Correlation Matrices Design in the Spatial Multiplexing Systems
Sunil Chinnadurai, Poongundran Selvaprabhu, Abdul Latif Sarker^{*}
Department of Electronics Engineering, Chonbuk National University, Jeonju, Republic of Korea
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To cite this article:
Sunil Chinnadurai, Poongundran Selvaprabhu, Abdul Latif Sarker, Poongundran Selvaprabhu. Correlation Matrices Design in the Spatial Multiplexing Systems. International Journal of Discrete Mathematics. Vol. 2, No. 1, 2017, pp. 20-30. doi: 10.11648/j.dmath.20170201.15
Received: January 13, 2017; Accepted: February 6, 2017; Published: February 24, 2017
Abstract: Channel correlation is closely related to the capacity of the multiple-input multiple-output (MIMO) correlated channel. Indeed, the high correlated channel degrades the system performance and the quality of wireless communication systems in terms of the capacity. Thus, we design an inverse-orthogonal matrix such as Toeplitz-Jacket matrix to design transmit and receive correlation matrices to mitigate the channel correlation of the MIMO systems. The numerical and simulation results are performed for both uncorrelated and correlated channel capacities in the case of single sided fading correlations.
Keywords: Transmit and Receive Correlation Matrices, The Correlated MIMO Channel, Inverse-Orthogonal Matrices Toeplitz -Jacket Matrices, The Channel Capacity, The Spatial Correlation
1. Introduction
MIMO communication offers significant capacity gains as well as improved diversity advantage [1], [2]. Multipath fading environment denotes a remarkable challenge in the implementation of reliable wireless MIMO systems. More recently, the development community has organized that using multiple antennas at the transmitter and the receiver can help overcome the detrimental effects of fading using a technique known as antenna diversity. In recent year studies report that in single-user, point-to-point links, using multiple-element arrays (MEAs) at both transmitter and receiver increases the capacity significantly over single-antenna systems [3], [4]. The information-theoretic capacity of MEA systems in a narrow-band Rayleigh-fading environment have analyzed in [3]. They consider independent and identically distributed (i.i.d.) fading at different antenna elements, and assume that the transmitter does not know the channel. Due to the channel variability of time and frequency where circumstances permit coding across many channel coherence intervals, the achievable rate scales as since and are the numbers of transmit and receive antennas, respectively and is the Signal to Noise Ratio.
Thus, the channel capacity of a multiple antenna system can be increased by the factor of without using additional transmits power or spectral bandwidth in [5].
However, on the Shannon capacity of multi-antenna wireless systems, the seminal work of Telatar in [6] has attracted a lot of attention. A single-user MIMO system has started the development with the investigation of the channel capacity. Many results on the capacity for different types of channel state information at the transmitter and/or receiver are known or unknown. Thus, we discuss in some detail about input and output correlation matrices such as transmit and receive correlation matrices and their effects on the channel capacity, with focus on the case of multiple numbers of antennas. The impact of correlation of the channel matrix on the achievable capacity in [4], [7], [8]. Most of the related works are using the assumption, that the channel covariance matrix is the Kronecker product of the covariance matrices of transmit and receive antennas [9], [10]. Many publications dealing with MIMO channel modeling aim at describing the spatial correlation properties of MIMO channels directly, e.g., [11], [12], [13], [14], [15], [16]. Their common approach is to model the correlation at the receiver and the transmitter independently, neglecting the statistical interdependence of both link ends. It may be desirable to study capacity and error rate performance accounting for spatial-correlation effects, due to the propagation channel and the transmit/receive arrays. Typically derived of spatially correlated MIMO channels under certain assumptions about the scattering in the propagation environment. One fashionable correlation model like the stochastic model has been used to investigate the capacity of MIMO radio channels in [11].
The assumption of narrowband channel [17] can be easily extended to include the discrete Fourier transform (DFT) unitary matrices [18] whose defining property is being unitarily equivalent (same singular values) to a channel with independent non-identically distributed entries. The unitary independent unitary channel model in [7] can also revert to the separable correlation model [10, 13, and 19]. In [20], [21], the transmit and receive antennas with orthogonal polarizations may provide low levels of correlation include minimum or antenna spacing while making communication link robust to polarization rotations in the channel. Therefore, we are interested in the correlated MIMO channel with inverse-orthogonal transmit and receive correlation matrices like Toeplitz-Jacket matrices in this paper.
