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 N<sup>t</sup> ¼ 2. The capacity C (40) of the proposed MIMO system is shown in Figure 3. We assume that the coefficients of the matrix D are denoted as exp ð Þ <sup>j</sup><sup>Θ</sup> <sup>=</sup> ffiffiffiffiffiffi N<sup>p</sup> <sup>p</sup> where <sup>Θ</sup> is a uniform random variable in the interval 0, 2 ½ Þ <sup>π</sup> . In the figure, the number of antenna patterns or virtual antenna outputs N<sup>a</sup> ¼ 8. If the channel coefficient is constant while a symbol is transmitted, the code set satisfying Eq. (11) could not affect on the performance. Thus, we do not specify the code set in this study. We evaluate the capacities for the number of arrival paths N<sup>p</sup> ¼ 1, 2, 4, 8, 16, and 32 and show them with solid lines. For comparison, we also show the ergodic capacity of the conventional MIMO systems with black dashed lines marked with dots ('�'). The pairs of transmit and receive antennas are 2 � 1 MIMO and 2 � 2 MIMO systems.

The capacities of the MIMO systems with APM are between those of the conventional 2 � 1 and 2 � 2 MIMO systems. In lower SNR region, the capacity of MIMO with APM for various N<sup>p</sup> is close to that of the conventional 2 � 1 MIMO. On the other hand, in higher SNR region, the values and also the slopes of the capacities converge to those of the conventional 2 � 2 MIMO. When the number of arrival paths N<sup>p</sup> increases, the capacities also increased and converged towards the capacity of the conventional 2 � 2 MIMO system. Since the slope of the capacity relates to diversity order, the proposed APM technique can achieve the same order as the conventional 2 � 2 MIMO systems even if N<sup>p</sup> ¼ 1. Because of the increased

<sup>4</sup> We use the same variable character as code matrix. Since they are used in different contexts, they might be easily distinguishable.

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

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.

<sup>C</sup> <sup>¼</sup> E log <sup>2</sup>det IN<sup>t</sup> <sup>þ</sup> <sup>γ</sup>

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

> Nt GHG

(40)

<sup>n</sup>. As we can see from

<sup>E</sup> GG<sup>H</sup> � � <sup>¼</sup> <sup>N</sup>tIN<sup>a</sup> , (41) <sup>E</sup> <sup>G</sup><sup>H</sup><sup>G</sup> � � <sup>¼</sup> <sup>N</sup>aIN<sup>t</sup> (42)

� � � �

where det is the determinant of a matrix and γ is the average signal-to-noise ratio

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 N<sup>t</sup> ¼ 2. The capacity C (40) of the proposed MIMO system is shown in

<sup>p</sup> where <sup>Θ</sup> is a uniform random variable in the interval 0, 2 ½ Þ <sup>π</sup> . In the

Figure 3. We assume that the coefficients of the matrix D are denoted as

figure, the number of antenna patterns or virtual antenna outputs N<sup>a</sup> ¼ 8. If the channel coefficient is constant while a symbol is transmitted, the code set satisfying Eq. (11) could not affect on the performance. Thus, we do not specify the code set in this study. We evaluate the capacities for the number of arrival paths N<sup>p</sup> ¼ 1, 2, 4, 8, 16, and 32 and show them with solid lines. For comparison, we also show the ergodic capacity of the conventional MIMO systems with black dashed lines marked with dots ('�'). The pairs of transmit and receive antennas are 2 � 1 MIMO and 2 � 2

The capacities of the MIMO systems with APM are between those of the conventional 2 � 1 and 2 � 2 MIMO systems. In lower SNR region, the capacity of MIMO with APM for various N<sup>p</sup> is close to that of the conventional 2 � 1 MIMO. On the other hand, in higher SNR region, the values and also the slopes of the capacities converge to those of the conventional 2 � 2 MIMO. When the number of arrival paths N<sup>p</sup> increases, the capacities also increased and converged towards the capacity of the conventional 2 � 2 MIMO system. Since the slope of the capacity relates to diversity order, the proposed APM technique can achieve the same order as the conventional 2 � 2 MIMO systems even if N<sup>p</sup> ¼ 1. Because of the increased

