On the Ergodic Capacity of Distributed MIMO Antenna Systems

Abstract

We consider distributed multiple-input–multiple-output (MIMO) antenna systems, along with their certain generalizations. We show that distributed MIMO configuration can be mapped to a semicorrelated (one side correlated) Wishart model. For a given set of large-scale fading parameters, associated with the path loss and shadow fading, we derive exact and closed-form results for the marginal density of eigenvalues of $\mathbf{H}^\dag \mathbf{H}$ (or $\mathbf{H} \mathbf{H} ^\dag$), where $\mathbf{H}$ is the channel matrix. We also obtain exact and closed-form expressions for the ergodic channel capacity with the aid of Meijer G-function. The ergodic capacity of semicorrelated Rayleigh fading channel follows as a special case. All analytical results are validated by comparison with Monte-Carlo simulations.

Appendices

Proofs for Eqs. (27) and (31)

As pointed out in Sect. 5, the MIMO DAS model and weighted sum of Wishart matrices, both can be mapped to a semicorrelated Wishart scenario. The eigenvalue statistics for weighted sum of Wishart matrices has already been worked out in [46]. As a consequence the proofs in the present case run parallel to those in [46]. However, for the sake of completeness, we reproduce the derivations here as well.

Exact result for the marginal density of eigenvalues for semicorrelated Wishart matrices is available from the notable works of several authors [27–33]. We employ here the determinantal expression obtained in [32, 33].

Consider $n\times m$ dimensional complex matrices W taken from the distribution

$$\begin{aligned} P_\mathbf{W }({\mathbf{W}} )=\pi ^{-mn} \det \!\,^{-m}\hat{\mathbf{V }}\,\exp [-{\text{tr}}({\mathbf{W}} ^\dag \hat{\mathbf{V }}^{-1}{\mathbf{W}} )], \end{aligned}$$

(35)

where the $n\times n$ dimensional covariance matrix is $\hat{\mathbf{V }}={\text{diag}}(\hat{v}_1,\ldots ,\hat{v}_n)$. We assumed here that there are no multiplicities in the entries of $\hat{\mathbf{V }}$, i.e., $\hat{v}_1,\ldots ,\hat{v}_n$ are distinct. In such a scenario the marginal density of nonzero-eigenvalues of ${\mathbf{W}} {\mathbf{W}} ^\dag$ or ${\mathbf{W}} ^\dag {\mathbf{W}}$ is given by

$$\begin{aligned} p_\lambda (\lambda )=\frac{1}{\nu \Delta _m(\{\hat{v}^{-1}\}) } \det \left[ \begin{array}{ccc} 0 &{} \left[ \frac{\exp (-\lambda /\hat{v}_k)}{\hat{v}_k^m}\right] _{k=1,\ldots ,n} \\ \left[ \frac{\lambda ^{m-j}}{\varGamma (m-j+1)}\right] _{j=1,\ldots ,n} &{} \left[ \hat{v}_k^{-j+1}\right] _{j,k=1,\ldots ,n} \end{array}\right] . \end{aligned}$$

(36)

Here $\nu =\min (m,n)$, and $\Delta _m(\{\hat{v}^{-1}\})=\prod _{j>k}(\hat{v}_j^{-1}-\hat{v}_k^{-1})$ is the Vandermonde determinant. Equation (27) follows from (36) by setting $m=M$ and $n=\mathsf{L}$ and using the well known result $\Delta (\{r\})=\det [r_j^{k-1}]$.

We now proceed to derive the ergodic channel capacity using the relation (13). To this end we expand (27) using the first column and obtain

$$\begin{aligned} p_\lambda (\lambda )=c\, \sum _{\mu =1}^{\mathsf{L}}(-1)^{\mu }g_\mu (\lambda )\det \,\left[ \begin{array}{ccc} \left[ f_k(\hat{v},\lambda )\right] _{k=1,\ldots ,L_1+\cdots +L_N} \\ \mathop {{\left[ h_{j,k}(\hat{v})\right] }_{{j,k=1, \ldots ,\mathsf{L}}}}\limits _{{\qquad (j\ne \mu )}}\end{array}\right] . \end{aligned}$$

(37)

