Category Archives: Education

Commentary, Education, Technical insights

When Normalization is Dangerous

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The signal-to-noise ratio (SNR) generally depends on the transmit power, channel gain, and noise power:

Since the spectral efficiency (bit/s/Hz) and many other performance metrics of interest depend on the SNR, and not the individual values of the three parameters, it is a common practice to normalize one or two of the parameters to unity. This habit makes it easier to interpret performance expressions, to select reasonable SNR ranges, and to avoid mistakes in analytical derivations.

There are, however, situations when the absolute value of the transmitted/received signal power matters, and not the relative value with respect to the noise power, as measured by the SNR. In these situations, it is easy to make mistakes if you use normalized parameters. I see this type of errors far too often, both as a reviewer and in published papers. I will give some specific examples below, but I won’t tell you who has made these mistakes, to not point the finger at anyone specifically.

Wireless energy transfer

Electromagnetic radiation can be used to transfer energy to wireless receivers. In such wireless energy transfer, it is the received signal energy that is harvested by the receiver, not the SNR. Since the noise power is extremely small, the SNR is (at least) a billion times larger than the received signal power. Hence, a normalization error can lead to crazy conclusions, such as being able to transfer energy at a rate of 1 W instead of 1 nW. The former is enough to keep a wireless transceiver on continuously, while the latter requires you to harvest energy for a long time period before you can turn the transceiver on for a brief moment.

Energy efficiency

The energy efficiency (EE) of a wireless transmission is measured in bit/Joule. The EE is computed as the ratio between the data rate (bit/s) and the power consumption (Watt=Joule/s). While the data rate depends on the SNR, the power consumption does not. The same SNR value can be achieved over a long propagation distance by using high transmit power or over a short distance by using a low transmit power. The EE will be widely different in these cases. If a “normalized transmit power” is used instead of the actual transmit power when computing the EE, one can get EEs that are one million times smaller than they should be. As a rule-of-thumb, if you compute things correctly, you will get EE numbers in the range of 10 kbit/Joule to 10 Mbit/Joule.

Noise power depends on the bandwidth

The noise power is proportional to the communication bandwidth. When working with a normalized noise power, it is easy to forget that a given SNR value only applies for one particular value of the bandwidth.

Some papers normalize the noise variance and channel gain, but then make the SNR equal to the unnormalized transmit power (measured in W). This may greatly overestimate the SNR, but the achievable rates might still be in the reasonable range if you operate the system in an interference-limited regime.

Some papers contain an alternative EE definition where the spectral efficiency (bit/s/Hz) is divided by the power consumption (Joule/s). This leads to the alternative EE unit bit/Joule/Hz. This definition is not formally wrong, but gives the misleading impression that one can multiply the EE value with any choice of bandwidth to get the desired number of bit/Joule. That is not the case since the SNR only holds for one particular value of the bandwidth.

Knowing when to normalize

In summary, even if it is convenient to normalize system parameters in wireless communications, you should only do it if you understand when normalization is possible and when it is not. Otherwise, you can make embarrassing mistakes, such as submitting a paper where the results are six orders of magnitude wrong. And, unfortunately, there are several such papers that have been published and these create a bad circle by tricking others into making the same mistakes.

Education, News

30% Discount on “Massive MIMO Networks” Book

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The hardback version of the massive new book Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency (by Björnson, Sanguinetti, Hoydis) is currently available for the special price of $70 (including worldwide shipping). The original price is $99.

This price is available until the end of April when buying the book directly from the publisher through the following link:

https://www.nowpublishers.com/Order/BuyBook?isbn=978-1-68083-985-2

Note: The book’s authors will give a joint tutorial on April 15 at WCNC 2018. A limited number of copies of the book will be available for sale at the conference and if you attend the tutorial, you will receive even better deal on buying the book!

Commentary, Education, Technical insights

How Distortion from Nonlinear Massive MIMO Transceivers is Radiated Spatially

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While the research literature is full of papers that design wireless communication systems under constraints on the maximum transmitted power, in practice, it might be constraints on the equivalent isotropically radiated power (EIRP) or the out-of-band radiation that limit the system operation.

Christopher Mollén recently defended his doctoral thesis entitled High-End Performance with Low-End Hardware: Analysis of Massive MIMO Base Station Transceivers. In the following video, he explains the basics of how the non-linear distortion from Massive MIMO transceivers is radiated in space.

Education, Technical insights

Are Link Simulations Needed Anymore?