The Toeplitz matrix is introduced in [22], [23] and the Jacket matrix is currently proposed by M. H. Lee in [24], [25]. A Jacket matrix in which all entries are of modulus 1 is called a complex Hadamard matrix in [26], [27]. The Jacket matrices are a generalization of complex Hadamard matrices. A Hadamard matrix is a square matrix whose entries are either +1 or -1 and whose rows are mutually orthogonal. A class of Jacket matrices are motivated by the center weighted Hadamard matrices available in signal processing, image compression, numerical analysis and communications. The researchers have made a considerable amount of effort to develop various kinds of orthogonal transforms. Since the orthogonal transform with the independent parameters can carry many different characterizations of digital signals, it is interesting to investigate the possibility of generalization of Hadamard and DFT and so on.
This paper is organized as follows:
In Section 2 and Section 3 gives some details about the system model and Toeplitz structure of transmit and receive correlation matrices design, respectively. In Section 4 we design Toeplitz-Jacket structure of transmit and receive correlation matrices. In Section 5, we describe the capacity of the MIMO deterministic channel. We analyze the numerical result in Section 6. Finally, we investigate the simulation results and conclusions are presented in Section 7 and Section 8.
2. System Model
Let us consider a narrowband point-to-point MIMO system with transmit and receive antennas which are modeled as:
(1)
where and are the received and transmitted signal vectors at time , is the energy of the transmitted signals and is the zero-mean independent and identically distributed (i.i.d.) complex Gaussian noise vector at time include covariance matrix. The matrix is assumed MIMO channel matrix to be Rayleigh fading i.e. .
The channel matrix for the cases in which we have correlated transmit and outputs receive antennas is approximately modeled as [7, 8, 9, 10, 11].
(2)
where and are the deterministic transmit and receive correlation matrices, respectively while is i.i.d., Rayleigh-faded channel. The operator is called Hermitian. The deterministic transmit and receive side correlation matrices can be defined as:
(3)
(4)
and
(5)
Therefore, we can write a MIMO channel correlation matrix as follows:
(6)
where the Kronecker product of of and is defined as:
(7)
whereas, we can be defined the transmit and receive matrices, and are
,
And
For an example 1: If transmit and receive correlation matrices as follows [11]:
Let, and,
So the MIMO correlation matrix becomes matrix as:
(8)
However, the general treatment of matrix we refer the reader to [7]. Under the virtual channel condition in [28], the use of uniform linear arrays (ULAs) at the transmitter and the receiver makes and are an approximately Toeplitz. Thus, we will discuss about an approximately Toeplitz structure of transmit and receive correlation matrices, and are in the next Section 3.
3. Toeplitz Structure of Transmit and Receive Correlation Matrices Design
Definition of Toeplitz Matrix: It is well known that a Toeplitz matrix is a matrix where for very, i.e., form in [22, 23]
(9)
By the definition of (9), we recall Theorem 1 as
Theorem 1: A square correlation matrix is an approximately Toeplitz.
Proof of Theorem 1: Let and the observation vector denote the elements of the time series,,…,. To show the composition of the vector explicitly, we write,
(10)
Now we define the correlation matrix of a stationary discrete-time stochastic process denoted by the time series as the expectation of the outer product of the observation vector with itself. Let is a correlation matrix, reflecting the correlations between the rows or column vectors of and define in this way. Thus we obtain from [29]
(11)
where the operator represents Hermitian transposition, i.e., the operation of transposition combined with complex conjugation. By substituting "(10)" in "(11)"
(12)
Hence, we claim "(12)" is an approximately Toeplitz matrix. Similarly we can rearranged backward of an observation vector is given by
(13)
In "(12)" and "(13)" shows that the element on the main diagonal is always real valued. For complex valued data, the remaining elements of assume complex values. The definition of "(9)" and "(12)", we can claim transmit and receive correlation matrix, and is an approximately Toeplitz i.e.,
(14)
(15)
Therefore, the correlation matrix plays a key role in the statistical analysis and design of discrete-time filters in [29]. It is therefore important that we understand its various properties and their applications. Especially, the definition of "(11)", we find that the correlation matrix of a stationary discrete-time stochastic process the following properties as [29]:
Case1: Correlation matrix of a stationary discrete-time stochastic process is Hermitian.
