21 *3.2.1. Multipath fading channel model*

22 Let the transfer function for user *k*'s channel be

$$\mathcal{W}\_k(\mathbf{Z}) = \sum\_{i=1}^{M} a\_{k,i} Z^{-\tau\_{k,i}}.\tag{29}$$

24 As we can see from (29), the number of paths is *M* and the channel power and delay for *i*-th channel path are *<sup>k</sup>*,*i* and *<sup>k</sup>* ,*<sup>i</sup>* 25 , respectively.

26 We use two vectors to represent these parameters:

$$\boldsymbol{\pi}\_{k} = \begin{bmatrix} \boldsymbol{\pi}\_{k,1}, \boldsymbol{\pi}\_{k,2}, \dots, \boldsymbol{\pi}\_{k,L} \end{bmatrix}^{\mathrm{T}} \tag{30}$$

28 and

$$\boldsymbol{\alpha}\_{k} = \left[ \boldsymbol{\alpha}\_{k,1}, \boldsymbol{\alpha}\_{k,2}, \dots, \boldsymbol{\alpha}\_{k,L} \right]^{\top}. \tag{31}$$

1 Let

14 BookTitle

$$
\tau\_{k,1} \le \tau\_{k,2} \le \dots \le \tau\_{k,L} \tag{32}
$$

3 and the channel power is normalized

$$\sum\_{i=1}^{L} a\_{k,i}^2 = 1.\tag{33}$$

Without loss of generality, we may assume that 0 5 *<sup>k</sup>*,1 for each user and *L* is the maximum possible number of paths. When a user's path number, say *M*<sup>1</sup> 6 , is less than *M*, we can let all the elements in *<sup>k</sup>* ,*<sup>i</sup>* and *<sup>k</sup>*,*<sup>i</sup>* 7 be zero if the following condition is satisfied

$$M\_1 + 1 \le i \le M. \tag{34}$$

9 We may also assume that the maximum delay is much smaller than the processing gain *N* [46]. Before our formulation, we first define a (2*N* 1) *L* composite signature matrix **A***<sup>k</sup>* 10 in 11 the following form

$$\mathbf{A}\_{k} = [\tilde{\mathbf{a}}\_{k,1}, \tilde{\mathbf{a}}\_{k,2}, \dots, \tilde{\mathbf{a}}\_{k,\perp}] \tag{35}$$

where *k,i* **a** <sup>~</sup> 13 is a vector containing *i*-th delayed spreading code for user *k*. It is defined as

$$\tilde{\mathbf{a}}\_{k,i} = \overbrace{\overbrace{0, \dots, 0}^{\tau\_{\tilde{k},i}}, \mathbf{a}\_{k'}^T, \overbrace{0, \dots, 0}^{N - \tau\_{\tilde{k},i} - 1}}^{N - \tau\_{\tilde{k},i} - 1}, \tag{36}$$

15 Since a multipath fading channel is involved, the current received bit signal will be 16 interfered by previous bit signal. As mentioned above, the maximum path delay is much 17 smaller than the processing gain. The interference will not be severe and for simplicity, we 18 may ignore this effect. Let us denote the channel gain for multipath fading as

$$\mathbf{h}\_k = \mathbf{a}\_k \mathbf{A}\_k. \tag{37}$$

#### 20 *3.2.2. Selection/maximal-ratio combining*

21 We assume that there are *N* diversity branches experiencing slow and flat Rayleigh fading, 22 and all of the fading processes are independent and identically distributed (i.i.d.). During analysis, we consider only the hypothesis *H*<sup>1</sup> 23 "a yes" signal in the input stochastic process. 24 Then the equivalent received baseband signal for the *k*-th diversity branch takes the 25 following form:

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 15 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 93

$$\mathbf{x}\_{k}(t) = h\_{k}(t)\mathbf{s}(t - \tau\_{k}) + w\_{k}(t) \quad , \quad k = 1, \ldots, N\_{\prime} \tag{38}$$

where ( ) *<sup>k</sup>* 2 *s t* is a 1-D baseband transmitted signal that without loss of generality, is assumed to be real, *h* (*t*) *<sup>k</sup>* 3 is the complex channel gain for the *k*-th branch subjected to Rayleigh fading, *<sup>k</sup>* 4 is the propagation delay along the *k-*th path of received signal, and *w* (*t*) *<sup>k</sup>* is the zero-mean complex AWGN with two-sided power spectral density 2 *N*<sup>0</sup> 5 with 6 the dimension *W Hz* . At GR front end, for each diversity branch, the received signal is split 7 into its in-phase and quadrature signal components. Then, the conventional HS/MRC 8 scheme is applied over all of these quadrature branches, as shown in Fig.2.

