*3.3.2. MGF of QBHS MRC q* / 16

20 BookTitle

2 where

7 where

5 following form:

10 to SNR per symbol.

12 integrations, is given by [54]

*s*

98 Contemporary Issues in Wireless Communications

/ / 0.5

> 2 <sup>3</sup> .

/ /

. 0

, 0

 

*<sup>d</sup> <sup>x</sup> <sup>x</sup>*

*<sup>d</sup> <sup>x</sup> <sup>x</sup>*

 

2 2

2 2

2 2

*M PAM M PAM*

*<sup>M</sup> <sup>g</sup> P q q d<sup>θ</sup> M π θ* 1 (64)

0 2( 1) ( ) exp , sin ( ) *M PAM QBHS MRC QBHS MRC π*

<sup>1</sup> *M PAM <sup>g</sup> <sup>M</sup>* 3 (65)

Averaging (64) over *QBHS MRC q* / 4 the SER of *M-*ary PAM system is determined in the

2( 1) 2( 1) exp ( ) , sin sin ( ) ( ) *QBHS MRC QBHS MRC*

*M M g g <sup>P</sup> q f q dqdθ φ <sup>d</sup><sup>θ</sup> M π θ θ M π*

6 (66)

*φ* (*s*) *E* {exp(*sq*)} *<sup>q</sup> <sup>q</sup>* 8 (67)

is the MGF of random variable *q*, {} *Eq* 9 is the mathematical expectation of MGF with respect

11 A finite-limit integral for the Gaussian *Q*-function, which is convenient for numerical

 

2sin exp

13 *Q x* (68)

0

0

16 The complementary error function is defined as *erfc*(*x*) 1 *erf* (*x*) so that

*x*

0

0.5

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

 

2sin exp

2

<sup>1</sup> ( ) or ( ) 2 ( 2 ), <sup>2</sup> <sup>2</sup> *<sup>x</sup> Q x erfc erfc x Q x*

*erf x t dt Q x*

15 (69)

17 (70)

<sup>2</sup> ( ) exp( ) 1 2 ( 2 ).

0.5 0.5

*s qq*

 

 

1

( ) 0.5

14 The error function can be related to the Gaussian *Q*-function by

*π π*

0 0 0

2

Since all of the 2*N* quadrature branches are i.i.d., the MGF of *QBHS MRC q* / 17 takes the following 18 form [35]:

$$\varphi\_{q\_{\text{QHS}\lesssim\text{MMC}}} = 2L \binom{2N}{2L} \stackrel{\text{\tiny{}}}{\text{\tiny{}}} \exp(sq) f(q) [q\wp(s,q)]^{2L-1} [F(q)]^{2(N-L)} dq \,\tag{73}$$

20 where *f* (*q*) and *F*(*q*) are, respectively, the probability density function (pdf) and the 21 cumulative distribution function (cdf) of *q*, the SNR per symbol, for each quadrature branch, 22 and

$$\rho(s,q) = \bigcap\_{q}^{\alpha} \exp(s\mathbf{x})f(\mathbf{x})d\mathbf{x} \tag{74}$$

24 is the marginal moment generating function (MMGF) of SNR per symbol of a single 25 quadrature branch.

Since *<sup>k</sup> g* and *gk<sup>N</sup>* 1 (*k* 1,,*N*) follow the zero-mean Gaussian distribution with the variance <sup>2</sup> *<sup>h</sup>* given by (44), one can show that *<sup>k</sup> q* and *<sup>k</sup> <sup>N</sup> q* 2 follow the Gamma distribution 3 with pdf given by [49]

$$f(q) = \left\{ \frac{1}{\sqrt{q}} \exp\{-\frac{q}{\overline{q}}\} \sqrt{\pi q} \right\}, \quad q \ge 0 \tag{75}$$
 
$$q \le 0 \quad ,$$

5 where

22 BookTitle

$$\overline{q} = \frac{E\_b \sigma\_h^2}{\sigma\_w^2} \tag{76}$$

7 is the average SNR per symbol for each diversity branch. The MMGF of SNR per symbol of 8 a single quadrature branch can be determined in the following form:

$$q(s,q) = \frac{1}{\sqrt{1-s\overline{q}}} \operatorname{erfc}\left(\sqrt{\frac{1-s\overline{q}}{\overline{q}}}q\right). \tag{77}$$

10 Moreover, the cdf of *q* becomes

$$F(q) = 1 - q\varphi(0, q) = 1 - \text{erfc}\left(\sqrt{\frac{q}{\overline{q}}}\right),\tag{78}$$

12 where *erfc*(*x*) is the complimentary error function.

#### 13 **3.4. PCF determination**

#### 14 *3.4.1. AWGN channel*

15 In this section, we determine the PCF at the GR output with the first stage of PPIC. From 16 [43], the linear minimum mean-square error (MMSE) solution of PCF for the first stage of 17 PPIC is given by

$$p\_{1, \text{opt}} = \frac{\sigma\_{2,0}^2 - \rho\_1 \sigma\_{1,1} \sigma\_{2,0}}{\sigma\_{1,1}^2 + \sigma\_{2,0}^2 - 2\rho\_1 \sigma\_{1,1} \sigma\_{2,0}},\tag{79}$$

19 where

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 23 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 101

$$
\sigma\_{1,1}^2 = E\left\{ (I\_k + \zeta\_k' - \hat{I}\_k)^2 \right\} \tag{80}
$$

2 is the power of residual MAI plus the total noise component forming at the GR output at the 3 first stage;

$$
\sigma\_{2,0}^2 = E\{(I\_k + \zeta\_k')^2\}\tag{81}
$$

5 is the power of true MAI plus the total noise component forming at the GR output (also 6 called the 0-th stage);

$$
\rho\_l \sigma\_{1,l} \sigma\_{2,0} = E\left\{ (I\_k + \zeta\_k - \hat{I}\_k)(I\_k + \zeta\_k) \right\} \tag{82}
$$

8 is a correlation between these two MAI terms. It can be rewritten as

22 BookTitle

5 where

variance <sup>2</sup> 

3 with pdf given by [49]

100 Contemporary Issues in Wireless Communications

10 Moreover, the cdf of *q* becomes

13 **3.4. PCF determination**

14 *3.4.1. AWGN channel*

17 PPIC is given by

19 where

12 where *erfc*(*x*) is the complimentary error function.

