On the capacity of MIMO correlated Nakagami- m fading channels using copula
- Mohammad Hossein Gholizadeh^{1},
- Hamidreza Amindavar^{1}Email author and
- James A Ritcey^{2}
https://doi.org/10.1186/s13638-015-0369-3
© Gholizadeh et al.; licensee Springer. 2015
Received: 8 January 2015
Accepted: 20 April 2015
Published: 19 May 2015
Abstract
In this paper, a novel approach is proposed based on the probability density function (PDF) concept to achieve the capacity of a correlated ergodic multi-input multi-output (MIMO) channel with Nakagami-m fading. In our proposed method, channel parameters are unknown, and they are initially estimated by using the PDF of the received samples in the receiving antennas. The copula theory is employed to estimate the parameters of the channel in the proposed PDF-based approach. By appealing to copula, the notion of PDF estimation is simplified in the computation technique when we are faced with correlated signals. Since we are working on a correlated channel, the copula concept results in a powerful estimation approach for the PDF of the signals in the receivers. Accurate PDF estimation leads to having a precise calculation for channel parameters. Hence, the new approach guarantees that the capacity of a correlated ergodic channel is predicted reliably. In the previous works, either the capacity of simple uncorrelated Nakagami-m channels is presented or an asymptotic formulation is suggested for a correlated Nakagami-m channel. However, our proposed method introduces an analytic expression for the capacity of the MIMO correlated Nakagami-m fading channel relying on copula. All the results in both channel parameter estimation and channel capacity prediction are validated with some simulations.
Keywords
MIMO correlated Nakagami-m fading channel PDF estimation Copula theory Channel capacity1 Introduction
The transmission over multi-input multi-output (MIMO) channels offers significant increases in data throughput and link range without additional bandwidth or increased transmit power and results in higher capacity [1,2]. It is often supposed that the channel state information (CSI) is perfectly known at the receiver. However, in the actual environment, the channel has to be estimated. Precise estimation of the channel parameters critically helps in obtaining an appropriate design for the communication systems. Since there are more channel parameters in MIMO channels, a more powerful approach is required for the estimation.
There is another idealized assumption about channel coefficients that are considered to be independent and identically distributed (i.i.d) [3,4]. However, the mentioned assumption is not practical, on the other hand, in many practical situations, there exists a correlation among the antennas. This is due to poor scattering conditions or physical vicinity among the antennas [5]. Thus, the investigation about the behavior of MIMO systems in correlated fading environments is of interest [6].
Since the Rayleigh model is a reasonable assumption for the fading in many wireless communication systems, it is often supposed that the MIMO channel fading is Rayleigh distributed [6,7]. Nevertheless, the measurements [8] conclude that the Nakagami-m model presents a better fitting to the fading channel distribution. Achieving more similarity to the actual environment, the usual uniform probability density function (PDF) assumption for the phase of the Nakagami-m model is not adopted in this paper, and a more reliable PDF is considered [9].
In this paper, a 2×2 MIMO channel is considered in which the transmitting antennas are close and correlated, and the receiving antennas are far and independent [7,10]. Note that it is able to be generalized to arbitrary number of transmitters and receivers. The Nakagami-m model is also assumed for the fading environment. In addition to changing the amplitude of the transmitting signal due to the fading, it also results in a nonuniform phase shifting in the transmitting signal. A new PDF-based approach is applied to derive the Nakagami-m parameters related to two different paths from transmitters to two isolated receivers. This method is also capable of estimating the correlation parameter between the signals sent from two near transmitters.
Since we are faced with the correlated signals and our method is PDF-based, the powerful concept of copula effectively improves the proposed estimation method. The copula theory is suitable when two or several random variables are dependent. Thus, to calculate the total PDF of the received signal in the receivers, which includes some correlated parts, the copula theory helps us to attain a more precise PDF, and it results in having more reliable estimated parameters.
After estimation, the MIMO channel capacity is predicted relying on the estimated parameters. Since we have a correlated Nakagami-m channel, the copula theory is again employed to achieve the PDF of eigenvalues of the channel matrix, and by using the obtained PDF, the capacity is calculated.
Symbols and mathematical notations
Notation | Meaning |
---|---|
(.)^{ T } | Transpose of matrix |
(.)^{ † } | Complex conjugate transpose of matrix |
|R| | Determinant of matrix R |
tr(.) | Trace operator |
Expectation value | |
C | Copula function |
c | Copula density function |
F | Cumulative distribution function |
f | Probability density function |
I | Identity matrix |
H | Channel matrix |
C _{ t } | Channel capacity |
ρ | Linear correlation parameter |
α | Clayton copula parameter |
ν | t copula parameter |
2 Copula
One of the popular methods in modeling the dependencies is the copula approach. The copula was first employed by Sklar in mathematical and statistical problems [11]. Copula is a mathematical function that combines univariate PDFs to produce a joint PDF with a particular dependency structure. In this paper, the estimation of fading parameters is done by using the PDF of the received signal, given that the received signal is one of the MIMO system outputs including the sum of several correlated signals transmitted through the MIMO channel. Moreover, this signal is corrupted by an independent noise. Due to the correlated nature of the received signal, we are required to determine the PDF of a signal that is composed of several dependent components. Thus, the copula concept is a powerful tool that is suitable for our problem, and it facilitates the PDF estimation procedure. The fundamental theorem for copula was given by Sklar. Based on the Sklar theorem, for a given joint multivariate PDF and the relevant marginal PDFs, there exists a copula function that relates them. In a multivariate case, Sklar’s theorem is as follows:
Conversely, if C is a copula and F _{1} ,…, F _{ n } are CDFs, then the function F defined by ( 1 ) is an n-dimensional CDF with margins F _{1} ,…, F _{ n }.