10 **Figure 2.** Block diagram receiver based on GR with quadrature subbranch HS/MRC and HS/MRC 11 schemes.

We can present *h* (*t*) *<sup>k</sup>* 12 given by (29)–(37) as i.i.d. complex Gaussian random variables 13 assuming that each of the *L* branches experiences the slow and flat Rayleigh fading

$$h\_k(t) = \alpha\_k(t) \exp\{-j\phi\_k(t)\} = \alpha\_k \exp\{-j\phi\_k\},\tag{39}$$

where *<sup>k</sup>* is a Rayleigh random variable and *<sup>k</sup>* 15 is a random variable uniformly distributed 16 within the limits of the interval [0,2) . Owing to the fact that the fade amplitudes are Rayleigh distributed, we can present *h* (*t*) *<sup>k</sup>* 17 as

$$h\_k(t) = h\_{kl}(t) + jh\_{k\underline{Q}}(t)\tag{40}$$

and *w* (*t*) *<sup>k</sup>* 19 as

9

14 BookTitle

*<sup>k</sup>* ,1 *<sup>k</sup>* ,2 *<sup>k</sup>* ,*<sup>L</sup>* 2

Without loss of generality, we may assume that 0 5

 and 

3 and the channel power is normalized

92 Contemporary Issues in Wireless Communications

20 *3.2.2. Selection/maximal-ratio combining*

the elements in *<sup>k</sup>* ,*<sup>i</sup>*

11 the following form

where *k,i* **a**

25 following form:

 

*<sup>k</sup>*,*<sup>i</sup>* 7 be zero if the following condition is satisfied

2 , 1

*k i*

*L*

*i* 

 <sup>4</sup> (33)

possible number of paths. When a user's path number, say *M*<sup>1</sup> 6 , is less than *M*, we can let all

<sup>1</sup> 8 *M* 1 . *i M* (34)

9 We may also assume that the maximum delay is much smaller than the processing gain *N* [46]. Before our formulation, we first define a (2*N* 1) *L* composite signature matrix **A***<sup>k</sup>* 10 in

,1 ,2 , [ , ,, ] *k k k kL* 12 *A* **aa a** (35)

, , 1 , [ ] 0, ,0, ,0, ,0 , . *k i N k i*

*T T*

 

<sup>~</sup> 13 is a vector containing *i*-th delayed spreading code for user *k*. It is defined as

*k i k* 

14 (36)

15 Since a multipath fading channel is involved, the current received bit signal will be 16 interfered by previous bit signal. As mentioned above, the maximum path delay is much 17 smaller than the processing gain. The interference will not be severe and for simplicity, we

. *k kk* 19 **h α A** (37)

21 We assume that there are *N* diversity branches experiencing slow and flat Rayleigh fading, 22 and all of the fading processes are independent and identically distributed (i.i.d.). During analysis, we consider only the hypothesis *H*<sup>1</sup> 23 "a yes" signal in the input stochastic process. 24 Then the equivalent received baseband signal for the *k*-th diversity branch takes the

**a** *a*

18 may ignore this effect. Let us denote the channel gain for multipath fading as

1.

*<sup>k</sup>*,1 for each user and *L* is the maximum

(32)

1 Let

$$
\pi v\_k(t) = \pi v\_{k1}(t) + \dot{\gamma} w\_{kQ}(t). \tag{41}
$$

16 BookTitle

The in-phase signal component *x* (*t*) *kI* and quadrature signal component *x* (*t*) 1 *kQ* of the received signal *x* (*t*) *<sup>k</sup>* 2 are given by

$$
\propto\_{kl}(t) = h\_k(t)a(t - \tau\_k) + \varpi\_{kl}(t), \tag{42}
$$

$$\mathbf{x}\_{kQ}(t) = h\_{kQ}(t)a(t - \tau\_k) + w\_{kQ}(t). \tag{43}$$

Since *h* (*t*), (*k* 1, , *K*) 5 *<sup>k</sup>* are subjected to i.i.d. Rayleigh fading, we can assume that the inphase *h* (*t*) *kI* and quadrature *h* (*t*) 6 *kQ* channel gain components are independent zero-mean 7 Gaussian random variables with the same variance [41]

