**2. System model**

In cellular system, the same carrier frequency/frequencies will be reused in neighboring cells to effectively utilize the limited spectrum. Cellular structures with frequency reuse factors (FRFs) of 1, 3, 4 and 7 are shown in Fig. 1 as examples of frequency reuse. And FRF=1, 3, 4, 7, 9 and 12 will be considered in this study. The commonly used first layer CCI model is used here, i.e., only the CCI from the first layer neighboring cells will be considered and the number of CCI cells will be *B* =6. As stated in the Introduction, we are going to use both CAN and DAN in each cell. CAN and DAN structures are shown in Fig. 2. There are totally *Nr* centralized / distributed antennas. In the DAN system, those antennas are connected to the DAN central processor by optical fibers. The received signals will be transmitted to the DAN central processor and in order to lower the cost, signal processing will be carried out by the DAN central processor.

Multi-User Interference Suppression by Using Frequency Domain Adaptive Antenna Array http://dx.doi.org/10.5772/57132 95

**Figure 1.** Frequency reuse in cellular system.

occurs when multiple users transmit simultaneously within the same cell (the MUI and CCI together is called multi-access interference (MAI)). Therefore, interference cancellation is

Recently, distributed antenna network (DAN) [7] has been proposed to solve the transmit power problem in broadband signal transmissions. As the data rate increases, impractical‐ ly large transmit power will be required to realize the high data rate if cell coverage is kept unchanged. Otherwise, the cell coverage has to be reduced if the transmit power is kept unchanged. DAN was proposed as a solution to increase the cell coverage while maintain‐ ing the low transmit power. In the DAN, a number of antennas are distributed in each cell and those antennas are connected with the DAN central processor (which is similar to the BS in conventional cellular system) through optical cables. A mobile user can communi‐ cate with its' nearby located antennas even when it is at the cell edge. Therefore, the transmit power in DAN can be kept low while the coverage of the cell can be greatly increased.

In the previous studies [8, 9], a SC frequency domain adaptive antenna array (SC-FDAAA) for the uplink transmission has been proposed and it has been shown that the SC-FDAAA can effectively suppress MAI in a severely frequency selective fading chan‐ nel. In this article, we will present the performance of DAN SC-FDAAA and compare SC-FDAAA in DAN and in conventional cellular system with centralized antennas at the BS

The rest of the article is organized as follows. The system model is given in Section II. SC-FDAAA for DAN and CAN will be described in Section III. The post SC-FDAAA signal to interference plus noise (SINR) will be given in Section IV. The performance of SC-FDAAA will be shown in Section V, both bit error rate (BER) distribution and the system

In cellular system, the same carrier frequency/frequencies will be reused in neighboring cells to effectively utilize the limited spectrum. Cellular structures with frequency reuse factors (FRFs) of 1, 3, 4 and 7 are shown in Fig. 1 as examples of frequency reuse. And FRF=1, 3, 4, 7, 9 and 12 will be considered in this study. The commonly used first layer CCI model is used here, i.e., only the CCI from the first layer neighboring cells will be considered and the number of CCI cells will be *B* =6. As stated in the Introduction, we are going to use both CAN and DAN in each cell. CAN and DAN structures are shown in Fig. 2. There are totally *Nr* centralized / distributed antennas. In the DAN system, those antennas are connected to the DAN central processor by optical fibers. The received signals will be transmitted to the DAN central processor and in order to lower the cost, signal processing will be carried out by the DAN

capacity will be presented. Finally, the article will be concluded by Section VI.

necessary in uplink transmissions.

94 Recent Trends in Multi-user MIMO Communications

(referred to as CAN system hereafter).

**2. System model**

central processor.

**Figure 2.** CAN system and DAN system.

It is assumed that there are *U* users within each cell and each user is equipped with one omni antenna. A block fading channel between each user and each antenna is assumed, i.e., the channel remains unchanged during the transmission period of a block. In this article, the symbol-spaced discrete time representation of the signal is used.

Assuming an *L* − path channel, the impulse response of the channel between the *u th* user and the *mth* antenna can be expressed as

$$h\_{u,m}\left(\boldsymbol{\pi}\right) = \sum\_{l=0}^{L-1} h\_{u,m,l} \mathcal{S}\left(\boldsymbol{\pi} - \boldsymbol{\pi}\_l\right) \tag{1}$$

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 0, 0 , , , 1 1 0

*i*

 p

è ø

.

 p

è ø

*i i*

(*k*) *Hu*,1

(*k*) *<sup>T</sup>* with ⋅ *<sup>T</sup>* representing the transpose operation.

