26 **4.2. Problem statement and system model**

29

We consider a MIMO DS-CDMA wireless communication system with *NT* 27 transmit and *NR* 28 receive antennas. The modulated symbols *b*[*k*] are taken from a scalar symbol alphabet F such as quadriphase-shift keying (QPSK) or quadrature amplitude modulation (QAM), 30 and have the following variance

$$
\sigma\_b^2 = E\{\|\mathcal{U}[k]\|^2\} = 1,\tag{139}
$$

32 where *E*{} denotes the mathematical expectation. The coefficients of the FIR beamforming filters of length *Lg* at the transmit antenna *<sup>t</sup> n* , *<sup>t</sup> NT* 1 *n* are denoted by *g* [*k*] *<sup>t</sup> <sup>n</sup>* 33 , where 0 1 *Lg* 34 *k* and their energy is normalized to

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 39 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 117

$$\sum\_{n\_r=1}^{N\_T} \sum\_{k=0}^{L\_g-1} \|g\_{n\_r}[k]\|^2 = 1. \tag{140}$$

The signal transmitted over antenna *<sup>t</sup>* 2 *n* at time *k* is given by

38 BookTitle

1 results show that for typical Global System for Mobile Communications/Enhanced Data 2 Rates for GSM Evolution (GSM/EDGE) channels the performance of short GR FIR 3 beamforming filters can be closely approached with infinite impulse response beamforming

6 Under consideration of channel estimation and correlation part, we investigate the 7 minimum mean square error (MMSE) GR. It takes an error of maximum likelihood (ML) 8 multiple-input multiple-output (MIMO) channel estimation and GR spatially correlation 9 into consideration in the computation of MMSE GR and log-likelihood ratio (LLR) of each

11 It is well-known [68–70] that the existing soft-output MMSE vertical Bell Lab Space 12 Time (V-BLAST) detectors have been designed based on perfect channel estimation. 13 Unfortunately, the estimated MIMO channel coefficient matrix is noisy and imperfect in 14 practical application environment, [71,72]. Therefore, these soft-output MMSE V-BLAST 15 detectors will suffer from performance degradation under practical channel estimation.

16 The ML symbol detection scheme is investigated in [72]. It takes into consideration the 17 channel estimation error under condition that the MIMO channel estimation is 18 imperfect. MMSE based on V-BLAST symbol detection algorithm addressing the impact 19 of the channel estimation error is discussed in [73]. However, the channel coding, 20 decision error propagation compensation, and spatially channel correlation are not

22 In the present section, we derive the MMSE GR for detecting each transmitted symbol and 23 provide a method to compute the LLR of each coded bit. When compared with the detection 24 scheme discussed in [73], our simulation results show that the MMSE GR can obtain sizable

We consider a MIMO DS-CDMA wireless communication system with *NT* 27 transmit and *NR* 28 receive antennas. The modulated symbols *b*[*k*] are taken from a scalar symbol alphabet

2 2 { } | | [ ] 1, *<sup>b</sup>* <sup>31</sup>*<sup>σ</sup> E bk* (139)

32 where *E*{} denotes the mathematical expectation. The coefficients of the FIR beamforming filters of length *Lg* at the transmit antenna *<sup>t</sup> n* , *<sup>t</sup> NT* 1 *n* are denoted by *g* [*k*] *<sup>t</sup> <sup>n</sup>* 33 , where

such as quadriphase-shift keying (QPSK) or quadrature amplitude modulation (QAM),

4 at the GR and significant gains over single antenna transmission can be achieved.

10 coded bit. Our GR analysis and investigation are based on the following statements:

5 *4.1.2. Channel estimation and spatial correlation*

116 Contemporary Issues in Wireless Communications

21 discussed and considered.

26 **4.2. Problem statement and system model**

0 1 *Lg* 34 *k* and their energy is normalized to

30 and have the following variance

25 performance gain.

29

F

$$a\_{n\_i}[k] = g\_{n\_i}[k] \otimes b[k].\tag{141}$$

4 where denotes a linear discrete-time convolution. The discrete-time received signal at the receive antenna *<sup>r</sup> n* , *<sup>r</sup> NR* 5 1 *n* can be modeled in the following manner

$$s\_{n\_r}[k] = \sum\_{n\_l=1}^{N\_T} h\_{n\_l n\_r}[k] \otimes a\_{n\_l}[k] + w\_{n\_r}[k] \tag{142}$$

where *w* [*k*] *<sup>r</sup> <sup>n</sup>* 7 denotes the spatially and temporally AWGN with zero mean and variance 8 given by

$$
\sigma\_w^2 = E\{\|\pi v\_{n\_r} \|k\|^2\} = 0.5N\_{0^\circ} \tag{143}
$$

where <sup>0</sup> 10 0.5*N* denotes the two-sided power spectral density of the underlying continuous in 11 time passband noise process.

The notation *h* [*k*] *<sup>t</sup> <sup>r</sup> <sup>n</sup> <sup>n</sup>* 12 , where 0 *k L* 1 , represents the overall channel impulse response between the transmit antenna *<sup>t</sup> n* and the receive antenna *<sup>r</sup>* 13 *n* of length *L*. In our model, *h* [*k*] *<sup>t</sup> <sup>r</sup> <sup>n</sup> <sup>n</sup>* 14 contains the combined effects of transmit pulse shaping, wireless channel, receive 15 filtering, and sampling. We assume an existence of block fading model, i.e., the channel is 16 constant for the duration of at least one data burst before it changes independently to a new realization. In general, *h* [*k*] *<sup>t</sup> <sup>r</sup> <sup>n</sup> <sup>n</sup>* 17 are spatially and temporally correlated because of 18 insufficient antenna spacing and transmit/receive filtering, respectively. Substituting (141) 19 into (142), we obtain

$$s\_{n\_r}[k] = h\_{n\_r}^{eq}[k] \otimes b[k] + w\_{n\_r}[k].\tag{144}$$

where the equivalent channel impulse response *h* [*k*] *eq nr* 21 corresponding to the receiving antenna *nr* 22 is defined as

$$h\_{n\_r}^{eq}[k] = \sum\_{n\_l=1}^{N\_T} h\_{n\_l n\_r}[k] \otimes \mathbf{g}\_{n\_l}[k] \tag{145}$$

40 BookTitle

and has the length 1 *Leq L Lg* 1 . Equation (144) shows that the MIMO DS-CDMA wireless 2 communication system with beamforming can be modeled as an equivalent single-input 3 multiple-output (SIMO) system. Therefore, the GR can use the same equalization, channel 4 estimation, and channel tracking techniques as for a single antenna transmission. We 5 assume that the GR employs receive diversity zero forcing or MMSE linear equalization [74].

6 Let us rewrite the main statements and definitions mentioned above in the matrix form for 7 our convenience in subsequent analysis of channel estimation. Thus, the received signal can 8 be expressed in the following form:

$$\mathbf{s} = \mathbf{H}\mathbf{a} + \mathbf{w} = \sum\_{k=1}^{N\_T} \mathbf{h}\_k a\_k + \mathbf{w}\_\prime \tag{146}$$

where *<sup>T</sup> NR* [*<sup>s</sup>* ,*<sup>s</sup>* , ,*<sup>s</sup>* ] **<sup>s</sup>** <sup>1</sup> <sup>2</sup> is the received signal vector; *<sup>T</sup> NT* [ , , , ] **H h**<sup>1</sup> **h**<sup>2</sup> **h** is the *NR NT* <sup>10</sup> MIMO channel coefficient matrix with elements *h* [*k*] *<sup>t</sup> <sup>r</sup> <sup>n</sup> <sup>n</sup>* 11 denoting the channel fading coefficient between the *<sup>t</sup> n* -th transmit antenna and the *<sup>r</sup>* 12 *n* -th receive antenna.