$$
\sigma\_h^2 = 0.5E\left[ \mid h\_k^2(t) \mid \right] \tag{44}
$$

where *E*[] is the mathematical expectation. Further, the in-phase *w* (*t*) *kI* 9 and quadrature *w* (*t*) *kQ* 10 noise components are also the independent zero-mean Gaussian random processes, each with two-sided power spectral density <sup>0</sup> 11 0.5*N* with the dimension *W Hz* [36]. Due to the independence of the in-phase *h* (*t*) *kI* and quadrature and quadrature *h* (*t*) 12 *kQ* channel gain components and the in-phase *w* (*t*) *kI* and quadrature *w* (*t*) *kQ* 13 noise components, the 2*N* 14 quadrature branch received signal components conditioned on the transmitted signal are i.i.d.

We can reorganize the in-phase and quadrature components of the channel gains *<sup>k</sup>* 15 *h* and Gaussian noise *w* (*t*) *<sup>k</sup>* when *k* 1,, *N* as *<sup>k</sup> g* and *<sup>k</sup>* 16 *v* , given, respectively by

$$\mathbf{g}\_{k}(t) = \begin{cases} h\_{kI}(t), & k = 1, \ldots, N \\ h\_{(k-N)Q}(t), & k = N+1, \ldots, 2N \end{cases} \tag{45}$$

$$\mathbf{w}\_{k}(t) = \begin{cases} \mathbf{w}\_{kI}(t), & k = 1, \ldots, N \\ \mathbf{w}\_{(k-N)Q}(t), & k = N+1, \ldots, 2N \end{cases} \tag{46}$$

19 The GR output with quadrature subbranch HS/MRC and HS/MRC schemes according to 20 GASP [2, 3, 6–9] is given by:

$$Z\_{\rm QBUS/MRC}^{\rm CR}(t) = s^2(t) \sum\_{k=1}^{2N} b\_k^2 \mathcal{g}\_k^2(t) + \sum\_{k=1}^{2N} b\_k^2 \mathcal{g}\_k^2(t) \left[ \upsilon\_{k\_{\rm AI}}^2(t) - \upsilon\_{k\_{\rm II}}^2(t) \right] \tag{47}$$

where ( ) ( ) <sup>2</sup> <sup>2</sup> *<sup>v</sup> <sup>t</sup> <sup>v</sup> <sup>t</sup> kAF kPF* <sup>22</sup> is the background noise forming at the GR output for the *k*-th branch; {0,1} *<sup>k</sup> b* and 2*L* of the *<sup>k</sup>* 23 *b* equal 1.

#### 1 *3.2.3. Synchronous DS-CDMA wireless communication system*

16 BookTitle

received signal *x* (*t*) *<sup>k</sup>* 2 are given by

94 Contemporary Issues in Wireless Communications

( ) ( ) ( ) ( ), *kI k k kI* 3 *x t h tat w t*

( ) ( ) ( ) ( ). *kQ kQ Q k k* 4 *x t h tat w t*

7 Gaussian random variables with the same variance [41]

2 2 0.5 ( ) [| |], *h k* <sup>8</sup>

The in-phase signal component *x* (*t*) *kI* and quadrature signal component *x* (*t*) 1 *kQ* of the

Since *h* (*t*), (*k* 1, , *K*) 5 *<sup>k</sup>* are subjected to i.i.d. Rayleigh fading, we can assume that the inphase *h* (*t*) *kI* and quadrature *h* (*t*) 6 *kQ* channel gain components are independent zero-mean

where *E*[] is the mathematical expectation. Further, the in-phase *w* (*t*) *kI* 9 and quadrature *w* (*t*) *kQ* 10 noise components are also the independent zero-mean Gaussian random processes, each with two-sided power spectral density <sup>0</sup> 11 0.5*N* with the dimension *W Hz* [36]. Due to the independence of the in-phase *h* (*t*) *kI* and quadrature and quadrature *h* (*t*) 12 *kQ* channel gain components and the in-phase *w* (*t*) *kI* and quadrature *w* (*t*) *kQ* 13 noise components, the 2*N* 14 quadrature branch received signal components conditioned on the transmitted signal are i.i.d.

