# Pilot decontamination in joint angle and power domain

In the previous post Pilot decontamination in angular domain I have introduced a pilot contamination reduction method which relies on the AoA distinction between the desired user and the interference users. The proposed MMSE-based estimator leads to full pilot decontamination under the strict condition that the desired and interference channels do not overlap in their AoA regions.

When this non-overlapping condition does not hold (owing to the random user location and scattering effects), the MMSE-based estimator will only alleviate pilot contamination, not eliminate. In this post, I will show a much enhanced pilot decontamination method [1] that combines the merits of the MMSE estimator and the known amplitude based projection method [2]. Our asymptotic analysis indicates that the new method requires weaker conditions compared to the previously proposed methods to achieve full pilot decontamination.

Before introducing the new method, let’s first have a brief review of the key results of the MMSE-based estimator and the amplitude based projection method.

• MMSE-based estimator

When the interference channels do not overlap with the desired channel in angular domain, then pilot contamination vanishes. A typical example is shown in Fig. 1.

More information about the asymptotical result can be found here.

• Amplitude based projection method [2]

The basic idea of this method is to project the received signal onto the subspace spanned by the desired channels. This subspace is obtained by EVD/SVD of the received data signals. If we plot the histogram of the eigenvalues of the empirical instantaneous covariance matrix built from the received data, we can identify three bulks of eigenvalues as in Fig. 2. The bunch of weakest eigenvalues corresponds to the noise. The bulk of dominant eigenvalues are the signals of interest. The bulk in between denotes the inter-cell interference.

When the interference channels are weaker than the desired channels, i.e., the bulk of desired signals and the bulk of interference are separable in power domain, then pilot contamination vanishes asymptotically with the number of antennas $M$ and the coherence time $C$. A simple example is shown in Fig. 3, where user $11$ and user $21$ share the same pilot sequence.

The EVD/SVD-based blind channel estimator achieves the following result. If the following condition holds

$|\mathbf{h}_{11}|^2 > |\mathbf{h}_{21}|^2$, (1)

then pilot contamination vanishes as $M, C \rightarrow \infty$.

In reality, the non-overlapping conditions may hold in only one domain, angular or power, depending on the user’s location. Moreover, in some cases, the interference may overlap with the desired signal in both angular domain and power domain, like the two blue users in Fig. 4. In such situations, neither the MMSE-based estimator nor the EVD/SVD based estimator could eliminate pilot contamination.

So, is there a simple approach to cope with all these situations?

The answer is YES. With our newly proposed Covariance-aided Amplitude based Projection method in [3], we can achieve full decontamination even in the case shown in Fig. 4.

• Covariance-aided Amplitude based Projection

The basic idea is to apply a statistical filter $\mathbf{\Xi}$ on the received data signal before performing the EVD/SVD-based amplitude based projection. This filter is a function of the channel’s second-order statistics. The purpose of this filter is to bring down the power of interference while preserving the power of the desired signal. Our asymptotical analysis shows that if the filtered desired signal is stronger than the filtered interference, then the channel estimation error vanishes completely with the number of antennas $M$ and the coherence time $C$. In other words, if the following condition holds,

$|\mathbf{\Xi h}_{11}|^2 > |\mathbf{\Xi h}_{21}|^2$, (2)

then

$\mathop {\lim }\limits_{M, C \to \infty } \frac{|{\bf{\widehat {{h}}}}_{11}^{\text{CA}} - {{\mathbf{h}}}_{11}|_2^2}{|{{\mathbf{h}}}_{11}|_2^2} = 0.$

The condition (2) is in general less restrictive than the condition (1), which is required by the amplitude-based projection method [2].

This Covariance-aided Amplitude based Projection method outperforms the MMSE-based estimator in [1] when the non-overlapping condition in angular domain is not satisfied. A demonstration is shown in Fig. 5 for a ULA base station.

