3. Modelling of antenna pattern multiplexing

In this section, we build a model of the receiver with APM and mathematically derive the received signals in MIMO applications.

#### 3.1 Signals to change antenna pattern

As we mentioned in the previous section, the antenna patterns can be changed by applying periodically time variable voltages to the VREs connected to parasitic antenna elements. Since the applied voltages are periodic function of time, we assume that the appeared antenna patterns are also periodic functions of time.

We define a periodic function of time, amð Þt , whose period is Ts. The function is assumed to be the weight for m-th antenna element for array antenna implementation and the reactance values of m-th VREs for ESPAR antenna implementation (see Figure 1). The function for a duration of the period 0 ≤ t <T<sup>s</sup> is given by

$$a\_m(t) = \sum\_{k=1}^{N\_1} b\_{m,k} \cdot f\_k(t),\tag{1}$$

where f <sup>k</sup>ð Þt is the k-th function of a set of N<sup>a</sup> orthonormal functions and bm, <sup>k</sup> is a complex-valued coefficient of f <sup>k</sup>ð Þt for m-th element. Since the functions f <sup>k</sup>ð Þt are orthogonal to each other, they have the following property:

$$\frac{1}{T\_s} \int\_0^{T\_s} f\_k(t) \cdot f\_l^\*\left(t\right) dt = \begin{cases} 1 & (k=l) \\ 0 & (k \neq l) \end{cases} \tag{2}$$

where f <sup>∗</sup> <sup>l</sup> ð Þt is the complex conjugate of fl ð Þt . From the orthogonality shown in Eq. (2), we can derive another property for k ¼ l and assume a property for k 6¼ l as

$$\left(f\_k(t)\cdot f\_l^\*\right)(t) = \begin{cases} 1 & (k=l) \\ \exp\left\{j\Theta(t)\right\} & (k\neq l) \end{cases},\tag{3}$$

Antenna Pattern Multiplexing for Enhancing Path Diversity DOI: http://dx.doi.org/10.5772/intechopen.89098

voltage applying to the VREs. Thus, multiplexed antenna patterns can be generated by changing the reactance values jX1ð Þt and jX2ð Þt which are both time-varying functions. Since the relationship between reactance values and generated antenna patterns is nonlinear, even two reactance functions can make several or more than

Note that, in this study, the objective of varying the weights of the antennas is not to control the antenna pattern or form a pattern that satisfies some criteria. We need to simply have the functionality of periodically time-varying antenna patterns. As can be seen from the figure, by ESPAR antennas, we can reduce the size related to antenna elements and the number of cables. Hence, we have selected the ESPAR antennas as a good candidate for utilising APM [10–13, 15, 16]. However, one of the problems relevant to using ESPAR antennas is in its difficulty of designing antenna patterns and time-varying voltage waveform applying to VREs. The difficulty comes from the nonlinear processes of the conversions from voltage to reactance and from reactance to antenna pattern and their time-varying properties. Therefore, to find the optimal set of voltage waveform applying to VREs is an open

In this section, we build a model of the receiver with APM and mathematically

As we mentioned in the previous section, the antenna patterns can be changed by applying periodically time variable voltages to the VREs connected to parasitic antenna elements. Since the applied voltages are periodic function of time, we assume that the appeared antenna patterns are also periodic functions of time. We define a periodic function of time, amð Þt , whose period is Ts. The function is assumed to be the weight for m-th antenna element for array antenna implementation and the reactance values of m-th VREs for ESPAR antenna implementation (see Figure 1). The function for a duration of the period 0 ≤ t <T<sup>s</sup> is given by

amðÞ¼ <sup>t</sup> <sup>X</sup>

orthogonal to each other, they have the following property:

<sup>f</sup> <sup>k</sup>ðÞ� <sup>t</sup> <sup>f</sup> <sup>∗</sup>

<sup>l</sup> ðÞ¼ t

1 Ts

where f <sup>∗</sup>

78

T ðs

0

<sup>l</sup> ð Þt is the complex conjugate of fl

<sup>f</sup> <sup>k</sup>ð Þ� <sup>t</sup> <sup>f</sup> <sup>∗</sup>

N<sup>a</sup>

k¼1

where f <sup>k</sup>ð Þt is the k-th function of a set of N<sup>a</sup> orthonormal functions and bm, <sup>k</sup> is a complex-valued coefficient of f <sup>k</sup>ð Þt for m-th element. Since the functions f <sup>k</sup>ð Þt are

Eq. (2), we can derive another property for k ¼ l and assume a property for k 6¼ l as

�

<sup>l</sup> ð Þ<sup>t</sup> <sup>d</sup><sup>t</sup> <sup>¼</sup> <sup>1</sup> ð Þ <sup>k</sup> <sup>¼</sup> <sup>l</sup>

1 ð Þ k ¼ l exp f g jΘð Þt ð Þ k 6¼ l

�

0 ð Þ k 6¼ l

bm, <sup>k</sup> � f <sup>k</sup>ð Þt , (1)

ð Þt . From the orthogonality shown in

, (3)

(2)

three multiplexed antenna patterns.