13 We adopt the following GR spatially correlated MIMO channel model

$$\mathbf{H} = \sqrt{\mathbf{R}\_\mathbf{r}} \,\mathbf{H}\_\mathbf{w} \tag{147}$$

with **H** *<sup>w</sup>* 15 denoting an independent and identically distributed (i.i.d.) matrix with entries obeying the Gaussian law with zero mean and unit variance, and **Rr** is the *NR NR* 16 receive 17 array correlation matrix determined by

$$\mathbf{R}\_{\mathbf{r}} = \sqrt{\mathbf{R}\_{\mathbf{r}}} \left(\sqrt{\mathbf{R}\_{\mathbf{r}}}\right)^{H}.\tag{148}$$

19 Then, we have

$$E\{\mathbf{H}\mathbf{H}^{H}\} = \mathcal{N}\_{\mathrm{T}}\mathbf{R}\_{\mathrm{r}}.\tag{149}$$

The channel is considered to be flat fading with coherence time of ( ) *NP N <sup>D</sup>* 21 MIMO vector symbols, where *NP* symbol intervals are dedicated to pilot matrix **S** *<sup>p</sup>* 22 and the remaining *<sup>N</sup> <sup>D</sup>* to data transmission, where *<sup>T</sup> NT* [*a* ,*a* , ,*a* ] 23 **a** <sup>1</sup> <sup>2</sup> is the transmitted complex signal 24 vector whose element given by (141) is taken from the complex modulation constellation F 25 , because the modulated symbols *b*[*k*] are taken from the scalar symbol alphabet F , such as QPSK signal, by mapping the coefficient of FIR beamforming filters *g* [*k*] *<sup>t</sup> <sup>n</sup>* 26 like the coded 27 bit vector

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 41 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 119

$$\mathbf{g}\_{n\_l} = \begin{bmatrix} \mathbf{g}\_{n\_l}^1, \mathbf{g}\_{n\_l}^2, \dots, \mathbf{g}\_{n\_l}^{\log 2} \,^M \end{bmatrix}^T \tag{150}$$

to one modulation symbol belonging to F , i.e., map( ) F *<sup>t</sup> <sup>t</sup> <sup>n</sup> <sup>n</sup>* 2 *a* **g** .

3 Meanwhile, we assume that each transmitted symbol is independently taken from the same 4 modulation constellation Fand has the same average energy, i.e.,

$$E\{\mathbf{a}\mathbf{a}^H\} = E\_b \mathbf{I}\_{\mathcal{N}\_T}.\tag{151}$$

Finally, *<sup>T</sup> NR* [*<sup>w</sup> <sup>w</sup>* , ,*<sup>n</sup>* ] <sup>1</sup> <sup>2</sup> <sup>6</sup>**<sup>w</sup>** , is the AWGN vector with covariance matrix determined by

$$\mathbf{K}\_{\mathbf{w}} = E\{\mathbf{w}\mathbf{w}^{H}\} = \sigma\_{w}^{2} \mathbf{I}\_{N\_{R}}.\tag{152}$$

*NT* **I** and *NR* 8 **I** are the identity matrices.

40 BookTitle

8 be expressed in the following form:

118 Contemporary Issues in Wireless Communications

where *<sup>T</sup>*

17 array correlation matrix determined by

19 Then, we have

27 bit vector

and has the length 1 *Leq L Lg* 1 . Equation (144) shows that the MIMO DS-CDMA wireless 2 communication system with beamforming can be modeled as an equivalent single-input 3 multiple-output (SIMO) system. Therefore, the GR can use the same equalization, channel 4 estimation, and channel tracking techniques as for a single antenna transmission. We 5 assume that the GR employs receive diversity zero forcing or MMSE linear equalization [74].

6 Let us rewrite the main statements and definitions mentioned above in the matrix form for 7 our convenience in subsequent analysis of channel estimation. Thus, the received signal can

<sup>9</sup>**s Ha w h w** (146)

*NR* [*<sup>s</sup>* ,*<sup>s</sup>* , ,*<sup>s</sup>* ] **<sup>s</sup>** <sup>1</sup> <sup>2</sup> is the received signal vector; *<sup>T</sup> NT* [ , , , ] **H h**<sup>1</sup> **h**<sup>2</sup> **h** is the *NR NT* <sup>10</sup> MIMO channel coefficient matrix with elements *h* [*k*] *<sup>t</sup> <sup>r</sup> <sup>n</sup> <sup>n</sup>* 11 denoting the channel fading

**H Rr H***<sup>w</sup>* 14 (147)

with **H** *<sup>w</sup>* 15 denoting an independent and identically distributed (i.i.d.) matrix with entries obeying the Gaussian law with zero mean and unit variance, and **Rr** is the *NR NR* 16 receive

( ) . *<sup>H</sup>* **R RR r rr** 18 (148)

The channel is considered to be flat fading with coherence time of ( ) *NP N <sup>D</sup>* 21 MIMO vector symbols, where *NP* symbol intervals are dedicated to pilot matrix **S** *<sup>p</sup>* 22 and the remaining

*NT* [*a* ,*a* , ,*a* ] 23 **a** <sup>1</sup> <sup>2</sup> is the transmitted complex signal 24 vector whose element given by (141) is taken from the complex modulation constellation

as QPSK signal, by mapping the coefficient of FIR beamforming filters *g* [*k*] *<sup>t</sup> <sup>n</sup>* 26 like the coded

25 , because the modulated symbols *b*[*k*] are taken from the scalar symbol alphabet

{ } . *<sup>H</sup> <sup>T</sup> E N* **HH R <sup>r</sup>** 20 (149)

F

F, such

coefficient between the *<sup>t</sup> n* -th transmit antenna and the *<sup>r</sup>* 12 *n* -th receive antenna.

13 We adopt the following GR spatially correlated MIMO channel model

*<sup>N</sup> <sup>D</sup>* to data transmission, where *<sup>T</sup>*

1

*k k*

*a*

*NT*

*k*

,

#### 9 **4.3. FIR beamforming for GR with linear equalization**

10 According to [75], the unbiased *SNR* for linear equalization with the optimum infinite 11 impulse response equalizer filters at the GR back end is given by

$$\text{SNR(g)} = \frac{\sigma\_b^4}{\sigma\_\epsilon^4} - \chi\_\epsilon \tag{153}$$

where <sup>2</sup> *<sup>b</sup> σ* is given by (139) and the linear equalization error variance <sup>2</sup> *<sup>e</sup>* 13 *σ* will be defined 14 below. We note that the assumption of infinite impulse response linear equalization filters at 15 the GR back end is not a major restriction, since typically FIR linear equalization filters of length equal to *LF Leq* 16 4 can approach closely the performance of infinite impulse response 17 linear equalization filters. In (153) we consider the constant *χ* 0 for the case of zero forcing 18 linear equalization and *χ* 1 for the case of MMSE linear equalization, respectively [74]. In 19 (153) the beamforming filter vector

$$\mathbf{g} = \begin{bmatrix} \mathbf{g}\_1(\mathbf{0})\mathbf{g}\_1(\mathbf{l}) \cdots \mathbf{g}\_1(L\_g - \mathbf{l})\mathbf{g}\_2(\mathbf{0}) \cdots \mathbf{g}\_{N\_T}(L\_g - \mathbf{l}) \end{bmatrix}^T \tag{154}$$

21 consists of the coefficients of all beamforming filters.