We can reorganize the in-phase and quadrature components of the channel gains *<sup>k</sup>* 15 *h* and

19 The GR output with quadrature subbranch HS/MRC and HS/MRC schemes according to

2 2

where ( ) ( ) <sup>2</sup> <sup>2</sup> *<sup>v</sup> <sup>t</sup> <sup>v</sup> <sup>t</sup> kAF kPF* <sup>22</sup> is the background noise forming at the GR output for the *k*-th branch;

1 1

*QBHS MRC kk kk k k k k Z t s t bg t bg t v t v t* <sup>21</sup> (47)

*k N Q kI*

( ) , 1, , ( ) *<sup>h</sup>*( ) *<sup>t</sup> <sup>k</sup> <sup>N</sup> <sup>N</sup> h t k N*

( ) , 1, , ( ) *<sup>w</sup>*( ) *<sup>t</sup> <sup>k</sup> <sup>N</sup> <sup>N</sup> <sup>w</sup> <sup>t</sup> <sup>k</sup> <sup>N</sup> <sup>v</sup> <sup>t</sup> k N Q kI*

*<sup>k</sup>* 17 (45)

*<sup>k</sup>* 18 (46)

( ) , 1, ,2 .

2 22 22 2 2

() () () () () (), [ ] *AF PF*

( ) , 1, ,2 ;

Gaussian noise *w* (*t*) *<sup>k</sup>* when *k* 1,, *N* as *<sup>k</sup> g* and *<sup>k</sup>* 16 *v* , given, respectively by

 

 

*N N GR*

*g t*

/

20 GASP [2, 3, 6–9] is given by:

{0,1} *<sup>k</sup> b* and 2*L* of the *<sup>k</sup>* 23 *b* equal 1.

*Eht* (44)

(42)

(43)

2 Consider a synchronous DS-CDMA system employing the GR with *K* users, the processing gain *N*, the number of frame *L*, the chip duration *Tc* , the bit duration *<sup>R</sup> NT <sup>T</sup> <sup>c</sup>* <sup>3</sup>*<sup>b</sup>* with 4 information bit encoding rate *R*. The signature waveform of the user *k* is given by

$$a\_k(t) = \sum\_{i=1}^{N} a\_{ki} p\_{T\_c}(t - iT\_c),\tag{48}$$

where { , ,..., } *ak*<sup>1</sup> *ak* <sup>2</sup> *akN* 6 is a random spreading code with each element taking value on 1 *N* equiprobably, *p* (*t*) *cT* is the unit amplitude rectangular pulse with duration *Tc* 7 . The 8 baseband signal transmitted by the user *k* is given by

$$s\_k(t) = A\_k(t) \sum\_{i=1}^{L} b\_{k,i} a\_k(t - iT\_b)\_i \tag{49}$$

where *A* (*t*) *<sup>k</sup>* 10 is the transmitted signal amplitude of the user *k*. The following form can 11 present the received baseband signal:

$$\mathbf{x}(t) = \sum\_{k=1}^{K} h\_k(t) s\_k(t) + \boldsymbol{\nu}(t) = \sum\_{k=1}^{K} \sum\_{i=1}^{L} S\_k(t) b\_{k,i} a\_k (t - iT\_b) + \boldsymbol{\nu}(t) \quad , \quad t \in [0, T\_b] \tag{50}$$

13 where taking into account (29)–(37) and (39) and as it was shown in [13]

$$S\_k(t) = h\_k(t)A\_k(t) = \alpha\_k^2 A\_k(t) \tag{51}$$

15 is the received signal amplitude envelope for the user *k*, *w*(*t*) is the complex Gaussian noise 16 with zero mean with

$$\mathbb{E}[\boldsymbol{w}\_k(t)(\boldsymbol{w}\_j(t))^\*] = \begin{cases} 2\boldsymbol{a}\_k^2 \sigma\_w^2 & \text{if} \quad j=k\\ 2\boldsymbol{a}\_k^2 \sigma\_w^2 \rho\_{kj} & \text{if} \quad j \neq k \end{cases} \tag{52}$$

 *kj* 18 is the coefficient of correlation. Using GR based on the multistage PPIC for DS-CDMA 19 systems and assuming the user *k* is the desired user, we can express the corresponding GR 20 output according to GASP and the main functioning condition of GR expressed by *<sup>s</sup>*mod [*k*] *<sup>s</sup>* [*k*] *<sup>i</sup> <sup>i</sup>* 21 as the first stage of the PPIC GR:

18 BookTitle

11

$$Z\_{k}(t) = \prod\_{0}^{T\_{b}} [\mathcal{Z}\boldsymbol{x}\_{k}(t)\mathbf{s}\_{k}^{\*}(t-\tau\_{k}) - \mathbf{x}\_{k}(t)\mathbf{x}\_{k}(t-\tau\_{k})]dt + \int\_{0}^{T\_{b}} a\_{k}^{2} [\boldsymbol{w}\_{k\_{\mathcal{M}}}(t)\boldsymbol{w}\_{k\_{\mathcal{M}}}(t-\tau\_{k}) - \boldsymbol{w}\_{k\_{\mathcal{M}}}(t)\boldsymbol{w}\_{k\_{\mathcal{M}}}(t-\tau\_{k})]dt,\quad(53)$$

where *s* (*t*) *<sup>k</sup>* is the model of the signal transmitted by the user *k* ( *<sup>s</sup>*mod [*k*] *<sup>s</sup>* [*k*] *<sup>i</sup> <sup>i</sup>* ); *<sup>k</sup>* <sup>2</sup>*τ* is the 3 delay factor that can be neglected for simplicity of analysis. For this case, we have

$$Z\_k = \mathbf{S}\_k(t)\mathbf{b}\_k + \sum\_{j=1, j\neq k}^{K} \mathbf{S}\_j(t)\mathbf{b}\_j\mathbf{o}\_{kj} + \boldsymbol{\zeta}\_k(t) = \mathbf{S}\_k(t)\mathbf{b}\_k + I\_k(t) + \boldsymbol{\zeta}\_k(t) = \mathbf{b}\_k(t)A\_k(t)\mathbf{b}\_k + I\_k(t) + \boldsymbol{\zeta}\_k(t), \tag{54}$$

5 where the first term in (54) is the desired signal;

$$\rho\_{kj} = \int\_{0}^{T\_b} \mathbf{s}\_k(t)\mathbf{s}\_f(t)dt\tag{55}$$

7 is the coefficient of correlation between signature waveforms of the *k-*th and *j-*th users; the 8 third term in (54)

$$\zeta\_k(t) = \int\_0^{T\_b} \alpha\_k^2 \left[ \boldsymbol{w}\_{k\_{AF}}^2(t) - \boldsymbol{w}\_{k\_{PF}}^2(t) \right] dt \tag{56}$$

10 is the total noise component at the GR output; and the second term in (54)

$$I\_k = \sum\_{j=1, j \neq k}^{K} S\_j b\_j \rho\_{kj} = \sum\_{j=1, j \neq k}^{K} h\_j A\_j b\_j \rho\_{kj} = \sum\_{j=1, j \neq k}^{K} \alpha\_j^2 A\_j b\_j \rho\_{kj} \tag{57}$$

is the MAI. The conventional GR makes a decision based on *Zk* 12 . Thus, MAI is treated as 13 another noise source. When the number of users is large, MAI will seriously degrade the 14 system performance. GR with partial interference cancellation, being a multiuser detection 15 scheme [31], is proposed to alleviate this problem.

16 Denoting the soft and hard decisions at the GR output for the user *k* by

$$
\hat{b}\_k^{(0)} = \mathbf{Z}\_k \text{ and } \hat{b}\_k^{(0)} = \text{sgn}(\mathbf{Z}\_k) \tag{58}
$$

18 respectively, the output of the GR with the first PPIC stage with a partial cancellation factor equal to <sup>1</sup> 19 *p* can be written by [43]

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 19 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 97

$$
\tilde{b}\_k^{(1)} = p\_1(Z\_k - \hat{I}\_k) + (1 - p\_1)\tilde{b}\_k^{(0)} = Z\_k - p\_1 \hat{I}\_{k'} \tag{59}
$$

where ~(1) *<sup>k</sup>* 2 *b* denotes the soft decision of user *k* at the GR output with the first stage of PPIC 3 and

$$\hat{I}\_k = \sum\_{j=1, j \neq k}^{K} \hat{S}\_j \hat{b}\_j^{(0)} \rho\_{kj} = \sum\_{j=1, j \neq k}^{K} h\_j A\_j \hat{b}\_j^{(0)} \rho\_{kj} = \sum\_{j=1, j \neq k}^{K} \alpha\_j^2 A\_j \hat{b}\_j^{(0)} \rho\_{kj} \tag{60}$$

5 is the estimated MAI using a hard decision.