Advances in Array Optimization

3. Modelling of antenna pattern multiplexing

derive the received signals in MIMO applications.

3.1 Signals to change antenna pattern

problem.

where Θð Þt is a uniform random process in the interval ½0, 2πÞ. The conventional APM methods use the DC and sinusoids of one or several frequencies as f <sup>k</sup>ð Þt in Eq. (1). In comparison, in this study, we consider the function f <sup>k</sup>ð Þt to be a signal that is spread by using the spreading code sequence used in direct-sequence spread spectrum (DSSS) systems or code-division multiple access (CDMA) systems. Thus, we assume the function f <sup>k</sup>ð Þt can be expressed as

$$f\_k(t) = \sum\_{l=1}^{N\_\varepsilon} c\_{lk} \cdot \mathbf{g}(t - (l-1)T\_\varepsilon),\tag{4}$$

where clk is the l-th chip of the waveform; f <sup>k</sup>ð Þt is assumed to have a complex value with a constant amplitude, j j clk <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffi N<sup>c</sup> <sup>p</sup> ; <sup>N</sup><sup>c</sup> is the number of chips in a period Ts; and g tð Þ is the pulse waveform of a chip. In this paper, we assume that g tð Þ is a rectangular pulse with duration T<sup>c</sup> for simplicity. That is, g tð Þ is shown as follows:

$$\mathbf{g}(t) = \begin{cases} 1 & (0 \le t < T\_c) \\ 0 & \text{otherwise} \end{cases} \tag{5}$$

The product of two functions in Eq. (3) for 0 ≤ t<T<sup>s</sup> can be rewritten as follows:

$$\left(f\_{k}(t)\cdot f\_{l}^{\*}\right)(t) = \sum\_{m=1}^{N\_{\varepsilon}} c\_{mk}\cdot c\_{ml}^{\*}\cdot \mathbf{g}\left(t - (m-1)T\_{\varepsilon}\right)\cdot \mathbf{g}^{\*}\left(t - (m-1)T\_{\varepsilon}\right) \tag{6}$$

$$\mathbf{x} = \begin{cases} 1 & (k=l) \\ \sum\_{m=1}^{N\_c} \mathbf{c}\_{mk} \cdot \mathbf{c}\_{ml}^\* \cdot \mathbf{g} (t - (m-1)T\_c) & (k \neq l) \end{cases} \tag{7}$$

As we can see from Eq. (7), the product can be shown by the product of only chips consisting of f <sup>k</sup>ð Þt and fl ð Þt . Then, we consider a discrete time expression of f <sup>k</sup>ð Þt by introducing vector c<sup>k</sup> whose components are the chips of f <sup>k</sup>ð Þt . The vector can be given as

$$\mathbf{c}\_{k} = \begin{pmatrix} \mathbf{c}\_{1k} \ \mathbf{c}\_{2k} \ \cdots \ \mathbf{c}\_{N\_{c}k} \end{pmatrix}^{\mathrm{T}},\tag{8}$$

where <sup>T</sup> is a transpose operator. Then, we obtain a code matrix, C, by aligning the vectors as follows:

$$\mathbf{C} = \begin{pmatrix} \mathbf{c}\_1 & \mathbf{c}\_2 & \cdots & \mathbf{c}\_{N\_\mathbf{s}} \end{pmatrix} \tag{9}$$

$$= \begin{pmatrix} c\_{11} & c\_{12} & \cdots & c\_{1N\_s} \\ c\_{21} & c\_{22} & \cdots & c\_{2N\_s} \\ \vdots & \vdots & \ddots & \vdots \\ c\_{N\_c1} & c\_{N\_c2} & \cdots & c\_{N\_cN\_s} \end{pmatrix} \tag{10}$$

Since the orthogonality between two functions shown in Eq. (2) is satisfied, the following property of C can be derived:

$$\mathbf{C}^{\mathsf{H}}\mathbf{C} = I\_{N\_{\sf a}} \tag{11}$$

where <sup>H</sup> is an Hermitian transpose operator and IN<sup>a</sup> is the identity matrix of size N<sup>a</sup> � Na.

The waveform amð Þt of Eq. (1) can be shown in a discrete time expression in matrix form as <sup>C</sup><sup>b</sup> by setting a vector, <sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>0</sup> <sup>b</sup>1⋯bN<sup>a</sup> ð ÞT.

## 3.2 Received signals at receiver with APM

In the proposed APM, we apply signal amð Þt given in Eq. (1) to the antenna pattern changing units. Here, we assume that the mapping from the signals amð Þt to the antenna patterns is a linear map<sup>2</sup> . In other words, the generated antenna patterns can be shown in a linear combination of fl ð Þt for l ¼ 1, …, Na.

