On Achieving Optimal Channel Capacity of MIMO- PSK Systems over Rayleigh Fading Channel
Sabuj Sarkar Institute of Information and Communication Technology Khulna University of Engineering &
Technology Khulna-9203, Bangladesh
Saikat Adhikary Department of Physics
Adamas University Kolkata-700126, India
Md. Mostafizur Rahman Department of Electronics and
Communication Engineering Khulna University of Engineering &
Technology Khulna-9203, Bangladesh [email protected] Abstract—Continuous demand of higher bandwidth is an
emerging topic in high speed wireless multiple input multiple output (MIMO) systems. Channel capacity is an important paradigm for achieving higher bandwidth as well as higher data rates. This paper mainly focuses on ergodic channel capacity as well as outage capacity of wireless MIMO combined with phase shift keying (PSK) systems over Rayleigh fading channels. First, ergodic channel capacity and distribution capacity in terms of outage channel capacity is analyzed for variable MIMO-PSK systems without knowing the transmitter channel state information (CSI). Finally, joint MIMO-PSK channel capacity in the presence of antenna correlation effect is estimated and consequently the antenna distribution that optimizes the channel capacity is determined.
Keywords- MIMO (Multi input Multi output), PSK (phase shift keying), ergodic channel capacity, Outage channel capacity, antenna correlation.
I. INTRODUCTION
Recent developments on wireless communications is mainly concentrated on MIMO system that provides high spectral efficiency using spatial diversity and multiplexing at both the transmitter and the receiver end contrary to the single-input single-output (SISO) systems. Research on [1-2] shows that MIMO channel capacity increases linearly for minimum transmit-receive antennas. The channel capacity of fading channels are analyzed in [3] using multiple transmission phenomena by emphasizing coding structures as well as equalization of multipath fading channels. A special hyper- geometric matrix termed as moment generating functions for unique propagation scenarios on MIMO arbitrary fading correlation at the transmitter and the receiver is dealt in [4]. An upper-bound on the ergodic sum-capacity are analyzed in [5]
for decentralized multiple-access channel knowing full receiver CSI and distributed CSI. An exclusive spectrum sharing technique is introduced in [6] through a novel receiver that improves ergodic capacity as well as outage capacity and reduces the problem of optimal power efficiently under average transmits and interference power phenomena. By using the frequency division multiple access, optimal bandwidth and power allocation in a cognitive radio has been studied in [7].
Physical layer security in a random wireless network has been investigated in [8] to calculate the probability of secure
connection and ergodic secrecy capacity. The usefulness of various channel capacity namely ergodic capacity, outage capacity and expected source distortion for different outage probabilities over Rayleigh fading channels has been studied in [9]. In [10], a simplified approach termed as dual MIMO configurations is used to analyze the ergodic capacity within an analytical upper bound applying the normalized power. Spatial correlation is studied in [11] by means of two popular MIMO channel models to predict the ergodic MIMO capacity of the reverberation chamber. An exact closed-form expression for maximizing the ergodic capacity of MIMO channels in Rayleigh fading has been studied in [12] to estimate optimal antenna allocation in case of transmitter has no channel knowledge and channel distribution side information (CDIT).
Ergodic, the zero-outage as well as the outage capacity with nonzero outage have been studied in [13]. The ergodic capacity of underlay cognitive (secondary) dual-hop relaying systems is analytically investigated in [14] with the amplify-and-forward transmission protocol over multipath fading and shadowing environments. Research on [15] investigates the capacity of multiple relay channels in different fading and shadowing environments under spectrum-sharing constraints. A novel ergodic capacity formula for decode-and-forward (DF) relaying over a very general fast fading channel model is proposed in [16]. In [17], the sum-capacity of the ergodic fading Gaussian overlay cognitive interference channel (EGCIFC) has been characterized. Ergodic Capacity of Cognitive Radio under Imperfect Channel State Information has been addressed in [18]. The ergodic capacity has been analyzed in [19] of a dual- hop amplify-and-forward relaying system where the relay is equipped with multiple antennas and subject to co-channel interference (CCI) and the additive white Gaussian noise.
Delay-limited capacities of a fixed data rate slow-fading MIMO channel with a long-term power constraint P at the transmitter have been studied in [20].
This paper mainly focuses on ways to increase data rates in order to optimize the channel capacity of MIMO wireless communications by applying variable MIMO-PSK antenna characteristics. The random channel knowledge is adequate to achieve optimal ergodic channel capacity as well as outage capacity.
II. MIMO SYSTEM MODEL
Let us consider a MIMO system with transmit and receive antennas, as shown in Figure 1. A narrowband time- invariant wireless channel can be represented as deterministic matrix H∈ × and transmitted symbol vector ∈ × , which is composed of independent input symbols , ,…, . Then, the received signal ∈ × can be rewritten in a matrix form as follows:
y = + z (1)
1 1
2 2
Figure 1 × MIMO system model
where z=( , , … , ) ∈ × is a noise vector, which is assumed to be zero-mean circular symmetric complex Gaussian. The noise vector z is referred to as circular symmetric when has the same distribution as z for any . The autocorrelation of transmitted signal vector is defined as
= { } (2) In case transmission power for each transmit antenna is assumed to be 1 then Tr( )= .
