The Common SINR Mistake

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 uplink bound 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 uplink bound 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.

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