Then, we consider the antenna pattern for a given direction. Suppose that a ball surrounds the entire receive antenna. On the ball, the p-th received signal path sent by <sup>l</sup>-th transmit antenna arrives at point <sup>ϕ</sup>lp, <sup>θ</sup>lp � �, where <sup>ϕ</sup>lp is an azimuth and <sup>θ</sup>lp is an elevation from the origin of the ball, respectively. We assume that a periodically time-varying far-field antenna pattern, dlp ϕlp, θlp, t � �, which the arrival path experiences, in an equivalent baseband expression can be given as

$$d\_{lp}\left(\phi\_{lp}, \theta\_{lp}, t\right) = \sum\_{k=1}^{N\_z} d\_{klp}\left(\phi\_{lp}, \theta\_{lp}\right) \cdot b\_k \cdot f\_k(t), \tag{12}$$

where dklp <sup>ϕ</sup>lp, <sup>θ</sup>lp � � is the complex-valued coefficient of <sup>f</sup> <sup>k</sup>ð Þ<sup>t</sup> for the direction of arrival path, which could be determined by the direction of the received signal, the structure of the antenna, and the waveforms applied to the antenna. Since the direction can change for each received signal, we assume dklp <sup>ϕ</sup>lp, <sup>θ</sup>lp � � is a random variable, whose amplitude and phase follow a distribution that can be determined by the structure of the antenna and the waveforms applied to the antenna. In discrete time matrix form, Eq. (12) can be shown as CBdlp, where

$$B = \text{diag}(\mathbf{b})\tag{13}$$

The p-th path sent from l-th transmit antenna received at the direction of

where hpl means a channel coefficient of a link between the l-th transmit antenna and p-th direction for the antenna and is a complex Gaussian random

Now we consider the received signals from N<sup>t</sup> transmit antennas. Since N<sup>p</sup> paths per transmit antenna arrive at the receiver, the output from the antenna with APM

where n tð Þ is an additive white Gaussian noise (AWGN) component. As shown in Figure 2, the received signal rpl is multiplied by N<sup>a</sup> multiplexed antenna patterns

Replacing x tð Þ with the corresponding vector x, we have the received signal in

X N<sup>p</sup>

p¼1

dklp <sup>ϕ</sup>lp, <sup>θ</sup>lp � �. Since <sup>N</sup><sup>p</sup> paths are transmitted from the transmit antenna and antenna patterns are orthogonal to each other, the received components for N<sup>p</sup>

rpl ¼ hpl � sl (14)

� � � rpl <sup>þ</sup> n tð Þ (15)

dklp <sup>ϕ</sup>lp, <sup>θ</sup>lp � � � bk � <sup>f</sup> <sup>k</sup>ðÞ� <sup>t</sup> hpl � sl <sup>þ</sup> n tð Þ (16)

CBdlphplsl þ n (17)

<sup>ϕ</sup>lp, <sup>θ</sup>lp � � is given as follows:

Receiving process of the receiver with APM.

Figure 2.

can be shown as follows:

matrix form as

81

x tðÞ¼ <sup>X</sup> Nt

l¼1

<sup>¼</sup> <sup>X</sup> Nt

l¼1

variable with zero mean and variance of unity.

Antenna Pattern Multiplexing for Enhancing Path Diversity

DOI: http://dx.doi.org/10.5772/intechopen.89098

X N<sup>p</sup>

dlp ϕlp, θlp, t

p¼1

X N<sup>p</sup>

X N<sup>a</sup>

k¼1

paths are added in each antenna pattern domain separately.

<sup>x</sup> <sup>¼</sup> <sup>X</sup> Nt

l¼1

p¼1

and diagð Þ b is a diagonal matrix whose diagonal components are given by b and <sup>d</sup>lp <sup>¼</sup> <sup>d</sup>0lp <sup>ϕ</sup>lp, <sup>θ</sup>lp � �d1lp <sup>ϕ</sup>lp, <sup>θ</sup>lp � �… <sup>d</sup>ð Þ <sup>N</sup>c�<sup>1</sup> lp <sup>ϕ</sup>lp, <sup>θ</sup>lp � � � � <sup>T</sup> .

The receiving process of the proposed MIMO receiver with APM is illustrated in Figure 2. We consider that the number of transmit antennas at the transmitter is Nt. Suppose that the channel coefficient is constant during a transmitted symbol. In addition, we assume that the signals transmitted from N<sup>t</sup> antennas suffer independent fading. Also, the transmitter is assumed to have no channel state information. Thus, the average transmit power of each transmitted symbol is assumed to be equivalent to each other. When we show the transmitted symbol from the l-th transmit antenna as sl (Figure 2), then, we can have E sl j j<sup>2</sup> h i <sup>¼</sup> 1 for <sup>l</sup> <sup>¼</sup> 1, …, <sup>N</sup><sup>t</sup> without loss of generality. Besides, the symbol is assumed to be an independent and identically distributed (i.i.d.) random variable. The number of arrival paths per transmit antenna is Np.