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One reason for why capacity lower bounds are so useful is that they are accurate proxies for link-level performance with modern coding. But this fact, well known to information and coding theorists, is often contested by practitioners. I will discuss some possible reasons for that here.

The recipe is to compute the capacity bound, and depending on the code blocklength, add a dB or a few, to the required SNR. That gives the link performance prediction. The coding literature is full of empirical results, showing how far from capacity a code of a given block length is for the AWGN channel, and this gap is usually not extremely different for other channel models – although, one should always check this.

But there are three main caveats with this:

  1. First, the capacity bound, or the “SINR” that it often contains, must be information-theoretically correct. A great deal of papers get this wrong. Emil explained in his blog post last week some common errors. The recommended approach is to map the channel onto one of the canonical cases in Figure 2.9 in Fundamentals of Massive MIMO, verify that the technical conditions are satisfied, and use the corresponding formula.
  2. When computing expressions of the type E[log(1+”SINR”)], then the average should be taken over all quantities that are random within the duration of a codeword. Typically, this means averaging over the randomness incurred by the noise, channel estimation errors, and in many cases the small-scale fading. All other parameters must be kept fixed. Typically, user positions, path losses, shadow fading, scheduling and pilot assignments, are fixed, so the expectation is conditional on those. (Yet, the interference statistics may vary substantially, if other users are dropping in and out of the system.) This in turn means that many “drops” have to be generated, where these parameters are drawn at random, and then CDF curves with respect to that second level of randomness needs be computed (numerically).Think of the expectation E[log(1+”SINR”)] as a “link simulation”. Every codeword sees many independent noise realizations, and typically small-scale fading realizations, but the same realization of the user positions. Also, often, neat (and tight) closed-form bounds on E[log(1+”SINR”)] are available.
  3. Care is advised when working with relatively short blocks (less than a few hundred bits) and at rates close to the constrained capacity with the foreseen modulation format. In this case, many of the “standard” capacity bounds become overoptimistic.As a rule of thumb, compare the capacity of an AWGN channel with the constrained capacity of the chosen modulation at the spectral efficiency of interest, and if the gap is small, the capacity bounds will be useful. If not, then reconsider the choice of modulation format! (See also homework problem 1.4.)

How far are the bounds from the actual capacity typically? Nobody knows, but there are good reasons to believe they are extremely close. Here (Figure 1) is a nice example that compares a decoder that uses the measured channel likelihood, instead of assuming a Gaussian (which is implied by the typical bounding techniques). From correspondence with one of the authors: “The dashed and solid lines are the lower bound obtained by Gaussianizing the interference, while the circles are the rate achievable by a decoder exploiting the non-Gaussianity of the interference, painfully computed through days-long Monte-Carlo. (This is not exactly the capacity, because the transmit signals here are Gaussian, so one could deviate from Gaussian signaling and possibly do slightly better — but the difference is imperceptible in all the experiments we’ve done.)”

Concerning Massive MIMO and its capacity bounds, I have met for a long time with arguments that these capacity formulas aren’t useful estimates of actual performance. But in fact, they are: In one simulation study we were less than one dB from the capacity bound by using QPSK and a standard LDPC code (albeit with fairly long blocks). This bound accounts for noise and channel estimation errors. Such examples are in Chapter 1 of Fundamentals of Massive MIMO, and also in the ten-myth paper:

(I wrote the simulation code, and can share it, in case anyone would want to reproduce the graphs.)

So in summary, while capacity bounds are sometimes done wrong; when done right they give pretty good estimates of actual link performance with modern coding.

(With thanks to Angel Lozano for discussions.)

Commentary, Education, Technical insights

The Common SINR Mistake

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We are used to measuring performance in terms of the signal-to-interference-and-noise ratio (SINR), but this is seldom the actual performance metric in communication systems. In practice, we might be interested in a function of the SINR, such as the data rate (a.k.a. spectral efficiency), bit-error-rate, or mean-squared error in the data detection. When the receiver has perfect channel state information (CSI), the aforementioned metrics are all functions of the same SINR expression, where the power of the received signal is divided by the power of the interference plus noise. Details can be found in Examples 1.6-1.8 of the book Optimal Resource Allocation in Coordinated Multi-Cell Systems.