22 The GR linear equalization error variance defined based on results discussed in [76] is given 23 by

42 BookTitle

$$
\sigma\_{\epsilon}^{4} = 4 \sigma\_{w}^{4} \int\_{-0.5}^{0.5} \frac{1}{\sum\_{n\_{r}=1}^{N\_{R}} |H\_{n\_{r}}^{eq}(f)|^{2} + \mu} df\_{\prime} \tag{155}
$$

where <sup>4</sup> <sup>4</sup> 0 and 4 *<sup>w</sup> <sup>b</sup>* 2 *μ μ σ σ* are valid for the case of zero-forcing linear equalization and for 3 the case of MMSE linear equalization, respectively. Furthermore, the frequency response *H* ( *f* ) {*h* (*k*)} *eq n eq nr <sup>r</sup>* <sup>4</sup> Gof the equivalent channel can be defined in the following form

$$H\_{n\_r}^{\*q}(f) = \mathbf{q}^H(f)\mathbf{H}\_{n\_r}\mathbf{g}\_{r'} \tag{156}$$

6 where the subscript *H* means the Hermitian transpose,

$$\mathbf{q}(f) = \left\{ 1 \, \exp(j2\pi f) \cdots \exp\left[j2\pi f (L\_{eq} - 1)\right] \right\}^{\mathrm{T}},\tag{157}$$

$$\mathbf{H}\_{n\_r} = \begin{bmatrix} \mathbf{H}\_{1n\_r} \mathbf{H}\_{2n\_r} \cdots \mathbf{H}\_{N\_TR\_r} \end{bmatrix}^T \tag{158}$$

and *<sup>t</sup> <sup>r</sup>* **<sup>H</sup>***<sup>n</sup> <sup>n</sup>* is a *Leq Lg* column-circulant matrix with the vector *<sup>T</sup> <sup>T</sup> ntnr ntnr Lg* [*<sup>h</sup> <sup>h</sup> <sup>L</sup>* ]1 (0) ( 1) <sup>9</sup> **<sup>0</sup>** 10 in the first column. Therefore, the GR *SNR* with the zero-forcing linear equalization and 11 MMSE linear equalization can be presented in the following form:

$$\text{SNR}(\mathbf{g}) = \frac{\sigma\_b^4}{4\sigma\_w^4 \int\_{-0.5}^{0.5} \frac{1}{\mathbf{g}^H \mathbf{G}(f)\mathbf{g} + \zeta} df} - \chi \tag{159}$$

with the *NT Lg NT Lg* 13 matrix

$$\mathbf{G}(f) = \sum\_{n\_r=1}^{N\_R} \mathbf{H}\_{n\_r}^H \mathbf{d}(f) \mathbf{d}^H(f) \mathbf{H}\_{n\_r} \,. \tag{160}$$

The optimum beamforming filter vector *opt* 15 **g** shall maximize *SNR*(**g**) subject to power constraint **g g** 1 *<sup>H</sup>* 16 . Unfortunately, this optimization problem is not convex, i.e. the standard 17 tools from convex optimization cannot be applied. Nevertheless, the Lagrangian of the 18 optimization problem can be formulated in the following form:

$$L(\mathbf{g}) = \text{SNR}(\mathbf{g}) + \mu \mathbf{g}^H \mathbf{g},\tag{161}$$

where *μ* denotes the Lagrange multiplier. The optimum vector *opt* 20 **g** has to satisfy the 21 following equality

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 43 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 121

$$\frac{\partial \mathbf{L(g)}}{\partial \mathbf{g}} = \mathbf{0}\_{N\_{\rm I} L\_{\rm g}} \, \tag{162}$$

2 which leads to the nonlinear eigenvalue problem, namely,

42 BookTitle

*eq nr <sup>r</sup>* <sup>4</sup>

*H* ( *f* ) {*h* (*k*)} *eq n*

G

120 Contemporary Issues in Wireless Communications

with the *NT Lg NT Lg* 13 matrix

21 following equality

0.5

0.5 1

1 (155)

where <sup>4</sup> <sup>4</sup> 0 and 4 *<sup>w</sup> <sup>b</sup>* 2 *μ μ σ σ* are valid for the case of zero-forcing linear equalization and for 3 the case of MMSE linear equalization, respectively. Furthermore, the frequency response

*e w N eq*

<sup>1</sup> 4 , | | ( ) *<sup>R</sup> r r*

of the equivalent channel can be defined in the following form

*n n σ σ df H f μ*

() () , *r r eq <sup>H</sup> Hf f <sup>n</sup> <sup>n</sup>* <sup>5</sup> **q Hg** (156)

( ) 1 exp( 2 ) exp[ 2 ( 1)] , { }*<sup>T</sup> eq* 7 **q** *f j πf j πf L* (157)

*nr nr nr NT nr* [ ] 8 **H H**<sup>1</sup> **H**<sup>2</sup> **H** (158)

*σ*

4

*w H*

<sup>1</sup> <sup>4</sup>

0.5

1

The optimum beamforming filter vector *opt* 15 **g** shall maximize *SNR*(**g**) subject to power constraint **g g** 1 *<sup>H</sup>* 16 . Unfortunately, this optimization problem is not convex, i.e. the standard 17 tools from convex optimization cannot be applied. Nevertheless, the Lagrangian of the

() () , *<sup>H</sup>* 19 *L SNR* **g g gg** *μ* (161)

where *μ* denotes the Lagrange multiplier. The optimum vector *opt* 20 **g** has to satisfy the

*r*

*n*

*N*

() () () . *<sup>R</sup>*

*f ff* <sup>14</sup>**G Hd d H** (160)

*H H n n*

 

0.5

12 **g** (159)

*<sup>σ</sup> SNR*

( )

and *<sup>t</sup> <sup>r</sup>* **<sup>H</sup>***<sup>n</sup> <sup>n</sup>* is a *Leq Lg* column-circulant matrix with the vector *<sup>T</sup> <sup>T</sup> ntnr ntnr Lg* [*<sup>h</sup> <sup>h</sup> <sup>L</sup>* ]1 (0) ( 1) <sup>9</sup> **<sup>0</sup>** 10 in the first column. Therefore, the GR *SNR* with the zero-forcing linear equalization and

4

( )

*r r*

**g G g ζ**

2

*T*

*df <sup>f</sup>*

*<sup>b</sup>*

*χ*

4 4

6 where the subscript *H* means the Hermitian transpose,

11 MMSE linear equalization can be presented in the following form:

18 optimization problem can be formulated in the following form:

$$\left[\int\_{-0.5}^{0.5} \frac{\mathbf{G}(f)}{\left[\mathbf{g}\_{opt}^{H}\mathbf{G}(f)\mathbf{g}\_{opt} + \mathbf{J}\right]^{2}} df \right] \mathbf{g}\_{opt} = \bar{\mu}\mathbf{g}\_{opt} \tag{163}$$

with the eigenvalue *μ* <sup>~</sup> 4 . Unfortunately, (163) does not seem to have a closed-form solution.