<sup>2</sup> In particular, in the case of the ESPAR antenna, the conversions from the applied voltage to the reactance and from the reactance to the antenna pattern could be nonlinear. Then, the assumption might be optimistic in reality. However, in some cases, we have shown for the conversion from the reactance to the antenna pattern that the effect of the nonlinearity can be suppressed by considering the conversion characteristics [15, 16].

Antenna Pattern Multiplexing for Enhancing Path Diversity DOI: http://dx.doi.org/10.5772/intechopen.89098

The waveform amð Þt of Eq. (1) can be shown in a discrete time expression in

In the proposed APM, we apply signal amð Þt given in Eq. (1) to the antenna pattern changing units. Here, we assume that the mapping from the signals amð Þt

is an elevation from the origin of the ball, respectively. We assume that a periodi-

Then, we consider the antenna pattern for a given direction. Suppose that a ball surrounds the entire receive antenna. On the ball, the p-th received signal path sent

� �

dklp ϕlp, θlp � �

� �

is the complex-valued coefficient of f <sup>k</sup>ð Þt for the direction of

B ¼ diagð Þ b (13)

.

. In other words, the generated antenna

ð Þt for l ¼ 1, …, Na.

, where ϕlp is an azimuth and θlp

� bk � f <sup>k</sup>ð Þt , (12)

� �

¼ 1 for l ¼ 1, …, N<sup>t</sup>

is a random

, which the arrival path

matrix form as <sup>C</sup><sup>b</sup> by setting a vector, <sup>b</sup> <sup>¼</sup> <sup>b</sup><sup>0</sup> <sup>b</sup>1⋯bN<sup>a</sup> ð ÞT.

3.2 Received signals at receiver with APM

patterns can be shown in a linear combination of fl

by l-th transmit antenna arrives at point ϕlp, θlp

dlp ϕlp, θlp, t � �

d1lp ϕlp, θlp � �

� � � � <sup>T</sup>

transmit antenna as sl (Figure 2), then, we can have E sl j j<sup>2</sup> h i

where dklp ϕlp, θlp

dlp ¼ d0lp ϕlp, θlp

transmit antenna is Np.

characteristics [15, 16].

80

� �

� �

cally time-varying far-field antenna pattern, dlp ϕlp, θlp, t

experiences, in an equivalent baseband expression can be given as

<sup>¼</sup> <sup>X</sup> N<sup>a</sup>

direction can change for each received signal, we assume dklp ϕlp, θlp

discrete time matrix form, Eq. (12) can be shown as CBdlp, where

k¼1

arrival path, which could be determined by the direction of the received signal, the structure of the antenna, and the waveforms applied to the antenna. Since the

variable, whose amplitude and phase follow a distribution that can be determined by the structure of the antenna and the waveforms applied to the antenna. In

and diagð Þ b is a diagonal matrix whose diagonal components are given by b and

The receiving process of the proposed MIMO receiver with APM is illustrated in Figure 2. We consider that the number of transmit antennas at the transmitter is Nt. Suppose that the channel coefficient is constant during a transmitted symbol. In addition, we assume that the signals transmitted from N<sup>t</sup> antennas suffer independent fading. Also, the transmitter is assumed to have no channel state information. Thus, the average transmit power of each transmitted symbol is assumed to be equivalent to each other. When we show the transmitted symbol from the l-th

… dð Þ <sup>N</sup>c�<sup>1</sup> lp ϕlp, θlp

without loss of generality. Besides, the symbol is assumed to be an independent and identically distributed (i.i.d.) random variable. The number of arrival paths per

<sup>2</sup> In particular, in the case of the ESPAR antenna, the conversions from the applied voltage to the reactance and from the reactance to the antenna pattern could be nonlinear. Then, the assumption might be optimistic in reality. However, in some cases, we have shown for the conversion from the reactance to the antenna pattern that the effect of the nonlinearity can be suppressed by considering the conversion

to the antenna patterns is a linear map<sup>2</sup>

Advances in Array Optimization

Figure 2. Receiving process of the receiver with APM.

The p-th path sent from l-th transmit antenna received at the direction of <sup>ϕ</sup>lp, <sup>θ</sup>lp � � is given as follows:

$$r\_{pl} = h\_{pl} \cdot s\_l \tag{14}$$

where hpl means a channel coefficient of a link between the l-th transmit antenna and p-th direction for the antenna and is a complex Gaussian random variable with zero mean and variance of unity.