#### 6 **3.3. Performance analysis**

18 BookTitle

where *s* (*t*) *<sup>k</sup>*

8 third term in (54)

11

2

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

*k k k j j kj k k k k k k k k k k*

<sup>4</sup> (54)

 *Tb kj <sup>k</sup> <sup>j</sup> ρ s t s t dt* 0 6 ( ) ( ) (55)

7 is the coefficient of correlation between signature waveforms of the *k-*th and *j-*th users; the

 

*<sup>k</sup> <sup>j</sup> <sup>j</sup> kj I S b ρ h A b ρ A b ρ*

*K*

*j j k*

is the MAI. The conventional GR makes a decision based on *Zk* 12 . Thus, MAI is treated as 13 another noise source. When the number of users is large, MAI will seriously degrade the 14 system performance. GR with partial interference cancellation, being a multiuser detection

(0) ˆ(0) and ( ) *kk k k b Z b sgn Z* 17 (58)

18 respectively, the output of the GR with the first PPIC stage with a partial cancellation factor

*<sup>k</sup> t <sup>k</sup> wk t wk t dt*

*AF PF*

*j j j kj*

*b*

0

*T*

*AF AF PF PF*

*t S tb I t t h tA tb I t t*

[ ( ) ( )] (56)

*j j j kj*

2

*K*

*j j k*

1,

 

(57)

() 2 () ( ) () ( ) [ ] [ ] () ( ) () ( ) ,

*Z t x ts t k k k k k k k kk k k k k k τ x tx t τ dt <sup>w</sup> tw t <sup>τ</sup> w tw t <sup>τ</sup> dt* 1 (53)

is the model of the signal transmitted by the user *k* ( *<sup>s</sup>*mod [*k*] *<sup>s</sup>* [*k*] *<sup>i</sup> <sup>i</sup>* ); *<sup>k</sup>* <sup>2</sup>*τ* is the

3 delay factor that can be neglected for simplicity of analysis. For this case, we have

0 0

1,

*j jk Z S tb S tb ρ*

5 where the first term in (54) is the desired signal;

<sup>2</sup> <sup>2</sup> <sup>2</sup> 9

( )

*K*

*j j k*

15 scheme [31], is proposed to alleviate this problem.

equal to <sup>1</sup> 19 *p* can be written by [43]

10 is the total noise component at the GR output; and the second term in (54)

1, 1,

16 Denoting the soft and hard decisions at the GR output for the user *k* by

*K*

96 Contemporary Issues in Wireless Communications

*b b*

*T T*

#### 7 *3.3.1. Symbol error rate expression*

Let *<sup>k</sup>* 8 *q* denote the instantaneous SNR per symbol of the *k*-th quadrature branch 9 (*k* 1,,2*N*) at the GR output under quadrature subbranch HS/MRC and HS/MRC schemes. In line with [3, 46] and (29)–(37) and (39), the instantaneous SNR *<sup>k</sup>* 10 *q* can be defined 11 in the following form:

$$q\_k = \frac{E\_b a\_k^2}{2\sigma\_w^2} \,'\,\tag{61}$$

where *Eb* 13 is the average symbol energy of the transmitted signal *s*(*t*) .

14 Assume that we choose 2*L* (1 *L N*) quadrature branched out of the 2*N* branches. Then, 15 the SNR per symbol at the GR output under quadrature subbranch HS/MRC and HS/MRC 16 schemes may be presented as

$$q\_{QBIIS/MRC} = \sum\_{k=1}^{21} q\_{(k)'} \tag{62}$$

where (*<sup>k</sup>* ) *q* are the ordered instantaneous SNRs *<sup>k</sup>* 18 *q* and satisfy the following condition

$$q\_{(1)} \ge q\_{(2)} \ge \cdots \ge q\_{(2N)}.\tag{63}$$

20 When *L N* , we obtain the MRC, as expected. Using the MGF method discussed in [33, 41], SER of *M-*ary pulse amplitude modulation (PAM) system conditioned on *QBHS MRC q* / 21 is given 22 by

$$P\_s(q\_{\text{QBHS/MRC}}) = \frac{2(M-1)}{M} \int\_0^{0.5\pi} \exp\left(-\frac{\mathcal{G}\_{M-\text{PAM}}}{\sin^2\theta} q\_{\text{QBHS/MRC}}\right) d\theta,\tag{64}$$

