18 **5.3. System model**

19 In this section, we think that to assess accurately the effects of multipath fading and 20 interference components from other users on the performance of DS-CDMA downlink 21 wireless communication system it is enough to consider a single-cell environment system 22 model. In particular, we analyze a complex baseband-equivalent model with the binary 23 phase-shift keying (BPSK) data and complex signature sequences over a multipath fading 24 channel for the DS-CDMA downlink wireless communication system. The baseband 25 representation of the total signal transmitted on the downlink can be presented in the 26 following form:

$$a(t) = \sum\_{k=0}^{K-1} a\_k(t) = \sum\_{k=0}^{K-1} \sqrt{P\_{a\_k}} b\_k(t) c\_k(t),\tag{202}$$

28 where *K* is the number of users;

$$a\_k(t) = \sqrt{P\_{a\_k}} \, b\_k(t) c\_k(t) \tag{203}$$

is the transmitted signal of the *k*-th user; *<sup>k</sup> Pa* 30 is the power of the *k*-th transmitted signal;

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 57 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 135

$$b\_k(t) = \sum\_{n=-\alpha}^{n} b\_k^{(n)} p\_T(t - nT) \tag{204}$$

2 is the data signal of the *k*-th user;

56 BookTitle

in the matrix *A*<sup>1</sup> 5 .

sequences, i.e., *a j* or *j* <sup>3</sup> 17 .

18 **5.3. System model**

26 following form:

28 where *K* is the number of users;

1 where indicates the complex conjugate. Hence, the unified complex Hadamard transform 2 matrix is orthogonal. Furthermore, the unified complex Hadamard transform matrices

4 *a*<sup>1</sup> *a*<sup>2</sup> *a*<sup>3</sup> 1 (201)

6 The unified complex Hadamard transform matrices have two categories of 32 basic matrices, depending on whether <sup>3</sup> *a* in (197) is imaginary or not [97]. If <sup>3</sup> 7 *a* is imaginary, the matrix 8 group is called the half-spectrum property unified complex Hadamard transform. 9 Otherwise, the group is called the non-half-spectrum property unified complex Hadamard 10 transform. The unified complex Hadamard transform spreading sequence *ck* , *k* 1,, *N* is 11 defined by the *k*-th row of the unified complex Hadamard transform matrix. It has been 12 shown in [97] that the non-half-spectrum property unified complex Hadamard transform 13 sequences have very similarly poor autocorrelation properties as WH sequences, and some 14 of the half-spectrum property unified complex Hadamard transform sequences exhibit a 15 reasonable compromise between the autocorrelation and cross-correlation functions. In this 16 section, we just consider the half-spectrum property unified complex Hadamard transform

19 In this section, we think that to assess accurately the effects of multipath fading and 20 interference components from other users on the performance of DS-CDMA downlink 21 wireless communication system it is enough to consider a single-cell environment system 22 model. In particular, we analyze a complex baseband-equivalent model with the binary 23 phase-shift keying (BPSK) data and complex signature sequences over a multipath fading 24 channel for the DS-CDMA downlink wireless communication system. The baseband 25 representation of the total signal transmitted on the downlink can be presented in the

1 1

*at a t P b tc t* 

*K K*

*k k*

0 0 ( ) ( ) ( ) ( ), *<sup>k</sup>*

*a* (*t*) *P b* (*t*)*c* (*t*) *<sup>k</sup> <sup>a</sup> <sup>k</sup> <sup>k</sup> <sup>k</sup>* 29 (203)

is the transmitted signal of the *k*-th user; *<sup>k</sup> Pa* 30 is the power of the *k*-th transmitted signal;

 <sup>27</sup> (202)

*k ak k*

3 contain a WH transform matrix as a special case, with

134 Contemporary Issues in Wireless Communications

$$b\_k^{(n)} \in \{-1, +1\} \tag{205}$$

denotes the *n*-th data bit value of the *k*-th user; the function *p* (*t*) *<sup>T</sup>* 4 is the rectangular pulse of symbol duration *T*; *c* (*t*) *<sup>k</sup>* 5 is the complex spreading signal defined by

$$c\_k(t) = \sum\_{m = -\infty}^{\infty} c\_k^{(m)} \phi(t - mT\_c);\tag{206}$$

and (*m*) *<sup>k</sup>* 7 *c* denotes the *m*-th complex chip value of the *k*-th user. The function (*t*) is a chip waveform that is time-limited to [0, ) *Tc* 8 with

$$\int\_{t}^{T\_{\varepsilon}} \phi^2(t)dt = T\_{\varepsilon'} \tag{207}$$

including the rectangular pulse of duration *Tc* , and *Tc* 10 is called the chip duration. 11 Throughout this section, we assume that

$$T = NT\_c.\tag{208}$$

13 Power control is assumed to be perfect, and we suppose that the transmitted signal power *<sup>k</sup> Pa* 14 is assumed to be known. We also assume

$$c\_k^{(m)} = d^{(m)} a\_k^{(m)} \,\,\,\,\,\tag{209}$$

16 where

$$d = \{d^{(m)}\} \quad \text{with} \quad d^{(m)} \in \{+1, -1\} \tag{210}$$

18 is the random scrambling code commonly used by all users, and

$$a^{(k)} = \{a\_k^{(m)}\} \tag{211}$$

58 BookTitle

1 is the user-specific orthogonal unified complex Hadamard transform spreading sequence 2 with period *N*. Thus,

$$\mathcal{c}^{(k)} = \{\mathcal{c}\_k^{(m)}\} \tag{212}$$

4 is the random sequence with

$$E\left\{\mathbf{c}\_{k}^{(m)}\left(\mathbf{c}\_{k'}^{(n)}\right)^{\*}\right\} = 0, \ m \neq n \tag{213}$$

6 for all *k* and *k* , where *E*{} denotes the mathematical expectation.

7 The final results of our analysis in this section are applicable to the DS-CDMA downlink 8 wireless communication systems that use long PN scrambling sequences such as *m*-9 sequences and Gold sequences [95]. This is because the periods of these long scrambling 10 codes are much larger than that of the spreading factor *N*, and have correlation properties 11 similar to those of the random scrambling sequences.