Now we consider the received signals from N<sup>t</sup> transmit antennas. Since N<sup>p</sup> paths per transmit antenna arrive at the receiver, the output from the antenna with APM can be shown as follows:

$$\mathbf{x}(t) = \sum\_{l=1}^{N\_t} \sum\_{p=1}^{N\_p} d\_{lp} \left( \phi\_{lp}, \theta\_{lp}, t \right) \cdot r\_{pl} + n(t) \tag{15}$$

$$=\sum\_{l=1}^{N\_t}\sum\_{p=1}^{N\_p}\sum\_{k=1}^{N\_u}d\_{klp}\left(\phi\_{lp},\theta\_{lp}\right)\cdot b\_k\cdot f\_k(t)\cdot h\_{pl}\cdot s\_l + n(t)\tag{16}$$

where n tð Þ is an additive white Gaussian noise (AWGN) component. As shown in Figure 2, the received signal rpl is multiplied by N<sup>a</sup> multiplexed antenna patterns dklp <sup>ϕ</sup>lp, <sup>θ</sup>lp � �. Since <sup>N</sup><sup>p</sup> paths are transmitted from the transmit antenna and antenna patterns are orthogonal to each other, the received components for N<sup>p</sup> paths are added in each antenna pattern domain separately.

Replacing x tð Þ with the corresponding vector x, we have the received signal in matrix form as

$$\mathbf{x} = \sum\_{l=1}^{N\_t} \sum\_{p=1}^{N\_p} \mathbf{C} \mathbf{d}d\_{lp} h\_{pl} s\_l + \mathbf{n} \tag{17}$$

Advances in Array Optimization

$$=\sum\_{l=1}^{N\_t} \text{CBD}\_l \mathbf{h}\_l \mathbf{s}\_l + \mathbf{n} \tag{18}$$

<sup>¼</sup> <sup>B</sup>�<sup>1</sup>

<sup>n</sup>0<sup>H</sup> h i <sup>¼</sup> <sup>E</sup> <sup>B</sup>�<sup>1</sup>

<sup>¼</sup> <sup>E</sup> <sup>B</sup>�<sup>1</sup>

<sup>n</sup>0<sup>H</sup> h i <sup>¼</sup> <sup>B</sup>�<sup>1</sup>

¼ σ2 n N<sup>c</sup> B�<sup>1</sup>

> ¼ σ2 n N<sup>c</sup> B�<sup>1</sup>

absolute value is unity as j j bk <sup>¼</sup> 1, the <sup>k</sup>-th diagonal element of <sup>B</sup>�<sup>1</sup> is <sup>b</sup><sup>∗</sup>

<sup>n</sup>0<sup>H</sup> h i <sup>¼</sup> <sup>σ</sup><sup>2</sup>

¼ σ2 n N<sup>c</sup> B�<sup>1</sup>

¼ σ2 n N<sup>c</sup>

The process of Eq. (27) can be implemented by multiplying bkf <sup>k</sup>ð Þt by x tð Þ in parallel and integrating them over the interval T<sup>s</sup> or with a correlator as shown in Figure 2. Since N<sup>t</sup> transmit antennas are assumed, N<sup>t</sup> components are added in each antenna pattern domain. Note that the process divides a single signal output

If we recognise the matrix DH in Eq. (30) as an equivalent channel matrix G ¼ DH that is equivalent to that of the conventional MIMO systems, we can

Since the length of y is Na, the proposed MIMO with APM seems equivalent to conventional N<sup>t</sup> � N<sup>a</sup> MIMO systems [19]. The number N<sup>a</sup> shows the number of

<sup>3</sup> The orthogonality in time domain does not guarantee the orthogonality in space domain or in terms of directivity. It is a challenging problem to develop a set of orthogonal functions in both time and space

the number of virtual receive antennas in the context of MIMO receivers. The

Thus, the autocorrelation matrix of n<sup>0</sup> is equivalent to that of n.

into N<sup>a</sup> outputs or N<sup>a</sup> antenna pattern domains.

orthogonal antenna patterns in time domain3

E n<sup>0</sup>

E n<sup>0</sup>

Antenna Pattern Multiplexing for Enhancing Path Diversity

DOI: http://dx.doi.org/10.5772/intechopen.89098

E n<sup>0</sup>

where <sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>B</sup>�<sup>1</sup>

modify Eq. (35) as

rewrite Eq. (30) as

domains.