In most cases, the receiver only has imperfect CSI and then it is harder to measure the performance. In fact, it took me years to understand this properly. To explain the complications, consider the uplink of a single-cell Massive MIMO system with K single-antenna users and M antennas at the base station. The received M-dimensional signal is

    $$\mathbf{y} = \sum_{i=1}^{K} \mathbf{h}_{i} x_{i} + \mathbf{n}$$

where $x_{i}$ is the unit-power information signal from user $i$, $\mathbf{h}_{i} \in \mathbb{C}^{M}$ is the fading channel from this user, and $\mathbf{n}\in \mathbb{C}^{M}$ is unit-power additive Gaussian noise. In general, the base station will only have access to an imperfect estimate $\hat{\mathbf{h}}_{i} \in \mathbb{C}^{M}$ of $\mathbf{h}_{i}$, for $i=1,\ldots,K.$

Suppose the base station uses  $\hat{\mathbf{h}}_{1},\ldots,\hat{\mathbf{h}}_{K}$ to select a receive combining vector $\mathbf{v}_k\in \mathbb{C}^{M}$ for user $k$. The base station then multiplies it with $\mathbf{y}$ to form a scalar that is supposed to resemble the information signal $x_{k}$:

    $$\mathbf{v}_k^H \mathbf{y} = \underbrace{\mathbf{v}_k^H \mathbf{h}_{k} x_{k}}_\textrm{Desired signal} + \underbrace{\sum_{i=1, i \neq k}^{K} \mathbf{v}_k^H\mathbf{h}_{i} x_{i}}_\textrm{Interference} + \underbrace{\mathbf{v}_k^H \mathbf{w}}_\textrm{Noise}.$$

From this expression, a common mistake is to directly say that the SINR is

    $$\mathrm{SINR}_k^\textrm{wrong} = \frac{| \mathbf{v}_k^H \mathbf{h}_{k}|^2}{ \sum_{i=1, i \neq k}^{K} | \mathbf{v}_k^H \mathbf{h}_{i}|^2 + \| \mathbf{v}_k \|^2},$$

which is obtained by computing the power of each of the terms (averaged over the signal and noise), and then claim that $\mathbb{E}\{\log_2(1+\mathrm{SINR}_k^\textrm{wrong} )\}$ is an achievable rate (where the expectation is with respect to the random channels). You can find this type of arguments in many papers, without proof of the information-theoretic achievability of this rate value. Clearly, $\mathrm{SINR}_k^\textrm{wrong} $ is an SINR, in the sense that the numerator contains the total signal power and the denominator contains the interference power plus noise power. However, this doesn’t mean that you can plug $\mathrm{SINR}_k^\textrm{wrong} $ into “Shannon’s capacity formula” and get something sensible. This will only yield a correct result when the receiver has perfect CSI.

A basic (but non-conclusive) test of the correctness of a rate expression is to check that the receiver can compute the expression based on its available information (i.e., estimates of random variables and deterministic quantities). Any expression containing $\mathrm{SINR}_k^\textrm{wrong}$ fails this basic test since you need to know the exact channel realizations \mathbf{h}_{1},\ldots,\mathbf{h}_{K} to compute it, although the receiver only has access to the estimates.

What is the right approach?

Remember that the SINR is not important by itself, but we should start from the performance metric of interest and then we might eventually interpret a part of the expression as an effective SINR. In Massive MIMO, we are usually interested in the ergodic capacity. Since the exact capacity is unknown, we look for rigorous lower bounds on the capacity. There are several bounding techniques to choose between, whereof I will describe the two most common ones.

The first lower bound on the uplink capacity can be applied when  the channels are Gaussian distributed and $\hat{\mathbf{h}}_{1}, \ldots, \hat{\mathbf{h}}_{K}$ are the MMSE estimates with the corresponding estimation error covariance matrices $\mathbf{C}_{1},\ldots,\mathbf{C}_{K}$. The ergodic capacity of user $k$ is then lower bounded by

$$R_k^{(1)} = \mathbb{E} \left\{ \log_2 \left( 1 + \frac{| \mathbf{v}_k^H \hat{\mathbf{h}}_{k}|^2}{ \sum_{i=1, i \neq k}^{K} | \mathbf{v}_k^H \hat{\mathbf{h}}_{i}|^2 + \sum_{i=1}^{K} \mathbf{v}_k^H \mathbf{C}_{i} \mathbf{v}_k + \| \mathbf{v}_k \|^2} \right) \right\}.

Note that this expression can be computed at the receiver using only the available channel estimates (and deterministic quantities). The ratio inside the logarithm can be interpreted as an effective SINR, in the sense that the rate is equivalent to that of a fading channel where the receiver has perfect CSI and an SNR equal to this effective SINR. A key difference from $\mathrm{SINR}_k^\textrm{wrong}$ is that only the part of the desired signal that is received along the estimated channel appears in the numerator of the SINR, while the rest of the desired signal appears as $\mathbf{v}_k^H \mathbf{C}_{k} \mathbf{v}_k$ in the denominator. This is the price to pay for having imperfect CSI at the receiver, according to this capacity bound, which has been used by Hoydis et al. and Ngo et al., among others.