The proof of the theorem could be seen in [12]. Function C has some inherent properties, a description of which can be found thoroughly in [12]. Based on the copula properties, we can state that a copula is itself a CDF, defined on [0,1]^{ n }, with uniform margins.
where f _{1}(x _{1}),…,f _{ n }(x _{ n }) are the marginal PDFs and c(.) is the copula density function. It is shown in (3) that a multivariate PDF is constructed by multiplying a copula density function and a set of marginal PDFs in which the copula density function can be selected independent of the margins.
The copulas are divided into two groups. The first one is the family of elliptical copulas. The most prominent elliptical copulas are normal and Student’s t. We can specify different levels of dependency between the margins in an elliptical copula, and it is a suitable feature of this group. The second class of copulas is known as the Archimedean copulas. The ease with which they are constructed, the great variety of copulas that belong to this class, and modeling the dependence in arbitrarily high dimensions with only one parameter are the popular properties of this family [12].
where R is the correlation matrix, u is the vector u=[u _{1,}…,u _{ n }] in which the ith element is u _{ i }=Φ ^{−1}(F _{ i }(x _{ i })) that Φ ^{−1} is the inverse of the univariate standard normal CDF. |.| and (.)^{ T } denote the determinant and transpose of the matrix, respectively.
It is called the normal copula because similar to normal distribution, it also enforces dependency by using pairwise correlations among the variables. However, in the normal copula, the marginal distributions are arbitrary. After discussing the copula concept and correlation modeling, a correlated channel is presented in the next section, and the parameters of the mentioned channel are estimated by using the copula function.
3 MIMO system model
4 Channel parameter estimation
where n _{ k }(t) is an independent zero-mean normally distributed random process, and ℓ and k are the numbers of the transmitter and the receiver, respectively. The envelope R _{ k ℓ }(t) and phase Θ _{ k ℓ }(t) of Nakagami fading model include the shape parameter m _{ k ℓ } and scale parameter Ω _{ k ℓ } that should be estimated.
The second-order moment of the Nakagami-m fading envelope is equal to the scale parameter. Thus, it is obtained simply. However, the estimation of the shape parameter is not straightforward and should be noticed more. We focus on estimating it, and we call it the fading parameter.
Therefore, the estimation of the channel parameters is reduced to obtaining the parameters m _{ k } for k=1,2.
As a result, the PDF of signal \(q'_{_{k1}}\left (t\right)\) is obtained. Similar statistical behavior for the two signals \(q'_{_{k1}}\left (t\right)\) and \(q'_{_{k2}}\left (t\right)\) results in an identical PDF for the second signal.
\(f_{q_{_{k1}}'}\left (q_{_{k1}}'\right)\) and \(f_{q_{_{k2}}'}\left (q_{_{k2}}'\right)\) are the marginal CDFs of the signals \(q'_{_{k1}}\left (t\right)\) and \(q'_{_{k2}}\left (t\right)\), respectively, and ρ _{ k } is the linear correlation parameter between these two signals. The linear correlation is a measure of dependency in this paper and is also called Pearson’s correlation. Note that since the signals in two transmitters are produced independently, the linear correlation between \(q'_{_{k1}}\left (t\right)\) and \(q'_{_{k2}}\left (t\right)\) are the same as the linear correlation related to the channel. Thus, the estimation of this parameter leads to specifying the channel correlation parameter.
As previously mentioned, three kinds of copula, i.e., normal, Clayton, and t copula, are applied for the estimation. Note that the linear correlation parameter ρ _{ k } in (27) is not exactly the copula parameter, and we should obtain the copula parameter from ρ _{ k } based on the related copula.
For the normal copula, the entries of the correlation matrix R in (4) are normal copula parameters, and fortunately, these parameters are almost the same as linear correlation parameters that present pairwise correlations among the variables.
In t copula, there are two parameters, one of which is the degrees of freedom and is considered equal to 2 in our simulations. The other one is exactly the same as the normal copula parameter and therefore is identical to the linear correlation parameter.
For generalizing (28) to the multivariate case, one can calculate α for each pair separately and consider the average of all obtained α values as the main Clayton copula parameter.
Until now, the PDF \(f_{q_{_{k}}'}\left (q_{_{k}}'\right)\) is estimated, and thus, the PDF \(f_{q_{_{k}}}\left (q_{_{k}}\right)\) is obtained analytically by using (17). Thus, the PDF of the received signals in both receivers, \(f_{q_{_{1}}}\left (q_{_{1}}\right)\) and \(f_{q_{_{2}}}\left (q_{_{2}}\right)\), are at hand. Using the obtained analytic PDF of the received signal in the kth receiver, the parameters m _{ k } and ρ _{ k } in the route between the transmitters and the kth receiver could be estimated as follows.