*R k H kS k H kS k H kS k N k* (3)


*U B U m m um u u m iu m u i u*

( )

ì ï ï ï ï ï ï ï ï í ï ï ï ï ï

, , 0 1

, , 0

*m m c*

(*k*)⋯*RNr*−<sup>1</sup>

(*k*) ⋯ *NNr*−<sup>1</sup>

( )

and the last term is the noise component.

(*k*), *R*<sup>1</sup>

where *R*(*k*)= *R*<sup>0</sup>

**3. SC-FDAAA**

(*k*) *N*<sup>1</sup>

*N* (*k*)= *N*<sup>0</sup>

where

= = = =+ + + å åå*<sup>i</sup>*

1

æ ö <sup>=</sup> ç ÷ -


0

=

å

*c*

*N*

*<sup>t</sup> Sk P st j k <sup>N</sup> <sup>N</sup>*

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

*<sup>t</sup> S k P s t jk <sup>N</sup> <sup>N</sup>*

a


exp 2

æ ö <sup>=</sup> ç ÷ -

*t c*

exp 2

*t c*

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 0 0 , , 1 1 0

(*k*) *<sup>T</sup>* , *Hu*(*k*)= *Hu*,0

The structure of the SC-FDAAA transceiver in both CAN system and DAN system can be generalized and shown in Fig. 3. Binary data sequence is modulated and divided into a sequence of blocks of *Nc* data symbols. The last *Ng* symbols in each block are copied and inserted as CP into the guard interval (GI) and placed at the beginning of each block. The received signal is transformed by an *Nc* -point fast Fourier transform (FFT) into the frequency domain signal and SC-FDAAA weight control is then performed on each frequency as


*U B U*

*u i u*

= = = =+ + + å åå*<sup>i</sup>*

æ ö <sup>=</sup> ç ÷ -

*t c*

1

*c t c N*

æ ö <sup>=</sup> ç ÷ -

*c t c*


0

=

å

*c*

p

è ø

p

è ø

The first term in (3) is the desired signal, the second term is the MUI, the third term is the CCI,

p

*i*

**RH H** *k kS k kS k* **H N** *kS k k* (5)

*u u iu iu*

,


d

d

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

a

, ,, ,

*i u u iu m iu*

*<sup>t</sup> H k h jk <sup>N</sup>*

*<sup>t</sup> H k h jk <sup>N</sup>*

*i ii i*

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

0

=


*Nc*

= -

*<sup>t</sup> Nk nt j k <sup>N</sup> <sup>N</sup>* 1

<sup>ï</sup> æ ö <sup>ï</sup> ç ÷ ïî è ø <sup>å</sup>

The received signals {*R* m(*k*); *m*=0~*N* r} are then expressed in a matrix form as

1

*u u um u*


= -

å

*c*

*N u m u m*

=

å

*c i i*

*N u m u m* ,

http://dx.doi.org/10.5772/57132

,

(*k*) *<sup>T</sup>* , and

(*k*) ⋯ *Hu*,*Nr*−<sup>1</sup>

(4)

97

*i i*

Multi-User Interference Suppression by Using Frequency Domain Adaptive Antenna Array

where *h <sup>u</sup>*,*m*,*<sup>l</sup>* and *τ<sup>l</sup>* are the path gain and time delay of the *<sup>l</sup> th* path, respectively. *<sup>h</sup> <sup>u</sup>*,*m*,*<sup>l</sup>* follows complex Gaussian distribution and satisfies ∑*<sup>l</sup>*=0 *<sup>L</sup>* <sup>−</sup><sup>1</sup> *<sup>E</sup>*{ <sup>|</sup>*<sup>h</sup> <sup>u</sup>*,*m*,*<sup>l</sup>* <sup>|</sup> 2} =1 , where *E*{ <sup>⋅</sup> } represents the expectation. It is assumed that the time delay is a multiple integer of the symbol duration and *τ<sup>l</sup>* =*l* is used. The cyclic-prefixed (CP) block signal transmission is used to make the received symbol block to be a circular convolution of the transmitted symbol block and the channel impulse response as well as to avoid inter block interference (IBI). It is also assumed that the CP is longer than the maximum path delay of the signal. In the following, we omit the insertion and removal of the CP for the simplicity.