12 For instance, at the base station transmitter in mobile communication system, the signals of 13 all *K* users are symbol synchronously added before passing through a frequency-selective 14 multipath-fading channel. The complex baseband-equivalent impulse response of the 15 multipath channel can be presented in the following form:

$$h(t) = \sum\_{l=0}^{L-1} \alpha\_l \exp(j\theta\_l) \delta(t - \pi\_l),\tag{214}$$

where *L* is the number of resolvable propagation paths, and exp( ) *<sup>l</sup> <sup>l</sup> α jθ* and *<sup>l</sup>* 17 *τ* are the complex fading factor and propagation delay of the *l*-th path, respectively. Note, *α<sup>l</sup>* 18 may be 19 Rayleigh-, Rician-, or Nakagami-distributed, depending on a specific channel model. All 20 random variables in (214) are assumed to be independent for *l*. The channel parameters, such as *α<sup>l</sup>* , *<sup>l</sup> θ* , and *<sup>l</sup>* 21 *τ* are here assumed to be known in the dispreading and demodulation 22 process, although in practice, the impulse response of the channel is typically estimated 23 using the pilot symbol or pilot channel.

24 Moreover, we assume that the multipaths at the GR input are resolvable and chip-25 synchronized, i.e., they are spaced, at least one chip duration apart in time and the relative 26 delays are multiples of the chip duration. Without loss of generality, the resolved paths are 27 assumed to be numbered such that

$$0 \le \tau\_0 < \tau\_1 < \dots < \tau\_{L-1} < T. \tag{215}$$

29 Hence, the baseband complex representation of the signal at the GR input (the input of GR 30 linear system) of any user is given by

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 59 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 137

$$\mathbf{x}(t) = \sum\_{k=0}^{K-1} \sum\_{l=0}^{L-1} \sqrt{P\_{a\_k}} \alpha\_l \exp(j\Theta\_l) b\_k(t - \tau\_l) c\_k(t - \tau\_l) + \mathbf{w}(t), \tag{216}$$

2 where *w*(*t*) is the complex background AWGN with zero mean and one-sided power spectral density *N*<sup>0</sup> 3 .

5 **Figure 11.** GR structure with *L* 1 fingers.

4

58 BookTitle

2 with period *N*. Thus,

4 is the random sequence with

136 Contemporary Issues in Wireless Communications

1 is the user-specific orthogonal unified complex Hadamard transform spreading sequence

*<sup>k</sup>* 3 *c c* (212)

7 The final results of our analysis in this section are applicable to the DS-CDMA downlink 8 wireless communication systems that use long PN scrambling sequences such as *m*-9 sequences and Gold sequences [95]. This is because the periods of these long scrambling 10 codes are much larger than that of the spreading factor *N*, and have correlation properties

12 For instance, at the base station transmitter in mobile communication system, the signals of 13 all *K* users are symbol synchronously added before passing through a frequency-selective 14 multipath-fading channel. The complex baseband-equivalent impulse response of the

1

*L*

*l*

 <sup>16</sup> (214)

( ) exp( ) ( ),

*h t α jθ δ t τ*

*lll*

0

where *L* is the number of resolvable propagation paths, and exp( ) *<sup>l</sup> <sup>l</sup> α jθ* and *<sup>l</sup>* 17 *τ* are the complex fading factor and propagation delay of the *l*-th path, respectively. Note, *α<sup>l</sup>* 18 may be 19 Rayleigh-, Rician-, or Nakagami-distributed, depending on a specific channel model. All 20 random variables in (214) are assumed to be independent for *l*. The channel parameters, such as *α<sup>l</sup>* , *<sup>l</sup> θ* , and *<sup>l</sup>* 21 *τ* are here assumed to be known in the dispreading and demodulation 22 process, although in practice, the impulse response of the channel is typically estimated

24 Moreover, we assume that the multipaths at the GR input are resolvable and chip-25 synchronized, i.e., they are spaced, at least one chip duration apart in time and the relative 26 delays are multiples of the chip duration. Without loss of generality, the resolved paths are

01 1 0 . *<sup>L</sup> ττ τ T* 28 (215)

29 Hence, the baseband complex representation of the signal at the GR input (the input of GR

6 for all *k* and *k* , where *E*{} denotes the mathematical expectation.

11 similar to those of the random scrambling sequences.

15 multipath channel can be presented in the following form:

23 using the pilot symbol or pilot channel.

27 assumed to be numbered such that

30 linear system) of any user is given by

{ } ( ) (*m*) *k*

*E c c m n <sup>n</sup> k m*

*<sup>k</sup>* { ( ) } 0, ( ) ( ) 5 (213)

> 6 In order to mitigate the multipath fading effect, the GR with coherent demodulation is 7 implemented. The GR structure is presented in Fig.11, where the number of fingers is equal 8 to the number of resolvable paths.

Since the symbols (*n*) *<sup>k</sup>* 9 *b* are i.i.d. from one symbol duration to another and from one user to 10 another, without loss of generality, we focus our attention on the GR output of the user 0 for 11 the zero-th transmitted symbol. The complex GR output of the *i*-th finger of user 0 in 12 accordance with the GASP in noise [1–3,6–9,11] is

$$z\_{i} = \% \epsilon \left\{ \int\_{\tau\_{i}}^{T+\tau\_{i}} 2\mathbf{x}(t) \sqrt{P\_{a\_{0}}^{\bullet}} c\_{0}^{\bullet}(t-\tau\_{i}) \exp(-j\theta\_{i}) dt - \int\_{\tau\_{i}}^{T+\tau\_{i}} \mathbf{x}(t) \mathbf{x}^{\star}(t-\tau\_{i}) dt + \int\_{\tau\_{i}}^{T+\tau\_{i}} \eta(t) \eta(t-\tau\_{i}) dt \right\}, \tag{217}$$

where ( ) exp( ) <sup>0</sup> <sup>0</sup> *<sup>a</sup> <sup>i</sup> <sup>i</sup> <sup>P</sup> <sup>c</sup> <sup>t</sup> <sup>τ</sup> <sup>j</sup><sup>θ</sup>* 14 is the model of transmitted signal for user 0, i.e., the reference signal generated by the GR, [3,6]; *<sup>i</sup>* 15 *τ* is the delay factor that can be neglected for simplicity 16 of analysis; *η*(*t*) is the noise forming at the GR AF output given by (4).