Since *<sup>k</sup> g* and *gk<sup>N</sup>* 1 (*k* 1,,*N*) follow the zero-mean Gaussian distribution with the

*<sup>h</sup>* given by (44), one can show that *<sup>k</sup> q* and *<sup>k</sup> <sup>N</sup> q* 2 follow the Gamma distribution

4 *f q q* (75)

6 (76)

7 is the average SNR per symbol for each diversity branch. The MMGF of SNR per symbol of

9 (77)

<sup>1</sup> <sup>1</sup> (,) . <sup>1</sup> ( ) *sq <sup>φ</sup> s q erfc q sq q*

( ) 1 (0, ) 1 , ( )*<sup>q</sup> F q <sup>φ</sup> q erfc*

11 (78)

15 In this section, we determine the PCF at the GR output with the first stage of PPIC. From 16 [43], the linear minimum mean-square error (MMSE) solution of PCF for the first stage of

> 2 2,0

 18 (79)

,opt 2 2

1 1,1 2,0

, <sup>2</sup>

1,1 2,0 1 1,1 2,0

*σ ρσ σ*

*σ σ ρσ σ*

1

*p*

*q q*

> 2 2

*w b h σ E q*

0 , 0 ,

exp , <sup>0</sup> <sup>1</sup>

*q*

*q*

*πq q*

 

8 a single quadrature branch can be determined in the following form:

 

( ) ( )

1,opt <sup>2</sup> <sup>2</sup> (0) (0) , 2 (0) (0) (0) , , 2 22 2 1, ˆ {( ) } 1 ˆ{ } <sup>1</sup> ˆ ˆ { } <sup>1</sup> ˆ ˆ <sup>ˆ</sup> (1 2 ) () ( { } {} [ { } *AF PF k kk K KK <sup>k</sup> <sup>u</sup> u v ul vl u v u lv lu K K K K u eu u v ul vl u v v vl l v u l u lv lu v l K j j j kj k k k j jk u l E I ζ I <sup>p</sup> E I S SSE ρ ρ b b <sup>N</sup> A P SSE ρ ρ b b SE ρ ζ b N E Ab ρ w tw* 2 (0) 0 1, 2 2 (0) 1, 4 2 2 2 (0) (0) , 4 2 2 2 (0) , ˆ ) ˆ 1 1 ˆ ˆ <sup>1</sup> ˆ ˆ (1 2 ) ] { } { { *<sup>b</sup> T K j j j kj j jk K j j j kj j jk K KK i i i i j j ik jk i j i k i kj ki i i ei i i j j ik jk i j t dt A b ρ E Ab ρ A A AE ρ ρ b b N A P A AE ρ ρ b b N* (0) 2 (0) 2 2 2 , 0 <sup>ˆ</sup> } [] { } () () }, *<sup>b</sup> AF PF T K K K K j j jk j k k k i k i kj ki j k A E ρ b w t w t dt* 9 (83)

where *Pe*,*<sup>i</sup>* 10 is the BER of user *i* at the corresponding GR output;

$$E\{\hat{b}\_i^{(0)}\hat{b}\_i^{(0)}\} = 1 - 2P\_{e,i} \quad \text{and} \qquad E\{\rho\_{ik}^2\} = \mathcal{N}^{-1}.\tag{84}$$

The PCF 1,opt 12 *p* can be regarded as the normalized correlation between the true MAI plus the 13 total noise component forming at the GR output and the estimated MAI. Assume that

$$\mathbf{b} = \{b\_k\}\_{k=1}^K \tag{85}$$

1 is the data set of all users;

24 BookTitle

$$\mathbf{\uprho} = \{\rho\_{ik}\}\_{i,k=1}^{K} \tag{86}$$

3 is the correlation coefficient set of random sequences;

$$f\_{\tilde{b}\_{\parallel}^{(0)} \cdot \mathbf{b}, \mathbf{\pmb}}(\tilde{b}\_{\perp}^{(0)} \mid \mathbf{b}, \mathbf{\pmb}) = \mathsf{A}^{\prime} \mathsf{A}^{\prime} (\mathsf{E}[\tilde{b}\_{\parallel}^{(0)} \mid \mathbf{b}, \mathbf{\pmb}], 4a^{4} \sigma\_{w}^{4}) \tag{87}$$

is the conditional normal pdf of ~(0) *<sup>i</sup> b* given **b** and and ( <sup>|</sup> , ) <sup>~</sup> , ~(0) (0) <sup>|</sup> , <sup>~</sup> , ~(0) (0) **<sup>b</sup> <sup>b</sup>** *<sup>i</sup> <sup>j</sup> <sup>b</sup> <sup>b</sup> <sup>f</sup> <sup>b</sup> <sup>b</sup> <sup>i</sup> <sup>j</sup>* 5 is the conditional joint normal pdf of ~(0) *<sup>i</sup> <sup>b</sup>* and ~(0) *<sup>j</sup>* 6 *b* given **b** and . Following the derivations in 7 [43], the expectation terms with hard decisions in (83) can be evaluated based on Price's 8 theorem [49] as follows

$$E\{\rho\_{ik}\rho\_{jk}\hat{b}\_i^{(0)}\hat{b}\_j^{(0)}\} = E\{E\{E\{\rho\_{ik}\rho\_{jk}\hat{b}\_i^{(0)}\hat{b}\_j^{(0)} \mid \mathbf{b}, \pm\} \mid \pm\}\} = E\{E\{\rho\_{ik}\rho\_{jk}\hat{b}\_i^{(0)}(2Q\_j - 1) \mid \pm\}\};\tag{88}$$

$$E\{\rho\_{jk}\mathbb{L}\_k\hat{b}^{(0)}\_{\rangle}\} = E\{E\{E\{\rho\_{jk}\mathbb{L}\_k\hat{b}^{(0)}\_{\rangle}\mid \mathbf{b}, \mathbf{p}\}\mid \mathbf{p}\}\} = 4\alpha^4\sigma^4\_w E\{E\{\rho^2\_{jk}f\_{\hat{b}^{(0)}\_{\hat{b}^{(0)}\_{\hat{b}}\mid \mathbf{b}, \mathbf{p}}(0\mid b, \mathbf{p})\mid \mathbf{p}\}\};\tag{89}$$

$$\begin{split} \mathrm{E}\{\rho\_{\mathrm{ul}}\rho\_{\mathrm{u}}\hat{b}^{(0)}\_{\mathrm{u}}\hat{b}^{(0)}\_{\mathrm{v}}\} &= \mathrm{E}\{\mathrm{E}\{\mathrm{E}\{\rho\_{\mathrm{il}}\rho\_{\mathrm{j}}\hat{b}^{(0)}\_{\mathrm{i}}\hat{b}^{(0)}\_{\mathrm{j}}\|\mathbf{b},\mathbf{p}\}\mid\mathbf{p}\} \right) = \mathrm{E}\{\mathrm{E}\{\rho\_{\mathrm{il}}\rho\_{\mathrm{j}}\left[16\rho\_{\mathrm{j}}a^{4}\sigma\_{\mathrm{w}}^{4}f\_{\hat{\mathbb{B}}^{(0)}\_{\mathrm{i}}\hat{\mathbb{B}}^{(0)}\_{\mathrm{i}}\mathbf{b},\mathbf{p}\right]} \mathrm{(0,0\,\mathsf{b}\,\mathsf{p}\text{)}} + \\ &\quad + (2\underline{Q}\_{\mathrm{i}} - 1)(2\underline{Q}\_{\mathrm{j}} - 1)\|\,\mathsf{p}\|\,\mathsf{p}\} \,\mathrm{J} \end{split} \tag{90}$$

12 where

10

$$Q\_k = Q\left(-\frac{M\{\tilde{b}\_k^{(0)} \mid \mathbf{b}, \mathbf{\rho}\}}{2a^2 \sigma\_w^2}\right) \tag{91}$$

14 and

$$\operatorname{Var}(\zeta\_k) = 4\alpha\_k^4 \sigma\_w^4 \tag{92}$$

16 is the total background noise variance forming at the GR output taking into account multipath fading channel; <sup>2</sup> *<sup>w</sup>* 17 *σ* is the additive Gaussian noise variance forming at the PF and 18 AF outputs of GR linear tract; the Gaussian *Q*- function is given by (68).