We note at this point that $g_j(\lambda )=\lambda ^{M-j}/\varGamma (M-j+1)=0$ if $j> M$ because of the diverging gamma function in the denominator. If $\mathsf{L}\le M$ this situation is not encountered. However, if $\mathsf{L}> M$ then $g_j(\lambda )$ is nonzero only for $1\le j\le M$. Thus in both cases we see that the nonzero terms in the summation in (37) involve $j=1,\ldots ,\nu$, with $\nu =\min (M,\mathsf{L})$. We incorporate this observation by changing the upper limit of the summation. We now bring in the $g_\mu (\lambda )$ occurring before the determinants to the respective first rows, i.e., with $f_k(\hat{v},\lambda )$, and obtain

$$\begin{aligned} p_\lambda (\lambda )=c\, \sum _{\mu =1}^{\nu }(-1)^{\mu }\det \,\left[ \begin{array}{ccc} \left[ g_\mu (\lambda )f_k(\hat{v},\lambda )\right] _{k=1,\ldots ,\mathsf{L}} \\ \mathop {{\left[ h_{j,k}(\hat{v})\right] }_{{j,k=1,\ldots ,\mathsf{L}}}}\limits _ {{\qquad (j\ne \mu )}}\end{array}\right] . \end{aligned}$$

(38)

This equation serves as yet another expression for the marginal density. We now use (13) and obtain the following expression for the ergodic capacity by interchanging the $\lambda$-integral and the summation:

$$\begin{aligned} C=\nu \,c \sum _{\mu =1}^{\nu }(-1)^{\mu } \int _0^\infty d\lambda \, \left( \det \,\left[ \begin{array}{ccc} \left[ g_\mu (\lambda )f_k(\hat{v},\lambda )\right] _{k=1,\ldots ,\mathsf{L}} \\ \mathop {{\left[ h_{j,k}(\hat{v})\right] }_{{j,k=1,\ldots ,\mathsf{L}}}}\limits _ {{\qquad (j\ne \mu )}}\end{array}\right] \right) \, \log _2 \left( 1+\frac{\rho }{M}\lambda \right) . \end{aligned}$$

(39)

The $\lambda$-integral can be introduced in the first row of the determinant, along with the term $\log _2 \left( 1+\frac{\rho }{M}\lambda \right)$ to yield

$$\begin{aligned} C=\nu \,c \sum _{\mu =1}^{\nu }(-1)^{\mu } \det \,\left[ \begin{array}{ccc} \left[ \mathcal{G}_{\mu ,k}(\hat{v})\right] _{k=1,\ldots ,\mathsf{L}} \\ \mathop {{\left[ h_{j,k}(\hat{v})\right] }_{{j,k=1,\ldots ,\mathsf{L}}}}\limits _ {{\qquad (j\ne \mu )}}\end{array}\right] , \end{aligned}$$

(40)

where

$$\begin{aligned} \mathcal{G}_{\mu ,k}(\hat{v})=\int _0^\infty d\lambda \, g_\mu (\lambda )f_k(\hat{v},\lambda )\log _2 \left( 1+\frac{\rho }{M}\lambda \right) . \end{aligned}$$

(41)

This integral can be expressed in a closed form in terms of Meijer G-function or exponential-integral-functions as in (33) and (34), respectively. To obtain (33) we identify the following special cases of Meijer G-functions [47]:

$$\begin{aligned}&G^{1,0}_{0,1} \left( \begin{array}{c} \_ \\ \beta \end{array}\bigg |\, z \right) = z^{\beta }e^{-z}, \end{aligned}$$

(42)

$$\begin{aligned}&\quad G^{1,2}_{2,2} \left( \begin{array}{c} 1,\, 1 \\ 1,\, 0 \end{array}\bigg |\, z \right) = \ln (1+z), \end{aligned}$$

(43)

We also employ the convolution integral for Meijer G-function:

$$\begin{aligned}&\int _0^\infty dz\,G^{m,n}_{p,q} \left( \begin{array}{c} a_1,\,\ldots \, a_p \\ b_1,\,\ldots \, b_q \end{array}\bigg |\, \eta z \right) \, G^{\mu ,\nu }_{\sigma ,\tau } \left( \begin{array}{c} c_1,\,\ldots \, c_\sigma \\ d_1,\,\ldots \, d_\tau \end{array}\bigg |\, \omega z \right) \nonumber \\&\quad =\frac{1}{\eta }\,G^{n+\mu ,m+\nu }_{q+\sigma ,p+\tau } \left( \begin{array}{c} -b_1,\ldots ,-b_m,\,c_1,\,\ldots \, c_\sigma ,\,-b_{m+1},\ldots ,\,-b_q \\ -a_1,\ldots ,-a_n,\,d_1,\,\ldots \, d_\tau ,\,-a_{n+1},\ldots ,\,-a_p \end{array}\Bigg |\, \frac{\omega }{\eta } \right) \nonumber \\&\quad =\frac{1}{\omega }\,G^{m+\nu ,n+\mu }_{p+\tau ,q+\sigma } \left( \begin{array}{c} a_1,\ldots ,a_n,\,-d_1,\,\ldots \, -d_\tau ,\,a_{n+1},\ldots ,\,a_p \\ b_1,\ldots ,b_m,\,-c_1,\,\ldots \, -c_\sigma ,\,b_{m+1},\ldots ,\,b_q \end{array}\Bigg |\,\frac{\eta }{\omega } \right) . \end{aligned}$$

