2. Types of antennas to achieve APM

Before mathematically analysing APM, we discuss the antennas that could realise the proposed APM concept. In APM, several antenna patterns, which are orthogonal to each other in time domain, should be multiplexed in similar manner to code-division multiplexing (CDM) or OFDM. To do so, it is essential that such antennas can generate time-varying antenna patterns. As such antennas, we consider array antennas or ESPAR antennas are good candidates because both antennas can change the antenna pattern from moment to moment.

We show a conceptual figure illustrating conventional array antenna, array antenna with APM, and ESPAR antenna with APM from left to right for comparative purposes in Figure 1. In the figure, we set the number of antenna elements at three as an example. Each antenna model consists of four parts: antenna elements,

<sup>1</sup> A part of the derivation is given in the our previous papers for limited cases of antenna pattern multiplexing [11, 17].

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

#### Figure 1.

diversity reception. The virtually rotating antenna was proposed also to diversify the received signal components in the frequency domain [7, 8]. The principle of setting diversity branches in this domain is similar to that of the modulated scattering array antenna. In the rotating antenna, a combination of reactance values, which generates a desired antenna pattern, is applied sequentially to the multiple reactance elements of the antenna to rotationally change the directivity of the antenna. Also investigated was an ESPAR antenna based on the diversity receiver whose antenna patterns are time variable in a sinusoidal manner and are suitable for the MIMO-orthogonal frequency-division multiplexing (OFDM) receiver [9]. Other researches have been studied on diversity and MIMO receivers with ESPAR antennas having periodically variable antenna patterns by both theoretical and experimental investigations [10–14]. They also investigated the reactance time sequence, which generates sinusoidal antenna patterns with suppressed higher-

In the studies shown above, it can be seen that, instead of using a fixed antenna pattern that may satisfy some criteria, they constantly changed the antenna pattern to generate multiple received signal components in the frequency domain. In this study, we propose a concept of antenna pattern multiplexing (APM) for setting multiple virtual antennas at the same location without additional physical antenna elements. Since the proposed APM also periodically varies antenna patterns to build multiple diversity branches or virtual antennas, it may be possible to consider the APM as a generalised method of the previously mentioned related studies. In APM, instead of sinusoidal waveforms or a sum of sinusoidal waveforms with different frequencies, we apply the sum of a set of orthogonal code sequences as the waveform to change antenna patterns. Therefore, the received signal can be separated into code domains to exploit path diversity instead of using only the narrow fre-

We introduce an antenna pattern matrix that consists of coefficients for each code sequence for each direction of received paths. With the matrix, we can derive the received signals of MIMO systems that use APM-based receivers in a form similar to the signals of the conventional MIMO systems. The ergodic capacity for

that the capacity can be improved by increasing the number of arrival paths and the number of virtual antennas when the coefficients of APM are randomly distributed.

Before mathematically analysing APM, we discuss the antennas that could real-

We show a conceptual figure illustrating conventional array antenna, array antenna with APM, and ESPAR antenna with APM from left to right for comparative purposes in Figure 1. In the figure, we set the number of antenna elements at three as an example. Each antenna model consists of four parts: antenna elements,

<sup>1</sup> A part of the derivation is given in the our previous papers for limited cases of antenna pattern

ise the proposed APM concept. In APM, several antenna patterns, which are orthogonal to each other in time domain, should be multiplexed in similar manner to code-division multiplexing (CDM) or OFDM. To do so, it is essential that such antennas can generate time-varying antenna patterns. As such antennas, we consider array antennas or ESPAR antennas are good candidates because both antennas

. Numerical results show

quency domain, which is the case for the previous studies.

the MIMO systems with APM technique is also derived<sup>1</sup>

can change the antenna pattern from moment to moment.

2. Types of antennas to achieve APM

multiplexing [11, 17].

76

order harmonics [15, 16].

Advances in Array Optimization

The receiver models employing the conventional array antenna, array antenna with APM, and ESPAR antenna with APM of three antenna elements.

antenna pattern hanging units, cables between antenna elements and receivers, and receivers which receive signals through the cables.