The baseband equivalent received signal block {*rm*(*t*);*<sup>t</sup>* =0~ *Nc*} of *Nc* symbols at the *mth* antenna is given by

$$\begin{split} r\_m(t) &= \sqrt{P\_0 \delta\_{0,m}^{-\alpha}} \sum\_{l=0}^{L-1} h\_{0,m,l} s\_0 \left( t - l \right) + \sum\_{u=1}^{U-1} \sqrt{P\_u \delta\_{u,m}^{-\alpha}} \sum\_{l=0}^{L-1} h\_{u,m,l} s\_u \left( t - l \right) \\ &+ \sum\_{i=1}^{B} \sum\_{u\_i=0}^{U\_i-1} \sqrt{P\_{i,u\_i} \delta\_{i,u\_i,m}^{-\alpha}} \sum\_{l=0}^{L-1} h\_{u\_i,m,l} s\_{u\_i} \left( t - l \right) + n\_m \left( t \right), \end{split} \tag{2}$$

where *su*(*t*) and *Pu* are respectively the transmit signal and transmit signal power of the *<sup>u</sup> th* user (*u* =0~*U* −1) ; *sui* and *Pi*,*ui* are respectively the transmit signal and transmit signal power of the *ui th* user in the *<sup>i</sup> th* co-channel cell; *δ*0,*<sup>m</sup>* represents the distance between the desired user and the *mth* antenna; *δi*,*<sup>m</sup>* represents the distance between the *<sup>i</sup> th* interfering user and the *mth* antenna; *δi*,*ui* ,*m* and *h ui* ,*m*,*<sup>l</sup>* are respectively the distance and channel gain between the CCI user and the *mth* antenna; *α* represents the path loss exponent in dB; and *nm*(*t*) is the additive white Gaussian noise (AWGN). To simplify the analysis, no shadowing loss is assumed.

Let the transmit signal from the *u* =0*th* user be the desired signal and the transmit signals from the other users be the interfering signals. The frequency domain representation of (2) is given by

Multi-User Interference Suppression by Using Frequency Domain Adaptive Antenna Array http://dx.doi.org/10.5772/57132 97

$$R\_m(k) = H\_{0,m}(k)S\_0(k) + \sum\_{u=1}^{U-1} H\_{u,m}(k)S\_u(k) + \sum\_{i=1}^{B} \sum\_{u\_i=0}^{U\_i-1} H\_{u\_i,m}(k)S\_{i,u\_i}(k) + N\_m(k),\tag{3}$$

where

It is assumed that there are *U* users within each cell and each user is equipped with one omni antenna. A block fading channel between each user and each antenna is assumed, i.e., the channel remains unchanged during the transmission period of a block. In this article, the

Assuming an *L* − path channel, the impulse response of the channel between the *u th* user and

( ) ( ) 1 , , , 0

<sup>=</sup> å -


*L u m uml l l*

=

 dt t

and *τ<sup>l</sup>* are the path gain and time delay of the *<sup>l</sup> th* path, respectively. *<sup>h</sup> <sup>u</sup>*,*m*,*<sup>l</sup>*

expectation. It is assumed that the time delay is a multiple integer of the symbol duration and *τ<sup>l</sup>* =*l* is used. The cyclic-prefixed (CP) block signal transmission is used to make the received symbol block to be a circular convolution of the transmitted symbol block and the channel impulse response as well as to avoid inter block interference (IBI). It is also assumed that the CP is longer than the maximum path delay of the signal. In the following, we omit the insertion

The baseband equivalent received signal block {*rm*(*t*);*<sup>t</sup>* =0~ *Nc*} of *Nc* symbols at the *mth*

( ) ( ) ( )

å åå

= - + -

*iu iu m u ml u m*

*P h s tl nt*

where *su*(*t*) and *Pu* are respectively the transmit signal and transmit signal power of the *<sup>u</sup> th*

and the *mth* antenna; *δi*,*<sup>m</sup>* represents the distance between the *<sup>i</sup> th* interfering user and the *mth*

and the *mth* antenna; *α* represents the path loss exponent in dB; and *nm*(*t*) is the additive white

Let the transmit signal from the *u* =0*th* user be the desired signal and the transmit signals from the other users be the interfering signals. The frequency domain representation of (2) is given

Gaussian noise (AWGN). To simplify the analysis, no shadowing loss is assumed.