17 The main functioning condition under employment of GR in DS-CDMA wireless 18 communication systems discussed in detail in Section 2.1 takes the following form

60 BookTitle

11

$$P\_{a\_k} = P\_{a\_k}^\* \tag{218}$$

where *<sup>k</sup> Pa* is the power of information signal and *ak* 2 *P* is the power of model signal (the 3 reference signal). In practice, we can perform this matching implementing, for example, 4 tracking systems. These statements and possible ways to solve this problem and how we can 5 implement all this in practice are discussed in detail in [1–3,6–9,11]. Thus, the following 6 process, in a general case, is formed at the GR output for *k*-th user according to 7 implementation of GASP in DS-CDMA wireless communication systems.

8 *The case 1*: a "yes" signal in the input process –

$$\begin{split} Z &= \Re \left\{ \int\_{\tau\_{i}}^{\tau\_{i}} \sum\_{l=0}^{L-1} a\_{i}(t) \mathbf{a}\_{l} \exp(j\theta\_{l}) \mathbf{a}\_{k}^{\*}(t-\tau\_{i}) dt - \int\_{\tau\_{i}}^{\tau\_{i+1}} \sum\_{l=0}^{L-1} \xi\_{i}(t) \mathbf{a}\_{i} \exp(j\theta\_{l}) \boldsymbol{\xi}\_{k}^{\*}(t-\tau\_{i}) dt + \int\_{\tau\_{i}}^{\tau\_{i+1}} \sum\_{l=0}^{L-1} \eta\_{k}(t) \mathbf{a}\_{l} \exp(j\theta\_{l}) \boldsymbol{\eta}\_{k}^{\*}(t-\tau\_{i}) dt \right\} \\ &= \Re \left\{ \int\_{\tau\_{i}}^{\tau\_{i+1}} \sum\_{l=0}^{L-1} \Big[ \sqrt{P\_{\mathbf{a}\_{l}}} \sqrt{P\_{\mathbf{a}\_{l}}} \, b\_{l}(t) \mathbf{a}\_{l} \exp(j\theta\_{l}) \mathbf{b}\_{k}^{\*}(t-\tau\_{i}) \mathbf{c}\_{l}(t) \mathbf{c}\_{k}^{\*}(t-\tau\_{i}) - \boldsymbol{\xi}\_{i}(t) \mathbf{a}\_{l} \exp(j\theta\_{l}) \boldsymbol{\xi}\_{k}^{\*}(t-\tau\_{i}) + \eta\_{k}(t) \mathbf{a}\_{l} \exp(j\theta\_{l}) \boldsymbol{\eta}\_{k}^{\*}(t-\tau\_{i}) \right] dt \end{split} \tag{219}$$

10 *The case 2*: a "no" signal in the input process –

$$Z = \mathfrak{R} \circ \left\{ \int\_{\tau\_i}^{T+\tau\_i} \sum\_{l=0}^{L-1} \eta\_k(t) \alpha\_l \exp(j\theta\_l) \eta\_k^\*(t-\tau\_i) dt - \int\_{\tau\_i}^{T+\tau\_i} \sum\_{l=0}^{L-1} \xi\_k(t) \alpha\_l \exp(j\theta\_l) \xi\_k^\*(t-\tau\_i) dt \right\} = \Delta\_{i^\*} \tag{220}$$

where *<sup>i</sup>* 12 is the background noise forming at the GR output. Finally, the GR combiner 13 output that produces a decision statistic is

$$\mathbf{Z} = \sum\_{i=0}^{L-1} w\_i z\_{i\prime} \tag{221}$$

where the selection of the combining weights 's *wi* 15 determines the specific diversity-16 combining technique.

#### 17 **5.4. Performance analysis**

#### 18 *5.4.1. SINR at GR output*

19 In this section, we investigate SINR by considering the GR shown in Fig. 11. It follows from 20 (217)-(221) that the *i*-th GR finger output for user 0 can be presented in the following form

$$\mathbf{z}\_{i} = \mathbf{T} \mathbf{P}\_{a\_0} \mathbf{a}\_i \mathbf{b}\_0^{(0)} + \mathbf{I}\_{M1II}^{(i)} + \mathbf{I}\_{MP}^{(i)} + \boldsymbol{\Lambda}\_i \{\mathbf{t}\},\tag{222}$$

1 where the first term is the signal component; the second term is the multiple-user 2 interference component determined by

$$I\_{MUI}^{(l)} = \% \sigma \left| \sum\_{k=l}^{K-1} P\_{a\_k} a\_l b\_k^{(0)} \int\_{\tau\_l}^{T+\tau\_l} c\_k(t-\tau\_l) c\_0^\*(t-\tau\_l) dt \right| \tag{223}$$

4 the third term is the multipath interference component given by

$$I\_{\rm MP}^{(i)} = \% \epsilon \left\{ \sum\_{k=0}^{K-1} \sum\_{l=0, l \neq i}^{L-1} P\_{a\_i} a\_l \exp\left[j(\Theta\_l - \Theta\_i)\right] \int\_{\tau\_i}^{T+\tau\_i} b\_k(t-\tau\_i) c\_k(t-\tau\_i) c\_0^\*(t-\tau\_i) dt \right\};\tag{224}$$

and the fourth term (*t*) *<sup>i</sup>* 6 is the background noise at the GR output given for a general case 7 by (7).

We can see that the multiple-user interference component (*i*) *MUI* 8 *I* and the multipath interference component (*i*) *MP* 9 *I* are due to the interference from the *i*-th path of other users' 10 signals and from the remaining *L* 1 paths from all users' signals, respectively. The background noise (*t*) *<sup>i</sup>* 11 at the GR output is the i.i.d. random process obeying the asymptotic Gaussian distribution with zero mean and variance of <sup>4</sup> 4*T <sup>w</sup>* 12 *σ* [1–3,6–9,11].