In the part of antenna elements, the array antennas have three antenna elements connected to receivers, while ESPAR antenna has an element connected to the corresponding receiver and two parasitic elements which are connected to variable reactance components. In the figure, the shaded elements in ESPAR antenna show parasitic elements. In this part, the distance between neighbouring elements should be more than a half wavelength λ=2 for array antennas to reduce the correlation between the received signals obtained by the elements. On the other hand, since ESPAR antennas form antenna patterns by exploiting mutual coupling between antenna elements, the neighbouring elements need to be sufficiently close to each other. A study on ESPAR antenna with six parasitic elements describes that λ=4 is an appropriate distance [18]. In our previous work on two-element ESPAR antenna, appropriate distances between the elements are around λ=8 [16]. Hence, ESPAR antennas can reduce the space required for antenna elements less than a half of the space of array antennas in the case of three antenna elements. The increase in the number of antenna elements provides more gains in terms of reducing antenna sizes for ESPAR antennas.

The antenna elements are connected to AP changing units, which are weight multiplication for array antennas and variable reactance elements (VREs) for ESPAR antennas. In the conventional array antenna, constant weights w1, w2, and w<sup>3</sup> are multiplied to form an antenna pattern based on a criteria such as maximising the signal-to-interference-plus-noise ratio (SINR) or minimising the interference. In array antenna with APM, the weights are functions of time w1ð Þt , w2ð Þt , and w3ð Þt , which form multiplexed AP and make the received signals travelling through antenna elements separable. Thus, we can add the signals and carry them to the receiver by a single cable. That is, the array antenna with APM can reduce the number of cables between antenna elements and the receiver and decrease their calibration cost. Since the parasitic elements do not connect to the receiver in ESPAR antenna, the cable cost can be also minimised. In ESPAR antenna, the antenna pattern or the directivity can be changed by the reactance values contributed by the parasitic elements. The reactance values of VREs can be changed by the 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 three multiplexed antenna patterns.

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,

where clk is the l-th chip of the waveform; f <sup>k</sup>ð Þt is assumed to have a complex

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

> g tðÞ¼ 1 0ð Þ <sup>≤</sup> <sup>t</sup><T<sup>c</sup> 0 otherwise �

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

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

As we can see from Eq. (7), the product can be shown by the product of only

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

c<sup>k</sup> ¼ ð Þ c1<sup>k</sup> c2<sup>k</sup> ⋯ cNc<sup>k</sup>

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

c<sup>11</sup> c<sup>12</sup> ⋯ c1N<sup>a</sup> c<sup>21</sup> c<sup>22</sup> ⋯ c2N<sup>a</sup> ⋮ ⋮⋱ ⋮ cNc1 cNc2 ⋯ cNcN<sup>a</sup>

Since the orthogonality between two functions shown in Eq. (2) is satisfied, the

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

N<sup>c</sup>

clk � g tð Þ � ð Þ l � 1 T<sup>c</sup> , (4)

<sup>p</sup> ; <sup>N</sup><sup>c</sup> is the number of chips in a

ml � g t<sup>ð</sup> � ð Þ <sup>m</sup> � <sup>1</sup> <sup>T</sup>cÞ � <sup>g</sup> <sup>∗</sup> ð Þ <sup>t</sup> � ð Þ <sup>m</sup> � <sup>1</sup> <sup>T</sup><sup>c</sup> (6)

ð Þt . Then, we consider a discrete time expression of

T, (8)

ml � g tð Þ � ð Þ m � 1 T<sup>c</sup> ð Þ k 6¼ l

C ¼ c<sup>1</sup> c<sup>2</sup> ⋯ c<sup>N</sup><sup>a</sup> ð Þ (9)

1

CCCA

<sup>C</sup><sup>H</sup><sup>C</sup> <sup>¼</sup> IN<sup>a</sup> (11)

(5)

(7)

(10)

we assume the function f <sup>k</sup>ð Þt can be expressed as

Antenna Pattern Multiplexing for Enhancing Path Diversity

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

value with a constant amplitude, j j clk <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffiffiffiffiffi

follows:

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

chips consisting of f <sup>k</sup>ð Þt and fl

can be given as

N<sup>a</sup> � Na.

79

the vectors as follows:

<sup>l</sup> ðÞ¼ <sup>t</sup> <sup>X</sup> N<sup>c</sup>

m¼1

(

P<sup>N</sup><sup>c</sup>

¼

following property of C can be derived:

0

BBB@

cmk � <sup>c</sup> <sup>∗</sup>

<sup>m</sup>¼<sup>1</sup> cmk � <sup>c</sup> <sup>∗</sup>

<sup>f</sup> <sup>k</sup>ðÞ¼ <sup>t</sup> <sup>X</sup> N<sup>c</sup>

l¼1

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 problem.