*th* user in the *<sup>i</sup> th* co-channel cell; *δ*0,*<sup>m</sup>* represents the distance between the desired user


*rt P h stl P h stl*

*L UL m m ml u um uml u l ul*

1 1

d


a

1 0 0

= = =

*i u l*

åå å *<sup>i</sup>*

*B L U*

a

d

*i*

and *Pi*,*ui*

, ,, ,,

*ii i i*

+ - +

1 11 0 0, 0, , 0 , ,, 0 10

( ) ( )

 a

 d

,

are respectively the transmit signal and transmit signal power

,*m*,*<sup>l</sup>* are respectively the distance and channel gain between the CCI user

*h h* (1)

*<sup>L</sup>* <sup>−</sup><sup>1</sup> *<sup>E</sup>*{ <sup>|</sup>*<sup>h</sup> <sup>u</sup>*,*m*,*<sup>l</sup>* <sup>|</sup> 2} =1 , where *E*{ <sup>⋅</sup> } represents the

follows

(2)

symbol-spaced discrete time representation of the signal is used.

t

the *mth* antenna can be expressed as

96 Recent Trends in Multi-user MIMO Communications

complex Gaussian distribution and satisfies ∑*<sup>l</sup>*=0

and removal of the CP for the simplicity.

where *h <sup>u</sup>*,*m*,*<sup>l</sup>*

antenna is given by

user (*u* =0~*U* −1) ; *sui*

,*m* and *h ui*

of the *ui*

by

antenna; *δi*,*ui*

$$\begin{cases} S\_u(k) = \frac{1}{\sqrt{N\_c}} \sqrt{P\_u S\_{u,m}^{-\alpha}} \sum\_{t=0}^{N\_c - 1} s\_u(t) \exp\left(-j2\pi k \frac{t}{N\_c}\right) \\ S\_{i,u\_i}(k) = \frac{1}{\sqrt{N\_c}} \sqrt{P\_{u\_i} \delta\_{i,u\_i,m}^{-\alpha}} \sum\_{t=0}^{N\_c - 1} s\_{i,u\_i}(t) \exp\left(-j2\pi k \frac{t}{N\_c}\right) \\ H\_{u,m}(k) = \sum\_{t=0}^{N\_c - 1} h\_{u,m} \exp\left(-j2\pi k \frac{t}{N\_c}\right) \\ H\_{u\_i,m}(k) = \sum\_{t=0}^{N\_c - 1} h\_{u\_i,m} \exp\left(-j2\pi k \frac{t}{N\_c}\right) \\ N\_m(k) = \frac{1}{\sqrt{N\_c}} \sum\_{t=0}^{N\_c - 1} n\_m(t) \exp\left(-j2\pi k \frac{t}{N\_c}\right) \end{cases} \tag{4}$$

The first term in (3) is the desired signal, the second term is the MUI, the third term is the CCI, and the last term is the noise component.

The received signals {*R* m(*k*); *m*=0~*N* r} are then expressed in a matrix form as

$$\mathbf{R}\left(k\right) = \mathbf{H}\_0\left(k\right)\mathbf{S}\_0\left(k\right) + \sum\_{u=1}^{U-1} \mathbf{H}\_u\left(k\right)\mathbf{S}\_u\left(k\right) + \sum\_{i=1}^{B} \sum\_{u\_i=0}^{U\_i-1} \mathbf{H}\_{i,u\_i}\left(k\right)\mathbf{S}\_{i,u\_i}\left(k\right) + \mathbf{N}\left(k\right),\tag{5}$$

where *R*(*k*)= *R*<sup>0</sup> (*k*), *R*<sup>1</sup> (*k*)⋯*RNr*−<sup>1</sup> (*k*) *<sup>T</sup>* , *Hu*(*k*)= *Hu*,0 (*k*) *Hu*,1 (*k*) ⋯ *Hu*,*Nr*−<sup>1</sup> (*k*) *<sup>T</sup>* , and *N* (*k*)= *N*<sup>0</sup> (*k*) *N*<sup>1</sup> (*k*) ⋯ *NNr*−<sup>1</sup> (*k*) *<sup>T</sup>* with ⋅ *<sup>T</sup>* representing the transpose operation.