In the next section, a novel method is expressed to calculate the capacity of the proposed MIMO channel based on the parameters estimated in (30).
5 Capacity analysis
where \(C_{1} = {\left ({\frac {{{m^{m}}}}{{{2^{m - 1}}{\mkern 1mu} {\Omega ^{m}}{\mkern 1mu} {\Gamma ^{2}}\left ({\frac {m}{2}} \right)}}} \right)^{4}}\), tr(.) denotes the trace operator, and θ _{ ij } is the phase of h _{ ij }. It is supposed in (33) that all entries have the same m and Ω, but it is simply generalized to the situation in which we have different values for the parameters.
where 0≤μ≤2π, and 0≤γ≤π/2.
where λ ^{′} is one of the eigenvalues λ _{1} and λ _{2} which is not selected as λ. Since f(λ _{1},λ _{2}) is obtained in (49), the capacity is calculated.
Thus, the capacity is obtained for asymptotic values of SNR.
Although we discussed a MIMO system with two transmitters and two receivers, it is able to be generalized to arbitrary number of transmitters and receivers. For example, in a 3×3 MIMO channel, we should estimate two correlation parameters from the samples of each receiver in (30), and in capacity prediction, matrix H in (33) is 3×3, and the copula density function in (35) has nine variables. Note that the procedure is the same as the case 2×2 MIMO channel. However, more transmitters and receivers lead to complicated mathematical calculations that could be sometimes cumbersome. In the next section, there are some simulations to approve the results related to both parameter estimation and capacity calculation.
6 Simulation and result
It is essential to assess the proposed approach by employing some simulations. The simulations should cover both discussions, channel parameter estimation and channel capacity prediction. At first, the ability of the proposed algorithm in Nakagami-m and correlation parameter estimation is evaluated. Suppose we have a 2×2 MIMO channel as the communication system to transfer the cosine signal 2 cos(2π f _{ c } t) with f _{ c }=100 MHz. Two adjacent antennas send this signal to two receiving antennas which are far from each other. This arrangement for transmitters and receivers leads to having two different environments from the transmitters to each one of the receivers (Figure 1). Thus, we suppose the path to the first receiver has a Nakagami behavior with parameter m _{1}, and the second one is affected by a Nakagami model with parameter m _{2}. On the other hand, since at each receiver, we have the sum of two signals from two near transmitters, these two signals are correlated. In the simulation, we suppose the correlation parameter between two signals in the first receiver is ρ _{1} and in the second one is ρ _{2}.
The index of performance, in all four figures, is presented by mean square error (MSE). All estimations are done with three kinds of copula, i.e., normal, Clayton, and t copula. The comparison between the copulas guarantees that the simulation results are reliable based on all mentioned copulas. However, for example, in our simulation, the normal copula has almost better fit with the correlation model compared with other copulas. Thus, when more accuracy is required, copula goodness-of-fit testing is done and the optimized selection about the various kinds of copula is performed [18].
Figure 7 includes the same simulations for the correlation parameters ρ _{1}=0.5 and ρ _{2}=0.5, where there is also capacity increasing when either the total transmit power or fading parameter increases. It is also obvious that the capacity in Figure 7 is totaly less than the capacity in the same cases in Figure 6. This is because of the larger correlation between the transmitted signals. Fortunately, the proposed procedure presents the value of channel capacity in the Nakagami-m MIMO system by using the copula concept even when there is a large correlation between the signals.
7 Conclusions
In this paper, a new approach is proposed to estimate simultaneously the fading parameters in every route in a MIMO system and also the correlation parameter between these routes. The proposed method is based on the PDF estimation and the copula theory. The copula concept facilitates the PDF estimation when we are faced with the correlation between some parameters. Hence, the combination of PDF estimation and copula concept creates a novel method to identify a correlated MIMO system with Nakagami-m fading. Moreover, we calculate the capacity of the ergodic MIMO channel by using the estimated parameters. Precise estimated parameters result in a suitable prediction for the channel capacity. Some simulations are also presented to depict the validity of our proposed procedure in both fading parameter estimation and channel capacity prediction.
8 Appendix
9 A The copula relationships
where α>0 is the Clayton copula parameter.
where \(t_{\nu }^{- 1}\left (. \right)\) denotes the inverse function of a standard univariate t _{ ν } distribution, matrix R is the correlation matrix, x is a vector that is defined as \(\mathbf {x} \buildrel \Delta \over = \left [ {{x_{1}}, \ldots,{x_{n}}} \right ]\), and ν is the t copula parameter that is called degrees of freedom.
10 B The proof for (21)
where Θ _{ i }s are the real roots of the equation Q _{2}=g(Θ), g ^{′}(Θ) is the derivative of g(Θ), and f _{ Θ }(Θ) is the PDF of the signal phase that is presented in (10).
Declarations
Authors’ Affiliations
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