. 70 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2019, 3 (2): 70-74 ReseaRch aRticle Capacity Analysis of Multiple-input-multiple-output System Over Rayleigh and Rician Fading Channel Yazen Saifuldeen Mahmood*, Ghassan Amanuel Qasmarrogy Department of Communication and Computer Engineering, Cihan University-Erbil, Kurdistan Region, Iraq ABSTRACT This paper aims to analyze the channel capacity in terms of spectral efficiency of a multiple-input-multiple-output (MIMO) system when channel state information (CSI) is known using water-filling algorithm and unknown at the transmitter side which it has been shown that the knowledge of the CSI at the transmitter enhancing the performance, the random Rayleigh and Rician channel models are assumed. Ergodic capacity and outage probability are the most channel capacity definitions which are investigated in this study. MATLAB code is devised to simulate the capacity of MIMO system for different numbers of antenna nodes versus different signal-to-noise ratio (SNR) values. In addition, the outage capacity probabilities for vary transmission rate and SNR are discussed. Keywords: Channel state information, ergodic capacity, multiple-input-multiple-output, outage capacity, rayleigh, rician, water filling INTRODUCTION Recently, the evolving technologies demand for multiple-input-multiple-output (MIMO) system is explosively growing; MIMO system offers greater capacity than limited capacity of conventional single-input single-output (SISO). MIMO system can able to improve channel capacity, range, and reliability without requiring any additional bandwidth or transmit power. The large transmission rates associated with MIMO channels are due to the fact that a rich scattering environment provides independent transmission paths from each transmit antenna to each receive antenna, i.e., multipath fading channel is produced. Although fading is caused a performance poverty in wireless communication, MIMO channels use the fading to increase the capacity.[1,2] MIMO channel capacity is related to the Shannon theoretic sense. The Shannon capacity is the maximum mutual information of a single user time-invariant channel corresponds to the maximum data rate that can be transmitted over the channel with arbitrarily small error probability. In the time- varying channel, the channel capacity has multiple definitions, depending on what is known about the instantaneous channel state information (CSI) at both transmitter and receiver, and whether or not capacity is measured based on averaging the rate overall channel states or maintaining a fixed rate for most channel states. When CSI is known perfectly both at transmitter and receiver, the transmitter can adapt its transmission strategy relative to the channel using many power control algorithms for wireless networks.[3,4] Therefore, channel capacity is characterized by the ergodic or outage. Ergodic capacity can be defined as expected value of channel capacity which is suitable for fast varying channels. The outage probability is the maximum rate that can be maintained in all channel states with some probability of outage. When only the receiver has perfect knowledge of the CSI, then the transmitter must maintain a fixed rate transmission strategy based on knowledge of the channel statistics only, which can include the full channel distribution just its mean and variance are known. In this case, ergodic capacity defines the rate that can be achieved through this fixed rate strategy based on receiver averaging overall channel states.[5] Alternatively, the transmitter can send at a rate that cannot be supported by all channel states, in these poor channel states that the receiver declares an outage and the transmitted data are lost. The remainder of this paper is organized as follows. The next section presents the multipath fading channels (Rayleigh and Rician). Capacity of MIMO system will be studied in section 3. In section 4, ergodic capacity and outage capacity notions are explained. Section 5, the simulation results with discussions will detail. Finally, major conclusion of the paper is discussed. Corresponding Author: Yazen Saifuldeen Mahmood, Cihan University-Erbil, Kurdistan Region, Iraq. E-mail: yazen.mahmood@cihanuniversity.edu.iq.test-google-a.com Received: Apr 9, 2019 Accepted: Apr 23, 2019 Published: Aug 20, 2019 DOI: 10.24086/cuesj.v3n2y2019.pp70-74 Copyright © 2019 Yazen Saifuldeen Mahmood, Ghassan Amanuel Qasmarrogy. This is an open-access article distributed under the Creative Commons Attribution License. Cihan University-Erbil Scientific Journal (CUESJ) https://creativecommons.org/licenses/by-nc-nd/4.0/ https://creativecommons.org/licenses/by-nc-nd/4.0/ Mahmood and Qasmarrogy: Capacity analysis of MIMO system over rayleigh and rician fading channel 71 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2019, 3 (2): 70-74 FADING CHANNEL MODELS Rayleigh Channel The Rayleigh fading channel is composed from the effects of multipath embrace constructive and destructive interference, and phase shifting of the signal. The Rayleigh distribution is commonly used to describe the statistical time-varying nature of the received envelope of the flat fading signal, or the envelop of an individual multipath component, where there is no line of sight (LOS) path means no direct path between transmitter and receiver. The envelope of the sum of two quadrature Gaussian noise signals obeys a Rayleigh distribution.