#### **3. SC-FDAAA**

The structure of the SC-FDAAA transceiver in both CAN system and DAN system can be generalized and shown in Fig. 3. Binary data sequence is modulated and divided into a sequence of blocks of *Nc* data symbols. The last *Ng* symbols in each block are copied and inserted as CP into the guard interval (GI) and placed at the beginning of each block. The received signal is transformed by an *Nc* -point fast Fourier transform (FFT) into the frequency domain signal and SC-FDAAA weight control is then performed on each frequency as

$$
\widetilde{\mathbf{R}}(k) = \mathbf{W}^T(k)\mathbf{R}(k) \tag{6}
$$

where *Crr*

and

where *A*<sup>0</sup>

decision as

where *R<sup>s</sup>*

*by [14]*

(*k*)=*H*<sup>0</sup>

sent the interference plus noise.

**4. Post SC-FDAAA SINR**

(*k*) and *R<sup>N</sup>* '

interference plus noise, respectively.

*Property: if a matrixZcan be written asZ* =*T* <sup>−</sup><sup>1</sup> + *PQ* <sup>−</sup><sup>1</sup>

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

*<sup>C</sup>rd* (*k*)=*E*{*<sup>R</sup>* \*

(*k*)= *E*{*R* \*

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

=

**C RR**

*rr*

reference signal, and \* denotes complex conjugate operation.

( ) ( ) ( )

*kk k*

0 0 ,

= + ¢

**AA N**

(*k*) [12], *N* '

*d* ^(*t*)= <sup>1</sup>

= ∑ *u*=1 *U* −1 *Au* \*

> *Nc* ∑ *k*=0 *Nc*−1

The post SC-FDAAA SINR on the *k th* frequency can be evaluated by [13]

*k*

( ) ( ) ( ) ( )

**W RW**

G = , *H*

*H*

( ) ' ( ) ( )

*k kk*

*k kk*

*s*

*N*

( ) { ( ) ( )}

*kE k k*

\*

\*

(*k*)*R*(*k*)} is the correlation matrix of the received signal and

Multi-User Interference Suppression by Using Frequency Domain Adaptive Antenna Array

*i i*

*rd* = = 0 00 **CR A** *k E kS k kS k* (10)

(9)

99

(*k*) + *N*0*I* is used to repre‐

http://dx.doi.org/10.5772/57132

). (11)

(*k*)} is the cross-correlation vector between the received signal and the

( ) ( ) ( ) ( ) ( ) ( )

=+ + +

( ) { ( ) ( )} ( ) ( ) \*

(*k*)*Au*(*k*) <sup>+</sup> ∑

In the next, time domain signal block estimate is then obtained by an *Nc* - point IFFT for data

*<sup>R</sup>*˜(*k*)exp( *<sup>j</sup>*2*π<sup>k</sup> <sup>t</sup>*

*Nc*

(*k*) are the auto-correlation matrix of the received desired signal and the

**WRW** (12)

*P* <sup>∗</sup> , *then the inverse matrix ofZ can be obtained*

*i*=1 *B* ∑ *ui* =0 *Ui* −1 *Ai*,*ui* <sup>∗</sup> (*k*)*Ai*,*ui*

å åå*<sup>i</sup>*

*U B U*

*u i u*


0 0 ,, 0 1 1 0

**AA AA A A I**

*kk kk k kN*

*i*

*u u iu iu*

<sup>1</sup> <sup>1</sup> \* \*

where

$$\mathbf{W}(k) = \begin{bmatrix} W\_0(k) \bot \; W\_{N\_r-1}(k) \end{bmatrix}^T. \tag{7}$$

$$\begin{array}{c|c} \hline \multicolumn{1}{1}{\text{Data modulation}} & \multicolumn{1}{1}{\text{Data translation}} \\ \hline \multicolumn{1}{1}{\text{Pilot}} \\ \hline \text{Pilot} & \text{MUX} \\ & & \\ \hline \end{array} \qquad \begin{array}{c|c} \hline \multicolumn{1}{1}{\text{MUX}} \\ \hline \multicolumn{1}{1}{\text{MUX}} \\ \hline \multicolumn{1}{1}{\text{SAM}} \\ \hline \end{array} \qquad \begin{array}{c|c} \hline \multicolumn{1}{1}{\text{OAM}} \\ \hline \multicolumn{1}{1}{\text{OAM}} \\ \hline \end{array} \qquad \begin{array}{c|c} \hline \multicolumn{1}{1}{\text{OAM}} \\ \hline \multicolumn{1}{1}{\text{OAM}} \\ \hline \end{array}$$