Define a periodic correlation function *Rk*,*<sup>m</sup>* 13 by

$$R\_c^{(k,m)}(q) = \begin{cases} \sum\_{p=0}^{N-l-q} c\_k^{(p)} \left( c\_m^{(p+q)} \right)^\*, & 0 \le q \le N-1\\ \sum\_{p=0}^{N-l+q} c\_k^{(p-q)} \left( c\_m^{(p)} \right)^\*, & 1-N \le q < 0\\ 0, & |q| \ge N \end{cases} \tag{225}$$

15 Then, let

60 BookTitle

, *k k a a P P* <sup>1</sup> (218)

*ak* 2 *P* is the power of model signal (the 3 reference signal). In practice, we can perform this matching implementing, for example, 4 tracking systems. These statements and possible ways to solve this problem and how we can 5 implement all this in practice are discussed in detail in [1–3,6–9,11]. Thus, the following 6 process, in a general case, is formed at the GR output for *k*-th user according to

( ) exp( ) ( ) ( ) exp( ) ( ) ( ) exp( ) ( )

) ( ) ( ) exp( ) ( )

*k i k l lk i*

*ξ t τ η t α jθ η t τ dt*

*k l lk i k l lk i k l lk i*

 

1

*L*

*i Z wz* 

where the selection of the combining weights 's *wi* 15 determines the specific diversity-

19 In this section, we investigate SINR by considering the GR shown in Fig. 11. It follows from 20 (217)-(221) that the *i*-th GR finger output for user 0 can be presented in the following form

*i a i MUI MP i* 21 *z TP α bI I t* (222)

(0) ( ) ( ) <sup>0</sup> ( ), *i i*

0

 <sup>14</sup> (221)

,

*i i*

0

( ) exp( ) ( ) ( ) exp( ) ( ) ,

*k l lk i k l lk i i*

(220)

*Z at α jθ a t τ dt ξ t α jθ ξ t τ dt η t α jθ η t τ dt*

9 . (219)

*Z η t α jθ η t τ dt ξ t α jθ ξ t τ dt* 

where *<sup>i</sup>* 12 is the background noise forming at the GR output. Finally, the GR combiner

where *<sup>k</sup> Pa* is the power of information signal and

8 *The case 1*: a "yes" signal in the input process –

*k k*

138 Contemporary Issues in Wireless Communications

10 *The case 2*: a "no" signal in the input process –

13 output that produces a decision statistic is

Re

1 0

Re

*i i T τ L*

*τ l*

16 combining technique.

17 **5.4. Performance analysis**

18 *5.4.1. SINR at GR output*

Re

11

7 implementation of GASP in DS-CDMA wireless communication systems.

11 1 00 0

*ii i ii i*

*ττ τ ll l*

*a ak l lk ik k i k l l*

*P Pbt α jθ b t τ c tc t τ ξ t α jθ*

*T τ T τ T τ LL L*

( ) exp( ) ( ) ( ) ( ) ( ) exp(

1 1

*i i*

*T τ T τ L L*

*i i*

*τ τ l l*

0 0

$$
\boldsymbol{\pi}\_l - \boldsymbol{\pi}\_i = \boldsymbol{q}\_l \boldsymbol{T}\_c. \tag{726}
$$

With the assumption of chip synchronization, it can be obtained that *<sup>l</sup>* 17 *q* is an integer and the 18 multiple-user interference and multipath interference components can be presented in the 19 following form:

$$\boldsymbol{I}\_{MUI}^{(i)} = \mathfrak{R} \boldsymbol{\epsilon} \left\{ \boldsymbol{T}\_c \sum\_{k=1}^{K-1} \boldsymbol{P}\_{a\_k} \boldsymbol{\alpha}\_i \boldsymbol{b}\_k^{(0)} \boldsymbol{R}\_c^{(k,0)}(\mathbf{0}) \right\}\_{\prime} \tag{227}$$

$$I\_{MP}^{(i)} = \% \epsilon \left\{ T\_c \sum\_{l=0}^{i-1} \sum\_{k=0}^{K-1} P\_{a\_k} \alpha\_l \exp\left[ j \left( \Theta\_l - \Theta\_i \right) \right] \mathbf{R}\_{k0} + T\_c \sum\_{l=i+1}^{L-1} \sum\_{k=0}^{K-1} P\_{a\_k} \alpha\_l \exp\left[ j \left( \Theta\_l - \Theta\_i \right) \right] \hat{\mathbf{R}}\_{k0} \right\}, \tag{228}$$

3 where

2

62 BookTitle

$$R\_{k0} = b\_k^{(0)} R\_c^{(k,0)} (q\_l) + b\_k^{(1)} R\_c^{(k,0)} (\mathcal{N} + q\_l)\_{\prime} \tag{229}$$

$$
\hat{\mathcal{R}}\_{k0} = b\_k^{(-1)} \mathcal{R}\_c^{(k,0)} (q\_l - \mathcal{N}) + b\_k^{(0)} \mathcal{R}\_c^{(k,0)} (q\_l). \tag{230}
$$

6 In the following, we are going to compare the SINR at the GR output in the DS-CDMA 7 downlink wireless communication system implementing the unified complex Hadamard 8 transform spreading codes with that at the GR output in the DS-CDMA downlink wireless 9 communication system using the WH spreading sequences. Note, that under employment of 10 the orthogonal spreading codes in DS-CDMA downlink wireless communication system, 11 such as the WH real spreading codes and the orthogonal unified complex Hadamard 12 transform spreading codes considered in this section, we must take into consideration that

$$R\_c^{\{k,0\}}(\mathbf{0}) = \mathbf{0} \; , \; k \neq \mathbf{0}. \tag{231}$$

14 Consequently, the multiple-user interference component is equal to zero, i.e.,

$$I\_{MUI}^{(i)} = \mathbf{0}.\tag{232}$$

16 When the WH spreading codes are used in the DS-CDMA downlink wireless 17 communication system employing the GR, we can obtain that the multipath interference 18 component takes the following form:

$$I\_{\rm MP}^{(l)} = T\_c \sum\_{l=0}^{l-1} \sum\_{k=0}^{K-1} P\_{a\_k} a\_l \cos \left(\theta\_l - \theta\_i\right) \mathbf{R}\_{k0} + T\_c \sum\_{l=i+1}^{L-1} \sum\_{k=0}^{K-1} P\_{a\_k} a\_l \cos \left(\theta\_l - \theta\_i\right) \hat{\mathbf{R}}\_{k0}.\tag{233}$$

Owing to the mutually independent random variables (*n*) *<sup>k</sup>* 20 *b* for 0 *k K* 1 , the multipath interference component (*i*) *MP* 21 *I* has zero mean. With regard of (213), it can be easily shown via 22 straightforward computation that, for the WH spreading sequences we are able to obtain

$$E\{R\_{k0}^2\} = E\{\hat{R}\_{k0}^2\} = \mathcal{N}.\tag{234}$$

Therefore, the variance of the multipath interference component (*i*) *MP I* is denoted by <sup>2</sup> *W MP <sup>I</sup>* <sup>1</sup>*<sup>σ</sup>* 2 and can be determined in the following form:

$$
\sigma\_{I\_{\rm MP}^{\rm M}}^2 = \mathcal{N} T\_c^2 \sum\_{l=0, l \neq i}^{L-1} a\_l^2 \cos^2 \left(\Theta\_l - \Theta\_i\right) \sum\_{k=0}^{K-1} P\_{a\_i}^2. \tag{235}
$$