A. Channel Capacity when CSI is Known
The capacity of a deterministic channel is defined as C=max( ) ( ; ) bits/channel use in which f(x) is the probability density function (PDF) of the transmit signal vector x, and ( ; ) is the mutual information of random vectors x and y.
The channel capacity is the maximum mutual information that can be achieved by varying the PDF of the transmit signal vector. From the fundamental principle of the information theory, the mutual information of the two continuous random vectors, x and y; is given as
( ; ) = ( ) − ( | ) (3) in which ( ) is the differential entropy of y and ( | ) is the the conditional differential entropy of y when x is given.
Using the statistical independence of the two random vectors z and x in Equation (3), we can show the following relationship:
( | ) = ( ) (4) Using Equation (4), we can express Equation (3) as
( ; ) = ( ) − ( ) (5)
From Equation (5), given that ( )is a constant, we can see that the mutual information is maximized when ( ) is maximized. Using Equation (2), meanwhile, the auto- correlation matrix of y is given as
= { }=E{( +z)( + )}
= {( + }
= { + }
= { } + { }
= + (6) Where is the energy of the transmitted signals, and is the power spectral density of the additive noise { } . The differential entropy ( ) is maximized when y is ZMCSCG, which consequently requires x to be ZMCSCG as well. Then, the mutual information of y and z is respectively given as
( ) = log { det( )} (7) ( ) = log { det( )} (8) It has been shown that using Equation (7), the mutual information of Equation (5) is expressed as
( ; ) = log det( + ) bps/Hz (9)
Then, the channel capacity of deterministic MIMO channel is expressed as
= max
( ) log det( + ) (10)
z
Transmitter Channel Receiver
x y Figure 2 Decomposition model for unknown CSI at the
transmitter side
When channel state information (CSI) is available at the transmitter side, decomposition model can be performed as shown in Figure 2, in which a transmitted signal is pre- processed with K in the transmitter and then, a received signal is post-processed with in the receiver. Referring to the notations in Figure 2, the output signal in the receiver can be written as
= + ̌ (11)
K H
Tx Rx
where ̌ = z Using the singular value decomposition in Equation (1), we can rewrite Equation (11) as
= + ̌ (12) which is equivalent to the following r virtual SISO channels, that is,
= + ̌ i=1,2,3,……….r (13) If the transmit power for the ith transmit antenna is given by
= {| | }, the capacity of the i-th virtual SISO channel is
( ) = log (1 + ) i=1,2,3,……….r (14)
Assume that total available power at the transmitter is limited to
{ } = {| | } = (15) The MIMO channel capacity is now given by a sum of the capacities of the virtual SISO channels, that is,
= ( ) = log (1 + ) (16) where the total power constraint in Equation (14) must be satisfied. The capacity in Equation (16) can be maximized by solving the following power allocation problem:
= max
{ } log (1 + ) (17) subject to ∑ = . It can be shown that a solution to the optimization problem in equation (16) is given as
= ( − ), i=1,2,3,……….r (18)
= (19)
where is a constant and ( ) is defined as ( ) = , ≥ 0
0, < 0 (20) B. Channel Capacity when CSI is Not Available at the
Transmitter Side
When H is not known at the transmitter side, one can spread the energy equally among all the transmit antennas, that is, the autocorrelation function of the transmit signal vector x is given as
= (21) In this case, the channel capacity is given as
= log ( + ) (22) Using the eigen-decomposition = ∆ and the identity det( + )= det( + ), where ∈ × and
∈ × , the channel capacity in Equation (22) is expressed as
= log ( + ∆ )= log ( + ∆)
=∑ log (1 + ) (23) where r denotes the rank of H, that is, r= ≜ min ( , ).
From Equation (23), we can see that a MIMO channel is converted into r virtual SISO channels with the transmit power / for each channel and the channel gain of for the i-th SISO channel. Note that the result in Equation (23) is a special case of Equation (15) with = 1, i=1,2,… r, when CSI is not available at the transmitter and thus, the total power is equally allocated to all transmit antennas. If we assume that the total channel gain is fixed, for example ‖ ‖ = ∑ , H has a full rank, = =N, and r=N, then the channel capacity Equation (23) is maximized when the singular values of H are the same for all (SISO) parallel channels, that is,
= , i=1,2,…,N (24) Equation (24) implies that the MIMO capacity is maximized when the channel is orthogonal, that is,
= H= (25) which leads its capacity to N times that of each parallel channel, that is,
= log ( + ) (26) III. FADING MIMO CHANNEL CAPACITY
In Section 2, we have assumed that MIMO channels are deterministic. In general, MIMO channels change randomly.