The second lower bound on the uplink capacity is

$$R_k^{(2)} = \log_2 \left( 1 + \frac{ | \mathbb{E}\{ \mathbf{v}_k^H \mathbf{h}_{k} \} |^2}{ \sum_{i=1}^{K} \mathbb{E} \{ | \mathbf{v}_k^H \mathbf{h}_{i}|^2 \} - | \mathbb{E}\{ \mathbf{v}_k^H \mathbf{h}_{k} \} |^2+ \mathbb{E}\{\| \mathbf{v}_k \|^2\} } \right),

which can be applied for any channel fading distribution. This bound provides a value close to $R_k^{(1)}$ when there is substantial channel hardening in the system, while $R_k^{(2)}$ will greatly underestimate the capacity when $\mathbf{v}_k^H \mathbf{h}_{k}$ varies a lot between channel realizations. The reason is that to obtain this bound, the receiver detects the signal as if it is received over a non-fading channel with gain \mathbb{E}\{ \mathbf{v}_k^H \mathbf{h}_{k} \} (which is deterministic and thus known in theory, and easy to measure in practice), but there are no approximations involved so $R_k^{(2)}$ is always a valid bound.

Since all the terms in $R_k^{(2)} $ are deterministic, the receiver can clearly compute it using its available information. The main merit of $R_k^{(2)}$ is that the expectations in the numerator and denominator can sometimes be computed in closed form; for example, when using maximum-ratio and zero-forcing combining with i.i.d. Rayleigh fading channels or maximum-ratio combining with correlated Rayleigh fading. Two early works that used this bound are by Marzetta and by Jose et al..

The two uplink rate expressions can be proved using capacity bounding techniques that have been floating around in the literature for more than a decade; the main principle for computing capacity bounds for the case when the receiver has imperfect CSI is found in a paper by Medard from 2000. The first concise description of both bounds (including all the necessary conditions for using them) is found in Fundamentals of Massive MIMO. The expressions that are presented above can be found in Section 4 of the new book Massive MIMO Networks. In these two books, you can also find the right ways to compute rigorous lower bounds on the downlink capacity in Massive MIMO.

In conclusion, to avoid mistakes, always start with rigorously computing the performance metric of interest. If you are interested in the ergodic capacity, then you start from one of the canonical capacity bounds in the above-mentioned books and verify that all the required conditions are satisfied. Then you may interpret part of the expression as an SINR.

5G, Beyond 5G, Commentary, Education, News, Technical insights

I Never Thought It Would Happen So Fast

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I never thought it would happen so fast. When I started to work on Massive MIMO in 2009, the general view was that fully digital, phase-coherent operation of so many antennas would be infeasible, and that power consumption of digital and analog circuitry would prohibit implementations for the foreseeable future. More seriously, reservations were voiced that reciprocity-based beamforming would not work, or that operation in mobile conditions would be impossible.

These arguments, it turned out, all proved to be wrong. In 2017, Massive MIMO was the main physical-layer technology under standardization for 5G, and it is unlikely that any serious future cellular wireless communications system would not have Massive MIMO as a main technology component.

But Massive MIMO is more than a groundbreaking technology for wireless communications: it is also an elegant and mathematically rigorous approach to teaching wireless communications. In the moderately-large number-of-antennas regime, our closed-form capacity bounds become convenient proxies for the link performance achievable with practical coding and modulation.

These expressions take into account the effects of all significant physical phenomena: small-scale and large-scale fading, intra- and inter-cell interference, channel estimation errors, pilot reuse (also known as pilot contamination) and power control. A comprehensive analytical understanding of these phenomena simply has not been possible before, as the corresponding information theory has too complicated for any practical use.

The intended audiences of Fundamentals of Massive MIMO are engineers and students. I anticipate that as graduate courses on the topic become commonplace, our extensive problem set (with solutions) available online will serve as a useful resource to instructors. While other books and monographs will likely appear down the road, focusing on trendier and more recent research, Fundamentals of Massive MIMO distills the theory and facts that will prevail for the foreseeable future. This, I hope, will become its most lasting impact.

To read the preface of Fundamentals of Massive MIMO, click here. You can also purchase the book here.