(44)

The restrictions on the indices for this integration formula can be found in [47]. Next we perform row interchanges in the determinants to bring $\mathcal{G}_{\mu ,k}$ in the respective $\mu$th row. Consequently, we have the expression for ergodic channel capacity as provided in (31).

Proofs for Eqs. (15), (20), (24), and (26)

To arrive at Eqs. (24) and (26) we need to set $\hat{v_1}=\cdots =\hat{v}_{L_1}=v_1; \hat{v}_{L_1+1}=\cdots =\hat{v}_{L_2}=v_2\,;\ldots ; \hat{v}_{L_{(N-1)}+1}=\cdots =\hat{v}_{L_N}=v_N$ in (27) and (31). However, direct substitution of these values makes the determinant in the numerator, as well as the determinant in the denominator (contained in the normalization) to become zero. Therefore, we must invoke a limiting procedure to obtain the proper results, as described below.

Let us focus on the columns involving up to $L_1$ in (27). The ratio of the determinants appears as

$$\begin{aligned} \frac{\det \left[ \begin{array}{cccccc} 0 &{} (\hat{v}_1^{-1})^M e^{\hat{v}_1^{-1}\lambda } &{} (\hat{v}_2^{-1})^M e^{\hat{v}_2^{-1}\lambda } &{} \ldots &{} (\hat{v}_{L_1}^{-1})^M e^{\hat{v}_{L_1}^{-1}\lambda } &{} \ldots \\ g_j (\lambda ) &{} (\hat{v}_1^{-1})^{j-1} &{} (\hat{v}_2^{-1})^{j-1} &{} \ldots &{} (\hat{v}_{L_1}^{-1})^{j-1} &{} \ldots \end{array}\right] }{\det \left[ \begin{array}{ccccc} (\hat{v}_1^{-1})^{j-1}&(\hat{v}_2^{-1})^{j-1}&\ldots&(\hat{v}_{L_1}^{-1})^{j-1}&\ldots \end{array}\right] } \end{aligned}$$

(45)

We take $\hat{v}_k^{-1}=\hat{v}_1^{-1}+\epsilon _k$ with small $\epsilon _k$ for $k=2,3,\ldots$, and Taylor-expand up to the term $\epsilon _k^{k-1}$:

$$\begin{aligned} (\hat{v}_k^{-1})^M e^{\hat{v}_k^{-1}\lambda }\approx & {} \sum _{r=0}^{k-1}\frac{\epsilon _k^r}{r!}\frac{\partial ^r}{\partial (\hat{v}_1^{-1})^r} (\hat{v}_1^{-1})^M e^{\hat{v}_1^{-1}\lambda };\\ (\hat{v}_k^{-1})^{j-1}\approx & {} \sum _{r=0}^{k-1}\frac{\epsilon _k^r}{r!}\frac{\partial ^r}{\partial (\hat{v}_1^{-1})^r} (\hat{v}_1^{-1})^{j-1}. \end{aligned}$$