4 Hence, the SINR at the GR *i*-th finger output for the WH spreading codes is determined in 5 the following form:

$$\text{SINR}\_{\text{0H}}^{\{i\}} = \frac{E\_{a\_0} \alpha\_i^2}{\sqrt{\frac{1}{N} \sum\_{l=0, l \neq i}^{L-1} \alpha\_l^2 \cos^2 \left(\theta\_l - \theta\_i\right) \sum\_{k=0}^{K-1} E\_{a\_k}^2 + 4 \sigma\_w^4}}, \tag{236}$$

7 where

62 BookTitle

2

3 where

1 *MUI*

140 Contemporary Issues in Wireless Communications

Re

18 component takes the following form:

( )

*i*

interference component (*i*)

( )

*i*

1 ( ) (0) ( ,0) 1

*MP c a l l i k c a l l i k*

(0) ( ,0) (1) ( ,0) <sup>0</sup> ( ) ( ), *k k R bR q bR N q k kc l kc l* <sup>4</sup> (229)

6 In the following, we are going to compare the SINR at the GR output in the DS-CDMA 7 downlink wireless communication system implementing the unified complex Hadamard 8 transform spreading codes with that at the GR output in the DS-CDMA downlink wireless 9 communication system using the WH spreading sequences. Note, that under employment of 10 the orthogonal spreading codes in DS-CDMA downlink wireless communication system, 11 such as the WH real spreading codes and the orthogonal unified complex Hadamard 12 transform spreading codes considered in this section, we must take into consideration that

( ,0)(0) 0 , 0. *<sup>k</sup> R k <sup>c</sup>* <sup>13</sup> (231)

( ) 0. *<sup>i</sup> <sup>I</sup> MUI* 15 (232)

16 When the WH spreading codes are used in the DS-CDMA downlink wireless 17 communication system employing the GR, we can obtain that the multipath interference

1 1 1 1

*i K L K*

*l k li k*

Owing to the mutually independent random variables (*n*)

 <sup>19</sup> (233)

0 0 1 0

*<sup>k</sup>* 20 *b* for 0 *k K* 1 , the multipath

*MP* 21 *I* has zero mean. With regard of (213), it can be easily shown via 22 straightforward computation that, for the WH spreading sequences we are able to obtain

2 2 0 0 <sup>ˆ</sup> {}{} . *k k* <sup>23</sup>*ER ER N* (234)

*MP c al l i k c al l i k*

*IT P α θθ RT P α θθ R*

<sup>ˆ</sup> cos cos . *k k*

0 0

14 Consequently, the multiple-user interference component is equal to zero, i.e.,

( 1) ( ,0) (0) ( ,0) <sup>0</sup> <sup>ˆ</sup> ( ) ( ). *k k R b R q N bR q k k c l kc l* 5 (230)

*I TP α j θ θ RT P α j θ θ R*

*c a ik c*

*K i k*

Re

*k I TP α b R* 

1 1 1 1

*i K L K*

*l k li k*

0 0 1 0

(0) , *<sup>k</sup>*

<sup>ˆ</sup> exp exp , *k k*

(228)

(227)

0 0

$$E\_{a\_k} = P\_{a\_k} T, \quad k = 0, 1, \ldots, K - 1 \tag{237}$$

is the energy per data symbol of the *m*-th user and the variance <sup>2</sup> *<sup>w</sup>* 9 *σ* is given by (23).

10 Similarly, when the unified complex Hadamard transform spreading codes are employed by 11 the DS-CDMA downlink wireless communication system using the GR, we can obtain that the multipath interference component (*i*) *MP* 12 *I* takes the following form:

$$\begin{split} I\_{\text{MP}}^{(l)} &= T\_c \sum\_{l=0}^{i-1} \sum\_{k=0}^{K-1} P\_{a\_l} \alpha\_l \cos(\theta\_l - \theta\_i) \% \epsilon \{ R\_{k0} \} + T\_c \sum\_{l=i+1}^{L-1} \sum\_{k=0}^{K-1} P\_{a\_l} \alpha\_l \cos(\theta\_l - \theta\_i) \% \epsilon \{ \hat{R}\_{k0} \} \\ &+ T\_c \sum\_{l=0}^{i-1} \sum\_{k=0}^{K-1} P\_{a\_l} \alpha\_l \sin(\theta\_l - \theta\_i) \text{Im} \{ R\_{k0} \} + T\_c \sum\_{l=i+1}^{L-1} \sum\_{k=0}^{K-1} P\_{a\_l} \alpha\_l \sin(\theta\_l - \theta\_i) \text{Im} \{ R\_{k0} \}. \end{split} \tag{238}$$

Note that the multipath interference component (*i*) *MP* 14 *I* has zero mean. If the half-spectrum 15 property unified complex Hadamard transform spreading sequences are employed by the 16 DS-CDMA downlink wireless communication system using the GR, it can be easily shown 17 from (213) via straightforward computation that

$$E\left(\left\lceil \mathcal{R}\epsilon(\mathcal{R}\_{k0}) \right\rceil^2\right) = E\left(\left\lceil \operatorname{Im}(\mathcal{R}\_{k0}) \right\rceil^2\right) = \frac{N}{2},\tag{239}$$

$$E\left\{ \left[ \Re \epsilon \left( \hat{\mathcal{R}}\_{k0} \right) \right]^2 \right\} = E\left\{ \left[ \text{Im} \{ \hat{\mathcal{R}}\_{k0} \} \right]^2 \right\} = \frac{N}{2}. \tag{240}$$

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Hence, the variance of the multipath interference component (*i*) *MP I* is denoted by <sup>2</sup> *H MP <sup>I</sup>* <sup>1</sup>*σ* and 2 can be determined in the following form:

$$
\sigma\_{I\_{MP}^{II}}^2 = \frac{N}{2} T\_c^2 \sum\_{l=0, l \neq i}^{L-1} a\_l^2 \sum\_{k=0}^{K-1} P\_{a\_k}^2. \tag{241}
$$

