Posts Tagged ‘MIMO’

MIMO Log-Det Formula for Channel Capacity with Real Input / Output Signals

November 5, 2014

    Consider a MIMO (multiple input multiple output) system of {N_t} transmitters and {N_r} receivers and assume

  \displaystyle {\bf R} = {\bf H}_0{\bf S}+ {\bf Z}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

where {{\bf R}}, {{\bf S}} and {{\bf Z}} are column vectors of real random variables representing received signals, transmitted signals and noise.

    The capacity of the MIMO channel is given by [1]

\displaystyle C_{\rm MIMO} = \max_{f_{\bf S}\left({\bf s}\right)}I\left({\bf S}, {\bf R}\right) =  \max_{f_{\bf S}\left({\bf s}\right)}\left[h\left({\bf R}\right)-h\left({\bf R}|{\bf S}\right)\right], \ \ \ \ \ \ \ \ (2)

where {f_{\bf S}\left({\bf s}\right)} is the p.d.f. of the transmitted signal vector {{\bf S}}. It turns out [1, 2] that {h\left({\bf R}|{\bf S}\right) = h\left({\bf Z}\right)} where {h\left({\bf Z}\right)} is the differential entropy of the noise and satisfies

\displaystyle h\left({\bf Z}\right) = \frac{1}{2}\log_2\left[\left(2\pi e\right)^{N_r}\left|{\bf V}_{\bf Z}\right|\right]. \ \ \ \ \

Here {{\bf V_Z}} is the covariance matrix of the noise. Thus Eq. (2) becomes

\displaystyle C_{\rm MIMO} =  \max_{f_{\bf S}\left({\bf s}\right)}\left[h\left({\bf R}\right)-h\left({\bf Z}\right)\right]  = \max_{f_{\bf S}\left({\bf s}\right)}\left[h\left({\bf R}\right)\right]-h\left({\bf Z}\right), \ \ \ \ \ (3)

since the entropy of {{\bf Z}} does not depend on the probability density function of the input variable {{\bf S}}.

    In the SISO (single input single output) problem the mutual information was maximized while holding the variance of the output p.d.f. constant [3]. In the MIMO problem, the mutual information is maximized while holding the covariance matrix {{\bf V}_{\bf R}} of the output signal vector {{\bf R}} constant and setting the mean of {{\bf R}} to zero. The method for finding the maximum is outlined in the references [2, 3, 4].

    As has been shown [1, 2, 4] {h\left({\bf R}\right)} is maximized when {f_{\bf S}} is the p.d.f. of jointly-normal random variables. Using the jointly-normal p.d.f for the maximum indicated in Eq. (4) produces

\displaystyle C_{\rm MIMO} = h_{\rm norm}\left({\bf R}\right)-h\left({\bf Z}\right), \ \ \ \ \

\displaystyle = \frac{1}{2}\log_2\left|{\bf V}_{\bf R}\right|-\frac{1}{2}\log_2\left|{\bf V}_{\bf Z}\right| = \frac{1}{2}\log_2\left|{\bf V}_{\bf R}\right|-\frac{1}{2}\log_2\left|\sigma_{Z}^2{\bf I}_{N_r}\right|. \ \ \ \ \ (4)

Next, the covariance matrix {{\bf V}_{\bf R}} of the received signal {{\bf R}} is evaluated in terms of the covariance matrix {{\bf V}_{\bf S}} of the input signal {{\bf S}} and the covariance matrix {{\bf V}_{\bf Z}} of the noise {{\bf Z}} using the relationship of Eq. (1). Thus

\displaystyle {\bf V}_{\bf R} = E\left({\bf R}{\bf R}^T\right) = E\left[\left({\bf H}_0{\bf S}+{\bf Z}\right)  \left({\bf H}_0{\bf S}+{\bf Z}\right)^T\right], \ \ \ \ \