5 Therefore, we use the following gradient algorithm for calculation of the optimum FIR 6 beamforming filters at the GR, which recursively improves an initial beamforming filter vector <sup>0</sup> 7 **g** . The main statements of the gradient algorithm are:


$$\bar{\mathbf{g}}\_{i+1} = \mathbf{g}\_i + \delta \left[ \int\_{-0.5}^{0.5} \frac{\mathbf{G}(f)}{\left[ \mathbf{g}\_i^H \mathbf{G}(f) \mathbf{g}\_i + \mathbf{J} \right]^2} df \right] \mathbf{g}\_{i\prime} \tag{164}$$

where *<sup>i</sup>* 12 *δ* is a suitable adaptation step size.

13 3. Normalize the beamforming filter vector

$$\mathbf{g}\_{i+1} = \frac{\tilde{\mathbf{g}}\_{i+1}}{\sqrt{\tilde{\mathbf{g}}\_{i+1}^{H} \tilde{\mathbf{g}}\_{i+1}}}.\tag{165}$$


17 For the termination constant *ε* in Step 4 a suitably small value should be chosen, e.g. <sup>4</sup> <sup>10</sup> *<sup>ε</sup>* . Ideally the adaptation step size *<sup>i</sup>* 18 *δ* should be optimized to maximize the speed of convergence. Here, we follow [77] and choose *<sup>i</sup> <sup>δ</sup>* proportional to <sup>1</sup> *<sup>i</sup> λ* , where *<sup>i</sup>* 19 *λ* is the 20 maximal eigenvalue of the matrix

$$\left[\int\_{-0.9}^{0.5} \frac{\mathbf{G}(f)}{[\mathbf{g}\_i^H \mathbf{G}(f) \mathbf{g}\_i + \mathbf{J}\_i]^2} df\right] \tag{166}$$

in iteration *i*. In particular, we found empirically that <sup>1</sup> 0.01 *<sup>i</sup> <sup>i</sup>* 22 *δ λ* is a good choice. 44 BookTitle

1 Because of non-convexity of the underlying optimization problem, we cannot guarantee that 2 the gradient algorithm converges to the global maximum. However, adopting the 3 initialization procedure explained below, the solution found by this gradient algorithm seems to be close to optimum, i.e., if *Lg* 4 is chosen sufficiently large the FIR beamforming 5 filters obtained with the gradient algorithm approach and the performance of the optimal 6 infinite impulse response beamforming filters at the GR was discussed in [52].

7 We found empirically that a convergence to the optimum or a close to optimum solution is 8 achieved if the beamforming filter length is gradually increased. If the desired beamforming filter length is *Lg* , the gradient algorithm is executed *Lg* 9 times. The beamforming filter vector is initialized with the normalized all-ones vector of size *NT* 10 for the first execution (*υ* 1) of the gradient algorithm. For the *υ* -th execution, *Lg* 11 2 *υ* , the first (*υ* 1) 12 beamforming filters coefficients of each antenna are initialized with the optimum 13 beamforming filter coefficients for that antenna found in the (*υ* 1) -th execution of the 14 gradient algorithm and the *υ* -th coefficients are initialized with zero. In each execution step 15 *υ* , the algorithm requires typically less than 50 iterations to converge, i.e., the overall complexity of the algorithm are on the order of 50 *Lg* 16 .

#### 17 **4.4. MMSE GR**

#### 18 *4.4.1. Channel estimation*

19 It was proved that for ML MIMO channel estimator the optimal pilot matrix minimizing the 20 mean square estimation error is an orthogonal pilot matrix [71, 72]. Under the use of the 21 pilot matrix, i.e.,

$$\mathbf{S}\_p \mathbf{S}\_p^H = E\_P N\_P \mathbf{I}\_{N\_T \prime} \tag{167}$$

where *NP NT* and *EP* 23 is the energy of each pilot symbol, the estimated channel matrix 24 can be expressed as [71, 72] **H**ˆ **H H** , where

$$
\Delta \mathbf{H} = \mathbf{w} \mathbf{S}\_p^H \left( E\_P N\_P \right)^{-1} \tag{168}
$$

26 is the channel estimation error matrix, which is correlated with the matrix **H** and with 27 entries subjected to Gaussian distribution with zero mean and variance

2 2 <sup>1</sup> ( ), *h w PP σ σ E N* 28 (169)

29 which is determined independently of instantaneous channel realization. We can conclude that **H**ˆ 30 is a complex Gaussian matrix with zero mean and covariance matrix

Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems 45 Signal Processing by Generalized Receiver in DS-CDMA Wireless Communications Systems http://dx.doi.org/10.5772/58990 123

$$\text{Cov}[\hat{\mathbf{H}}] = E\{\hat{\mathbf{H}}\hat{\mathbf{H}}^H\} = N\_T \{\mathbf{R}\_\mathbf{r} + \sigma\_{\Lambda\mathbf{h}}^2 \mathbf{I}\_{N\_R}\}. \tag{170}$$

Let **<sup>h</sup>** *<sup>m</sup>* **<sup>h</sup>** *<sup>m</sup>* , and ,( 1,2, , ) <sup>ˆ</sup> **h***<sup>m</sup> m NT* denote the *m*-th column of matrices **H**, **H** and **H**<sup>ˆ</sup> 2 , 3 respectively. Then, by the important properties of complex Gaussian random vector [78] and 4 with some manipulations, we obtain

$$E\{\boldsymbol{\Lambda}\mathbf{h}\_m\mid\boldsymbol{\hat{h}}\_m\}=\sigma\_{\boldsymbol{\Lambda}h}^2\hat{\mathbf{h}}\_m(\mathbf{R}\_\mathbf{r}+\sigma\_{\boldsymbol{\Lambda}h}^2\mathbf{I}\_{N\_R})^{-1};\tag{171}$$

$$\text{Cov}[\Delta \mathbf{h}\_m \Delta \mathbf{h}\_m^H \mid \hat{\mathbf{h}}\_m] = \sigma\_{\Lambda \mathbf{h}}^2 \mathbf{I}\_{N\_R} - \frac{\sigma\_{\Lambda \mathbf{h}}^4}{\mathbf{R}\_\mathbf{r} + \sigma\_{\Lambda \mathbf{h}}^2 \mathbf{I}\_{N\_R}}.\tag{172}$$

#### 7 *4.4.2. Computation of MMSE GR*

44 BookTitle

122 Contemporary Issues in Wireless Communications

17 **4.4. MMSE GR**

21 pilot matrix, i.e.,

18 *4.4.1. Channel estimation*

1 Because of non-convexity of the underlying optimization problem, we cannot guarantee that 2 the gradient algorithm converges to the global maximum. However, adopting the 3 initialization procedure explained below, the solution found by this gradient algorithm seems to be close to optimum, i.e., if *Lg* 4 is chosen sufficiently large the FIR beamforming 5 filters obtained with the gradient algorithm approach and the performance of the optimal

7 We found empirically that a convergence to the optimum or a close to optimum solution is 8 achieved if the beamforming filter length is gradually increased. If the desired beamforming filter length is *Lg* , the gradient algorithm is executed *Lg* 9 times. The beamforming filter vector is initialized with the normalized all-ones vector of size *NT* 10 for the first execution (*υ* 1) of the gradient algorithm. For the *υ* -th execution, *Lg* 11 2 *υ* , the first (*υ* 1) 12 beamforming filters coefficients of each antenna are initialized with the optimum 13 beamforming filter coefficients for that antenna found in the (*υ* 1) -th execution of the 14 gradient algorithm and the *υ* -th coefficients are initialized with zero. In each execution step 15 *υ* , the algorithm requires typically less than 50 iterations to converge, i.e., the overall

19 It was proved that for ML MIMO channel estimator the optimal pilot matrix minimizing the 20 mean square estimation error is an orthogonal pilot matrix [71, 72]. Under the use of the

*H p p P PN* 22 **SS I** *E N* (167)

where *NP NT* and *EP* 23 is the energy of each pilot symbol, the estimated channel matrix

26 is the channel estimation error matrix, which is correlated with the matrix **H** and with

29 which is determined independently of instantaneous channel realization. We can conclude

27 entries subjected to Gaussian distribution with zero mean and variance

that **H**ˆ 30 is a complex Gaussian matrix with zero mean and covariance matrix

 *EPNP <sup>H</sup>* **<sup>H</sup> wS** *<sup>p</sup>* 25 (168)

2 2 <sup>1</sup> ( ), *h w PP σ σ E N* 28 (169)

, *T*

<sup>1</sup> ( )

6 infinite impulse response beamforming filters at the GR was discussed in [52].

complexity of the algorithm are on the order of 50 *Lg* 16 .