19 Although numerical integration in [43, 56] can be adopted for determining the optimal PCF 1,opt 20 *p* for the first stage based on (83)-(90), it requires huge computational complexity. To 21 simplify this problem, we assume that the total background noise forming at the GR output 22 can be considered as a constant factor and may be small enough such that the *Q* functions 23 in (88) and (90) are all constants and (89) can be approximated to zero. That is

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 25 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 103

$$4\alpha\_k^4 \sigma\_w^4 \ll \min\_{\{a\_k A\_k, \mathfrak{p}\}} \{ E[\tilde{b}\_i^{(0)} \mid \mathbf{b}, \mathfrak{p}] \}^2 = 4\alpha\_m^4 A\_m^2 N^{-2} \,\tag{93}$$

2 where [57]

1

24 BookTitle

1 is the data set of all users;

102 Contemporary Issues in Wireless Communications

4

8 theorem [49] as follows

(2 1)(2 1)

multipath fading channel; <sup>2</sup>

*i j*

*Q Q*

]| }

},

<sup>4</sup> <sup>4</sup> { } 4 *<sup>k</sup> <sup>k</sup> <sup>w</sup>* 15 *Var ζ*

10

12 where

14 and

is the conditional normal pdf of ~(0)

conditional joint normal pdf of ~(0)

3 is the correlation coefficient set of random sequences;

(0)

*<sup>j</sup>* 6 *b* given **b** and

*K ik <sup>i</sup> <sup>k</sup> ρ*

(0) (0) 4 4 | , ( ) ({ } ) | , | , ,4 *<sup>i</sup> <sup>b</sup> i iw f b E b*

 

*E E ρ ρ b Q*

 

*<sup>b</sup>* **b** *b*

(0)


(0) (0)

, |,

(92)

and ( <sup>|</sup> , ) <sup>~</sup>

**<sup>b</sup> <sup>b</sup>** *<sup>i</sup> <sup>j</sup> <sup>b</sup> <sup>b</sup> <sup>f</sup> <sup>b</sup> <sup>b</sup> <sup>i</sup> <sup>j</sup>*

<sup>|</sup> , <sup>~</sup> , ~(0) (0)

, ~(0) (0)

. Following the derivations in

 

(87)

(88)

(89)

{ } , <sup>1</sup> 2 (86)

> *<sup>b</sup> b b* N

> > *<sup>i</sup> b* given **b** and

5 is the

7 [43], the expectation terms with hard decisions in (83) can be evaluated based on Price's

 

 *E E ρ f <sup>b</sup>* **b** *b*

ˆ ˆ { } {{ | } } { , 4 |} { { ( | )| }}; 0 ,

ˆ ˆ ˆ ˆ , 16 0,0 ,

(0) 2 2 ,

 

> 

*w*

2 {| } ( ) *<sup>k</sup>*

**<sup>b</sup>**

{ } { { | }| } { [ (| )

11 (90)

*M b Q Q*

13 (91)

16 is the total background noise variance forming at the GR output taking into account

*<sup>w</sup>* 17 *σ* is the additive Gaussian noise variance forming at the PF and

19 Although numerical integration in [43, 56] can be adopted for determining the optimal PCF 1,opt 20 *p* for the first stage based on (83)-(90), it requires huge computational complexity. To 21 simplify this problem, we assume that the total background noise forming at the GR output 22 can be considered as a constant factor and may be small enough such that the *Q* functions

*<sup>i</sup> <sup>b</sup>* and ~(0)

ˆ ˆ (0) (0) ˆ ˆ (0) (0) ˆ(0) { } {{ { | , }| } { } { (2 1)| }}; *ik jk i j ik jk i j ik jk i j* <sup>9</sup>*<sup>E</sup> ρ ρ b b EEE ρ ρ b b* **<sup>b</sup>**

(0) (0) 44 2

 

*<sup>j</sup> jk k j jk k j w jk <sup>b</sup> <sup>E</sup> ρ ζ b EEE ρ ζ <sup>b</sup>*

(0) (0) (0) (0) 4 4

*i j ul vl u v ik jk i j ik jk ij w b b*

 

{ } {

*E ρ ρ b b EEE ρ ρ b b E E ρ ρ f*

*k*

18 AF outputs of GR linear tract; the Gaussian *Q*- function is given by (68).

23 in (88) and (90) are all constants and (89) can be approximated to zero. That is

*ρ*

$$\begin{aligned} \left| \alpha\_m^2 A\_m = \min \alpha\_k^2 A\_k \; ; \\ \sum\_{k=m}^K \alpha\_k^2 A\_k b\_k \rho\_{kl} &= -\alpha\_m^2 A\_m b\_m \rho\_{mk}^{\prime} \; ; \\ \min |\; \rho\_{mk} - \rho\_{mk}^{\prime}| &= \frac{2}{N} \; . \end{aligned} \tag{94}$$

4 With this, we can rewrite (88) and (90) as follows:

$$E\{E\{\rho\_{ik}\rho\_{jk}\hat{b}\_i^{(0)}(2Q\_j - 1) \mid \mathbf{p}\}\} = B\_1 E\{\rho\_{ik}\rho\_{jk}\} E\{\hat{b}\_i^{(0)} \mid \mathbf{p}\} = 0;\tag{95}$$

$$\begin{split} &E\{\mathbb{E}[\rho\_{ik}\rho\_{jk}]16\rho\_{ik}\sigma^{4}\sigma^{4}\_{w}f\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}}}}}(0,0\mid\mathbf{b},\pm)+(2Q\_{i}-1)(2Q\_{j}-1)\}\mid\mathfrak{g}\} \\ &=E\{\mathbb{E}[16\boldsymbol{\alpha}^{4}\sigma^{4}\_{w}\rho\_{ik}\rho\_{jl}f\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}}}}(0,0\mid\mathbf{b},\pm)\mid\mathfrak{g}\}\} + B\_{2}E\{\mathbb{E}[\rho\_{ik}\rho\_{jk}\mid\mathfrak{g}]\} \\ &=E\{\mathbb{E}\{4\boldsymbol{\alpha}^{4}\sigma^{4}\_{w}\rho\_{ik}\rho\_{jl}\rho\_{ij}f\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}^{(0)}\_{\bar{\mathcal{B}}}}}(0,0\mid\mathbf{b},\pm)\mid\mathfrak{g}\}\}, \end{split} \tag{96}$$

where *B*1 and *B*<sup>2</sup> 7 are constants. According to assumptions made above, (0,0 | , ) <sup>|</sup> , <sup>~</sup> , ~(0) (0) **<sup>b</sup>** *bu bv* **<sup>b</sup>** <sup>8</sup>*f* can be expressed by

$$f\_{\tilde{b}\_{\tilde{i}}^{(0)},\tilde{b}\_{\tilde{j}}^{(0)}|\mathbf{b},\mathbf{\bar{\sigma}}}(0,0\mid\mathbf{b},\mathbf{\bar{\sigma}}) = \frac{\exp\{-0.5m\_{b}^{T}\mathbf{B}\_{b}^{-1}m\_{b}\}}{8\pi\alpha^{4}\sigma\_{w}^{4}\sqrt{1-\rho\_{\tilde{ij}}^{2}}},\tag{97}$$