4 Therefore, the SINR at the GR *i*-th finger output for the half-spectrum property unified 5 complex Hadamard transform spreading sequences employed by the DS-CDMA downlink 6 wireless communication system using the GR is determined by

$$\text{SINR}\_{\text{ff}}^{(i)} = \frac{E\_{a\_0} a\_i^2}{\sqrt{\frac{1}{2N} \sum\_{l=0, l \neq i}^{L-1} a\_l^2 \sum\_{k=0}^{K-1} E\_{a\_k}^2 + 4 \sigma\_w^4}}. \tag{242}$$

#### 8 *5.4.2. GR finger weights*

9 There is a need to note that the interference plus the thermal noise power seen by different 10 fingers of the GR employed by DS-CDMA downlink wireless communication systems is 11 different. Assume that the multipath interference signals are uncorrelated from one finger to 12 another. In this case, the optimal weight in terms of the maximizing SINR at the GR output 13 is dependent on the multipath interference. This optimal weight is called the modified 14 maximal ratio combining (MMRC). There is a need to note that for the traditional maximal 15 ratio combining (MRC) the weights are chosen in the following manner

$$
\mathfrak{w}\_i = \mathfrak{a}\_i. \tag{243}
$$

17 Here, if the MMRC is employed as the combiner, the combining weights for DS-CDMA 18 downlink wireless communication system using the WH codes and DS-CDMA downlink 19 wireless communication system with the unified complex Hadamard transform spreading 20 sequences under employment of the GR are different. This is a direct consequence of the 21 difference in the SINR values at the GR finger output of the two systems.

22 For the WH spreading sequences used by the DS-CDMA downlink wireless communication 23 system employing the GR, MMRC weights take the following form:

$$w\_i^{\rm NH} = \sqrt{\frac{P\_{a\_0} \alpha\_i^2}{\frac{1}{N} \sum\_{l=0, l \neq i}^{L-1} \alpha\_l^2 \cos^2 \left(\theta\_l - \theta\_i\right) \sum\_{k=0}^{K-1} E\_{a\_k}^2 + 4\sigma\_w^4}}. \tag{244}$$

1 For the unified complex Hadamard transform spreading sequences used by the DS-CDMA 2 downlink wireless communication system employing the GR, MMRC weights have the 3 following form:

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8 *5.4.2. GR finger weights*

Hence, the variance of the multipath interference component (*i*)

6 wireless communication system using the GR is determined by

( )

*SINR*

15 ratio combining (MRC) the weights are chosen in the following manner

21 difference in the SINR values at the GR finger output of the two systems.

*N*

23 system employing the GR, MMRC weights take the following form:

*w*

2 can be determined in the following form:

142 Contemporary Issues in Wireless Communications

*MP <sup>I</sup>* <sup>1</sup>*σ* and

<sup>3</sup> (241)

4 Therefore, the SINR at the GR *i*-th finger output for the half-spectrum property unified 5 complex Hadamard transform spreading sequences employed by the DS-CDMA downlink

> *a i i <sup>H</sup> L K*

7 (242)

9 There is a need to note that the interference plus the thermal noise power seen by different 10 fingers of the GR employed by DS-CDMA downlink wireless communication systems is 11 different. Assume that the multipath interference signals are uncorrelated from one finger to 12 another. In this case, the optimal weight in terms of the maximizing SINR at the GR output 13 is dependent on the multipath interference. This optimal weight is called the modified 14 maximal ratio combining (MMRC). There is a need to note that for the traditional maximal

. *wi i* 16 *α* (243)

17 Here, if the MMRC is employed as the combiner, the combining weights for DS-CDMA 18 downlink wireless communication system using the WH codes and DS-CDMA downlink 19 wireless communication system with the unified complex Hadamard transform spreading 20 sequences under employment of the GR are different. This is a direct consequence of the

22 For the WH spreading sequences used by the DS-CDMA downlink wireless communication

24 (244)

*WH a i i L K*

 0 2

. <sup>1</sup> cos <sup>4</sup> *<sup>k</sup>*

*P α*

2 2 2 4

*α θθ E σ*

*l li a w*

1 1

0, 0

*l li k*

*N*

*N σ T α P*

2 2 22

*I c la l li k*

1 1

*L K*

0, 0 . <sup>2</sup> *<sup>H</sup> MP k*

0 2

*E α*

1 1

<sup>1</sup> <sup>4</sup> 2 *<sup>k</sup>*

0, 0

*l li k*

22 4

*α E σ*

*la w*

.

*MP I* is denoted by <sup>2</sup>

*H*

$$w\_i^H = \sqrt{\frac{P\_{a\_0} a\_i^2}{\frac{1}{2N} \sum\_{l=0, l \neq i}^{L-1} a\_l^2 \sum\_{k=0}^{K-1} E\_{a\_k}^2 + 4\sigma\_w^4}}. \tag{245}$$

5 After the MMRC scheme, the SINR at the GR output is equal to the sum of all fingers' 6 SINRs, as in (236) or (242). The use of the unified complex Hadamard transform spreading 7 sequences in DS-CDMA downlink wireless communication system employing the GR 8 ensures that the SINR at the GR output is independent of the phase offsets between different 9 paths. The use of the WH real sequences in DS-CDMA downlink wireless communication 10 systems employing the GR causes the SINR at the GR output to be related to the squared 11 cosine of the phase offsets between paths, as seen by comparing (236) and (242). Even 12 though the average SINR per finger at the GR output in the DS-CDMA downlink wireless 13 communication system under employment of the WH codes is the same as that at the GR 14 output in the DS-CDMA downlink wireless communication system under the use of the 15 unified complex Hadamard transform spreading sequences, owing to

$$E\{\cos^2(\theta\_l - \theta\_i)\} = 0.5,\tag{246}$$

the SINR distribution over the random variable cos ( ) <sup>2</sup> *<sup>l</sup> <sup>i</sup>* 17 *θ θ* can cause degradation in BER 18 performance under some conditions. This is analogous to the case of a transmission over flat 19 Rayleigh fading channels: even when the Rayleigh gain has a mean of 1, the performance is 20 far worse than in an AWGN channel.