The SC-FDAAA weight that minimizes the mean squared error (MSE) between *<sup>R</sup>*˜(*k*) and the reference signal *S*<sup>0</sup> (*k*) (the pilot signal will be used as the reference signal) is given by [10, 11]

$$\mathbf{W}(k) = \mathbf{C}\_{rr}^{-1}(k)\mathbf{C}\_{rd}(k),\tag{8}$$

where *Crr* (*k*)= *E*{*R* \* (*k*)*R*(*k*)} is the correlation matrix of the received signal and *<sup>C</sup>rd* (*k*)=*E*{*<sup>R</sup>* \* (*k*)*S*<sup>0</sup> (*k*)} is the cross-correlation vector between the received signal and the reference signal, and \* denotes complex conjugate operation.

$$\begin{split} \mathbf{C}\_{rr}\left(k\right) &= E\left\{\mathbf{R}^\*\left(k\right)\mathbf{R}\left(k\right)\right\} \\ &= \mathbf{A}\_0^\*\left(k\right)\mathbf{A}\_0\left(k\right) + \sum\_{u=1}^{U-1} \mathbf{A}\_u^\*\left(k\right)\mathbf{A}\_u\left(k\right) + \sum\_{i=1}^B \sum\_{u\_i=0}^{U\_i-1} \mathbf{A}\_{i,u\_i}^\*\left(k\right)\mathbf{A}\_{i,u\_i}\left(k\right) + N\_0\mathbf{I} \\ &= \mathbf{A}\_0^\*\left(k\right)\mathbf{A}\_0\left(k\right) + \mathbf{N}'\left(k\right), \end{split} \tag{9}$$

and

*<sup>R</sup>*˜(*k*)=*<sup>W</sup> <sup>T</sup>* (*k*)*R*(*k*) (6)

*T*

*<sup>N</sup>* **W** *k Wk W k* L (7)

( ) <sup>=</sup> <sup>é</sup> <sup>0</sup> ( ),, . -<sup>1</sup> ( )<sup>ù</sup> <sup>ë</sup> *<sup>r</sup>* <sup>û</sup>

The SC-FDAAA weight that minimizes the mean squared error (MSE) between *<sup>R</sup>*˜(*k*) and the

(*k*) (the pilot signal will be used as the reference signal) is given by [10, 11]

( ) ( ) ( ) <sup>1</sup> , - <sup>=</sup> *rr rd* **W CC** *k kk* (8)

where

98 Recent Trends in Multi-user MIMO Communications

**Figure 3.** SC-FDAAA transceiver structure.

reference signal *S*<sup>0</sup>

$$\mathbf{C}\_{rd}\left(k\right) = E\left\{\mathbf{R}^\*\left(k\right)\mathbf{S}\_0\left(k\right)\right\} = \mathbf{A}\_0\left(k\right)\mathbf{S}\_0\left(k\right) \tag{10}$$

where *A*<sup>0</sup> (*k*)=*H*<sup>0</sup> (*k*)*S*<sup>0</sup> (*k*) [12], *N* ' = ∑ *u*=1 *U* −1 *Au* \* (*k*)*Au*(*k*) <sup>+</sup> ∑ *i*=1 *B* ∑ *ui* =0 *Ui* −1 *Ai*,*ui* <sup>∗</sup> (*k*)*Ai*,*ui* (*k*) + *N*0*I* is used to repre‐ sent the interference plus noise.

In the next, time domain signal block estimate is then obtained by an *Nc* - point IFFT for data decision as

$$\hat{\vec{d}}(t) = \frac{1}{\sqrt{N\_c}} \sum\_{k=0}^{N\_c - 1} \tilde{R}(k) \exp\left(j2\pi k \frac{t}{N\_c}\right). \tag{11}$$

#### **4. Post SC-FDAAA SINR**

The post SC-FDAAA SINR on the *k th* frequency can be evaluated by [13]

$$\Gamma\left(k\right) = \frac{\mathbf{W}^H\left(k\right)\mathbf{R}\_s\left(k\right)\mathbf{W}\left(k\right)}{\mathbf{W}^H\left(k\right)\mathbf{R}\_{N^\cdot}\left(k\right)\mathbf{W}\left(k\right)},\tag{12}$$

where *R<sup>s</sup>* (*k*) and *R<sup>N</sup>* ' (*k*) are the auto-correlation matrix of the received desired signal and the interference plus noise, respectively.