[6] The Rayleigh distribution has a probability density function (pdf) given by:               2 22 2             0 0              0 r r e rP r r σ σ (1) Where, σ2 is the time-average power of the received signal before envelope detection. Rician Channel When there is LOS, direct path is normally the strongest component goes into deeper fade compared to the multipath components. This kind of signal is approximated by Rician distribution. Rayleigh fading with strong LOS content is said to have a Rician distribution or to be Rician fading.[7] The Rician distribution has a probability density function (pdf) given by:   2 2 2 ( ) 2 02 2 0, 0 0 0 r A r Ar e I A r P r r                  σ σ σ (2) Where, I 0 is the modified Bessel function of the first kind and zero order. The parameter A denotes the peak amplitude of dominant signal. CAPACITY OF MIMO SYSTEM CSI Not Available at the Transmitter MIMO system provides a significant capacity gain over a traditional SISO channel, given that the underlying channel is rich of scatters with independent spatial fading. MIMO systems offered increasing in channel capacity based on the diversity at both the transmitter and the receiver and also on the number of antennas at both sides. When the transmit power is equally distributed among the N transmit antennas in case of CSI is perfectly known at the receiver but unknown at the transmitter, MIMO capacity with N Tx and M Rx antennas can be given in terms of bit per second per Hertz by, * 2log det /MIMO MC I HH bps Hz N            (3) Where, ρ is the signal to noise ratio (SNR), I M is the identity matrix, (*) means transpose-conjugate, and H is the M × N channel transmission matrix. This equation can be reduced as follows:[8] 21 log 1 /   =  = +  ∑ m MIMO ii C bps Hz N (4) Where, m = min (M, N) λ i are the eigenvalues of H H*, known that the square root of λ i is the diagonal matrix, D. The “diagonalization” process has been achieved by performing the singular value decomposition of matrix H which put it in the form. H=U D V* (5) With D being a diagonal matrix, and U and V unitary orthonormal matrices. CSI Available at the Transmitter (Water- filling Algorithm) Water-filling algorithm is an optimum solution related to a channel capacity improvement. This algorithm makes CSI known at the transmitter side which can be derived by maximizing the MIMO channel capacity under the rule that more power is allocated to the channel that is in good condition and less or none at all to the bad channels. Algorithm steps:[9] 1. Take the inverse of the channel gains 2. Water filling has non-uniform step structure due to the inverse of the channel gain 3. At first, take the sum of the total power P t and the inverse of the channel gain. It gives the complete area in the water filling and inverse power gain 1 1n t i i P H  (6) 4. Decide the initial water level by the formula given below by taking the average power allocated (average water level) 1 1n t i i P H channels    (7) 5. The power values of each subchannel are calculated by subtracting the inverse channel gain of each channel 1 1 1 n t i i i P H power allocated channels H       (8) 6. In case, the power allocated value becomes negative stop the iteration process. The capacity of a MIMO system is algebraic sum of the capacities of all channels and is given by the formula below.   n 2i 1Capacity log 1 power allocated * H  (9) Mahmood and Qasmarrogy: Capacity analysis of MIMO system over rayleigh and rician fading channel 72 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2019, 3 (2): 70-74 ERGODIC CAPACITY AND OUTAGE CAPACITY Ergodic Capacity MIMO channels are random in nature. Therefore, H is a random matrix, which means that its channel capacity is also randomly time varying. By taking the ensemble average of the information rate over the distribution of the elements of the channel matrix H, we obtain the ergodic capacity of a MIMO channel. Hence, the MIMO channel capacity can be modeled as follows:[10] C = E{C(H)} (10) The ergodic channel capacity without using CSI at the transmitter side, from (6), is given as follows: m 2 ii 1C E log 1 N               (11) Similarly, the ergodic channel capacity for using CSI at the transmitter side is given as follows: opt m i 2 ii 1C E log 1 N              (12) Where, opt i is the power coefficient that corresponds to the amount of power assigned to the ith channel. Outage Capacity Outage capacity is used for slowly varying channels where the instantaneous SNR is assumed to be constant for a large number of transmitted symbols. Unlike the ergodic capacity definition, schemes designed to achieve outage capacity allow for channel errors. Hence, in deep fades, these schemes allow the data to be lost and a higher data rate can be thereby maintained than schemes achieving Shannon capacity, where the data need to be received without an error overall fading states.[11] The parameter P out indicates the probability that the system can be in outage, i.e., the probability that the system cannot successfully decode the transmitted symbols. Corresponding to this outage probability, there is a minimum received SNR, SNR min given by P out = P(SNR