83

<sup>C</sup>HCBDH<sup>s</sup> <sup>þ</sup> <sup>B</sup>�<sup>1</sup>

C<sup>H</sup>n (29)

¼ DHs þ n<sup>0</sup> (30)

<sup>C</sup><sup>H</sup><sup>n</sup> � �<sup>H</sup> h i (31)

CHE nn<sup>H</sup> � �CB�1H (33)

C<sup>H</sup>CB�1H (34)

B�1H (35)

B�1H (36)

B (37)

IN<sup>c</sup> : (38)

y ¼ Gs þ n<sup>0</sup> (39)

. However, the number corresponds to

<sup>k</sup> . Therefore,

C<sup>H</sup>nn<sup>H</sup>CB�1H � � (32)

C<sup>H</sup>n. The autocorrelation matrix of n<sup>0</sup> can be derived as follows;

C<sup>H</sup>n B�<sup>1</sup>

Here, since the code set C and the matrix B are fixed, and from Eq. (21), we have

where B�<sup>1</sup> is a diagonal matrix because B is a diagonal matrix. If we use bk whose

n N<sup>c</sup> B�<sup>1</sup>

we can derive the relation <sup>B</sup>�1H <sup>¼</sup> <sup>B</sup>. With the relation between <sup>B</sup>�1H and <sup>B</sup>, we can

where Dl is an antenna pattern matrix for l-th transmitted symbol whose size is N<sup>a</sup> � N<sup>p</sup> and is given as

$$D\_l = \begin{pmatrix} \mathbf{d}\_{l1} & \mathbf{d}\_{l2} & \dots & \mathbf{d}\_{lN\_p} \end{pmatrix},\tag{19}$$

and h<sup>l</sup> shows a channel vector whose length is N<sup>p</sup> and can be given as

$$h\_l = \begin{pmatrix} h\_{1l} & h\_{2l} & \dots & h\_{N\_\mathbb{P}l} \end{pmatrix}^\mathrm{T},\tag{20}$$

and n is a noise vector whose length is N<sup>c</sup> and whose element nk is an i.i.d. white Gaussian random variable with zero mean and variance σ<sup>2</sup> <sup>n</sup>=Nc. Then, the autocorrelation matrix of n can be defined as follows:

$$\mathbb{E}\left[\mathfrak{m}^{\rm H}\right] = \frac{\sigma\_n^2}{N\_c} I\_{N\_c} \tag{21}$$

Eq. (18) can be further simplified as

$$
\mathfrak{x} = \text{CBDHs} + \mathfrak{n},
\tag{22}
$$

by introducing the antenna pattern matrix D defined as

$$D = (D\_1 \quad D\_2 \quad \dots \quad D\_{N\_t}),\tag{23}$$

and the channel matrix H, which is a block matrix of hl, defined as

$$H = \begin{pmatrix} h\_1 & \mathbf{0} & \dots & \mathbf{0} \\ \mathbf{0} & h\_2 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \mathbf{0} \\ \mathbf{0} & \dots & \mathbf{0} & h\_{N\_t} \end{pmatrix},\tag{24}$$

where 0 is a zero and column vector of length N<sup>p</sup> and s is a vector of transmitted symbols defined as

$$\mathfrak{s} = \begin{pmatrix} \mathfrak{s}\_1 & \mathfrak{s}\_2 & \dots & \mathfrak{s}\_{N\_t} \end{pmatrix}^T,\tag{25}$$

and its autocorrelation function is given as follows from the assumption:

$$\mathbb{E}\left[\mathbf{s}\mathbf{s}^{\mathsf{H}}\right] = I\_{N\_{\mathsf{t}}} \tag{26}$$

The output signal x of the antenna is multiplied by the complex conjugate of the applied waveform. This signal process can be achieved by multiplying B�<sup>1</sup> C<sup>H</sup> by x from the left-hand side, that is, from Eq. (22) to Eq. (11) as

$$\mathbf{y} = \mathbf{B}^{-1} \mathbf{C}^{\mathbf{H}} \mathbf{x} \tag{27}$$

$$=\boldsymbol{B}^{-1}\boldsymbol{C}^{\mathrm{H}}(\boldsymbol{C}\boldsymbol{B}\boldsymbol{D}\boldsymbol{H}\boldsymbol{s}+\boldsymbol{\mathfrak{n}})\tag{28}$$

Antenna Pattern Multiplexing for Enhancing Path Diversity DOI: http://dx.doi.org/10.5772/intechopen.89098

<sup>¼</sup> <sup>X</sup> Nt

N<sup>a</sup> � N<sup>p</sup> and is given as

Advances in Array Optimization

symbols defined as

82

l¼1

where Dl is an antenna pattern matrix for l-th transmitted symbol whose size is

Dl ¼ d<sup>l</sup><sup>1</sup> d<sup>l</sup><sup>2</sup> … dlN<sup>p</sup> � �

h<sup>l</sup> ¼ h1<sup>l</sup> h2<sup>l</sup> … hNp<sup>l</sup> � �<sup>T</sup>

and n is a noise vector whose length is N<sup>c</sup> and whose element nk is an i.i.d.