Therefore, H is a random matrix, which means that channel capacity is also randomly time-varying. In other words, the MIMO channel capacity can be given by its time average. In practice, we assume that the random channel is an ergodic process. Then, we should consider the following statistical notion of the MIMO channel capacity:
̅ = { ( )} = { max
( ) log (
+ )} (27) Which is frequently known as an ergodic channel capacity.
For example, the ergodic channel capacity for the open-loop system without using CSI at the transmitter side, from Equation (23), is given as
= { log (1 + )} (28)
Similarly, the ergodic channel capacity for the closed-loop (CL) system using CSI at the transmitter side, from Equation (23), is given as
= { max
∑ log (1 + )
= {∑ log (1 + )} (29) Another statistical notion of the channel capacity is the outage channel capacity. Define the outage probability as
( ) = ( ( ) < ) (30) In other words, the system is said to be in outage if the decoding error probability cannot be made arbitrarily small with the transmission rate of R bps/Hz. Then, the -outage channel capacity is defined as the largest possible data rate such that the outage probability in Equation (30) is less than . In other words, it is corresponding to such that
( ( ) ≤ ) = (31)
IV. PSKMODULATION
One of the simplest forms of PSK is QPSK which is widely used in wireless communication that uses four points on the constellation diagram, equi-spaced around a circle. With four phases, QPSK can encode two bits per symbol, shown in the Fig. 3 with gray coding to minimize the BER. By quadrature means the signal shifts between phase states which are separated by 90 degrees. There are four states because 2 = 4.
As it has four states so it is a more bandwidth-efficient type of modulation than BPSK, possibly twice as efficient. For PSK modulations we use m-phases depending on angular separation.
Q
01 11
I
00 10
Figure 3 QPSK constellation diagram
V. RAYLEIGH FADING CHANNEL
Rayleigh fading is an acceptable model in multi scattering cases to recover the radio signal arrives at the receiver. The channel impulse response will be well-modelled as a Gaussian process irrespective of the distribution of the individual components. If there is no dominant component to the scatter, then such a process will have zero mean and phase evenly distributed between 0 and 2π radians. The envelope of the channel response will therefore be Rayleigh distributed.
( ) =
Ω , ≥ 0 (31) Where, Ω = ( )
The gain and phase elements of a channel's distortion are conveniently represented as a complex number.
VI. PERFORMANCEANALYSISANDDISCUSSIONS The simulations studies have been carried out using MATLAB toolbox. Figure 4 shows the ergodic channel capacity vs SNR characteristics for variable MIMO-PSK at unknown CSI, under the conditions expressed in table 1. From the figureit is obvious that for higher MIMO-PSK combinations, the ergodic channel capacity increases significantly for higher values of SNR.
Figure 4 Ergodic channel capacity for variable MIMO-PSK at unknown CSI
The simulation parameters those are used for simulation purpose throughout implementing and evaluating optimum channel capacity are given in the table 1.
Table 1: Simulation Parameters
Figure 5 MIMO-PSK channel distribution capacity at unknown CSI
0 5 10 15 20 25 30
0 10 20 30 40 50 60 70
SNR(dB)
Ergodic Channel Capacity (bps/Hz)
Shannon Capacity 2×2 MIMO 3×3 MIMO 4×4 MIMO 5×5 MIMO 6×6 MIMO 7×7 MIMO
Tx-Rx Antenna
Configurations 1~7
SNR (dB) 0~30, 0~20
Channel Characteristics Rayleigh fading Modulation and
Demodulation PSK
Outage capacity
Figure 5 shows the cumulative distribution function (CDF) of the random MIMO-PSK channel capacities when SNR is 10dB, in which =0.1-outage capacity is indicated. It clearly states that the MIMO-PSK channel capacity improves with increasing number of transmit and receive antennas.
Figure 6 Comparison of correlated channel with proposed random MIMO-PSK Rayleigh fading channel
Figure 6 shows comparison of correlated channel with proposed random MIMO-PSK Rayleigh fading channel. From the comparison, it is evident that for proposed random MIMO- PSK Rayleigh fading channel the channel capacity is higher than correlated MIMO-PSK channels for all values of SNR.
So the proposed random MIMO-PSK Rayleigh fading channel provides optimal performance.
VII. CONCLUSIONS
The channel capacity in terms of ergodic as well as outage capacity is studied under unknown CSI conditions and it clearly states that channel capacity significantly enhanced for higher MIMO-PSK combinations over Rayleigh fading channel. The channel capacity improves with higher transmit- receive antennas. The channel capacity for the proposed random MIMO-PSK channels is also estimated through simulation. From all of the analysis, it is concluded that the higher order MIMO-PSK combinations with proposed random distribution provides optimal capacity over Rayleigh fading channel.
ACKNOWLEDGMENTS
The authors would like to acknowledge the support of communication laboratory, Department of Electronics and Communication Engineering, KUET, Bangladesh.
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Correlated Proposed random
distributed