24 can be expressed as [71, 72] **H**ˆ **H H** , where

Let {1,2, , } *<sup>i</sup> NT* 8 *k* be the index of *i*-th detected spatial data stream according to the maximal post-detection *SINR* ordering rule. Denote *<sup>a</sup> <sup>j</sup> <sup>μ</sup>* and <sup>2</sup> *<sup>j</sup> <sup>a</sup>* 9 *σ* as the mean and variance of the signal *<sup>j</sup>* 10 *a* , respectively, which can be determined by *a posteriori* symbol probability 11 estimation as in [70]. By performing the soft interference cancellation (SIC) [70] and 12 considering channel estimation error, the corresponding interference-cancelled received signal vector *<sup>i</sup> <sup>k</sup>* **s** <sup>~</sup> 13 can be determined in the following form:

$$\tilde{\mathbf{s}}\_{k\_{\hat{j}}} = \mathbf{H}\mathbf{a} - \sum\_{j=k\_{\hat{1}}}^{k\_{\hat{j}-1}} \hat{\mathbf{h}}\_{j} \boldsymbol{\mu}\_{a\_{\hat{j}}} + \mathbf{w}\_{PF} = \sum\_{j=k\_{\hat{1}}}^{k\_{N\_{\hat{T}}}} (\hat{\mathbf{h}}\_{j} - \Delta \mathbf{h}\_{j}) \boldsymbol{a}\_{j} + \sum\_{j=k\_{\hat{1}}}^{k\_{\hat{i}-1}} \hat{\mathbf{h}}\_{j} (\boldsymbol{a}\_{j} - \boldsymbol{\mu}\_{a\_{\hat{j}}}) - \sum\_{j=k\_{\hat{1}}}^{k\_{\hat{j}-1}} \Delta \mathbf{h}\_{j} \boldsymbol{a}\_{j} + \mathbf{w}\_{PF} \tag{173}$$

where **w***PF* 15 is the noise forming at the PF output of GR front end linear system.

Then, conditionally on **H**ˆ 16 , the MMSE GR output is given as [3, 51]

$$\tilde{\mathbf{Y}}\_{i} = \frac{E\{2a\_{k\_{\hat{l}}}\bar{\mathbf{s}}\_{k\_{\hat{l}}}^{H}|\,\hat{\mathbf{H}}\} - E\{\bar{\mathbf{s}}\_{k\_{\hat{l}}}\bar{\mathbf{s}}\_{k\_{\hat{l}}}^{H}|\,\hat{\mathbf{H}}\} + E\{\mathbf{w}\_{AF\_{\hat{l}}}\mathbf{w}\_{AF\_{\hat{l}}}^{H}\}}{E\{\bar{\mathbf{s}}\_{k\_{\hat{l}}}\bar{\mathbf{s}}\_{k\_{\hat{l}}}^{H}|\,\hat{\mathbf{H}}\}}\tag{174}$$

where **w** *AF* 18 is the reference zero mean Gaussian noise with *a priori* information **a "no"**  19 **signal** and with the following covariance matrix in a general case [1, 3]

$$E\{\mathbf{w}\_{PF}\mathbf{w}\_{PF}^{H}\} = E\{\mathbf{w}\_{AF}\mathbf{w}\_{AF}^{H}\} = \sigma\_w^2 \mathbf{I}\_{N\_R \ \prime} \tag{175}$$

46 BookTitle

1 because the AF and PF of GR front end linear system do not change the statistical 2 parameters of input process (Gaussian noise, for example). Thus, according to (173) and 3 (175), we can write

$$\begin{split} \boldsymbol{E} \{ \bar{\mathbf{s}}\_{k\_{\hat{j}}} \bar{\mathbf{s}}\_{k\_{\hat{j}}}^{H} \| \hat{\mathbf{H}} \} &= \Big\{ \sum\_{j=k\_{\hat{j}}}^{k\_{N\_{\hat{T}}}} \Big[ \hat{\mathbf{h}}\_{j} \hat{\mathbf{h}}\_{j}^{H} - \hat{\mathbf{h}}\_{j} \boldsymbol{E} \{ \Delta \mathbf{h}\_{j}^{H} \, \| \, \hat{\mathbf{H}} \} - \boldsymbol{E} \{ \Delta \mathbf{h}\_{j} \, \| \, \hat{\mathbf{H}} \} \bar{\mathbf{h}}\_{j}^{H} \Big] + \sum\_{j=1}^{k\_{N\_{\hat{T}}}}^{k\_{N\_{\hat{T}}}} \boldsymbol{E} \{ \Delta \mathbf{h}\_{j} \Delta \mathbf{h}\_{j}^{H} \, \| \, \hat{\mathbf{H}} \} \Big\} \boldsymbol{E}\_{b} \\ &+ \sum\_{j=k\_{\hat{l}}}^{k\_{\hat{r}-1}} \Big[ \hat{\mathbf{h}}\_{j} \hat{\mathbf{h}}\_{j}^{H} - \hat{\mathbf{h}}\_{j} \boldsymbol{E} \{ \Delta \mathbf{h}\_{j}^{H} \, \| \, \hat{\mathbf{H}} \} - \boldsymbol{E} \{ \Delta \mathbf{h}\_{j} \, \| \, \hat{\mathbf{H}} \} \bar{\mathbf{h}}\_{j}^{H} \Big] \sigma\_{a\_{j}}^{2} + \sigma\_{w}^{2} \mathbf{I}\_{N\_{\hat{R}}}. \end{split} \tag{176}$$

Based on results discussed in the previous Section, it is evidently that **h** *<sup>j</sup>* 5 is only correlated with **<sup>h</sup>** *<sup>j</sup>* <sup>ˆ</sup> 6 . Then, we have

$$E\{\boldsymbol{\Delta}\mathbf{h}\_{\neq}\boldsymbol{\Delta}\mathbf{h}\_{\neq}^{H}\mid\hat{\mathbf{H}}\}=E\{\boldsymbol{\Delta}\mathbf{h}\_{\neq}\boldsymbol{\Delta}\mathbf{h}\_{\neq}^{H}\mid\hat{\mathbf{h}}\_{\neq}\}.\tag{177}$$