10 where

9

$$\mathbf{m}\_b = \left[ E\{\widetilde{b}\_i^{(0)} \mid \mathbf{b}, \pmb{\rho}\}, E\{\widetilde{b}\_j^{(0)} \mid \mathbf{b}, \pmb{\rho}\} \right]^T \tag{98}$$

12 and

$$\mathbf{B}\_{b} = E\left\{ (\widetilde{\mathbf{b}} - \mathbf{m}\_{b})(\widetilde{\mathbf{b}} - \mathbf{m}\_{b})^{\top} \right\} \tag{99}$$

14 with

$$\tilde{\mathbf{b}} = [\tilde{b}\_i^{(0)}, \tilde{b}\_j^{(0)}]^\mathrm{T}. \tag{100}$$

Since <sup>1</sup> **B***<sup>b</sup>* 2 is a positive semi-definite matrix, i.e.

$$m\_b^T B\_b^{-1} m\_b \ge 0,\tag{101}$$

4 we can have

5

26 BookTitle

$$0 < f\_{\vec{b}\_i^{(0)}, \vec{b}\_j^{(0)} \mid \mathbf{b}, \mathfrak{g}}(0, 0 \mid \mathbf{b}, \mathfrak{g}) \le \max\_{\substack{\rho\_{\vec{y}} \\ \rho\_{\vec{y}} \ast \pm 1}} \frac{1}{8\pi a^4 \sigma\_w^4 \sqrt{1 - \rho\_{\vec{y}}^2}}.\tag{102}$$

6 With the above results,

$$\min\_{\rho\_{\vec{\eta}}, \rho\_{\vec{\eta}}, \pi \in \mathbb{V}} \sqrt{1 - \rho\_{\vec{\eta}}^2} = \frac{2\sqrt{N-1}}{N}. \tag{103}$$

8 where [57]

$$
\rho\_{\bar{y}} = \mathrm{l} - 2\mathcal{N}^{-1} \quad \text{or} \quad -\mathrm{l} + 2\mathcal{N}^{-1} \tag{104}
$$

10 and

$$E\{\rho\_{ik}\rho\_{jk}\rho\_{ij}\} = \sum\_{m=1}^{N}\sum\_{p=1}^{N}\sum\_{q=1}^{N}E\{c\_{im}c\_{km}c\_{jp}c\_{kp}c\_{iq}c\_{jq}\} = \sum\_{m=1}^{N}N^{-3} = N^{-2}.\tag{105}$$

Thus, we can derive a range of 1,opt 12 *p* as follows

$$\frac{\sum\_{i \neq k}^{K} a\_i^4 A\_i^2 \{1 - 2P\_{\varepsilon, \mu}\}}{\sum\_{i \neq k}^{K} a\_i^4 A\_i^2 + \frac{1}{\pi \sqrt{N - 1}} \sum\_{i \neq k}^{K} \sum\_{j \neq l, j}^{K} a\_i^2 A\_i a\_j^2 A\_j} \le p\_{1, \text{opt}} < 1 - \frac{2 \sum\_{i \neq k}^{K} a\_i^4 A\_i^2 P\_{\varepsilon, \mu}}{\sum\_{i \neq k}^{K} a\_i^4 A\_i^2}. \tag{106}$$

14 If the power control is perfect, i.e.

$$
\alpha\_i^2 A\_i = \alpha\_j^2 A\_j = \alpha^2 A \quad \text{and} \quad P\_{e\beta} = P\_e \tag{107}
$$

and *Pe* is approximated by the BER of high SNR case, i.e., the ( ) *<sup>K</sup>*<sup>1</sup> *<sup>Q</sup> <sup>N</sup>* 16 function [58, 59], 17 (83) can be rewritten as

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 27 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 105

$$\frac{1 - 2Q\left(\sqrt{\frac{N}{K-1}}\right)}{1 + \frac{K-2}{\pi\sqrt{N-1}}} \le p\_{1, \text{opt}} < 1 - 2Q\left(\sqrt{\frac{N}{K-1}}\right). \tag{108}$$

2 It is interesting to see that the lower and upper boundary values can be explicitly calculated 3 from the processing gain *N* and the number of users *K*.

#### 4 *3.4.2. Multipath channel*

26 BookTitle

4 we can have

8 where [57]

10 and

6 With the above results,

5

Since <sup>1</sup> **B***<sup>b</sup>* 2 is a positive semi-definite matrix, i.e.

104 Contemporary Issues in Wireless Communications

Thus, we can derive a range of 1,opt 12 *p* as follows

<sup>2</sup> <sup>2</sup> <sup>2</sup> 15

14 If the power control is perfect, i.e.

17 (83) can be rewritten as

*f*

(0) (0) [ ] , . *<sup>T</sup> i j* **<sup>b</sup>** *b b* 1 (100)

<sup>1</sup> 0, *<sup>T</sup> bb b* 3 *mB m* (101)

(0) (0) , |, 44 2

<sup>1</sup> <sup>0</sup> (0,0| , ) max . 8 1 *i j ij ij*

**<sup>b</sup> <sup>b</sup>**

2

*<sup>ρ</sup> <sup>N</sup>* <sup>7</sup> (103)

<sup>1</sup> <sup>1</sup> 1 2 or 1 2 <sup>9</sup>*ρij N N* (104)

<sup>11</sup> (105)

2 1 min 1 .

111 1 {} { } . *N NN N*

4 2 4 2

*i i eu i i eu*

*A P A P p*

1 2 2

, ,

*mpq m E ρρρ Ec c c c c c N N* 

1,opt 4 2 2 2 4 2

*A A A A*

*<sup>i</sup> Ai <sup>j</sup> Aj A Pe*,*<sup>i</sup> Pe*

and *Pe* is approximated by the BER of high SNR case, i.e., the ( ) *<sup>K</sup>*<sup>1</sup> *<sup>Q</sup> <sup>N</sup>* 16 function [58, 59],

*i i i ij j i i i k i kj li i k*

*K K*

*i k i k K K K K*

*b b ρ*

, 1

*ik jk ij im km jp kp iq jq*

,

 

 13 (106)

( )

1 1

*π N*

*ij ij ij ρ ρ*

1

*N*

*ρ w ij*

*π ρ* 

3 2

1 .

and (107)

  (102)

5 Based on representation in (37), we can obtain the received signal vector in the following 6 form:

$$\mathbf{x}(t) = \sum\_{k=1}^{K} A\_k(t) b\_k \mathbf{h}\_k + \mathbf{w}(t). \tag{109}$$

8 Introduce the following notation for the correlation coefficient

$$
\sigma\_{jk} = \mathbf{h}\_j^\top \mathbf{h}\_k \qquad \text{and} \qquad \sigma\_k = \sigma\_{kk}. \tag{110}
$$

10 In commercial DS-CDMA wireless communication systems, the users' spreading codes are 11 often modulated with another code having a very long period. As far as the received signal 12 is concerned, the spreading code is not periodic. In other words, there will be many possible 13 spreading codes for each user. If we use the result derived above, we then have to calculate 14 the optimum PCFs for each possible code and the computational complexity will become 15 very high. Since the period of the modulating code is usually very long, we can treat the 16 code chips as independent random variables and approximate the correlation coefficient *jk* 17 given by (110) as a Gaussian random variable. In this case, the GR output for the first 18 stage can be presented in the following form:

$$Z\_{k}(t) = A\_{k}(t)b\_{k}h\_{k}^{T}h\_{k} + \sum\_{j=1, j\neq k}^{K} A\_{j}(t)b\_{j}h\_{j}^{T}h\_{k} + \mathcal{L}\_{k}(t) = A\_{k}(t)b\_{k}\varpi\_{k} + \sum\_{j=1, j\neq k}^{K} A\_{j}(t)b\_{j}\varpi\_{k} + \mathcal{L}\_{k}(t),\tag{111}$$

where the background noise (*t*) *<sup>k</sup>* 20 forming at the GR output is given by (56).