2 where

20 BookTitle

$$g\_{M-PAM} = \frac{\mathfrak{Z}}{M^2 - \mathbf{1}}.\tag{65}$$

Averaging (64) over *QBHS MRC q* / 4 the SER of *M-*ary PAM system is determined in the 5 following form:

$$P\_s = \frac{2(M-1)}{M\pi} \int\_0^{0.5\pi} \int\_0^{\pi} \exp\{-\frac{\mathcal{S}\_{M-\text{PAM}}}{\sin^2\theta} q\} f\_{\boldsymbol{q}\_{\text{QPSK/MLC}}}(\boldsymbol{q}) d\boldsymbol{q} d\boldsymbol{\theta} \ = \frac{2(M-1)}{M\pi} \int\_0^{0.5\pi} \boldsymbol{\rho}\_{\boldsymbol{q}\_{\text{QPSK/MLC}}} \left(-\frac{\mathcal{S}\_{M-\text{PAM}}}{\sin^2\theta}\right) d\boldsymbol{\theta} \ (\boldsymbol{\Theta}\ \boldsymbol{\theta})^{-1}$$

7 where

$$\varphi\_q(\mathbf{s}) = E\_q \left\langle \exp(\mathbf{s}q) \right\rangle \tag{67}$$

is the MGF of random variable *q*, {} *Eq* 9 is the mathematical expectation of MGF with respect 10 to SNR per symbol.

11 A finite-limit integral for the Gaussian *Q*-function, which is convenient for numerical 12 integrations, is given by [54]

$$Q(\mathbf{x}) = \begin{cases} \frac{1}{\pi} \int\_0^{0.5\pi} \exp\left\{-\frac{\mathbf{x}^2}{2\sin^2\theta}\right\} d\theta & \mathbf{x} \ge \mathbf{0} \\\\ 1 - \frac{1}{\pi} \int\_0^{0.5\pi} \exp\left\{-\frac{\mathbf{x}^2}{2\sin^2\theta}\right\} d\theta & \mathbf{x} < \mathbf{0} \end{cases} \tag{68}$$

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

$$\text{erf}(\mathbf{x}) = \frac{2}{\sqrt{\pi}} \int\_0^{\mathbf{x}} \exp(-t^2) dt = 1 - 2Q(\sqrt{2}\mathbf{x}).\tag{69}$$

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

$$\mathbf{Q(x)} = \frac{1}{2} \text{erfc}\left(\frac{\mathbf{x}}{\sqrt{2}}\right) \qquad \text{or} \qquad \text{erfc(x)} = 2\mathbf{Q(\sqrt{2}x)},\tag{70}$$

1 which is convenient for computing values using MATLAB since **erfc** is a subprogram in 2 MATLAB but the Gaussian *Q*-function is not (unless you have a *Communications Toolbox*). 3 Note that the Gaussian *Q*-function is the tabulated function.

4 Now, let us compare (64) and (68) to obtain the closed form expression for the SER of *M*-ary 5 PAM wireless communication system employing the GR with quadrature subbranch 6 HS/MRC and HS/MRC schemes. We can easily see that taking into account (44), (45), (61), 7 (62), and (65), the SER of *M*-ary PAM system employing the GR with quadrature subbranch 8 HS/MRC and HS/MRC schemes can be defined in the following form

$$P\_s(q\_{Q\text{BHS}/MRC}) = \frac{2M-1}{M} \mathcal{Q}\left(\sqrt{\frac{6}{M^2 - 1} q\_{Q\text{BHS}/MRC}}\right). \tag{71}$$

10 Thus, we obtain the closed form expression for the SER of *M*-ary PAM system employing 11 the GR with quadrature subbranch HS/MRC and HS/MRC schemes that agrees with (8.136) 12 and (8.138) in [55]. If *M* 2 , the average BER performance of coherent binary phase-shift 13 keying (BPSK) wireless communication system using the quadrature subbranch HS/MRC 14 and HS/MRC schemes under GR implementation can be determined in the following form:

$$P\_b = \frac{1}{\pi} \int\_0^{0.5\pi} \varphi\_{q\_{\text{QHS} \lesssim \text{MDC}}} \left( -\frac{1}{\sin^2 \theta} \right) d\theta. \tag{72}$$