\displaystyle = E\left[{\bf H}_0{\bf S}\left({\bf H}_0{\bf S}\right)^T+{\bf H}_0{\bf S}{\bf Z}^T + {\bf Z}  \left({\bf H}_0{\bf S}\right)^T+{\bf Z}{\bf Z}^T\right], \ \ \ \ \

\displaystyle = E\left[{\bf H}_0{\bf S}{\bf S}^T{\bf H}_0^T+{\bf H}_0{\bf S}{\bf Z}^T+{\bf Z}{\bf S}^T{\bf H}_0^T  +{\bf Z}{\bf Z}^T\right], \ \ \ \ \

\displaystyle = {\bf H}_0{\bf V}_{\bf S}{\bf H}_0^T + {\bf H}_0E\left({\bf S}{\bf Z}^T\right)  + E\left({\bf Z}{\bf S}^T\right){\bf H}_0^T + {\bf V}_{\bf Z}, \ \ \ \ \ \ \ (5)

So

\displaystyle {\bf V}_{\bf R} = {\bf H}_0{\bf V}_{\bf S}{\bf H}_0^T + {\bf V}_{\bf Z}  = {\bf H}_0{\bf V}_{\bf S}{\bf H}_0^T +\sigma_Z^2{\bf I}_{N_r} , \ \ \ \ \ \ \ \ \ (6)

where two terms have been dropped from Eq. (5) because the noise and the transmitted signal are uncorrelated and the noise is assumed to have zero mean. Using Eq. (6) in Eq. (4) shows

\displaystyle C_{\rm MIMO} = \frac{1}{2}\log_2\left|{\bf H}_0{\bf V}_{\bf S}{\bf H}_0^T+\sigma_Z^2{\bf I}_{N_r}\right| -\frac{1}{2}\log_2\left|\sigma_Z^2{\bf I}_{N_r}\right|, \ \ \ \ \

\displaystyle = \frac{1}{2}\log_2\left[\left|{\bf H}_0{\bf V}_{\bf S}{\bf H}_0^T + \sigma_Z^2{\bf I}_{N_r}\right|\left|  \sigma_Z^2{\bf I}_{N_r}\right|^{-1}\right], \ \ \ \ \

\displaystyle = \frac{1}{2}\log_2\left|{\bf I}_{N_r}+\frac{1}{\sigma_Z^2}{\bf H}_0{\bf V}_{\bf S}{\bf H}_0^T\right|, \ \ \ \ \

the capacity per channel use or capacity in bits per sample. The capacity per unit time (or bits/second) is therefore [3]

\displaystyle C_{t{\rm MIMO}} = B\log_2\left|{\bf I}_{N_r}+\frac{1}{\sigma_Z^2}{\bf H}_0{\bf V}_{\bf S}{\bf H}_0^T  \right|, \ \ \ \ \

where {B} is the signal bandwidth and {2B} is the sampling rate.
In the SISO case, this reduces to

\displaystyle C_t = B\log_2\left(1+\frac{\sigma_S^2}{\sigma_Z^2}\right) = B\log_2\left(1+\frac{P_S}{P_Z}\right), \ \ \ \ \

where {\sigma_S^2} is the variance of the input signal and {P_S/P_Z} is the ratio of the input signal power to noise power. Thus the result for {C_t} in the MIMO case reduces to {C_t} for the SISO case [3] as expected.

    A more detailed derivation is given in [2] and is available by clicking below.

References

[1] J. R. Hampton, Introduction to MIMO Communications\/, Cambridge University Press, N.Y. (2014).

[2] H. L. Rappaport, “Derivation of MIMO Log-Det Formula for Channel Capacity with Real Input / Output Signals,” 7G Communications, 7GCTN05, November 5, 2014. real

[3] H. L. Rappaport, “Notes on Information Theory II and the Geometric Interpretation of the Shannon Channel Capacity,” 7G Communications, 7GCTN02, October 2014. infoII

[4] I. E. Telatar, “Capacity of Multi-Antenna Gaussian Channels,” Euro. Trans. Telecommun., 10 (6), (1999), pp. 585 – 595.