<sup>4</sup> We use the same variable character as code matrix. Since they are used in different contexts, they

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

3.3 Capacity of MIMO systems with APM

Advances in Array Optimization

(SNR) per transmit antenna and is defined as <sup>γ</sup> <sup>¼</sup> <sup>1</sup>=σ<sup>2</sup>

channel matrix of the conventional MIMO systems:

4. Numerical results

N<sup>p</sup>

exp ð Þ <sup>j</sup><sup>Θ</sup> <sup>=</sup> ffiffiffiffiffiffi

MIMO systems.

might be easily distinguishable.

84

Figure 3. Ergodic capacity of MIMO systems with APM technique versus average SNR for various number of arrival paths Np. (Nt ¼ 2, Na ¼ 8).

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 hardware cost.

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 MIMO systems are also drawn.

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

#### Figure 4.

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 distribution. (Nt ¼ 2, Np ¼ 16).

The property deduces the proposed APM, which can obtain diversity gain from

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

Future work is a development of efficient multiplexed antenna patterns, which have larger number of orthogonal antenna patterns than the number of antenna

This work was carried out by the joint usage/research programme of the Insti-

tute of Materials and Systems for Sustainability (IMaSS), Nagoya University.

path diversity and diversity reception based on the virtual antennas.

Antenna Pattern Multiplexing for Enhancing Path Diversity

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

antenna elements.

Acknowledgements

Abbreviations

AP antenna pattern

PSK phase-shift keying

RF radio frequency

87

APM antenna pattern multiplexing CDM code-division multiplexing CDMA code-division multiple access DS/SS direct-sequence spread spectrum

MIMO multiple-input multiple-output MISO multiple-input single-output MSE modulated scattering element

QAM quadrature amplitude modulation

SINR signal-to-interference-plus-noise ratio

SIMO single-input multiple-output

SISO single-input single-output SNR signal-to-noise ratio VRE variable reactance element

ESPAR electronically steerable passive array radiator i.i.d. independent and identically distributed

OFDM orthogonal frequency-division multiplexing

elements equipped with a cable.

Figure 5. Ergodic capacity of MIMO systems with APM technique for various number of orthogonal antenna patterns. Average SNR per antenna pattern is a constant (Nt ¼ 2, Np ¼ 16).

of the proposed MIMO systems with APM is equivalent to that of the conventional 2 � 1 MIMO system. Hence, diversity gain cannot be obtained even if the number of arrival paths is sufficiently large.

Then we consider the case that the average SNR per antenna pattern or virtual antenna is given as γ. In this case, the coefficients in the matrix D are random variables shown as exp ð Þ jΘ where Θ is a uniform random variable in the interval ½ Þ 0, 2π . The ergodic capacities of the proposed MIMO systems with APM are shown for N<sup>a</sup> ¼ 1, 2, 4, and 8 for N<sup>p</sup> ¼ 16 in Figure 5. In the figure, we also illustrate the capacities of the conventional MIMO systems for 2 � 1, 2 � 2, 2 � 4, and 2 � 8 in terms of the numbers of transmit and receive antennas. As we can see from the figure, the capacities are equivalent to each other between N<sup>a</sup> for APM and the same number of receive antennas for the conventional MIMO. For example, when N<sup>a</sup> ¼ 8, the capacity is the same as that of 2 � 8 MIMO systems. Therefore, when the average SNR per antenna pattern is same as the average SNR per the number of receive antennas, the proposed APM-based MIMO systems with N<sup>a</sup> antenna patterns achieve almost equivalent capacity to the conventional N<sup>t</sup> � N<sup>a</sup> MIMO systems.
