}(l) = \mathbb{E}[\exp(l\mathbf{x})] = \int\_{-\eta}^{\eta} \exp(l\mathbf{x})\,\mathbf{p}(\mathbf{x})d\mathbf{x} = c \iint\_{\mathbf{0}} \exp(l\mathbf{x})\mathbf{x}^{0.5\nu - 1} \exp(-0.5\mathbf{x})\,d\mathbf{x}\,\tag{12}$$

10 where E[] is the mathematical expectation. At 1, the constant

$$c = \frac{1}{2^{0.5} \Gamma(0.5\nu)} = \frac{1}{\sqrt{2\pi}}\ . \tag{13}$$

Assume that [ ] [ ] <sup>2</sup> <sup>1</sup>*z k k <sup>i</sup> <sup>i</sup>* and [ ] [ ] <sup>2</sup> <sup>2</sup>*z k k <sup>i</sup> <sup>i</sup>* . The pdfs for the random variables *<sup>i</sup> z*1 and *<sup>i</sup> z*<sup>2</sup> 12 are defined by the chi-square <sup>2</sup> 13 distribution law with one degree of freedom [3]:

$$p(\mathbf{z}\_{\rm th}) = \frac{1}{\sqrt{2\pi |\mathbf{z}\_{\rm th}|} \ \sigma\_w} \exp\left\{-\frac{\mathbf{z}\_{\rm th}}{2\sigma\_w^2}\right\} \ \ , \ \ \mathbf{z}\_{\rm th} > 0 \ \tag{14}$$

$$p(\mathbf{z}\_{2i}) = \frac{1}{\sqrt{2\pi \ \mathbf{z}\_{2i}} \ \sigma\_w} \exp\left\{-\frac{\mathbf{z}\_{2i}}{2\sigma\_w^2}\right\} \quad , \ \mathbf{z}\_{2i} > 0 \ \text{ .} \tag{15}$$

Introduce a new random variable *<sup>i</sup> <sup>i</sup> <sup>i</sup> z z z* <sup>1</sup> <sup>2</sup> . The MGF of the random variable *<sup>i</sup>* 16 *z* is given 17 using the following formula:

$$\begin{split} \mathcal{M}\_{\boldsymbol{z}\_{1}}(\boldsymbol{l}) &= \mathbb{E}\left[\exp(\boldsymbol{l}\boldsymbol{z}\_{1})\right] = \mathbb{E}\left\{\exp[\boldsymbol{l}(\boldsymbol{z}\_{1i} - \boldsymbol{z}\_{2i})]\right\} = \mathbb{E}\left[\exp(\boldsymbol{l}\boldsymbol{z}\_{1i})\exp(-\boldsymbol{l}\boldsymbol{z}\_{2i})\right] \\ &= \mathbb{E}\left[\exp(\boldsymbol{l}\boldsymbol{z}\_{1i})\right] \mathbb{E}\left[\exp(-\boldsymbol{l}\boldsymbol{z}\_{2i})\right] = \mathcal{M}\_{\boldsymbol{z}\_{1i}}(\boldsymbol{l})\mathcal{M}\_{\boldsymbol{z}\_{2i}}(-\boldsymbol{l}). \end{split} \tag{16}$$

10 BookTitle

2

The MGF of the random variable <sup>1</sup> 1 *z* is defined in the following form:

$$\mathcal{M}\_{\mathbf{z}\_{l\_l}}(l) = \frac{1}{\sqrt{2\pi}} \int\_0^{\eta} \exp(l\boldsymbol{\varepsilon}\_{l\_l}) \boldsymbol{z}\_{l\_l}^{-0.5} \exp\left\{-\frac{\boldsymbol{\varepsilon}\_{l\_l}}{2\sigma\_{\mathbf{w}}^2}\right\} \, d\boldsymbol{\varepsilon}\_{l\_l} = \frac{1}{\sqrt{2\pi}} \int\_0^{\eta} \boldsymbol{z}\_{l\_l}^{-0.5} \exp\left\{-\left(\frac{1}{2\sigma\_{\mathbf{w}}^2} - l\right) \boldsymbol{z}\_{l\_l}\right\} d\boldsymbol{\varepsilon}\_{l\_l}. \tag{17}$$

Introducing the variable *<sup>i</sup> <sup>w</sup> <sup>i</sup> g l z*<sup>1</sup> <sup>2</sup> (0.5 ) 3 , we can obtain:

$$\begin{split} \mathcal{M}\_{\mathbf{\tilde{z}}\_{\rm u}}(l) &= \frac{2\sigma\_{w}^{2}}{\sqrt{2\pi}\sigma\_{w}} \Bigg[ \Big[ 2\sigma\_{w}^{2} (1 - 2\sigma\_{w}^{2}l)^{-1} \right]^{-\frac{1}{2}} g\_{i}^{-0.5} \exp(-g\_{i}) (1 - 2\sigma\_{w}^{2}l)^{-1} dg\_{i} \\ &= \frac{1}{\sqrt{2\pi}\sigma\_{w}} \Bigg[ 2\sigma\_{w}^{2} (1 - 2\sigma\_{w}^{2}l)^{-1} \Bigg]^{-\frac{1}{2}} \Bigg[ g\_{i}^{-0.5} \exp(-g\_{i}) dg\_{i} = \Big[ \pi (1 - 2\sigma\_{w}^{2}l) \Bigg]^{-\frac{1}{2}} \Bigg[ g\_{i}^{-0.5} \exp(-g\_{i}) dg\_{i}. \end{split} \tag{18}$$