8 From the basic relationship between the autocorrelation and covariance functions, we have

$$E\{\boldsymbol{\Delta}\mathbf{h}\_{\dot{j}}\boldsymbol{\Delta}\mathbf{h}\_{\dot{j}}^{H}\|\hat{\mathbf{h}}\_{\dot{j}}\} = \text{Cov}\{\boldsymbol{\Delta}\mathbf{h}\_{\dot{j}}\boldsymbol{\Delta}\mathbf{h}\_{\dot{j}}^{H}\|\hat{\mathbf{h}}\_{\dot{j}}\} + E\{\boldsymbol{\Delta}\mathbf{h}\_{\dot{j}}\|\hat{\mathbf{h}}\_{\dot{j}}\}E\{\boldsymbol{\Delta}\mathbf{h}\_{\dot{j}}^{H}\|\hat{\mathbf{h}}\_{\dot{j}}\}.\tag{178}$$

10 Substituting (171) and (172) into (178), we can write

$$E\{\boldsymbol{\Delta}\mathbf{h}\_{\boldsymbol{\beta}}\boldsymbol{\Delta}\mathbf{h}\_{\boldsymbol{\beta}}^{H}\mid\hat{\mathbf{h}}\_{\boldsymbol{\beta}}\} = \sigma\_{\boldsymbol{\Delta}\mathbf{h}}^{2}\mathbf{I}\_{N\_{R}} - \sigma\_{\boldsymbol{\Delta}\mathbf{h}}^{4}\{\mathbf{R}\_{\mathbf{r}} + \sigma\_{\boldsymbol{\Delta}\mathbf{h}}^{2}\mathbf{I}\_{N\_{R}}\}^{-1} + \sigma\_{\boldsymbol{\Delta}\mathbf{h}}^{4}\{\mathbf{R}\_{\mathbf{r}} + \sigma\_{\boldsymbol{\Delta}\mathbf{h}}^{2}\mathbf{I}\_{N\_{R}}\}^{-1}\mathbf{h}\_{\boldsymbol{\beta}}\mathbf{h}\_{\boldsymbol{\beta}}^{H}\{\mathbf{R}\_{\mathbf{r}} + \sigma\_{\boldsymbol{\Delta}\mathbf{h}}^{2}\mathbf{I}\_{N\_{R}}\}.\tag{179}$$

12 Introduce the following notations

$$\boldsymbol{\Lambda} = \left(\mathbf{R}\_{\mathbf{r}} + \sigma\_{\Lambda\mathbf{h}}^2 \mathbf{I}\_{N\_R}\right)^{-1},\tag{180}$$

$$\boldsymbol{\Xi} = \mathbf{I}\_{\mathcal{N}\_R} - \sigma\_{\Lambda \mathbf{t}}^2 \boldsymbol{\Lambda},\tag{181}$$

$$\mathbf{R}\_{aa} = \text{diag}\{\sigma\_{a\_{k\_1}}^2, \sigma\_{a\_{k\_2}}^2, \dots, \sigma\_{a\_{k\_{i-1}}}^2\}. \tag{182}$$

16 Substituting (171) and (179) into (176) and taking into consideration (180)–(182), we have

$$\begin{split} \mathbf{E} \{ \tilde{\mathbf{s}}\_{k\_{\hat{l}}} \tilde{\mathbf{s}}\_{k\_{\hat{l}}}^{H} \} &= \mathbf{E}\_{\theta} \mathbf{E} \hat{\mathbf{H}}\_{k\_{\hat{l}}:k\_{\hat{l}\_{\hat{l}}}} \hat{\mathbf{H}}\_{k\_{\hat{l}}:k\_{\hat{l}\_{\hat{l}}}}^{H} \mathbf{E} + \mathbf{E}\_{\theta} \sigma\_{\mathbf{M}}^{4} \mathbf{A} \hat{\mathbf{H}}\_{k\_{\hat{l}}:k\_{\hat{l}-1}} \hat{\mathbf{H}}\_{k\_{\hat{l}}:k\_{\hat{l}-1}}^{H} \mathbf{A} + \left( \mathbf{\hat{H}}\_{k\_{\hat{l}}:k\_{\hat{l}-1}} \mathbf{R}\_{\mathbf{a}} \hat{\mathbf{H}}\_{k\_{\hat{l}}:k\_{\hat{l}-1}}^{H} + N\_{\Gamma} E\_{\theta} \sigma\_{\mathbf{A}}^{2} \right) \mathbf{E} \\ &- \sigma\_{\mathbf{M}}^{2} \mathbf{A} \hat{\mathbf{H}}\_{k\_{\hat{l}}:k\_{\hat{l}-1}} \mathbf{R}\_{\mathbf{a}} \hat{\mathbf{H}}\_{k\_{\hat{l}}:k\_{\hat{l}-1}}^{H} + \sigma\_{w}^{2} \mathbf{I}\_{N\_{R}\prime} \end{split} \tag{183}$$

where the notation **H***<sup>n</sup>*:*<sup>m</sup>* 1 denotes the submatrix containing the *n*-th to *m*-th columns of the 2 matrix **H**.

3 Based on similar manipulations, we can write

$$E\{\mathbf{2}\_{k\_{\hat{l}}}\bar{\mathbf{s}}\_{k\_{\hat{l}}}^H|\,\hat{\mathbf{H}}\} - E\{\bar{\mathbf{s}}\_{k\_{\hat{l}}}\bar{\mathbf{s}}\_{k\_{\hat{l}}}^H|\,\hat{\mathbf{H}}\} + E\{\mathbf{w}\_{AF\_{\hat{l}}}\mathbf{w}\_{AF\_{\hat{l}}}^H\} = E\_{\mathbf{b}}\hat{\mathbf{h}}\_{k\_{\hat{l}}}^H\sigma\_{\Delta\mathbf{l}}^2 \{\mathbf{R}\_{\mathbf{r}} + \sigma\_{\Delta\mathbf{l}}^2\mathbf{I}\_{N\_R}\}^{-1} + \{\mathbf{w}\_{AF}\hat{\mathbf{h}}\_{k\_{\hat{l}}}^H\mathbf{w}\_{AF}^H - \mathbf{w}\_{PF}\hat{\mathbf{h}}\_{k\_{\hat{l}}}^H\mathbf{w}\_{PF}^H\}\mathbf{I}\_{N\_R}.\tag{184}$$

5 In the root mean-square sense, the second term in (184) representing the GR back end 6 background noise tends to approach zero. By this reason, finally we can write

$$E\{2a\_{k\_i}\bar{\mathbf{s}}\_{k\_i}^H\|\hat{\mathbf{H}}\} - E\{\bar{\mathbf{s}}\_{k\_i}\bar{\mathbf{s}}\_{k\_i}^H\|\hat{\mathbf{H}}\} + E\{\mathbf{w}\_{AF\_i}\mathbf{w}\_{AF\_i}^H\} = E\_b\hat{\mathbf{h}}\_{k\_i}^H\sigma\_{\Lambda b}^2 \left(\mathbf{R}\_\mathbf{r} + \sigma\_{\Lambda b}^2 \mathbf{I}\_{N\_R}\right)^{-1}.\tag{185}$$

Combining (183) and (185), we obtain the MMSE GR output **Y***<sup>i</sup>* <sup>~</sup> , conditionally on **H**<sup>ˆ</sup> 8 .