So a complex-valued matrix is Hermitian if it is equal to its conjugate transpose. We may thus express the Hermitian properties of the correlation matrix by writing
(16)
where this property follows directly from the definition of "(12)". Thus the Hermitian property of the is to write from "(12)"
(17)
where the operator is called conjugate transpose and is the auto-correlation function of the stochastic process for a lag of . Accordingly, for a wide-sense stationary process we only need or values of the auto-correlation function for in order to completely define the matrix. Thus, we may rewrite from "(12)" as follows:
(18)
Case2: Correlation matrix of a stationary discrete-time stochastic process is Toeplitz.
By "(12)", we can say, a correlation matrix is Toeplitz, if all elements on its main diagonal are equal and if the elements on any other diagonal parallel to the main diagonal are also equal. From the expanded form of given in "(18)", we see that all the elements on the main diagonal are equal to , all the elements on the first diagonal above the main diagonal are equal to , all the elements along the first diagonal below the main diagonal are equal to, and so on for the other diagonals. Therefore, we conclude that transmit and receive correlation matrices, and is an approximately Toeplitz which has shown in "(14)" and "(15)".
4. Design Toeplitz - Jacket Structure of Transmit and Receive Correlation Matrices
Let a square correlation matrix is called Jacket matrix in [30] or inverse-orthogonal Toeplitz matrix in [31, 32], or type II matrix in [33], if its inverse matrix satisfies:
(19)
i.e., the inverse matrix can be obtained by taking element-wise inverse and transposition up to a negligible constant factor. Equivalently these matrices satisfy the following relations [31], [32] as
(20)
where is the Kronecker delta- a function of two variables usually integers,
(21)
If is nonzero constant then the definition of Jacket matrix in [30] can be rewritten as follows: A square matrix
(22)
is called a Jacket matrix if it’s normalized element-inverse transposed
(23)
Satisfies,
(24)
where operator is called transpose inverse.
For an example 2: Let , and is nonzero complex numbers then Toeplitz-Jacket matrix of order 4 becomes from [31],
(25)
and the element-wise inverse ofis given by
(26)
where (see in [34]). Particularly, a real Hadamard matrix of order with the Toeplitz structure is either circulant or negacyclic in [31, corollary 3.1]. So we can write by "(12)" and [31]:
(27)
We see that when the "(27)" becomes,, then by "(12)", we get for i.e., then matrix is circulant. Otherwise, then by "(12)", we get hence i.e., the matrix is negacyclic. The well-known example of circulant complex Hadamard matrices, see e.g. [35]. Therefore, when be an arbitrary of a matrix and be a nonzero complex number, then matrix of order is given by
(28)
where the entry of reads.
Hence Hadamard matrices have both theoretical applications ranging from harmonic analysis [36] to quantum information theory [32]; as well as applications in signal processing [30,37]. Thus, we recall Theorem 2 as follows:
Theorem 2: Toeplitz-Jacket structure of correlation matrices is approximately circulant one.
Proof of Theorem 2: Let, from "(12)", where the operator represents the principal root, the matrix, is given by "(27)"
(29)
where and is adiagonal matrix, respectively and we claim Toeplitz-Jacket structure of correlation matrixis a circulant matrix as in [31]. Finally, setting "(29)" in "(2)", then the renovated channel matrix becomes
(30)
where , and.
5. Capacity Analysis of MIMO Deterministic Channel
The well-known formula of the capacity of a deterministic channel is defined as [38-39]
(31)
where is the probability density function (PDF) of the transmit signal vector, and is the mutual information of random vector and is given by
(32)
is the differential entropy of with covariance in [6], since is obtained by
,
is the conditional entropy of when is given and is a constant. In "(32)" shows that the mutual information is maximized when is maximized. Therefore, when CSI is present at transmitter side, the capacity of deterministic MIMO channel is given by
(33)
When the MIMO channel matrix is randomly changed, then the channel capacity is also randomly time-varying [39] since the random channel is an ergodic process. Therefore, the average capacity is calculated by "(33)" for known channel with CSI at the transmitter side,
(34)
5.1. Analysis of Uncorrelated Channel Capacity
For real channel realization, the transmitter has no CSI and its power is equally allocated to each transmitting element. Thus, the matrix is given by
(35)
In this case, the channel capacity "(33)" is obtained by "(35)"
(36)
The SVD expansion of a matrix is
(37)
where is an unitary matrices which means that
(38)
and the diagonal matrix, is given by
(39)
where . So the capacity (36), expressed in terms of the singular values
(40)
The "(40)" has lower and upper bound as follows [5]
(41)
Using "(5)" in "(41)" becomes
(42)
Therefore, for low SNR case whereas consider is a fixed SNR value, the capacity "(36)" becomes
(43)
In "(43)" expression is independent of, and thus, even under the most favorable propagation conditions the multiplexing gains are lost, and from the perspective of the capacity, multiple transmit antennas are of no value.