Suppose the desired user (green) and the interference user (red) have approximately the same signal strength at BS 1. The angular support of the desired user is the green sector while the angular support of the interference, i.e, the red sector, is half overlapping with that of the desired user. In this case, the two users are overlapping in both power domain and angular domains. Consequently neither the MMSE-based estimator nor the EVD/SVD based estimator will work. In contrast, our new method in [3] will. To give an intuitive interpretation, let’s decompose the interference channel into two parts, the $\mathbf{h}_{21}^{(i)}$ the multipath that fall into the desired AoA support and $\mathbf{h}_{21}^{(o)}$, the multipath falling outside. The statistical filter $\mathbf{\Xi}$ will filter out $\mathbf{h}_{21}^{(o)}|$, with the residual interference signal greatly weakened. Now this residual interference can be removed by the EVD/SVD based estimator, since it’s no longer overlapping with the desired signal in power domain.
For the choice of $\mathbf{\Xi}$, math proofs, please refer to [3], where you may also find the generalization to multi-user per cell setting and the low-complexity alternatives of this method.

References
[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.
[2] R. R. Müller, L. Cottatellucci, and M. Vehkaperä, “Blind pilot decontamination,” IEEE J. Sel. Topics Signal Process, vol. 8, no. 5, pp. 773–786, Oct. 2014.
[3] H. Yin, L. Cottatellucci, and D. Gesbert, R. R. Müller, and G. He, “Robust pilot decontamination based on joint angle and power domain discrimination,” IEEE Trans. Signal Process., Vol. 64, No. 11, pp. 2990-3003, Jun. 2016.

# Pilot decontamination in angular domain

Massive MIMO is widely believed to be one of the key technologies for the 5th generation (5G) wireless networks. The realization of the high energy efficiency and throughput of massive MIMO, especially in the downlink transmission, relies on the accurate knowledge of the CSI. In practical wireless system, CSI is acquired based on training sequences sent by user terminals. Due to limited time and frequency resources, non-orthogonal pilot sequences are typically used by user terminals in neighboring cells, resulting in the so-called “pilot contamination”. In fact, it was believed that pilot contamination constituted a performance bottleneck of massive MIMO.

In this post, I will show a way to solve this problem.

Consider a base station equipped with $M$ antennas which form a uniform linear array (ULA). In the uplink training phase, users send pilot sequences to the base station. Assuming the statistics of the channel (channel covariance matrices) are known by the base station, we can construct an MMSE channel estimator

$\widehat{{\mathbf{h}}}_j^{(j){\text{MMSE}}} = {\mathbf{R}}^{(j)}_j \left( \tau (\sum_{l=1}^{L}{\mathbf{R}}^{(j)}_l) + \sigma_n^2 {\mathbf{I}}_{M} \right)^{-1} {\mathbf{S}}^H \mathbf{y}^{(j)},$

where ${\mathbf{R}}^{(j)}_l$ is the channel covariance matrix, $\mathbf{S}$ is the training matrix, and $\mathbf{y}^{(j)}$ is the received signal at the base station. More details of this equation can be found in [1].

We have proved in [1] that with this MMSE estimator, pilot contamination can be made to vanish completely under a certain condition on the AoA distributions. This condition is what we call “non-overlapping” condition.
Theorem 1. If the AoA supports of the interference users are strictly non-overlapping with the AoA support of the desired user, then

$\mathop {\lim }\limits_{M \to \infty } \widehat{{{\mathbf{h}}}}_j^{(j){\text{MMSE}}} = {\mathbf{\widehat h}}_j^{(j){\text{no int}}},$

where ${\mathbf{\widehat h}}_j^{(j){\text{no int}}}$ denotes the interference-free estimate.

An example of this non-overlapping condition is shown in Fig. 1. The multipath AoAs of user 1 are distributed within sector 1, while the AoAs of user 2 (interference user) are within sector 2, which does not overlap with sector 1.

Under this condition, even if user 1 and user 2 have non-orthogonal pilots (or the same pilot), they have little interference with each other.

This finding leads to a coordinated pilot assignment (CPA) strategy: we assign a pilot sequence to a group of “compatible” users who have disjoint AoA supports. Of course we do not need to estimate the AoAs, as this pilot assignment is only based on the long-term statistics of users’ channels. The utility function is the channel estimation MSE, which is a function of the channel covariance matrix. For more details of the mathematical proofs and methods, 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 Journal on Selected Areas in Communications, Vol. 31, No. 2, pp. 264-273, Feb. 2013.

# 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.