#### 9 *4.4.3. Computation of LLR*

By applying **Y***<sup>i</sup>* <sup>~</sup> to *<sup>i</sup> <sup>k</sup>* **<sup>s</sup>** <sup>~</sup> , we have the process at the MMSE GR output [13,52,61] *<sup>i</sup> <sup>i</sup> <sup>k</sup> <sup>i</sup> <sup>k</sup>* **<sup>Z</sup> <sup>Y</sup> <sup>s</sup>** <sup>~</sup> <sup>~</sup> <sup>~</sup> <sup>10</sup> . 11 According to the Gaussian approximation of the MMSE GR back end process, we can write

$$
\tilde{\mathbf{Z}}\_{k\_i} \approx \tilde{\boldsymbol{\mu}}\_{k\_i} \boldsymbol{a}\_{k\_i}^2 + \tilde{\mathbf{r}}\_{\mathbf{k}\_i \prime} \tag{186}
$$

13 where

18

46 BookTitle

3 (175), we can write

1

*i*

*j k*

*k*

*i i*

1

with **<sup>h</sup>** *<sup>j</sup>* <sup>ˆ</sup> 6 . Then, we have

1 because the AF and PF of GR front end linear system do not change the statistical 2 parameters of input process (Gaussian noise, for example). Thus, according to (173) and

ˆ ˆ ˆ ˆ

4 (176)

Based on results discussed in the previous Section, it is evidently that **h** *<sup>j</sup>* 5 is only correlated

8 From the basic relationship between the autocorrelation and covariance functions, we have


{ } { }{ } { }

**s s H hh h h H h Hh h h H**

*E EE E E*

*N N <sup>T</sup> <sup>T</sup>*

[ ]

*k k H HH H H k k jj j j j j jj b j k j*

ˆˆ ˆ ˆ ˆ ˆ .


**hh h h H h Hh I**

{ }{ }

[ ]

10 Substituting (171) and (172) into (178), we can write

12 Introduce the following notations

1 1 1 1

**ΛH RH I**

ˆ ˆ ,

*H h k k aa k k w N*

*i i R*

2 2 : :

*σ σ*

*H H H jj j j j j a wN*

*E E σ σ*

*i*

124 Contemporary Issues in Wireless Communications

2 2

*j*

ˆ ˆ { } { }. | | *H H j j jj j* 7 *E E* **hh H hh h** (177)

ˆ ˆ ˆˆ { } { } { } { }. | | || *HH H jj j jj j jj j j* 9 *E* **hh h hh h hh h h** *Cov E E* (178)

ˆ 2 4 2 14 2 1 2 { } ( ) ( ) ( ). |

**Λ R I <sup>r</sup>** 13 (180)

*σ* 14 **Ξ I Λ** (181)

16 Substituting (171) and (179) into (176) and taking into consideration (180)–(182), we have

ˆˆ ˆˆ ˆ ˆ

*EE E σ N E σ*

*HH H H*

*<sup>i</sup> kk k aa aa a diag σσ σ* 15 **R** (182)

: : :: : :

*s s* **ΞH H Ξ ΛH H Λ H RH Ξ**

*k k b k k k k b h k k k k k k aa k k T b h*

{ } ( ) *i i i i N N T T i i <sup>i</sup> <sup>i</sup>*

17 (183)

2 1 ( ) , *<sup>R</sup> h N <sup>σ</sup>*

<sup>2</sup> , *N h <sup>R</sup>*

12 1 22 2 { , ,, }.

11 1 1 11 1 1

4 2

*H H j j j hN h hN h hN j j hN <sup>E</sup> σσσ σσ σ* **rrr** 11 **h h h I R I R I hh R I** (179)

*RR R R*

*R*

{ }

1

$$\widetilde{\mathbf{u}}\_{k\_l} = E\{\widetilde{\mathbf{Y}}\_l(\hat{\mathbf{h}}\_{k\_l} - \Delta \mathbf{h}\_{k\_l}) \mid \mathbf{H}\} = \widetilde{\mathbf{Y}}\_l \Xi \hat{\mathbf{h}}\_{k\_l} \tag{187}$$

and <sup>~</sup> <sup>2</sup> <sup>2</sup> *<sup>i</sup> AFki PFki* 15 **η***<sup>k</sup>* **w w** is the background noise at the MMSE GR output with zero-mean and variance <sup>2</sup> <sup>4</sup> <sup>~</sup> <sup>4</sup> *<sup>η</sup><sup>k</sup> <sup>w</sup> <sup>σ</sup> <sup>σ</sup> <sup>i</sup>* 16 .

Therefore, the LLR value of the coded bit *<sup>λ</sup> <sup>i</sup> <sup>k</sup>* 17 **g** can be approximated as [68, 69]

$$\mathcal{K}(\mathbf{g}\_{k\_i}^{\boldsymbol{\lambda}}) \approx \frac{1}{\sigma\_{\hat{\eta}\_{k\_i}}^2} \Big( \min\_{\boldsymbol{\alpha}\_i \in \mathcal{F}\_{\boldsymbol{\lambda}}^0} \|\mathbf{Z}\_i - \boldsymbol{\mu}\_i \boldsymbol{\alpha}\_i\|^2 - \min\_{\boldsymbol{\alpha}\_i \in \mathcal{F}\_{\boldsymbol{\lambda}}^1} \|\mathbf{Z}\_i - \boldsymbol{\mu}\_i \boldsymbol{\alpha}\_i\|^2 \Big), \tag{188}$$

where <sup>0</sup> F*<sup>λ</sup>* and <sup>1</sup> 19 F *<sup>λ</sup>* denote the modulation constellation symbols subset of F whose *λ* -th bit 20 equals 0 and 1, respectively.

#### 1 *4.4.4. Remarks*

48 BookTitle

When the channel estimation error is neglected, i.e., <sup>0</sup> <sup>2</sup> <sup>2</sup>*σ<sup>h</sup>* in (180), (181) and (183), the 3 MMSE GR output given by (174) reduces to that of the modified soft-output MMSE GR, in 4 which only decision error propagation is considered, [13, 61]. On the other hand, if *NR* **R I** 5 **<sup>r</sup>** and no residual interference cancellation error is assumed the MMSE GR output 6 given by (174) reduces to that of [50]. For the sake of simplicity, we call this detector as the 7 conventional soft-output MMSE GR hereafter if this detector is applied in channel coded 8 MIMO DS-CDMA wireless communication system. Meanwhile, if both decision error 9 propagation and channel estimation error are neglected, the MMSE GR output given by 10 (174) reduces to that of the conventional MMSE GR output of [51].

#### 11 **4.5. SER definition**

12 We continue a discussion of SER formula derivation presented in (61)-(72), subsection 3.3.1. 13 In the case of *M*-ary PSK system the exact expression for the SER takes the following form 14 [79]

$$P\_{SER} = \frac{1}{\pi} \int\_0^{\pi - \frac{\pi}{M}} \exp\left\{-\frac{E\_b}{N\_0} \sin^2 \frac{\pi}{M}\right\} d\theta. \tag{189}$$

16 Taking into account (61), (67), and (189), we can write the *SER* of QPSK system employed 17 the GR in the following form:

$$P\_{SER} = \frac{1}{\pi} \int\_0^{\pi - \frac{\pi}{M}} \phi\_{q\_{QMS, \text{MAC}}} \left( -\frac{1}{2 \sin^2 \theta} \right) d\theta. \tag{190}$$

19 There is a need to note that a direct comparison of QPSK and BPSK systems on the basis of 20 average symbol-energy-to-noise-spectral-density ratio indicates that the QPSK is 21 approximately 3 dB worse than the BPSK.