If and as a consequence [5]
(44)
Thus, the capacity "(36)" is given by
(45)
which matches the upper bound "(42)". Similarly, If and as a result
(46)
With the equality, combines "(32)" and "(46)", yields
(47)
which matches the upper bound "(42)".
5.2. Analysis of Correlated Channel Capacity
When spatial correlation is applied in "(36)", then using "(2)" in "(36)", the channel capacity is given by [39]
(48)
With the equality, this can be written to
(49)
By setting the condition and in "(49)", that means no correlation exists between the receive antennas. Thus, the capacity "(49)" becomes,
(50)
which matches as "(44)".
By setting another condition and in "(49)", that means no correlation exists between the transmit antennas. Then, the capacity "(49)" becomes,
(51)
If , and is full rank and when the SNR is high, the channel capacity "(49)" can be approximated as [39]
(52)
where term is always negative by the fact that for any correlation matrix. So the MIMO channel capacity has been reduced in "(52)". Since the determinate of a unitary matrix is unity, so the determinate of correlation matrix is given by
(53)
and the geometric mean is bounded by the arithmetic mean, that is,
(54)
From "(53)" and "(54)", it is obvious that,
(55)
The equality in "(55)" holds when the correlation matrix is identity matrix. Thus, the quantity in term, are all negative. As a result, we assume an approximately Toeplitz-Jacket structure of transmit and receive correlation matrices as "(29)" to reduce the reflection of input and output correlation matrices of the MIMO correlated channel as in "(30)". Similarly, by setting the values of "(29)" and "(30)" in "(50)", "(51)" and "(52)" and can be written:
(56)
(57)
and
(58)
6. The Numerical Analysis
The channel correlation of a MIMO system is mainly due to two components such as spatial correlation and the antenna mutual coupling. Except the spatial correlation will contribute to the correlation, antenna mutual coupling will also contribute for MIMO system in [40]. In the transmitter antenna mutual coupling causes the input signals being coupled into neighboring antennas. The antenna mutual coupling influences both the spatial correlation and SNR, is taken into account by means of the impedance matrix in [5], [40].
In this paper, we will ignore the antenna mutual coupling and assume the spatial correlation scheme for simple example of enumeration that the channels are Gaussian random channels with a unit variance and a zero mean. For a measured MIMO i.i.d channel has the following form:
(59)
In general, we define the spatial correlation coefficient between the channels as [40], [41], [42]
(60)
where and the operator is a called zero order Bessel function, is the transmit branch, is the receive branch, the antenna separations at the transmitter and receiver are and, respectively.
For example 3: Let us consider the antenna separations at the transmitter and receiver are, in [40] and the i.i.d. MIMO channel is designed a MIMO system where such as and equipped with dipole antenna aligned as uniform linear arrays (ULAs) as follows:
(61)
Since the transmit correlation matrix, with fix receiving antenna can be calculated as:
(62)
By using "(26)" and "(62)" is given by
(63)
which matches the negative values as "(26)" and the transmit correlation matrix "(62)" becomes negacyclic. Thus, we get, the operator represents the principal root of transmit correlation matrix. Then, the transmit correlation matrix "(62)" becomes a new transmit correlation matrices is given by:
(64)
where,
,
and which matches the "Jacket" conditions as in [30], [34].
Similarly, the antennas are dipoles, so the channel correlation matrix at receiver with fix transmitting antenna can be calculated as:
(65)
Similarly, by (65) and (26), we get
(66)
which matches the negative values as "(26)" and the receive correlation matrix "(89)" becomes negacyclic. Thus, we get, the operator represents the principal root of receive correlation matrix. Then, the receive correlation matrix "(65)" becomes new receive correlation matrices is given by:
(67)
where
,
and .
and which matches the "Jacket" conditions as in [30, 34].
