## Abstract

In this chapter, we show the concept of antenna pattern multiplexing (APM), which enhances path diversity gain and antenna pattern diversity reception in multipath rich fading environment. We discuss the types of antennas that achieve the APM, i.e., generating time-varying antenna pattern and the benefits of reducing antenna size and hardware cost. When electronically steerable passive array radiator (ESPAR) antenna is used, the benefits can be maximised. A model of receiving process is proposed for analysing the ergodic capacity of multiple-input multiple-output (MIMO) systems using APM. We derive a model of received signals to analyse the system performance. The received signal in matrix form includes an equivalent channel matrix, which is a product of antenna pattern matrix, the channel coefficient vector for each output. Numerical results in terms of ergodic capacity show the comparable performances of the proposed MIMO with APM to the conventional MIMO systems; in particular, the number of arrival paths and the number of antenna pattern are sufficiently large. Also the ergodic capacity can be equivalent to that of the conventional MIMO systems when the average SNR per antenna pattern is constant among the virtual antennas.

### Keywords

- antenna pattern multiplexing
- path diversity
- single-input multiple-output
- multiple-input multiple-output
- capacity
- multipath fading

## 1. Introduction

Multiple-input multiple-output (MIMO) systems have attracted much attention as a means to improve the capacity of wireless communications by increasing the number of antennas. However, implementing multiple antennas can be a problem, particularly in mobile terminals due to their space limitation. In this study, we focus on array antennas at the receiver to enhance the capacity.

To resolve the problem, several methods have been proposed to achieve multiple separate received signal components by using a single radio frequency (RF) front-end with electronically steerable passive array radiator (ESPAR) antennas [1]. The modulated scattering array antenna was proposed for diversity and MIMO receivers [2, 3, 4, 5, 6]. The antenna consists of an antenna element for receiving signals and several modulated scattering elements (MSEs) like ESPAR antennas. The impedance of an MSE can be modulated or changed by with an applied sinusoidal voltage of frequency

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

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 the MIMO systems with APM technique is also derived^{1}. Numerical results show 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.

## 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, 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

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

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, 11, 12, 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.

## 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,

where

where

where

where

The product of two functions in Eq. (3) for

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

where

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

where

The waveform

### 3.2 Received signals at receiver with APM

In the proposed APM, we apply signal ^{2}. In other words, the generated antenna patterns can be shown in a linear combination of

Then, we consider the antenna pattern for a given direction. Suppose that a ball surrounds the entire receive antenna. On the ball, the

where

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

The

where

Now we consider the received signals from

where

Replacing

where

and

and

Eq. (18) can be further simplified as

by introducing the antenna pattern matrix

and the channel matrix

where

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

The output signal

where

Here, since the code set

where

Thus, the autocorrelation matrix of

The process of Eq. (27) can be implemented by multiplying

If we recognise the matrix

Since the length of ^{3}. However, the number corresponds to the number of virtual receive antennas in the context of MIMO receivers. The 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

### 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 ^{4} as

where

## 4. Numerical results

In this section, we show the ergodic capacity of a MIMO system whose receiver uses the proposed APM technique. Through the section the number of transmit antennas is

The capacities of the MIMO systems with APM are between those of the conventional

We show the ergodic capacities versus average SNR of the proposed MIMO systems with APM for fixed number of arrival paths

The capacities for APM techniques increase in the number of antenna patterns

Then we consider the case that the average SNR per antenna pattern or virtual antenna is given as

## 5. Conclusions

In this chapter we propose a concept of APM for MIMO receiver to reduce the antenna size and hardware cost with keeping the availability of diversity gain. We discuss the types of antennas which achieve the APM, i.e., generating time-varying antenna pattern. Also, we discuss the benefits of the antennas, in particular, for ESPAR antenna-based structure. The number of virtual antennas or antenna patterns can be increased with the number of multiplexed orthogonal signals used to change the antenna patterns. A model of receiving process is proposed for analysing the capacity of systems using APM. We derive a model of received signals to analyse the system performance. The received signal in matrix form includes an equivalent channel matrix, which is a product of antenna pattern matrix, the channel coefficient vector for each output.

When the number of arrival paths and the number of antenna pattern are sufficiently large, the ergodic capacity approaches to that of

On the other hand, numerical results show that the ergodic capacity is equivalent to that of the conventional MIMO systems when the average SNR per antenna pattern is constant. Then, the proposed APM-based receiver can exploit path diversity gain and antenna pattern diversity maximally without additional physical antenna elements.

Future work is a development of efficient multiplexed antenna patterns, which have larger number of orthogonal antenna patterns than the number of antenna elements equipped with a cable.

## Acknowledgments

This work was carried out by the joint usage/research programme of the Institute of Materials and Systems for Sustainability (IMaSS), Nagoya University.

antenna pattern

antenna pattern multiplexing

code-division multiplexing

code-division multiple access

direct-sequence spread spectrum

electronically steerable passive array radiator

independent and identically distributed

multiple-input multiple-output

multiple-input single-output

modulated scattering element

orthogonal frequency-division multiplexing

phase-shift keying

quadrature amplitude modulation

radio frequency

single-input multiple-output

signal-to-interference-plus-noise ratio

single-input single-output

signal-to-noise ratio

variable reactance element

## Notes

- A part of the derivation is given in the our previous papers for limited cases of antenna pattern multiplexing [11, 17].
- 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].
- 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.
- We use the same variable character as code matrix. Since they are used in different contexts, they might be easily distinguishable.