22 Another signaling scheme that allows multiple signals to be transmitted using quadrature 23 carries is the QAM. In this case, the transmitted signal can be presented in the following 24 form:

$$a\_k(t) = \sqrt{\frac{2}{T\_s}} \left[ A\_i \cos(2\pi f\_c t) + B\_i \sin(2\pi f\_c t) \right], \quad 0 \le t \le T\_s \tag{191}$$

where *Ai* and *Bi* 1 take on the possible values *p*; 3*p*,,( *M* 1) *p* with equal probability, where *M* is an integer power of 4; *Ts* is the sampling interval, and *<sup>c</sup>* 2 *f* is the carrier frequency. The parameter *p* can be related to the average symbol energy *Eb* 3 as given 4 by

$$p = \sqrt{\frac{3E\_b}{2(M-1)}}.\tag{192}$$

6 Taking into consideration a definition of the *SER* derived in [80] for *M-*ary QAM system 7 employed the GR, we obtain

$$\begin{split} P\_{\text{SEP}} &= 1 - \frac{1}{M} \Biggl[ (\sqrt{M} - 2)^2 \left[ 1 - 2Q \left( \sqrt{\frac{3p\epsilon\_{\text{qmaxacc}}}{M - 1}} \right) \right]^2 + 4(\sqrt{M} - 2) \left[ 1 - 2Q \left( \sqrt{\frac{3p\epsilon\_{\text{qmaxacc}}}{M - 1}} \right) \right] \Biggr] 1 - Q \left( \sqrt{\frac{3p\epsilon\_{\text{qmaxacc}}}{M - 1}} \right) \Biggr] \\ &+ 4 \left[ 1 - Q \left( \sqrt{\frac{3p\epsilon\_{\text{qmaxacc}}}{M - 1}} \right) \right]^2, \end{split} \tag{193}$$

9 where *Q*(*x*) is the Gaussian *Q*-function given by

$$Q(\mathbf{x}) = \frac{1}{\sqrt{2\pi}} \int\_{\mathbf{x}}^{\mathbf{v}} \exp(-0.5t^2)dt. \tag{194}$$

#### 11 **4.6. Simulation results**

48 BookTitle

1 *4.4.4. Remarks*

126 Contemporary Issues in Wireless Communications

11 **4.5. SER definition**

17 the GR in the following form:

21 approximately 3 dB worse than the BPSK.

<sup>2</sup> <sup>25</sup>( ) [

14 [79]

24 form:

When the channel estimation error is neglected, i.e., <sup>0</sup> <sup>2</sup> <sup>2</sup>*σ<sup>h</sup>* in (180), (181) and (183), the 3 MMSE GR output given by (174) reduces to that of the modified soft-output MMSE GR, in 4 which only decision error propagation is considered, [13, 61]. On the other hand, if *NR* **R I** 5 **<sup>r</sup>** and no residual interference cancellation error is assumed the MMSE GR output 6 given by (174) reduces to that of [50]. For the sake of simplicity, we call this detector as the 7 conventional soft-output MMSE GR hereafter if this detector is applied in channel coded 8 MIMO DS-CDMA wireless communication system. Meanwhile, if both decision error 9 propagation and channel estimation error are neglected, the MMSE GR output given by

12 We continue a discussion of SER formula derivation presented in (61)-(72), subsection 3.3.1. 13 In the case of *M*-ary PSK system the exact expression for the SER takes the following form

0

0

*s*

 

18 (190)

19 There is a need to note that a direct comparison of QPSK and BPSK systems on the basis of 20 average symbol-energy-to-noise-spectral-density ratio indicates that the QPSK is

22 Another signaling scheme that allows multiple signals to be transmitted using quadrature 23 carries is the QAM. In this case, the transmitted signal can be presented in the following

*<sup>k</sup> A f t B f t t T <sup>T</sup> <sup>a</sup> <sup>t</sup>* cos(2 ) sin(2 ) , <sup>0</sup>

*M*

 

1

*SER*

2

] (191)

sin

 

exp . sin

/ 2

*i c i c s*

1 1 . 2sin ( ) *<sup>M</sup> QBHS MRC SER <sup>q</sup> P d*

 

0 2

*b*

*E*

*N M P d*

15 (189)

16 Taking into account (61), (67), and (189), we can write the *SER* of QPSK system employed

10 (174) reduces to that of the conventional MMSE GR output of [51].

#### 12 *4.6.1. FIR beamforming and MIMO wireless communication system*

13 For a definition of numerical results using simulation, we consider the typical urban channel 14 [81] of the GSM/EDGE system as a practical example. As is usually done for GSM/EDGE, the 15 transmit pulse shape is modeled as a linearized Gaussian minimum-shift keying pulse [82]. 16 The GR input linear system filter is a square-root raised-cosine filter with roll-off factor 0.3. 17 Furthermore, we assume *NT* 3 transmit and *NR* 3 receive antennas and a maximum 18 channel length of *L* 5 . The correlation coefficient between all pairs of transmit antennas is 19 *ρ* 0.5 .

Figure 9 shows the average *SNR* as a function of the *SNR* noted by *Eb N*<sup>0</sup> 20 for the GR with 21 MMSE linear equalization in the cases of FIR (the curves 2 and 3) and infinite impulse response (the curve 1) beamforming filter, respectively, where *Eb* 22 denotes the average 23 received energy per symbol. The curve 5 corresponds to the case for infinite impulse 50 BookTitle

7

8 9

1 response beamforming filter for receiver discussed in [67]. The *SNR* was obtained by 2 averaging the respective *SNR*s over 1000 independent realizations of the typical urban 3 channel. For this purpose, in the case of FIR beamforming filter at the GR, the *SNR* given by 4 (159) was used and the corresponding beamforming filters at the GR were calculated using 5 the gradient algorithm discussed in Section 4.3. For infinite impulse response beamforming 6 filter at the GR the result given in [52] was used.

10 **Figure 9.** Average *SNR* of MMSE linear equalization at the GR for beamforming with FIR and infinite 11 impulse response filters. The result for single antenna transmission is also indicated (the curve 4). IIR – 12 infinite impulse response beamforming filter.

13 For comparison, we also show simulation results with FIR linear equalization filters at the GR of length *LF* 4*L* for FIR beamforming filters at the GR with 1 *Lg* 14 (the curve 3) 15 optimized for infinite impulse response linear equalization filters at the GR. These 16 simulation results confirm that sufficiently long FIR linear equalization filters at the GR 17 closely approach performance of infinite impulse response linear equalization filters at the 18 GR, which are necessary for (178) to be valid. As expected, the beamforming with infinite 19 impulse response beamforming filters at the GR constitutes a natural performance upper 20 bound for the beamforming with FIR beamforming filters at the GR. However, interestingly, for the typical urban channel the FIR beamforming filter of length 3 *Lg* 21 (the curve 2) is

1 sufficient to closely approach the performance of the infinite impulse response beamforming 2 at the GR (the curve 1).

We note that for high values of *Eb N*<sup>0</sup> 3 even an FIR beamforming filter at the GR of length 1 *Lg* 4 achieves a performance gain of more than 4.5 dB compared to single antenna 5 transmission, i.e. *NT NR* 1 (the curve 4). Additional simulations not shown here for 6 other GSM/EDGE channel profiles have shown that, in general, the FIR beamforming filter at the GR of length 6 *Lg* 7 is sufficient to closely approach the performance of the infinite impulse response beamforming at the GR. Thereby, the value of *Lg* 8 required to approach 9 the performance of the infinite impulse response beamforming at the GR seems to be 10 smaller if the channel is less frequency selective.