21 Now, we compare the BER performance of DS-CDMA downlink wireless communication 22 system employing the GR under the use of the unified complex Hadamard transform 23 spreading sequences with that of DS-CDMA downlink wireless communication system 24 employing the GR under the use of the WH real sequences. To analyze the performance of 25 spreading sequences and the diversity-combining schemes considered, we adopt a Gaussian 26 approximation approach based on the central limit theorem [99]. The Gaussian 27 approximation is known not only to give accurate estimations of the probability of error in 28 the region of practical interest, but also to offer insights into the effects of various sequence 29 and system parameters and interference sources on the performance of the GR [1–3]. For 30 simplicity, here we only consider the single finger GR case, i.e., the GR has only one 31 demodulating finger, and the finger is locked onto an arbitrary path, say, the *i*-th multipath 32 component. Under the Gaussian approximation, for the WH spreading sequences a 33 straightforward derivation based on the decision process indicates that the conditional 34 symbol error probability (SEP) for a given

$$\boldsymbol{\theta} = (\theta\_0, \theta\_1, \dots, \theta\_{L-1}) \tag{247}$$

2 takes the following form

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$$\text{SEP}\_{i}^{\text{NH}}(\Theta) = \Phi\{\sqrt{\text{SINR}\_{\text{NH}}^{(i)}}\}\_{f} \tag{248}$$

where (*i*) *WH* 4 *SINR* is as in (236), and

$$\Phi(x) = \frac{1}{\sqrt{2\pi}} \int\_{x}^{x} \exp\left(-\frac{y^2}{2}\right) dy \tag{249}$$

6 is the error integral.

7 Averaging (248) with respect to the associated random variables, the average probability of 8 error may be determined in the following form:

$$\text{SEP}\_{l}^{\text{WH}\_{\text{av}}} = E\_{\theta} \left\{ \text{SEP}\_{l}^{\text{WH}} \left( \theta \right) \right\} \tag{250}$$

10 The averaging may most efficiently be carried out via the Monte Carlo or MatLab 11 techniques.

12 Under employment of the unified complex Hadamard transform spreading sequences in the 13 DS-CDMA downlink wireless communication system using the GR, the Gaussian 14 approximation leads us to the following form of the conditional SEP:

$$SEP\_i^H = \Phi\left(\sqrt{\text{SINR}\_H^{(i)}}\right) \tag{251}$$

where (*i*) *<sup>H</sup>* 16 *SINR* is as in (242). At this point, we should compare (250) and (251). For this 17 purpose, we use a procedure proposed in [99]. If any function *f* (*x*) is the convex function 18 and *X* is the random variable, then Jensen's inequality

$$E\{f(X)\} \ge f\{E\{X\}\}\tag{252}$$

20 is satisfied. To apply Jensen's inequality, first define

$$X = \cos^2(\theta\_l - \theta\_i). \tag{253}$$

1 Then since *θ<sup>i</sup>* , *i* 0,1, 2,, *L* 1 are uniformly distributed within the limits of the interval 2 [0,2*π*) , straightforward calculations give us the following result

$$E\{X\} = 0.5.\tag{254}$$

4 Moreover, the function *f* (*x*) here has the following form:

$$f(\mathbf{x}) = \Phi(\sqrt{\frac{1}{a+bx}}),\tag{255}$$

6 where *a* 0 and *b* 0 . Calculating the second derivative of the function *f* (*x*) with respect 7 to *x*, we found that *f* (*x*) is a convex function if the following condition is satisfied

$$a + bx \le \frac{1}{3} \tag{256}$$

9 is satisfied.

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2 takes the following form

where (*i*)

6 is the error integral.

11 techniques.

where (*i*)

*WH* 4 *SINR* is as in (236), and

144 Contemporary Issues in Wireless Communications

8 error may be determined in the following form:

( , , , ) <sup>0</sup> <sup>1</sup> *<sup>L</sup>*<sup>1</sup> 1 *θ θ θ θ* (247)

( ) ( ) ( ), *WH <sup>i</sup> <sup>i</sup> WH* 3 *SEP θ SINR* (248)

> *x*

5 (249)

7 Averaging (248) with respect to the associated random variables, the average probability of

*<sup>i</sup> av* 9 (250)

10 The averaging may most efficiently be carried out via the Monte Carlo or MatLab

12 Under employment of the unified complex Hadamard transform spreading sequences in the 13 DS-CDMA downlink wireless communication system using the GR, the Gaussian

*i H* 15 *SEP SINR* (251)

*<sup>H</sup>* 16 *SINR* is as in (242). At this point, we should compare (250) and (251). For this 17 purpose, we use a procedure proposed in [99]. If any function *f* (*x*) is the convex function

19 *E*{*f* (*X* )} *f E*{*X*} (252)

<sup>2</sup> cos ( ). *<sup>X</sup> l i* <sup>21</sup> *θ θ* (253)

( ) ( ), *H i*

*WH*

14 approximation leads us to the following form of the conditional SEP:

18 and *X* is the random variable, then Jensen's inequality

20 is satisfied. To apply Jensen's inequality, first define

*SEP E* {*SEP* (*θ*)} *WH θ i*

*<sup>π</sup> <sup>x</sup>* ( ) <sup>2</sup> exp 2 <sup>1</sup> ( )

*dy <sup>y</sup>*

2

10 A sufficient condition

$$E\_{a\_0} \ge \frac{3}{a\_l^2} \sqrt{\frac{1}{N} \sum\_{l=0, l \neq i}^{L-1} a\_l^2 \sum\_{k=0}^{K-1} E\_{a\_k}^2 + 4\sigma\_w^4} \tag{257}$$

12 satisfies the inequality (256) and allows applications of Jensen's inequality successively to 13 each component in *θ* by using (248) and (250). Then we obtain

$$SEP\_i^{\rm NH\_{uv}} \ge \Phi\left(\sqrt{\text{SINR}\_{\rm NH}^{\{i,\rm low\}}}\right) \tag{258}$$

15 where

$$\text{SINR}\_{\text{WH}}^{(i,low)} = \frac{E\_{a\_0} \mathbf{a}\_i^2}{\sqrt{\frac{1}{2N} \sum\_{l=0, l \neq i}^{L-1} \mathbf{a}\_l^2 \sum\_{k=0}^{K-1} E\_{a\_k}^2 + 4 \sigma\_w^4}}. \tag{259}$$

17 Comparing the results of (258) and (259) for the WH spreading sequences employed by the 18 DS-CDMA downlink wireless communication system under the use of the GR and the result 19 of (251) for the unified complex Hadamard transform spreading sequences employed by the 20 DS-CDMA downlink wireless communication system under the GR use, we found that