5 Based on the definition of the gamma function [26]

$$\Gamma(x) = \bigcap\_{\ell=0}^{\infty} l^{x-1} \exp(-l) dl \; , \tag{19}$$

7 we obtain that

$$\int\_{0}^{v} g\_i^{-0.5} \exp(-g\_i) dg\_i = \Gamma(0.5) = \sqrt{\pi}. \tag{20}$$

Finally, the MGF of the random variable *<sup>i</sup> z*<sup>1</sup> 9 is defined as

$$\mathcal{M}\_{\mathbf{2}\_{\mathbf{1}\_l}}(l) = \sqrt{\pi} \left[ \pi (1 - 2\sigma\_w^2 l) \right]^{-\frac{1}{2}} = \left( 1 - 2\sigma\_w^2 l \right)^{-\frac{1}{2}}.\tag{21}$$

The mean and the variance of the random variables *<sup>i</sup> z*<sup>1</sup> 11 can be determined in the following 12 form:

$$\mathbb{E}[z\_{1i}] = \left. \frac{\partial \mathcal{M}\_{\bar{z}\_{ii}}(l)}{\partial l} \right|\_{l=0} = \sigma\_{w}^{2} \, \, \, \, \tag{22}$$

$$\operatorname{Var}[\boldsymbol{z}\_{l,l}] = \operatorname{E}[\boldsymbol{z}\_{l,l}^{2}] - \operatorname{E}[\boldsymbol{z}\_{l,l}]^{2} = \frac{\partial^{2}\mathcal{M}\_{\boldsymbol{z}\_{l,l}}(l)}{\partial^{2}l} \bigg|\_{l=0} - \operatorname{E}[\boldsymbol{z}\_{l,l}]^{2} = 3\boldsymbol{\sigma}\_{\boldsymbol{w}}^{4} - \boldsymbol{\sigma}\_{\boldsymbol{w}}^{4} = 2\boldsymbol{\sigma}\_{\boldsymbol{w}}^{4} \,. \tag{23}$$

By the analogous way, we can find that the MGF of the random variable *<sup>i</sup> z*<sup>2</sup> 1 takes the 2 following form:

$$\mathcal{M}\_{2\_{\mathcal{Q}\_l}}(-l) = (1 + 2\sigma\_w^2 l)^{-0.5} \,. \tag{24}$$

Since *<sup>M</sup> <sup>i</sup> <sup>i</sup> s k* <sup>1</sup> { [ ]} 4 are spatially correlated for *i-*th antenna array elements, and according to [28], the MGF of *M <sup>i</sup> <sup>i</sup> <sup>s</sup> <sup>k</sup>* 1 <sup>2</sup> 5 [ ] is defined as

$$\mathcal{M}\_{\sum\_{i=1}^{M} \mathbf{s}\_i^2[k]}(l) = \prod\_{i=1}^{M} (1 - E\_s \sigma\_h^2 \beta\_i l)^{-1} \,\,\,\,\,\tag{25}$$

where *<sup>i</sup>* 7 is the eigenvalue of the *i-*th spatial channel of the correlation matrix **C** given by (2). Based on (21), (24), and (25), the MGF of the GD partial decision statistics ( ) *TGR* **X***<sup>k</sup>* 8 is 9 determined in the following form:

$$\begin{split} \mathcal{M}\_{\mathbb{T}\_{\mathcal{K}}(\mathcal{K}\_{l})}(l) &:= \prod\_{l=1}^{M} [1 - \mathbb{E}\_{s}\sigma\_{h}^{2}\beta\_{l}l]^{-1} \prod\_{l=1}^{M} \mathcal{M}\_{\mathbb{z}\_{1}}(l) \prod\_{l=1}^{M} \mathcal{M}\_{\mathbb{z}\_{\mathbb{z}\_{l}}}(-l) = \prod\_{l=1}^{M} [1 - \mathbb{E}\_{s}\sigma\_{h}^{2}\beta\_{l}l]^{-1} \prod\_{l=1}^{M} (1 - 2\sigma\_{w}^{2}l)^{-0.5} \prod\_{l=1}^{M} (1 + 2\sigma\_{w}^{2}l)^{-0.5} \\ &= \prod\_{l=1}^{M} [1 - \mathbb{E}\_{s}\sigma\_{h}^{2}\beta\_{l}l]^{-1} (1 - 2\sigma\_{w}^{2}l)^{-0.5M} (1 + 2\sigma\_{w}^{2}l)^{-0.5M} = (1 - 4\sigma\_{w}^{4}l^{2})^{-0.5M} \prod\_{l=1}^{M} [1 - \mathbb{E}\_{s}\sigma\_{h}^{2}\beta\_{l}l]^{-1}. \end{split} \tag{26}$$

11 Based on (26) and taking into consideration results discussed in [27], the mean and the variance of the test statistics (**X**) *TGR* under the hypothesis H<sup>1</sup> 12 take the following form, 13 respectively:

$$m\_{\mathfrak{M}}^{GR} \equiv \mathbb{E}\left[T\_{GR}(\mathbf{X}\_k) \middle| \mathfrak{H}\_l\right] = NM\left(E\_s \sigma\_h^2 + 2\sigma\_w^2\right),\tag{27}$$

$$Var\_{\mathfrak{H}\_1}^{GR} = Var\left[T\_{GR}(\mathbf{X}\_k) \middle| \mathfrak{H}\_1\right] = N \left[\sum\_{i=1}^{M} E\_s^2 \sigma\_h^4 \beta\_i^2 + 4M\sigma\_w^4\right].\tag{28}$$
