# The low-rankness of the channel covariance matrix

In this post, I will briefly talk about a property of massive MIMO channels – the low-rankness of the channel covariance matrix, which is defined as ${{\mathbf{R}}} \triangleq \mathbb{E}\{ {{{\mathbf{h}}}{\mathbf{h}}^H} \}$ where the expectation is taken over the random realizations of the channel vector ${\mathbf{h}}$.

Generally speaking, due to limited antenna spacing and angular spread, the channel covariance matrix in massive MIMO has a reduced rank. In other words, the wireless signal of a certain user lives in a reduced subspace. This feature can be exploited in application scenarios like pilot contamination reduction, spatial multiplexing, statistical interference mitigation, compressed sensing (as it reveals a certain sparsity of the signal), etc.

Intuition tells us that the rank is a function of the richness of the scattering environment, the antenna spacing, and the number of antennas. But how can we express it in closed-form?

Let’s consider a very common antenna placement: uniform linear array (ULA).

Using a general multipath channel model, the uplink channel vector $\mathbf{h}$ can be written as a linear combination of $\mathbf{P}$ steering vectors. ${\mathbf{h}} = \frac{\beta}{\sqrt{P}} \sum\limits_{p = 1}^P {{\mathbf{a}}({\theta _{p}}){e^{i\varphi_{p}}}}$,

where the p-th path has the angle of arrival (AoA) ${\theta _{p}}$ and the random phase $\varphi_{p}$.

Now consider the scenario when the multipath AoAs are randomly distributed within a bounded support $\Phi = [\theta_{\text{min}}, \theta_{\text{max}}]$, i.e., the AoA $\theta$ always satisfies $\theta_{\text{min}} \leq \theta \leq \theta_{\text{max}}$
as shown in Fig. 2.

When the number of antennas $M$ grows large, no matter how many paths we have in the channel, the rank of the channel covariance matrix yields $\frac{{rank} ({{\mathbf{R}}})} {M} \leq {\left( {\cos (\theta^{{\text{min}}}) - \cos (\theta^{{\text{max}}})} \right)} \frac{D}{\lambda }$,

where $\lambda$ is the signal wavelength and $D$ is the antenna spacing.

This low-rankness property indicates that in massive MIMO there exists a large null space ${null} ({{\mathbf{R}}})$, which can be exploited for the purpose of, for example, interference rejection.

For detailed mathematical proof, please refer to

 H. Yin, D. Gesbert, M. Filippou and Y. Liu, “A coordinated approach to channel estimation in large-scale multiple-antenna systems,” IEEE J. Sel. Areas Commun., Vol. 31, No. 2, pp. 264-273, Feb. 2013.

Apart from ULA and single AoA support, some generalizations of this result can be made to the case of random linear array, multiple clusters of AoA supports, and to two-dimensional distributed array. These generalized results can be found in

 H. Yin, D. Gesbert, and L. Cottatellucci, “Dealing with interference in distributed large-scale MIMO systems : A statistical approach,” IEEE J. Sel. Topics Signal Process., Vol. 8, No. 5, pp. 942-953, Oct. 2014.