MIMO System

A general model of a MIMO system is shown in Figure 7. The input data stream is encoded using vector encoder, modulated in the transmitter and transmitted by M transmitting antennas. The signal is affected by fading and AWGN noise. The receiver has N antennas, Each antenna receives the signals from all M transmit antennas as shown in Figure 7. And therefore the received signals exhibit inter-channel interference. The received signals are down converted to the base band and sampled once per symbol interval. The MIMO processing unit estimates the transmitted data streams from the sampled base-band signals. The vector decoder is a parallel-to serial converter, which combines the parallel input data streams to one output data stream.

In a system with M transmit and N receive antennas, there exist $M*N$ sub-channels between transmitter and receiver. In general, each sub-channel exhibits a selective fading and consequently it is modeled as a linear discrete time finite impulse response (FIR) filter with complex coefficients.

For better understanding, a 4*4 MIMO system is illustrated in Figure 8, consisting of 4 transmitting and 4 receiving antennas. And there exist 16 sub-channels or parallel paths.

In the case of flat fading, the signal in each sub-channel or path is only attenuated and phase shifted due to different propagation times between each receive and transmit antenna. The subchannel is reduced to one tap FIR filter, i.e., one complex coefficient. When the channel is constant during the whole time slot, the channel is quasistatic.

Capacity of MIMO System

For capacity investigation, the wireless channel is considered quasi-static and flat faded. Let, $x_i$ is the transmitted signal from $i^{th}$ antenna.

$h_{ji}$ be the complex channel coefficient from $i_{th}$ transmitting antenna to $j_{th}$ receiving antenna, where, $ i = 1, 2,…, M $ transmitting antennas and $j = 1, 2,…, N$ receiving antennas.

$y_j$ is received signal at $j_{th}$ antenna.

$n_j$ denotes samples of additive white Gaussian noise with variance $\sigma_{n}^2$ at $j_{th}$ receiver.

For simplicity, consider $M = N = 2$,

Receiving antenna 1 receives signals $x_1$ and $x_2$ from antennas 1 and 2 faded with the coefficient $h_{11}$ and $h_{12}$. Therefore received signal $y_1$ at antenna 1 can be represented as,

$y_1 = h_{11}x_1 + h_{12}x_2 + n_1$

Similarly, receiving antenna 2 receives signals $x_1$ and $x_2$ from antennas 1 and 2 faded with the coefficient $h_{21}$ and $h_{22}$. Therefore received signal $y_2$ at antenna 2 can be represented as,

$y_2 = h_{21}x_1 + h_{22}x_2 + n_2$

In matrix notation, received signal y can be represented as $Y= H X + n$ as follows,

Where, H is channel matrix representing channel coefficients between $i_{th}$ transmitting antennas and $j_{th}$ receiving antenna.

For $M = N_T$ and $N= N_R$ , the received signal can be represented as,

where Y is the column vector of the received signals, H is the channel matrix, X is the column vector of the transmitted signal, and n is a column vector of the additive White Gaussian noise. In order to perform analysis of capacity of MIMO channel, various analytical models are available in the literature, for example, Independent Identically Distributed (IID) channel model, separately correlated MIMO channel model, uncorrelated key hole MIMO channel model. The capacity of the system depends only on the transmitted signal power, noise, and channel characteristics. The channel capacity C for flat fading deterministic channel can be expressed as,

$C = log_2 [det(I + \frac{SNR}{M}HH^*)]$.

where,

$SNR = P/ \sigma_{n}^2$,

P: The cumulative power transmitted by all antennas,

$\sigma_{n}^2$: The noise power at each receive antenna,

H: The matrix describing quasi-static channel response,

I: The $ N_R\times N_R $ identity matrix,

The superscript * denotes transpose conjugate of channel matrix H.

With the knowledge of the channel coefficients $h_{ji}$ at the transmitter, we can determine the optimum power distribution at each transmit antenna to achieve maximum capacity.

It is demonstrated that the capacity of MIMO systems can increase linearly with the number of transmit antennas as long as the number of receive antennas is greater than or equal to the number of transmit antennas as illustrated in Figure 9.

Further, multiple parallel signal streams can be transmitted simultaneously in the same frequency band in to parallel subchannels (paths) and thus increase spectral efficiency and capacity. This technique is called spatial multiplexing as illustrated in Figure 7. The path can be viewed as independent radio channels. The column vectors of flat fading channel matrix H are usually nonorthogonal. However, by singular value decomposition (SVD), the channel matrix can be decomposed into diagonal matrix $\sum$ and two unitary matrices U and V,

$H = U \sum V^H$

The diagonal entries of $\sum$ are in fact the non-negative square roots of the eigenvalues of $HH^*$.The number of non - zero eigenvalues $\lambda_1, \lambda_2, \lambda_3, …, \lambda_k$ of $HH^*$ is equal to the rank of channel matrix H and also to the number of independent sub-channels, $ k=M*N$. The global capacity could be expressed as the sum of the sub-channel capacities. In fact, the singular values of channel matrix determine the gains of the independent parallel channels.

Example 1: Consider the transmission $Y= H X + n$ with perfect Channel information at the transmitter over a deterministic point to point MIMO channel whose channel matrix H is given as,

where, X is the transmitted signal vector and n is awgn noise vector. The power transmitted from each antenna is 1W and noise power is 0.1 W with a = b = 1.

Determine the i) Number of transmitting antennas and receiving antennas ii) $HH^*$, iii) Eigen values of the channel matrix and iv) Total capacity of the system v) Compare MIMO capacity with SISO system.

Solution:

i) As in channel matrix H, Number of rows represent number of receiving antennas and Number of columns represent number of transmitting antennas, therefore $N = 2$ and $M = 4$, It is a $4*2$ MIMO system.

ii) On taking the transpose of H and calculating $HH^*$, we get,

$HH^* = 2a^2 \quad 0$

$\quad \quad \quad \quad 0 \quad 2b^2$

iii) Eigen values are the diagonal elements of $HH^*$, so they are $2a^2$ and $2b^2$

iv) Capacity: It is given as,

$C = log_2 [det(I + \frac{SNR}{M}HH^*)]$

where, $SNR = P/ \sigma^2_n$

Since there are 4 transmitting antennas, the cumulative power P will be 4W and M is number of transmitting antennas = 4,

$ I \ is \ 2 \times2$ identity matrix, and $\rho = 4/ 0.1 = 40$. On substituting all values we get, $C = 10.66$ bits/sec/Hz.

v) In SISO, $SNR = 1/ 0.1 = 10$,

therefore $C = log_2(1+ 10) = 3.44$ bits/sec/Hz.

hence in MIMO, capacity is larger than SISO system.

Consider below figure, a data sequence 101 is to be sent over a MIMO system with three transmitters. In the diversity form of MIMO, same data, 101 is transmitted via three antennas over all paths, If each path is subjected to different fading then it is possible that from one of the path the data can be retrieved correctly. Here, this system has a diversity gain of 3. Thus diversity gain depends upon the number of paths available for transmission. Higher the number of transmission paths, higher is the diversity gain.

The second form uses spatial-multiplexing techniques, the data 1,0,1 is multiplexed on the three paths, each available path is considered as subchannel. Each subchannel carries different data, similar to the idea of an OFDM signal. Clearly, by multiplexing the data, throughput or the capacity of the channel is increased, but lost the diversity gain. The multiplexing has tripled the data rate, so the multiplexing gain is 3 but diversity gain is now 0.

Thus in a diversity system the gain comes in form of increased reliability, and in spatial multiplexing the gain comes in form of increased data rate.