**Singular Value Decomposition: Principles And Applications In Multiple Input Multiple Output Communication System**

Wael Abu Shehab^{1} and Zouhair Al-qudah^{2}

^{1}Department of Electrical Engineering, Al-Hussein Bin Talal University, Ma’an, Jordan

^{2}Department of Communication Engineering, Al-Hussein Bin Talal University, Ma’an, Jordan

*Abstract*

The authors discuss the importance of using the singular value decomposition (SVD) in computing the capacity of multiple input multiple output (MIMO) and in estimation the channel gain from the transmitter to the receiver. Examples that show how the SVD simplifies computing the MIMO channel capacity are discussed. Numerical results that show what factors determine the performance of using SVD in channel estimation are also discussed.

**1.Introduction**

One of the pioneering works in communication system is the use of multiple input multiple output (MIMO) which provides a very large spectral efficiency [1], [2]. MIMO transmission based on singular value decomposition (SVD) is an effective mathematical technique to obtain the MIMO channel capacity [1], [3]. Consider a MIMO channel with transmit antenna and receive antenna, modeled as

In more details, SVD was firstly used in [1] where it was shown that the SVD based MIMO transmission is capacity achieving. Ergodic capacity of MIMO-SVD systems have been investigated in [3] and [2] in the case that channel may be modeled as either Rayleigh or Rician fading, respectively. In [5], [6], the performance analysis of MIMO-SVD has been investigated in the context of un-coded transmission. Channel estimation for an MIMO-SVD system has been investigated in [7], [8]. In addition, the effect of channel estimation error on the performance of MIMO-SVD has been proposed in [9] and finally an iterative MIMO channel SVD estimation has been investigated in [10].

The channel capacity is the ultimate data rate that a channel can support without any error. Let us consider the model of fast fading channel where the transmitted signal is multiplied by a random fading coefficient and an additive white Gaussian noise (AWGN) is added as follows

In this paper, an introduction to SVD has been introduced in Section 2 where the basic definitions are discussed. The importance of using SVD in computing the capacity of MIMO systems has been addressed in Section 3. Examples that discuss how the SVD simplifies computing the MIMO channel capacity are also introduced. An iterative technique that is used to estimate the channel gain has been presented in Section 4. Further, many numerical examples that show the performance of this iterative technique is discussed in Section 5. Finally, the paper is concluded in Section 6.

**2.SVD: PRINCIPLES AND PROPERTIES**

**3.SVD FOR MIMO SYSTEMS**

**4.ITERATIVE MIMO-SVD CHANNEL ESTIMATION**

The receiver is designed so that it can estimate the channel state information(CSI). The performance of the MIMO communication system depends highly on the accuracy of the CSI. Any error even if it is small in the estimation of the CSI deteriorates the channel performance. In this section, an iterative MIMO channel SVD estimation technique is introduced.

We start from the channel model described in (1). The estimation procedure can be developed by minimizing the mean square error (MSE) criterion as follows

First, in figure 1, we study the performance of the iterative MIMO channel SVD estimation algorithm based on the NMSE criterio. Specifically, this figure shows the performance in the case that two different training sequence are used and when the number of iterations are . As shown, the performance of the estimator improves significantly when the number of iterations increases. Also, the NMSE and the number of training sequences are inversely proportional. In other words, the NMSE of the channel estimation decreases as the number of the training sequence increases.

Figure 1. The NMSE of the channel matrix estimation Vs. the signal-to-noise ratio (SNR).

Figure 2. The NMSE of estimating the channel matrix Vs. the number of iterations .

Figure 2 shows the performance of the estimation algorithm as a function of the number of iterations and the number of receive antennas . This figure clearly shows that the performance improvement is insignificant after the 4th iteration.

**6.CONCLUSIONS**

An introduction to the SVD has been introduced. The effect of using SVD in MIMO communication system has been discussed. It converts the MIMO system into parallel channel equal to the rank of the channel matrix,. An iterative SVD technique is presented which is used to estimate the channel matrix from the transmit antennas to the receive antennas. Simulation results that show the effect of the number of transmit/receive antennas, the length of the training sequence and number of iterations on the performance of the presented iterative technique have been drawn.

**References**

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