21 Evaluating the GR output process given by (111), based on the well-known results, for 22 example, discussed in [60], we can define the BER performance for the user *k* in the 23 following form:

$$P\_b^{(k)} = 0.5P(Z\_k \mid b\_k = 1) + 0.5P(Z\_k \mid b\_k = -1) = P(Z\_k \mid b\_k = 1). \tag{112}$$

28 BookTitle

In (112), we assume that the occurrence of probabilities for 1 *<sup>k</sup> b* and 1 *<sup>k</sup>* 1 *b* are equal, and that the error probabilities for 1 *<sup>k</sup> b* and 1 *<sup>k</sup>* 2 *b* are also equal. As we can see from (111), there are three terms. The first term corresponds to the desired user bit. If we let 1 *<sup>k</sup>* 3 *b* , it is 4 a deterministic value. The third term in (111) given by (56) corresponds to the GR 5 background noise interference which pdf is defined in [3, Chapter 3, pp. 250–263, 324–328]. 6 The second term in (111) corresponds to the interference from other users and is subjected to 7 the binomial distribution. Note that correlation coefficients in (111) are small and DS-CDMA 8 wireless communication systems are usually operated in low *SNR* environments. The 9 variance of the second term is then much smaller in comparison with the variance of the third term. Thus, we can assume that *Zk* conditioned on 1 *<sup>k</sup>* 10 *b* can be approximated by 11 Gaussian distribution, as shown in [3, Chapter 3, pp. 250–263, 324–328] and [13]. Then the 12 BER performance takes the following form

$$P(Z\_k) = Q \left\{ \sqrt{\frac{E\_{\mathcal{X}}\left\{ \mathcal{M}\_k^{(l)} \right\}}{E\_{\mathcal{X}}\left\{ \mathcal{Y}\_k^{(l)} \right\}}} \right\},\tag{113}$$

where {} *E*L denotes the expectation operator over the spreading code set L and (*l*) M*<sup>k</sup>* 14 and (*l*) V*<sup>k</sup>* are the expected squared mean and variance of *Zk* 15 , respectively, given the *l*-th possible 16 code in L. Letting

$$R\_k = \sum\_{j \neq k} q\_j \qquad \text{and} \qquad \Lambda\_k = \sum\_{j \neq k} \varpi\_{jk}^2 \tag{114}$$

where *<sup>j</sup> q* is defined in (61), considering *jk* 18 as a Gaussian random variable, we obtain

$$E\_{\mathcal{K}}\left[\mathcal{M}\_{k}^{(l)}\right] = A\_{k}^{2}\left\{E\_{\mathcal{K}}\left[\boldsymbol{\sigma}\_{k}^{(l)}\right] - p\_{k}E\_{\mathcal{K}}\left[\boldsymbol{\Lambda}\_{k}^{(l)}\right]\right\}^{2} = A\_{k}^{2}\left\{1 - p\_{k}E\_{\mathcal{K}}\left[\boldsymbol{\Lambda}\_{k}^{(l)}\right]\right\}^{2} \tag{115}$$

20 and the mathematical expectation of variance as

$$E\_{\mathcal{X}}\left[\mathcal{Y}\_{k}^{(l)}\right] = 4\sigma\_{w}^{4}\left\{E\_{\mathcal{X}}\left[\Omega\_{1,k}^{(l)}\right]p\_{k}^{2} - 2E\_{\mathcal{X}}\left[\Omega\_{2,k}^{(l)}\right]p\_{k} + E\_{\mathcal{X}}\left[\Omega\_{3,k}^{(l)}\right]\right\}.\tag{116}$$

22 Note that the expectations in (115) and (116) are operated on interfering user bits and noise using the correlation coefficient *jk* given by (110). The coefficients of [ ] (*l*) *E*L V*<sup>k</sup>* 23 are 24 represented by

$$\boldsymbol{\Omega}\_{1,k}^{(l)} = \boldsymbol{R}\_k \left[ \sum\_{j \neq k} \boldsymbol{\varpi}\_{jk} \boldsymbol{\varpi}\_j + \sum\_{j \neq k} \sum\_{m \neq j,k} \boldsymbol{\varpi}\_{jm} \boldsymbol{\varpi}\_{mk} \right]^2 + \left[ \sum\_{j \neq k} \boldsymbol{\varpi}\_{jk}^2 \boldsymbol{\varpi}\_j + \sum\_{j \neq k} \sum\_{m \neq j,k} \boldsymbol{\varpi}\_{jm} \boldsymbol{\varpi}\_{mk} \boldsymbol{\varpi}\_{jk} \right];\tag{117}$$

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 29 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 107

$$\Omega\_{2,k}^{(l)} = R\_k \Big[ \sum\_{j \neq k} \varpi\_{jk}^2 \varpi\_j + \sum\_{j \neq k} \sum\_{m \neq \ j,k} \varpi\_{jm} \varpi\_{mk} \varpi\_{jk} \Big] + \sum\_{j \neq k} \varpi\_{jk}^2;\tag{118}$$

$$
\Omega\_{3,k}^{(l)} = R\_k \sum\_{j \neq k} \sigma\_{jk}^2 + \sigma\_k. \tag{119}
$$

3 The optimal PCF for the user *k* can be found as

28 BookTitle

12 BER performance takes the following form

106 Contemporary Issues in Wireless Communications

where *<sup>j</sup> q* is defined in (61), considering

20 and the mathematical expectation of variance as

L

V

L M

using the correlation coefficient

*R*

1,

where {} *E*L

> L. Letting

*<sup>l</sup>* 19 *E*

24 represented by

21

(*l*) V

16 code in

In (112), we assume that the occurrence of probabilities for 1 *<sup>k</sup> b* and 1 *<sup>k</sup>* 1 *b* are equal, and that the error probabilities for 1 *<sup>k</sup> b* and 1 *<sup>k</sup>* 2 *b* are also equal. As we can see from (111), there are three terms. The first term corresponds to the desired user bit. If we let 1 *<sup>k</sup>* 3 *b* , it is 4 a deterministic value. The third term in (111) given by (56) corresponds to the GR 5 background noise interference which pdf is defined in [3, Chapter 3, pp. 250–263, 324–328]. 6 The second term in (111) corresponds to the interference from other users and is subjected to 7 the binomial distribution. Note that correlation coefficients in (111) are small and DS-CDMA 8 wireless communication systems are usually operated in low *SNR* environments. The 9 variance of the second term is then much smaller in comparison with the variance of the third term. Thus, we can assume that *Zk* conditioned on 1 *<sup>k</sup>* 10 *b* can be approximated by 11 Gaussian distribution, as shown in [3, Chapter 3, pp. 250–263, 324–328] and [13]. Then the

