16 **3. Signal processing with optimal combining and partial cancellation**

#### 17 **3.1. Brief review**

6

10 BookTitle

M

2

2 <sup>1</sup> ( )

1

M

7 we obtain that

12 form:

*i*

 

0

Introducing the variable *<sup>i</sup> <sup>w</sup> <sup>i</sup> g l z*<sup>1</sup> <sup>2</sup> (0.5 ) 3

0

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

 

*w*

2

*w*

2

*w*

*w*

88 Contemporary Issues in Wireless Communications

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

exp( ) exp

*i i*

2

*w i*

> 1 2

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

0.5

 

1

*z*

2 1

*<sup>i</sup> <sup>i</sup> <sup>i</sup> <sup>z</sup> <sup>l</sup>*

*Var z z z <sup>i</sup>*

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

0

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

1

2

*<sup>l</sup> wl wl <sup>z</sup> <sup>i</sup>* <sup>10</sup>M

*z w w i i wi*

*l l g g l dg*

2 21 2

4 (18)

[ ] []

0.5 1

[ ]

  1 2

<sup>1</sup> <sup>1</sup> <sup>1</sup> { } { ( ) }

*<sup>z</sup> dz <sup>z</sup> <sup>l</sup> <sup>z</sup> dz <sup>z</sup> <sup>l</sup> lz <sup>z</sup> <sup>i</sup>*

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

*i*

, we can obtain:

0.5

2 21 2 1

. <sup>2</sup>

1 1 2

(17)

1

*w*

exp

0.5 <sup>1</sup>

*i*

 

E[ ] <sup>3</sup> <sup>2</sup> . ( )

 

*i w w w*

1

(21)

2 1

1 1 2 2

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

( ) exp( ) ,

exp( ) (0.5) . *i ii g g dg*

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

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

2

13 (22)

<sup>2</sup> <sup>2</sup> [ ]

2 1

<sup>1</sup> 2

1

( ) E[ ] , *<sup>i</sup> <sup>z</sup> i w l*

 M

[ ] E[ ] E[ ] <sup>2</sup> <sup>4</sup> <sup>4</sup> <sup>4</sup>

*l*

2

14 (23)

M

*z*

*l*

*l*

0

0

*l*

8 (20)

0 <sup>1</sup> *<sup>x</sup> <sup>l</sup> <sup>l</sup> dl <sup>x</sup>* 6 (19)

0 0

*w w i ii w i ii*

 

 

0

*w*

0.5 0.5

*l g g dg l g g dg*

18 In this section, we investigate the generalized receiver (GR) under the quadrature subbranch 19 hybrid selection/maximal-ratio combining (HS/MRC) for 1-D modulations in multipath 20 fading channel and compare its symbol error rate (SER) performance with that of the 21 traditional HS/MRC scheme discussed in [29,30]. It is well known that the HS/MRC receiver 22 selects the *L* strongest signals from *N* available diversity branches and coherently combines 12 BookTitle

1 them. In traditional HS/MRC scheme, the strongest *L* signals are selected according to 2 signal-envelope amplitude [29–35]. However, some receiver implementations recover 3 directly the in-phase and quadrature components of the received branch signals. 4 Furthermore, the optimal maximum likelihood estimation (MLE) of the phase of a diversity 5 branch signal is implemented by first estimating the in-phase and quadrature branch signal 6 components and obtaining the signal phase as a derived quantity [36,37]. Other channel-7 estimation procedures also operate by first estimating the in-phase and quadrature branch 8 signal components [38–41]. Thus, rather than *N* available signals, there are 2*N* available 9 quadrature branch signal components for combining. In general, the largest 2*L* of these 2*N* 10 quadrature branch signal components will not be the same as the 2*L* quadrature branch 11 signal components of the *L* branch signals having the largest signal envelopes.

12 In this section, we investigate how much improvement in performance can be achieved 13 employing the GR with modified HS/MRC, namely, with the quadrature subbranch 14 HS/MRC and HS/MRC schemes, instead of the conventional HS/MRC combining scheme for 15 1-D signal modulations in multipath fading channel. At GR discussed in [42], the *N* 16 diversity branches are split into 2*N* in-phase and quadrature subbranches. Then the GR with 17 HS/MRC scheme is applied to these 2*N* subbranches. Obtained results show the better 18 performance is achieved by this quadrature subbranch HS/MRC scheme in comparison with 19 the traditional HS/MRC scheme for the same value of average signal-to-noise ratio (SNR) 20 per diversity branch.

21 Another problem discussed is the problem of partial cancellation factor (PCF) in DS-CDMA 22 wireless communication system with multipath fading channel. It is well known that the 23 multiple access interference (MAI) can be efficiently estimated by the partial parallel 24 interference cancellation (PPIC) procedure and then partially be cancelled out of the 25 received signal on a stage-by-stage basis for DS-CDMA wireless communication system 26 [43]. To ensure a high-quality performance, PCF for each PPIC stage needs to be chosen 27 appropriately, where the PCF should be increased as the reliability of the MAI estimates 28 improves. There are some papers on the selection of the PCF for a receiver based on the 29 PPIC. In [44–46], formulas for the optimal PCF were derived through straightforward 30 analysis based on soft decisions. In contrast, it is very difficult to obtain the optimal PCF for 31 a receiver based on PPIC with hard decisions owing to their nonlinear character. One 32 common approach to solve the nonlinear problem is to select an arbitrary PCF for the first 33 stage and then increase the value for each successive stage, since the MAI estimates may 34 become more reliable in later stages [43, 47, 48]. This approach is simple, but it might not 35 provide satisfactory performance.

36 In this section, we use the Price's theorem [49, 50] to derive a range of the optimal PCF for 37 the first stage in PPIC of DS-CDMA wireless communication system with multipath fading 38 channel employing GR based on GASP [1–3], where the lower and upper boundary values 39 of the PCF can be explicitly calculated from the processing gain and the number of users of 40 DS-CDMA wireless communication system in the case of periodic code scenario. Computer 1 simulation shows that, using the average of these two boundary values as the PCF for the 2 first stage, we are able to reach the bit error rate (BER) performance that is very close to the 3 potentially achieved one [51] and surpasses the BER performance of the real PCF for DS-4 CDMA wireless communication systems discussed, for example, in [43].

5 With this result, a reasonable PCF can quickly be determined without using any time-6 consuming Monte Carlo simulations. It is worth mentioning that the two-stage GR 7 considered in [52] based on the PPIC using the proposed PCF at the first stage achieves the 8 BER performance comparable to that of the three-stage GR based on the PPIC using an 9 arbitrary PCF at the first stage. In other words, at the same BER performance, the proposed 10 approach for selecting the PCF can reduce the GR complexity based on the PPIC. The PCF 11 selection approach is derived for multipath fading channel cases discussed in [42, 53].

12 In this section, we describe the multipath fading channel model and provide system models 13 for selection/maximal ratio combining and synchronous DS-CDMA wireless communication 14 systems; carry out the performance analysis obtaining a symbol error rate expression in the 15 closed-form and define a marginal moment generating function of SNR per symbol of a 16 single quadrature branch; determine the lower and upper PCF bounds based on the 17 processing gain *N* and the number of users *K* under multipath fading channel model in DS-18 CDMA wireless communication systems employing GR; discuss simulation results; and 19 make some conclusions.