Now, employing adequate column operations, we obtain

$$\begin{aligned} \frac{\det \,\left[ \begin{array}{cccccc} 0 &{} (\hat{v}_1^{-1})^M e^{\hat{v}_1^{-1}\lambda } &{} \frac{\epsilon _k}{1!}\frac{\partial }{\partial (\hat{v}_1^{-1})} (\hat{v}_1^{-1})^M e^{\hat{v}_1^{-1}\lambda } &{} \ldots &{} \frac{\epsilon _k^{k-1}}{(k-1)!}\frac{\partial ^{k-1}}{\partial (\hat{v}_1^{-1})^{k-1}} (\hat{v}_1^{-1})^M e^{\hat{v}_1^{-1}\lambda } &{} \ldots \\ g_j (\lambda ) &{} (\hat{v}_1^{-1})^{j-1} &{}\frac{\epsilon _k}{1!}\frac{\partial }{\partial (\hat{v}_1^{-1})} (\hat{v}_1^{-1})^{j-1} &{} \ldots &{} \frac{\epsilon _k^{k-1}}{(k-1)!}\frac{\partial ^{k-1}}{\partial (\hat{v}_1^{-1})^{k-1}}(\hat{v}_1^{-1})^{j-1} &{} \ldots \end{array}\right] }{\det \,\left[ \begin{array}{ccccc} (\hat{v}_1^{-1})^{j-1}&\frac{\epsilon _k}{1!}\frac{\partial }{\partial (\hat{v}_1^{-1})} (\hat{v}_1^{-1})^{j-1}&\ldots&\frac{\epsilon _k^{k-1}}{(k-1)!}\frac{\partial ^{k-1}}{\partial (\hat{v}_1^{-1})^{k-1}}(\hat{v}_1^{-1})^{j-1}&\ldots \end{array}\right] } \end{aligned}$$

(46)

The factors containing $\epsilon _k$ and factorial can be taken out of the columns, both from numerator and denominator, and cancelled out. This leaves us with

$$\begin{aligned} \frac{\det \,\left[ \begin{array}{cccccc} 0 &{} (\hat{v}_1^{-1})^M e^{\hat{v}_1^{-1}\lambda } &{} \frac{\partial }{\partial (\hat{v}_1^{-1})} (\hat{v}_k^{-1})^M e^{\hat{v}_1^{-1}\lambda } &{} \ldots &{} \frac{\partial ^{k-1}}{\partial (\hat{v}_1^{-1})^{k-1}} (\hat{v}_k^{-1})^M e^{\hat{v}_1^{-1}\lambda } &{} \ldots \\ g_j (\lambda ) &{} (\hat{v}_1^{-1})^{j-1} &{}\frac{\partial }{\partial (\hat{v}_1^{-1})} (\hat{v}_1^{-1})^{j-1} &{} \ldots &{} \frac{\partial ^{k-1}}{\partial (\hat{v}_1^{-1})^{k-1}}(\hat{v}_1^{-1})^{j-1} &{} \ldots \end{array}\right] }{\det \,\left[ \begin{array}{ccccc} (\hat{v}_1^{-1})^{j-1}&\frac{\partial }{\partial (\hat{v}_1^{-1})} (\hat{v}_1^{-1})^{j-1}&\ldots&\frac{\partial ^{k-1}}{\partial (\hat{v}_1^{-1})^{k-1}}(\hat{v}_1^{-1})^{j-1}&\ldots \end{array}\right] } \end{aligned}$$

(47)

To evaluate the derivatives of $(\hat{v}_k^{-1})^M e^{\hat{v}_1^{-1}\lambda }$ we use Rodrigues’ formula for the associated Laguerre polynomials,

$$\begin{aligned} \mathcal{L}_k^{(\beta )}(z)=\frac{z^{-\beta } e^z }{k!}\frac{\partial ^k}{\partial z^k}\left( z^{k+\beta } e^{-z} \right) , \end{aligned}$$

(48)

with adequate scaling of the variables. Similar steps are followed for rest of the columns and consequently we arrive at (24).

We employ a similar procedure in (31) to arrive at (26). To evaluate the derivative of Meijer G-function we use the result

$$\begin{aligned} z^r \frac{\partial ^r}{\partial z^r} G^{3,1}_{2,3} \left( \begin{array}{c} a_1,\, a_2 \\ b_1,\, b_2,\, b_3 \end{array}\Bigg |\, z \right) = G^{3,2}_{3,4} \left( \begin{array}{c} 0,\, a_1,\, a_2 \\ b_1,\, b_2,\, b_3,\, r \end{array}\Bigg |\, z \right) , \end{aligned}$$

(49)

which follows from the following more general expression [47]:

$$\begin{aligned} z^r \frac{\partial ^r}{\partial z^r}G^{m,n}_{p,q} \left( \begin{array}{c} a_1,\, \ldots ,\, a_p \\ b_1,\,\ldots ,\, b_q \end{array}\bigg |\, z \right) = G^{m,n+1}_{p+1,q+1}\left( \begin{array}{c} 0,\,a_1,\, \ldots ,\, a_p \\ b_1,\,\ldots ,\, b_q,\,r \end{array}\bigg |\, z \right) . \end{aligned}$$

(50)

Finally, (15) and (20) follow trivially from, respectively, (24) and (26) by setting identical number of antennas at each port, i.e., $L_1=\cdots =L_N=L$.