Pilot Contamination is Not Captured by the MSE

Pilot contamination used to be seen as the key issue with the Massive MIMO technology, but thanks to a large number of scientific papers we now know fairly well how to deal with it. I outlined the main approaches to mitigate pilot contamination in a previous blog post and since then the paper Massive MIMO has unlimited capacity has also been picked up by science news channels.

When reading papers on pilot (de)contamination written by many different authors, I’ve noticed one recurrent issue: the mean-squared error (MSE) is used to measure the level of pilot contamination. A few papers only plot the MSE, while most papers contain multiple MSE plots and then one or two plots with bit-error-rates or achievable rates. As I will explain below, the MSE is a rather poor measure of pilot contamination since it cannot distinguish between noise and pilot contamination.

A simple example

Suppose the desired uplink signal is received with power p and is disturbed by noise with power (1-a) and interference from another user with power a. By varying the variable a between 0 and 1 in this simple example, we can study how the performance changes when moving power from the noise to the interference, and vice versa.

By following the standard approach for channel estimation based on uplink pilots (see Fundamentals of Massive MIMO), the MSE for i.i.d. Rayleigh fading channels is

    $$\textrm{MSE} = \frac{p}{p+(1-a)+a} = \frac{p}{p+1}, $$

which is independent of a and, hence, does not care about whether the disturbance comes from noise or interference. This is rather intuitive since both the noise and interference are additive i.i.d. Gaussian random variables in this example. The important difference appears in the data transmission phase, where the noise takes a new independent realization and the interference is strongly correlated with the interference in the pilot phase, because it is the product of a new scalar signal and the same channel vector.

To demonstrate the important difference, suppose maximum ratio combining is used to detect the uplink data. The effective uplink signal-to-interference-and-noise-ratio (SINR) is

    $$\textrm{SINR} = \frac{p(1-\textrm{MSE}) M}{p+1+a M \cdot \textrm{MSE}}$$

where M is the number of antennas. For any given MSE value, it now matters how it was generated, because the SINR is a decreasing function of a. The term a M  \cdot \textrm{MSE} is due to pilot contamination (it is often called coherent interference) and is proportional to the interference power $a$. When the number of antennas is large, it is far better to have more noise during the pilot transmission than more interference!

Implications

Since the MSE cannot separate noise from interference, we should not try to measure the effectiveness of a “pilot decontamination” algorithm by considering the MSE. An algorithm that achieves a low MSE can potentially be mitigating the noise, leaving the interference unaffected. If that is the case, the pilot contamination term $a M  \cdot \textrm{MSE}$ will remain. The MSE has been used far too often when evaluating pilot decontamination algorithms, and a few papers (I found three while writing this post) did only consider the MSE, which opens the door for questioning their conclusions.

The right methodology is to compute the SINR (or some other performance indicator in the data phase) with the proposed pilot decontamination algorithm and with competing algorithms. In that case, we can be sure that the full impact of the pilot contamination is taken into account.

8 thoughts on “Pilot Contamination is Not Captured by the MSE”

  1. Hi,
    Thanks for the important point.
    According to my observation, pilot contamination causes floor in MSE curves in i.i.d Rayleigh fading channels while noise does not. If my observation is valid, then would it be true to say that if an algorithm mitigated this floor then it has mitigated pilot contamination for sure ?
    Thanks

    1. I’m not sure what it means to have a “floor in MSE curves”. I guess you are referring to the MSE as a function of the signal power (or SNR)? If you observe an error floor in that case, it is because both the power of the received desired signal and some interfering signal grow as you increase the signal power. That happens irrespective of whether the interfering signal is “pilot contamination” or some other random noise-like signal.

      Since the coherent interference is something that scales with M and the error floor (or lack thereof) is something that appear when the signal power (or SNR) grows, I don’t think there are any simple conclusions to be made.

      1. Dear Dr. Bjornson, Thanks a lot for your reply.
        I plotted the MSE curve as a function of the transmit SNR for M=1000 antenna (massive MIMO case) and different number of cells. For L=1 cell, no floor appears in MSE curve while for L>1 cells, the curves encounter floors and the level of these floors grows with increase in number of cells. In case of L=1 cell, there exists no pilot contamination and the corresponding curve does not suffer from floor therefore noise itself could not produce floor in MSE curve while for the mult-icell case, there exists pilot contamination and the curves suffer from floor which increases with in crease in the number of cells.
        If my simulation is bug free and the results are valid, then would it be true to claim that MSE curve vs TxSNR could somehow measure pilot contamination level?
        Thanks a lot

        1. The cause of this behavior is that with increasing SNR, the noise becomes negligible while the desired and interfering signals remain. The SNR goes to infinity while the SIR is constant. There is pilot contamination in the sense there exist interfering signals during the pilot transmission, but this doesn’t say anything about what will happen during data transmission. Suppose the interfering signals are turned off during the data transmission or that the interfering signals are independent between the pilot and data phase. In both these special cases, there won’t be any coherent interference.

          So, yes, what you observe is pilot contamination, but we cannot say how serious this pilot contamination will be without also studying the data phase. And it is not the high SNR regime that is of main interest in massive MIMO but the many-antennas regime.

          1. Yes, you are absolutely correct.

            As you mentioned, the point that I missed was that MSE curve measures the performance of channel estimation phase only, while what we truly need is the performance of MRC scheme during data transmission period which is measured by BER curve.

            The MSE floor I observed tells us that channel estimation error (MSE) caused by the pilot contamination during pilot transmission phase could be no longer improved by increasing the SINR, but this does not tell us how badly this estimation error impacts the performance of the MRC signal processing block in the receiver during the data transmission period. We may expect that in the many-antennas regime, the MRC processing scheme keep on eliminating the interfering data symbols even in presence of even weakly estimated channel gains giving excellent BER curve while the opposite case might also be the case. Hence both MSE and BER curves are required for deciding on the performance of overall system.

            Thanks a lot.

  2. Thanks a lot Dr Emil for the explanation. I have two questions:

    1. What are other relevant metrics other than SINR that can satisfy “(or some other performance indicator in the data phase)”?

    2. I did not see the same argument in papers considering general transceiver design in single-cell scenarios or multi-cell scenarios employing pilot reuse plan to design pilot allocation. I mean is this an inherent property in MMSE estimation, that it does not distinguish between noise and interference, or is it a property of the pilot contamination problem?

    1. 1. Other revelvant metrics are some decoding error rate (BER, SER, PER), outage probability/rate, and the MSE in the data detection.

      2. Pilot contamination has always existed in cellular networks, but it is first when start doing coherent beamforming from arrays with many antennas that the coherent interference problem becomes clearly visible. It is not the MMSE estimation itself that is the problem; in fact, MMSE estimation scheme is what can solve the pilot contamination problem by exploiting the spatial correlation that always exists in practice: https://arxiv.org/pdf/1705.00538.pdf

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