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$$\text{SINR}^{(i,low)}\_{\text{WH}} = \text{SINR}^{(i)}\_{\text{H}}.\tag{260}$$

2 Thus, (251) is a lower bound on the SEP when the WH spreading sequences are used in the 3 DS-CDMA downlink wireless communication system employing the GR if the condition 4 (257) is satisfied. This result implies that the DS-CDMA downlink wireless communication 5 system employing the GR with the unified complex Hadamard transform spreading 6 sequences is more resistant to MAI in comparison with the WH spreading sequences used 7 by the DS-CDMA downlink wireless communication system employing the GR in the case 8 where the only finger is selected in the GR. For more fingers and various combining 9 schemes, although simple closed-form bounds for the SEP of the DS-CDMA downlink 10 wireless communication system employing the GR with the WH spreading sequences are 11 difficult to obtain, we believe that the same conclusion can be made under some similar 12 conditions obtained by using Jensen's inequality and some intensive calculations. This 13 conclusion can be verified under discussion of numerical simulations made in the next 14 section. Furthermore, in view of (58) and (64)–(67), we observe that the DS-CDMA downlink 15 wireless communication system employing the GR with the unified complex Hadamard 16 transform spreading sequences can achieve high reliable performance at not only high SNR *Eb N*<sup>0</sup> 17 , as in the case of the Rake receiver, but at low SNR , too.

#### 18 **5.5. Simulation results**

19 In this section, we compare the BER performance of the DS-CDMA downlink wireless 20 communication system employing the GR for the cases when the WH spreading sequences 21 and the unified complex Hadamard transform spreading sequences are used under different 22 combining schemes for finger weights such as the traditional equal gain combining (EGC), 23 MRC, and MMRC. Also we present the comparative analysis of BER performance under 24 employment of the GR and Rake receiver by the DS-CDMA downlink wireless 25 communication system. Simulations are performed over the Rayleigh, Ricean, and AWGN 26 channels, respectively.

27 The spreading sequences of length *N* 64 are considered, and powers are chosen as

$$P\_{a\_0} = P\_{a\_1} = \dots = P\_{a\_{K-1}} = 1.\tag{261}$$

29 Unless stated otherwise, the default system under consideration contains *K* 10 active 30 users, and the number of paths *L* 4 in Rayleigh fading multipath channel with

$$E\{\alpha\_l^2\} = 1, \ l = 0, \ 1, \ \dots, \ l - 1,\tag{262}$$

32 and uses the MMRC technique for the GR. In accordance with Section *5.2*, we choose

$$
\mu\_1 = 1, \ \mu\_2 = -j, \quad \text{and} \quad \mu\_3 = j \tag{263}
$$

2 to construct a set of the half-spectrum property unified complex Hadamard transform 3 spreading sequences. Note that a different sequence assigned to the user-of-interest results 4 in different BER; the BER in the following examples is an average over the sequence subset.

#### 5 *5.5.1. Effect of GR finger weights*

17

19

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146 Contemporary Issues in Wireless Communications

18 **5.5. Simulation results**

26 channels, respectively.

(, ) (). *i low <sup>i</sup>*

*WH <sup>H</sup>* 1 *SINR SINR* (260)

2 Thus, (251) is a lower bound on the SEP when the WH spreading sequences are used in the 3 DS-CDMA downlink wireless communication system employing the GR if the condition 4 (257) is satisfied. This result implies that the DS-CDMA downlink wireless communication 5 system employing the GR with the unified complex Hadamard transform spreading 6 sequences is more resistant to MAI in comparison with the WH spreading sequences used 7 by the DS-CDMA downlink wireless communication system employing the GR in the case 8 where the only finger is selected in the GR. For more fingers and various combining 9 schemes, although simple closed-form bounds for the SEP of the DS-CDMA downlink 10 wireless communication system employing the GR with the WH spreading sequences are 11 difficult to obtain, we believe that the same conclusion can be made under some similar 12 conditions obtained by using Jensen's inequality and some intensive calculations. This 13 conclusion can be verified under discussion of numerical simulations made in the next 14 section. Furthermore, in view of (58) and (64)–(67), we observe that the DS-CDMA downlink 15 wireless communication system employing the GR with the unified complex Hadamard 16 transform spreading sequences can achieve high reliable performance at not only high SNR

19 In this section, we compare the BER performance of the DS-CDMA downlink wireless 20 communication system employing the GR for the cases when the WH spreading sequences 21 and the unified complex Hadamard transform spreading sequences are used under different 22 combining schemes for finger weights such as the traditional equal gain combining (EGC), 23 MRC, and MMRC. Also we present the comparative analysis of BER performance under 24 employment of the GR and Rake receiver by the DS-CDMA downlink wireless 25 communication system. Simulations are performed over the Rayleigh, Ricean, and AWGN

27 The spreading sequences of length *N* 64 are considered, and powers are chosen as

30 users, and the number of paths *L* 4 in Rayleigh fading multipath channel with

32 and uses the MMRC technique for the GR. In accordance with Section *5.2*, we choose

01 1 1. *<sup>K</sup> aa a PP P* 28 (261)

29 Unless stated otherwise, the default system under consideration contains *K* 10 active

<sup>2</sup> { } 1, 0, 1, , 1, *<sup>l</sup>* <sup>31</sup>*<sup>E</sup> <sup>α</sup> l L* (262)

*Eb N*<sup>0</sup> 17 , as in the case of the Rake receiver, but at low SNR , too.

6 Figure 12 presents different combining techniques in the GR, namely, the traditional EGC, 7 MRC, and MMRC. Additionally, a comparison of the GR with the Rake receiver is made. 8 From presented simulation results it is evident that the GR with MMRC technique has a 9 great superiority over EGC and MRC techniques, especially in the high SNR region. This 10 phenomenon can be explained by the GR finger weights in (244) and (245), where the effect 11 of interference is taken into consideration under MMRC technique, but it is not considered 12 under the traditional EGC and MRC techniques. Additionally, Fig.12 demonstrates a great 13 superiority under employment of the GR by the DS-CDMA downlink wireless 14 communication system with the unified complex Hadamard transform spreading sequences 15 and WH spreading sequences over the implementation of the Rake receiver in the DS-16 CDMA downlink wireless communication system under the same conditions.

**Figure 12.** BER performance as a function of SNR *Eb N*<sup>0</sup> 20 with different combining techniques in GR 21 over Rayleigh fading channel (*K* 10, *L* 4) .

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