<sup>E</sup> nn<sup>H</sup> � � <sup>¼</sup> <sup>σ</sup><sup>2</sup>

n N<sup>c</sup>

white Gaussian random variable with zero mean and variance σ<sup>2</sup>

by introducing the antenna pattern matrix D defined as

H ¼

and the channel matrix H, which is a block matrix of hl, defined as

0

BBB@

h<sup>1</sup> 0 … 0 0 h<sup>2</sup> ⋱ ⋮ ⋮ ⋱⋱ 0 0 … 0 h<sup>N</sup><sup>t</sup>

where 0 is a zero and column vector of length N<sup>p</sup> and s is a vector of transmitted

The output signal x of the antenna is multiplied by the complex conjugate of the

and its autocorrelation function is given as follows from the assumption:

applied waveform. This signal process can be achieved by multiplying B�<sup>1</sup>

<sup>y</sup> <sup>¼</sup> <sup>B</sup>�<sup>1</sup>

from the left-hand side, that is, from Eq. (22) to Eq. (11) as

<sup>¼</sup> <sup>B</sup>�<sup>1</sup>

autocorrelation matrix of n can be defined as follows:

Eq. (18) can be further simplified as

and h<sup>l</sup> shows a channel vector whose length is N<sup>p</sup> and can be given as

CBDlhlsl þ n (18)

, (19)

, (20)

IN<sup>c</sup> (21)

CCCA, (24)

C<sup>H</sup> by x

x ¼ CBDHs þ n, (22)

D ¼ D<sup>1</sup> D<sup>2</sup> … DN<sup>t</sup> ð Þ, (23)

1

<sup>s</sup> <sup>¼</sup> <sup>s</sup><sup>1</sup> <sup>s</sup><sup>2</sup> … sN<sup>t</sup> ð ÞT, (25)

<sup>E</sup> ss<sup>H</sup> � � <sup>¼</sup> IN<sup>t</sup> (26)

C<sup>H</sup>x (27)

<sup>C</sup><sup>H</sup>ð Þ CBDH<sup>s</sup> <sup>þ</sup> <sup>n</sup> (28)

<sup>n</sup>=Nc. Then, the

$$\mathbf{h} = \mathbf{B}^{-1}\mathbf{C}^{\mathrm{H}}\mathbf{C}\mathbf{B}\mathbf{H}\mathbf{s} + \mathbf{B}^{-1}\mathbf{C}^{\mathrm{H}}\mathbf{n} \tag{29}$$

$$\mathbf{s} = \mathbf{D} \mathbf{H} \mathbf{s} + \mathbf{n}' \tag{30}$$

where <sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>B</sup>�<sup>1</sup> C<sup>H</sup>n. The autocorrelation matrix of n<sup>0</sup> can be derived as follows;

$$\mathbb{E}\left[\boldsymbol{\mathfrak{n}'}\boldsymbol{\mathfrak{n}'}^{\rm H}\right] = \mathbb{E}\left[\boldsymbol{B}^{-1}\boldsymbol{\mathbf{C}}^{\rm H}\boldsymbol{\mathfrak{n}}\left(\boldsymbol{B}^{-1}\boldsymbol{\mathbf{C}}^{\rm H}\boldsymbol{\mathfrak{n}}\right)^{\rm H}\right] \tag{31}$$

$$\mathbf{h} = \mathbf{E}\left[\mathbf{B}^{-1}\mathbf{C}^{\mathbf{H}}\mathbf{m}\mathbf{m}^{\mathbf{H}}\mathbf{C}\mathbf{B}^{-1\mathbf{H}}\right] \tag{32}$$

Here, since the code set C and the matrix B are fixed, and from Eq. (21), we have

$$\mathbb{E}\left[\mathfrak{n}'\mathfrak{n}'^{\text{H}}\right] = B^{-1}\mathbb{C}^{\text{H}}\mathbb{E}\left[\mathfrak{m}\mathfrak{n}^{\text{H}}\right]\mathbb{C}B^{-1\text{H}}\tag{33}$$

$$\sigma = \frac{\sigma\_n^2}{N\_c} B^{-1} \mathbf{C}^\mathbf{H} \mathbf{C} B^{-1\mathbf{H}} \tag{34}$$

$$=\frac{\sigma\_n^2}{N\_c}B^{-1}B^{-1\text{H}}\tag{35}$$

where B�<sup>1</sup> is a diagonal matrix because B is a diagonal matrix. If we use bk whose absolute value is unity as j j bk <sup>¼</sup> 1, the <sup>k</sup>-th diagonal element of <sup>B</sup>�<sup>1</sup> is <sup>b</sup><sup>∗</sup> <sup>k</sup> . Therefore, we can derive the relation <sup>B</sup>�1H <sup>¼</sup> <sup>B</sup>. With the relation between <sup>B</sup>�1H and <sup>B</sup>, we can modify Eq. (35) as

$$\mathbb{E}\left[\mathfrak{n}'\mathfrak{n}'^{\text{H}}\right] = \frac{\sigma\_n^2}{N\_c} \mathcal{B}^{-1} \mathcal{B}^{-1\text{H}} \tag{36}$$

$$\sigma = \frac{\sigma\_n^2}{N\_c} B^{-1} B \tag{37}$$

$$I = \frac{\sigma\_n^2}{N\_c} I\_{N\_c}.\tag{38}$$

Thus, the autocorrelation matrix of n<sup>0</sup> is equivalent to that of n.

