# 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 envirment, 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

[1] 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

[2] 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.