( )

*A p E*

LL

*k k k*

*jk* given by (110). The coefficients of [ ] (*l*) *E*

L

 

L

 and (*l*) M

(115)

(116)

L V

{ } *l k*

*k*

M

V

( ) ( ) , { }

*E* L

2 and , *k j k jk j k j k*

 <sup>17</sup> (114)

( ) <sup>2</sup> ( ) ( ) <sup>2</sup> <sup>2</sup> ( ) <sup>2</sup> [ ] { [ ] [ ]} {1 [ ]} *<sup>l</sup>*

() 4 () 2 ( ) ( ) 1, 2, 3, [ ] { [ ] [ ] [ ]}. 4 2 *l l ll k w kk kk k E E p E pE*

[ ] [ ]; *<sup>l</sup> k k jk j jm mk jk j jm mk jk j k j km j k j k j km j k*

 <sup>25</sup> (117)

22 Note that the expectations in (115) and (116) are operated on interfering user bits and noise

*<sup>k</sup>* 23 are

, ,

 

L

*l k k*

L

*k l*

*<sup>E</sup> PZ Q*

13 (113)

denotes the expectation operator over the spreading code set

*<sup>k</sup>* 14 and

*<sup>k</sup>* are the expected squared mean and variance of *Zk* 15 , respectively, given the *l*-th possible

*jk* 18 as a Gaussian random variable, we obtain

*l*

 L

<sup>2</sup> ( ) <sup>2</sup>

 

*R q*

*k k*

L *p E*

*<sup>k</sup> A E*

$$p\_{k, \text{opt}} = \arg\max\_{p\_k} \left\{ \frac{E\_{\mathcal{X}} \left[ \mathcal{M}\_k^{(l)} \right]}{E\_{\mathcal{X}} \left[ \mathcal{V}\_k^{(l)} \right]} \right\} = \left\{ p\_{k, \text{opt}} : E\_{\mathcal{X}} \left[ \mathcal{V}\_k^{(l)} \right] \frac{dE\_{\mathcal{X}} \left[ \mathcal{M}\_k^{(l)} \right]}{dp\_k} - E\_{\mathcal{X}} \left[ \mathcal{M}\_k^{(l)} \right] \frac{dE\_{\mathcal{X}} \left[ \mathcal{V}\_k^{(l)} \right]}{dp\_k} = 0 \right\}. \tag{120}$$

5 Substituting (115)–(119) into (120) and simplifying the result, we obtain the following 6 equation

$$p\_{k, \text{opt}} = \frac{E\_{\mathcal{X}} \left[ \boldsymbol{\Omega}\_{2,k}^{(l)} \right] - E\_{\mathcal{X}} \left[ \boldsymbol{\Omega}\_{3,k}^{(l)} \right] \mathbb{E}\_{\mathcal{X}} \left[ \boldsymbol{\Lambda}\_{k}^{(l)} \right]}{E\_{\mathcal{X}} \left[ \boldsymbol{\Omega}\_{1,k}^{(l)} \right] - E\_{\mathcal{X}} \left[ \boldsymbol{\Omega}\_{2,k}^{(l)} \right] \mathbb{E}\_{\mathcal{X}} \left[ \boldsymbol{\Lambda}\_{k}^{(l)} \right]}. \tag{121}$$

8 Unlike that in AWGN channel, the result for the aperiodic code scenario is more difficult to 9 obtain because there are more correlation terms in (114)–(120) to work with. Before 10 evaluation of the expectation terms in (98), we define some function as follows:

$$
\alpha\_{jk}(m,n) = \alpha\_{j,m}\alpha\_{k,n'} \tag{122}
$$

$$
\pi\_{jk}(m,n) = \pi\_{j,m} - \pi\_{k,n'} \tag{123}
$$

$$
\boldsymbol{\nu}\_{j\boldsymbol{k}}(m,n) = \boldsymbol{\tilde{\mathbf{a}}}\_{j,m}^T \boldsymbol{\tilde{\mathbf{a}}}\_{k,n}.\tag{124}
$$

14 Thus, (122)–(124) define some relative figures between the *m*-th channel path of the *j*-th user and the *n*-th channel path of the *k*-th user. The notation (*m*, *n*) *jk* 15 denotes the path gain product, (*m*, *n*) *jk* is the relative path delay, and (*m*, *n*) *jk* 16 is the code correlation with the relative delay (*m*, *n*) *jk* 17 . Expanding (122)–(124), we have seven expectation terms to evaluate. For purpose of illustration, we show how to evaluate the first term, [ ] <sup>2</sup> *E*L *jk* 18 here. By definition, we have *jk* 19 as

$$\boldsymbol{\sigma}\_{jk} = \mathbf{h}\_{j}^{T} \mathbf{h}\_{k} = \left\{ \sum\_{m=1}^{L} \tilde{\mathbf{a}}\_{j,m} \boldsymbol{\alpha}\_{j,m} \right\}^{T} \left\{ \sum\_{n=1}^{L} \tilde{\mathbf{a}}\_{k,n} \boldsymbol{\alpha}\_{k,n} \right\} = \sum\_{m=1}^{L} \sum\_{n=1}^{L} \boldsymbol{\alpha}\_{j,m} \boldsymbol{\alpha}\_{k,n} \tilde{\mathbf{a}}\_{j,m}^{T} \tilde{\mathbf{a}}\_{k,n} = \sum\_{m=1}^{L} \sum\_{n=1}^{L} \boldsymbol{\alpha}\_{jk} (m,n) \boldsymbol{\nu}\_{jk} (m,n). \tag{125}$$

The expectation of *jk* 21 over all possible codes can be presented in the following form: 30 BookTitle

$$\begin{split} \mathbb{E}\_{\mathcal{X}}\{\boldsymbol{\sigma}\_{jk}^{2}\} &= \mathbb{E}\left\{\sum\_{m\_{1}=1}^{L}\sum\_{n\_{1}=1}^{L}\sum\_{m\_{2}=1}^{L}\alpha\_{jk}(m\_{1},n\_{1})\mathbb{1}\_{jk}(m\_{1},n\_{1})\alpha\_{jk}(m\_{2},n\_{2})\mathbb{1}\_{jk}(m\_{2},n\_{2})\right\} \\ &= \sum\_{m\_{1}=1}^{L}\sum\_{n\_{1}=1}^{L}\sum\_{m\_{2}=1}^{L}\sum\_{n\_{2}=1}^{L}\alpha\_{jk}(m\_{1},n\_{1})\alpha\_{jk}(m\_{2},n\_{2})\mathbb{E}\{\boldsymbol{\nu}\_{jk}(m\_{1},n\_{1})\boldsymbol{\nu}\_{jk}(m\_{2},n\_{2})\}. \end{split} \tag{126}$$