The process of Eq. (27) can be implemented by multiplying bkf <sup>k</sup>ð Þt by x tð Þ in parallel and integrating them over the interval T<sup>s</sup> or with a correlator as shown in Figure 2. Since N<sup>t</sup> transmit antennas are assumed, N<sup>t</sup> components are added in each antenna pattern domain. Note that the process divides a single signal output into N<sup>a</sup> outputs or N<sup>a</sup> antenna pattern domains.

If we recognise the matrix DH in Eq. (30) as an equivalent channel matrix G ¼ DH that is equivalent to that of the conventional MIMO systems, we can rewrite Eq. (30) as

$$\mathbf{y} = \mathbf{G}\mathbf{s} + \mathbf{n}'\tag{39}$$

Since the length of y is Na, the proposed MIMO with APM seems equivalent to conventional N<sup>t</sup> � N<sup>a</sup> MIMO systems [19]. The number N<sup>a</sup> shows the number of orthogonal antenna patterns in time domain3 . However, the number corresponds to the number of virtual receive antennas in the context of MIMO receivers. The

<sup>3</sup> The orthogonality in time domain does not guarantee the orthogonality in space domain or in terms of directivity. It is a challenging problem to develop a set of orthogonal functions in both time and space domains.

equation above realises that the received components obtained by the receiver with APM are similar to those of the conventional MIMO systems. We assume that the receiver has perfect knowledge of the equivalent channel matrix G. Note that the receiver does not need to know every element of D or H for decoding. In practice, it may even be impossible to separately evaluate the components of D and H.

### 3.3 Capacity of MIMO systems with APM

From the received signal in Eq. (39) and the autocorrelation matrix of the transmitted symbols in Eq. (26), we can derive the ergodic capacity C<sup>4</sup> as

$$\mathbf{C} = \mathbb{E}\left[\log\_2 \det\left(I\_{N\_t} + \frac{\chi}{N\_t} G^H G\right)\right] \tag{40}$$

where det is the determinant of a matrix and γ is the average signal-to-noise ratio (SNR) per transmit antenna and is defined as <sup>γ</sup> <sup>¼</sup> <sup>1</sup>=σ<sup>2</sup> <sup>n</sup>. As we can see from Eq. (40), the capacity depends on the property of the equivalent channel matrix G or DH. Here, the matrix G satisfies the following properties which are similar to the channel matrix of the conventional MIMO systems:

$$\mathbf{E}\left[\mathbf{G}\mathbf{G}^{\mathrm{H}}\right] = \mathbf{N}\_{\mathrm{t}}I\_{\mathrm{N}\_{\mathrm{s}}},\tag{41}$$

capacity due to the increase in the number of arrival paths, it can be recognised that the proposed APM obtains path diversity gain. As mentioned in Section 2, APM technique can reduce the antenna size and hardware cost. Thus, we can find that the proposed technique can provide similar capacities with reduced size and less hard-

Ergodic capacity of MIMO systems with APM technique versus average SNR for various number of arrival

Antenna Pattern Multiplexing for Enhancing Path Diversity

DOI: http://dx.doi.org/10.5772/intechopen.89098

We show the ergodic capacities versus average SNR of the proposed MIMO systems with APM for fixed number of arrival paths N<sup>p</sup> ¼ 16 and various number of antenna patterns, i.e., N<sup>a</sup> ¼ 1, 2, 4, 8, and 16, in Figure 4. The number N<sup>p</sup> ¼ 16 might be sufficiently large to obtain the path diversity gain according to Figure 3. For comparison purposes, the performances of the conventional 2 � 1 and 2 � 2

The capacities for APM techniques increase in the number of antenna patterns N<sup>a</sup> and converge to those of the conventional 2 � 1 MIMO systems in lower SNR region and the 2 � 2 MIMO systems in higher SNR region. When N<sup>a</sup> ¼ 16, the capacity almost overlaps the capacity of the conventional 2 � 2 MIMO systems in the region average SNR, which is more than 20 dB. In the case N<sup>a</sup> ¼ 1, the capacity

Ergodic capacity of MIMO systems using receiver with APM technique for various number of orthogonal antenna patterns. Antenna pattern matrix has constant amplitude and random phase with uniform

ware cost.

Figure 4.

85

distribution. (Nt ¼ 2, Np ¼ 16).

Figure 3.

paths Np. (Nt ¼ 2, Na ¼ 8).

MIMO systems are also drawn.

$$\mathbf{E}\left[\mathbf{G}^{\mathrm{H}}\mathbf{G}\right] = \mathbf{N}\_{\mathbf{a}}I\_{N\_{\mathrm{t}}} \tag{42}$$