*Property: if a matrixZcan be written asZ* =*T* <sup>−</sup><sup>1</sup> + *PQ* <sup>−</sup><sup>1</sup> *P* <sup>∗</sup> , *then the inverse matrix ofZ can be obtained by [14]*

$$\mathbf{Z}^{-1} = \mathbf{T} - \mathbf{T} \mathbf{P} \left(\mathbf{Q} + \mathbf{P}^\* \mathbf{T} \mathbf{P}\right)^{-1} \mathbf{P}^\* \mathbf{T}. \tag{13}$$

and CAN system when FRF=1. The x axis is the BER abscissa and y axis is the probability that BER<abscissa. It can be observed that DAN SC-FDAAA outperforms CAN SC-FDAAA by having better BER performance. It can also be observed that when the number of users increases, the BER performance of SC-FDAAA will degrade in both DAN and CAN systems, which can be intuitionally expected due to the reduction of degree of freedom. From the results shown in Figs. 6-8, it can be further observed that when FRF increases, the C.D.F. curves of BER performance "shift" right-side, which means that the BER performance improves due to the reduction of CCI power. In addition, DAN SC-FDAAA always achieves better BER performance than CAN SC-FDAAA no matter how FRF varies. The results of BER performance have shown that the distributed nature of DAN system can significantly improve the BER

Multi-User Interference Suppression by Using Frequency Domain Adaptive Antenna Array

http://dx.doi.org/10.5772/57132

101

R/ 2 R/ 2

R/ 2

R/ 2 R/ 2

R/ 2

...

performance of SC-FDAAA over CAN system.

**Figure 4.** Antenna distribution in DAN system.

Let *Z* =*Crr* (*k*) , *T* = *RNI* <sup>−</sup>1(*k*) , *<sup>P</sup>* <sup>=</sup> *<sup>A</sup>*<sup>0</sup> <sup>∗</sup>(*k*) where *A*<sup>0</sup> (*k*)=*H*<sup>0</sup> (*k*)*S*<sup>0</sup> (*k*) and *Q* = *I* , then the inverse ma‐ trix *Crr* <sup>−</sup>1(*k*) can be calculated by submitting *Z* , *T* , *P* and *I* into

$$\begin{split} \mathbf{C}\_{rr}^{-1} &= \mathbf{R}\_{N}^{-1}(k) - \mathbf{R}\_{N}^{-1}(k)\mathbf{A}\_{0}^{\star}(k) \left[ \mathbf{I} + \mathbf{A}\_{0}(k)\mathbf{R}\_{N}^{-1}\mathbf{A}\_{0}^{\star}(k) \right]^{-1} \mathbf{A}\_{0}(k)\mathbf{R}\_{N}^{-1}(k) \\ &= \mathbf{R}\_{N}^{-1}(k) \left[ \mathbf{I} - \frac{\mathbf{A}\_{0}^{\star}(k)\mathbf{A}\_{0}(k)\mathbf{R}\_{N}^{-1}(k)}{\mathbf{I} + \mathbf{A}\_{0}(k)\mathbf{R}\_{N}^{-1}\mathbf{A}\_{0}^{\star}(k)} \right] \\ &= \left[ \frac{1}{1 + \mathbf{A}\_{0}(k)\mathbf{R}\_{N}^{-1}\mathbf{A}\_{0}^{\star}(k)} \right] \mathbf{R}\_{N}^{-1}(k). \end{split} \tag{14}$$

The SC-FDAAA weight is then obtained by substituting (9) and (13) into (7), given by

$$\mathbf{W}(k) = \left[\frac{1}{1 + \mathbf{A}\_0(k)\mathbf{R}\_N^{-1}\mathbf{A}\_0^\*(k)}\right] \mathbf{R}\_N^{-1}(k)\mathbf{A}\_0(k)\mathbf{S}\_0(k). \tag{15}$$

Finally, the SINR after the weight control can be expressed, by substituting (14) into (9), as

$$
\Gamma \begin{pmatrix} k \\ \end{pmatrix} = \mathbf{A}\_0 \begin{pmatrix} k \\ \end{pmatrix} \mathbf{R}\_{N}^{-1} \begin{pmatrix} k \\ \end{pmatrix} \mathbf{A}\_0^\* \begin{pmatrix} k \\ \end{pmatrix}. \tag{16}
$$