2 Introduce the following function

$$G\_{jk}(m\_1, n\_1, m\_2, n\_2) = B^2 E[\nu\_{jk}(m\_1, n\_1)\nu\_{jk}(m\_2, n\_2)].\tag{127}$$

The coefficient <sup>2</sup> 4 *B* in (127) is only the normalization constant. Since the spreading codes are seen as random, only if ( , ) <sup>1</sup> <sup>1</sup> *m n jk* is equal to ( , ) <sup>2</sup> <sup>2</sup> *m n jk* will ( , , , ) <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> *G m n m n jk* 5 be nonzero. Consider a specific set of { , , , } <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> 6 *m n m n* such that

$$
\tau\_{jk}(m\_1, n\_1) = \tau\_{jk}(m\_2, n\_2) = \tau \quad , \qquad \tau \ge 0. \tag{128}
$$

8 In this case, we have

$$\mathcal{G}\_{jk}(m\_1, n\_1, m\_2, n\_2) = \mathcal{B}^2 \sum\_{\nu=0}^{N-\tau-1} \mathbb{E}\{a\_{j,\nu+\tau}^2 a\_{k,\nu}^2\} = \mathcal{N} - \tau. \tag{129}$$

10 At 0 , we have the same result except that the sign of in (129) is plus. We can conclude that the function ( , , , ) <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> *G m n m n jk* 11 in (127) can be written in the following form:

$$G\_{jk}(m\_1, n\_1, m\_2, n\_2) = \begin{cases} N - \lfloor \tau \rfloor & \text{,} \qquad \text{if} \qquad \tau\_{jk}(m\_1, n\_1) = \tau\_{jk}(m\_2, n\_2) = \tau \\ 0 & \text{otherwise} \end{cases} \tag{130}$$

Using (126), (127), and (129), we can evaluate [ ] <sup>2</sup> *E*L *jk* 13 in (117)–(119). The formulations from 14 the other six expectation terms can be obtained by mathematical transformation that is not 15 difficult.

16 We now provide a simple example to show the multipath effect on the optimal PCFs. 17 Introduce the following notations that are satisfied for all *k*:

$$\mathbf{r}\_k = \begin{bmatrix} 0, T \end{bmatrix}^T; \ \mathbf{a}\_k = \begin{bmatrix} \beta, \delta \end{bmatrix}^T; \quad \text{and} \quad \beta^2 + \delta^2 = 1. \tag{131}$$

19 Using (131) and taking into consideration that in the case of AWGN channel

$$E\_x \left[ \Lambda\_k^{(l)} \right] = \frac{K - 1}{N} \,\,\,\,\,\tag{132}$$

1 at given the *l*-th possible code in L, we can write for the case of the multipath channel

$$E\_x[\Lambda\_k^{(l)}] = \frac{K-1}{N} + \frac{2(N-T)\beta^2\delta^2(K-1)}{N^2};\tag{133}$$

$$\begin{split} \mathbb{E}\_{\mathcal{X}}[\Omega\_{\mathrm{L}}^{(l)}] &= \mathbb{E}\_{\mathcal{X}}\left[R\_{\mathrm{k}}^{(l)} \left\{\sum\_{j=1, j\neq k}^{\mathrm{K}} \rho\_{jk}^{(l)} + \sum\_{j=1, j\neq k}^{\mathrm{K}} \sum\_{m=1, m\neq j,k}^{\mathrm{K}} \rho\_{jm}^{(l)} \rho\_{mk}^{(l)} \right\}^2 + \sum\_{j=1, j\neq k}^{\mathrm{K}} (\rho\_{jk}^{(l)})^2 + \sum\_{j=1, j\neq k}^{\mathrm{K}} \sum\_{m=1, m\neq j,k}^{\mathrm{K}} \rho\_{jm}^{(l)} \rho\_{mk}^{(l)} \rho\_{jk}^{(l)} \right] \\ &+ 2(N-T)\beta^2 \delta^2 \{R\_{\mathrm{k}}N^{-4}[N^2+10N+4(N-T)\beta^2 \delta^2 + 2(K-2)(4N+3K+(N-T)\beta^2 \delta^2 + 1]} \\ &+ (K-1)N^{-3}(N+3K-2) + 4(N-2T)\beta^4 \delta^4 \{R\_{\mathrm{k}}N N^{-4} + 6K-12\} + R\_{\mathrm{k}}N^{-4}(6N-10T)\beta^4 \delta^4 \ddots \end{split} \tag{134}$$

$$\begin{split} \mathbb{E}\_{\mathcal{X}}[\boldsymbol{\Omega}\_{2,k}^{(l)}] &= \mathbb{E}\_{\mathcal{X}}\left\{ \mathbf{R}\_{k}^{(l)} \left\{ \sum\_{j=1, j\neq k}^{K} (\boldsymbol{\rho}\_{jk}^{(l)})^2 + \sum\_{j=1, j\neq k}^{K} \sum\_{\substack{m=1, m\neq j,k}}^{K} \boldsymbol{\rho}\_{jm}^{(l)} \boldsymbol{\rho}\_{mk}^{(l)} \boldsymbol{\rho}\_{jk}^{(l)} \right\} + \sum\_{j=1, j\neq k}^{K} (\boldsymbol{\rho}\_{jk}^{(l)})^2 \right\} \\ &+ 2(N-T)\boldsymbol{\rho}^2 \boldsymbol{\delta}^2 [\mathbf{R}\_{i} N^{-3} (\mathbf{N} + 3\mathbf{K} - 2) + (k-1)N^{-2}]; \end{split} \tag{135}$$

$$E\_{\mathcal{X}}\left[\Omega\_{3,k}^{(l)}\right] = E\_{\mathcal{X}}\left\{R\_k \sum\_{j=1, j\neq k}^{K} (\rho\_{jk}^{(l)})^2 + 1\right\} + 2(N-T)\rho^2 \delta^2 R\_k N^{-2}.\tag{136}$$

6 Note that the first terms in (133)–(136) correspond to the optimal PCFs in AWGN channel. 7 Other terms are due to the multipath effect. It is evident to see that if 0 we have the case 8 of AWGN channel.

#### 9 **3.5. Simulation results**

5

30 BookTitle

11 2 2

*L L LL*

11 1

*mnm n*

11 11 22 22

 

(128)

11 22

in (129) is plus. We can conclude

 

(131)

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

 

{ }.

11 22 11 22

 

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

*mn mnE mn mn*

*jk jk jk jk jk*

1 (126)

*E E mn mn mn mn*

*jk jk jk jk*

2 11 22 11 22 ( ,, ,) ( ,) ( ,) [ ]. *G m n m n BE m n m n jk* 

*jk jk* 3 (127)

The coefficient <sup>2</sup> 4 *B* in (127) is only the normalization constant. Since the spreading codes are

 

 is equal to ( , ) <sup>2</sup> <sup>2</sup> *m n jk* will ( , , , ) <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> *G m n m n jk* 5 be nonzero.