7. Simulation Results
In this section, all Tables and Figures show the channel average and total capacity among the transmitter and receiver side correlation at 20 [dBs] and 30 [dB] SNR including 4x8 MIMO system, respectively. The Table 1 illustrates the average channel capacity in three different channel model at one side correlation like transmitter side correlation and 20 [dBs] SNR include 4x8 MIMO systems. It is evident that the i.i.d., channel always provides higher channel capacity between correlated channels. When i.i.d., channel generates the average channel capacity 13.4026 bps/Hz at 20 [dBs] SNR, then the correlated channel produces 12.4258 bps/Hz and 13.1042 bps/Hz at the same power. The Table 1 also shows the percentage of average channel capacity between ordinary and proposed channel model at transmitter side correlation. The percentage of proposed correlated channel model is 5.45% larger than ordinary channel model. When the receiver side correlation is applied in 4x8 MIMO systems at the equal power, the average channel capacity is significantly decreased at correlated channel model. In Table 2 shows 3.51% of average channel capacity at receives side correlation and 20 [dBs] SNR values. Similarly, we can depict Table 3 and Table 4 using transmitter and receiver side correlation at 30 [dBs] SNR values. However, while 30 [dBs] SNR values apply in 4x8 MIMO systems, the percentage of average channel capacity (4.10%) inherently increases at transmitter side correlation moreover the remarkable degrades the percentage of average capacity (2.03%) at receiver side correlation which has been shown in Table 3 and Table 4 in this paper.
Channel | Transmitter Side | Percentage (%) |
Model | Correlation only | at Correlated Channel |
i.i.d. Channel"(36)" | 13.4026 bps/Hz | |
Conv. Corr. Channel "(50)" and "(62)" | 12.4258 bps/Hz | 5.45% |
Prop. Corr. Channel "(56)" and "(64)" | 13.1042 bps/Hz |
Channel | Receiver Side | Percentage (%) |
Model | Correlation only | at Correlated Channel |
i.i.d. Channel "(36)" | 13.3698 bps/Hz | |
Conv. Corr. Channel "(50)" and "(65)" | 9.7379 bps/Hz | 3.51% |
Prop. Corr. Channel "(56)" and "(67)" | 10.0800 bps/Hz |
Channel | Transmitter Side | Percentage (%) |
Model | Correlation only | at Correlated Channel |
i.i.d. Channel "(36)" | 19.4997 bps/Hz | |
Conv. Corr. Channel "(50)" and "(62)" | 18.3098 bps/Hz | 4.10% |
Prop. Corr. Channel "(56)" and "(64)" | 19.0611 bps/Hz |
Channel | Receiver Side | Percentage (%) |
Model | Correlation only | at Correlated Channel |
i.i.d. Channel "(36)" | 19.5300 bps/Hz | |
Conv. Corr. Channel "(50)" and "(65)" | 14.3781 bps/Hz | 2.03% |
Prop. Corr. Channel "(56)" and "(67)" | 14.6710 bps/Hz |
The line graph which has been generated by Mat-Lab program compares the channel capacity in [bps/Hz] at 20 [dBs] and 30 [dBs] SNR values in the three different channel models in Figure 1, Figure2, Figure 3 and Figure 4, respectively. The Figure 1 exhibits the channel capacity at 20 [dBs] SNR values in the case of transmitter side correlation. When the transmitter and receiver side correlation is applied at 20 [dBs] in two different correlated channel models, then the proposed channel gain performs 1.02 [bps/Hz] over than ordinary correlated channel has been shown in Figure 1 and Figure 2. Overall, it can be seen that in Figure 3 and Figure 4 represents channel gain at least 1 [bps/Hz] than general correlated channel model.
8. Conclusions
In this paper, we have mainly investigated the ordinary Toeplitz structure and the proposed Toeplitz-Jacket structure of transmit and receive correlation matrices over the MIMO correlated channel. There are two prospects in this paper. In one is lower order channel matrix and the other is higher order of channel matrix. When a lower order channel matrix is applied to MIMO system, the capacity and performance of an ordinary Toeplitz structure are better than proposed method. In contrast, a higher order of channel matrix is applied to MIMO system, the performance of proposed Toeplitz-Jacket structure is much better than ordinary method. Thus, we can say the proposed Toeplitz-Jacket structure of transmit and receive correlation matrices is suitable for the case of the high dimensional MIMO system. Future research works will more universal cases such as more than recent general scenarios.
References