> 

> >

*N mn mn*

*jk jk*

 

1 2 22

11 22 , , 0 ( ,, ,) { } . *N G m n m n B Ea a N jk j k* 

 <sup>9</sup> (129)


L 

( ) <sup>1</sup> [ ] , *<sup>l</sup> k <sup>K</sup> <sup>E</sup>*

*N*

12 (130)

 *jk* 13 in (117)–(119). The formulations from 14 the other six expectation terms can be obtained by mathematical transformation that is not

16 We now provide a simple example to show the multipath effect on the optimal PCFs.

 

L20 (132)

11 2 2

*L L LL*

11 1

*mnm n*

2

{ }

108 Contemporary Issues in Wireless Communications

L

2 Introduce the following function

seen as random, only if ( , ) <sup>1</sup> <sup>1</sup> *m n jk*

8 In this case, we have

10 At

15 difficult.

11 22 ( , ) ( , ) , 0. *jk mn mn jk* 7 

0 , we have the same result except that the sign of

11 22

Using (126), (127), and (129), we can evaluate [ ] <sup>2</sup> *E*

17 Introduce the following notations that are satisfied for all *k*:

<sup>T</sup> <sup>T</sup> 2 2 [0, ] ; [ , ] ; and 1. *k k* <sup>18</sup>**<sup>τ</sup>** *<sup>T</sup>* **<sup>a</sup>**

19 Using (131) and taking into consideration that in the case of AWGN channel

*G mnmn*

*jk*

that the function ( , , , ) <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> *G m n m n jk* 11 in (127) can be written in the following form:

 

Consider a specific set of { , , , } <sup>1</sup> <sup>1</sup> <sup>2</sup> <sup>2</sup> 6 *m n m n* such that

#### 10 *3.5.1. Selection/maximal-ratio combining*

11 In this section, we discuss some examples of GR performance with quadrature subbranch 12 HS/MRC and HS/MRC schemes and compare with the conventional HS/MRC receiver. The 13 average SER of coherent BPSK and 8-PAM signals under processing by the GR with the 14 quadrature subbranch HS/MRC and HS/MRC schemes as a function of average SNR per 15 symbol per diversity branch for various values of 2*L* and 2*N* 8 is presented in Fig.3. It is 16 seen that the GR SER performance with quadrature subbranch HS/MRC and HS/MRC 17 schemes at (*L*,*N*) (3,4) achieves virtually the same performance as the GR with traditional 18 MRC, and that the performance at (*L*,*N*) (2,4) is typically less than 0.5 dB in SNR poorer 19 than that of GR with traditional MRC in [42]. Additionally, a comparison with the 20 traditional HS/MRC receiver in [29, 30] is made. Advantage of GR implementation in DS-21 CDMA wireless communication systems is evident.

22 The average SER of coherent BPSK and 8-PAM signals under processing by GR with 23 quadrature subbranch HS/MRC and HS/MRC schemes as a function of average SNR per 32 BookTitle

5

1 symbol per diversity branch for various values of 2*N* at 2*L* 4 is shown in Fig.4. We note 2 the substantial benefits of increasing the number of diversity branches *N* for fixed *L*. 3 Comparison with the traditional HS/MRC receiver is made. Advantage of GR 4 implementation in DS-CDMA wireless communication systems is evident.

6 **Figure 3.** Average SER of coherent BPSK and 8-PAM for GR with quadrature subbranch HS/MRC and 7 HS/MRC schemes versus the average SNR per symbol per diversity for various values of 2*L* with 8 2*N* 8 .

9 Comparative analysis of average BER as a function of the average SNR per bit per diversity 10 branch of coherent BPSK signals employing GR with quadrature subbranch HS/MRC and 11 HS/MRC schemes and GR with traditional HS/MRC scheme for various values of *L* with 12 *N* 8 is presented in Fig. 5. To achieve the same value of average SNR per bit per diversity 13 branch, we should choose 2*L* quadrature branches for the GR with quadrature subbranch 14 HS/MRC, HS/MRC schemes, and *L* diversity branches for the GR with traditional HS/MRC 15 scheme. Figure 5 demonstrates that the GR BER performance with quadrature subbranch 16 HS/MRC and HS /MRC schemes is much better than that of the GR with traditional HS/MRC 17 scheme, about 0.5 dB to 1.2 dB, when *L* is less than one half *N*. This difference decreases with 18 increasing *L*. This is expected because when *L N* we obtain the same performance. Some 19 discussion of increases in GR complexity and power consumption is in order. We first note 20 that GR with quadrature subbranch HS/MRC and HS/MRC schemes requires the same 21 number of antennas as GR with traditional HS/MRC scheme. On the other hand, the former 22 requires twice as many comparators as the latter, to select the best signals for further 23 processing. However, GR designs that process the quadrature signal components will require 24 2*L* receiver chains for either the GR with quadrature subbranch HS/MRC and HS/MRC 1 schemes or the GR with traditional HS/MRC scheme. Such receiver designs will use only a 2 little additional power, as GR chains consume much more power than the comparators.

3

4 5

32 BookTitle

110 Contemporary Issues in Wireless Communications

5

8 2*N* 8 .

1 symbol per diversity branch for various values of 2*N* at 2*L* 4 is shown in Fig.4. We note 2 the substantial benefits of increasing the number of diversity branches *N* for fixed *L*. 3 Comparison with the traditional HS/MRC receiver is made. Advantage of GR

6 **Figure 3.** Average SER of coherent BPSK and 8-PAM for GR with quadrature subbranch HS/MRC and 7 HS/MRC schemes versus the average SNR per symbol per diversity for various values of 2*L* with

9 Comparative analysis of average BER as a function of the average SNR per bit per diversity 10 branch of coherent BPSK signals employing GR with quadrature subbranch HS/MRC and 11 HS/MRC schemes and GR with traditional HS/MRC scheme for various values of *L* with 12 *N* 8 is presented in Fig. 5. To achieve the same value of average SNR per bit per diversity 13 branch, we should choose 2*L* quadrature branches for the GR with quadrature subbranch 14 HS/MRC, HS/MRC schemes, and *L* diversity branches for the GR with traditional HS/MRC 15 scheme. Figure 5 demonstrates that the GR BER performance with quadrature subbranch 16 HS/MRC and HS /MRC schemes is much better than that of the GR with traditional HS/MRC 17 scheme, about 0.5 dB to 1.2 dB, when *L* is less than one half *N*. This difference decreases with 18 increasing *L*. This is expected because when *L N* we obtain the same performance. Some 19 discussion of increases in GR complexity and power consumption is in order. We first note 20 that GR with quadrature subbranch HS/MRC and HS/MRC schemes requires the same 21 number of antennas as GR with traditional HS/MRC scheme. On the other hand, the former 22 requires twice as many comparators as the latter, to select the best signals for further 23 processing. However, GR designs that process the quadrature signal components will require 24 2*L* receiver chains for either the GR with quadrature subbranch HS/MRC and HS/MRC

4 implementation in DS-CDMA wireless communication systems is evident.

6 **Figure 4.** Average SER of coherent BPSK and 8-PAM for GR with quadrature subbranch HS/MRC and 7 HS/MRC schemes versus the average SNR per symbol per diversity for various values of 2*N* with 8 2*L* 4 .

9 On the other hand, GR designs that implement co-phasing of branch signals without 10 splitting the branch signals into the quadrature components will require *L* receiver chains 11 for GR with traditional HS/MRC scheme and 2*L* receiver chains for GR with quadrature 12 subbranch HS/MRC and HS/MRC schemes, with corresponding hardware and power 13 consumption increases.
