Asymptomania

I am borrowing the title from a column written by my advisor two decades ago, in the array signal processing gold rush era.

Asymptotic analysis is a popular tool within statistical signal processing (infinite SNR or number of samples), information theory (infinitely long blocks) and more recently, [massive] MIMO wireless communications (infinitely many antennas).

Some caution is strongly advisable with respect to the latter. In fact, there are compelling reasons to avoid asymptotics in the number of antennas altogether:

  • First, elegant, rigorous and intuitively comprehensible capacity bound formulas are available in closed form.
    The proofs of these expressions use basic random matrix theory, but no asymptotics at all.
  • Second, the notion of “asymptotic limit” or “asymptotic behavior” helps propagate the myth that Massive MIMO somehow relies on asymptotics or “infinite” numbers (or even exorbitantly large numbers) of antennas.
  • Third, many approximate performance results for Massive MIMO (particularly “deterministic equivalents”) based on asymptotic analysis are complicated, require numerical evaluation, and offer little intuitive insight. (And, the verification of their accuracy is a formidable task.)

Finally, and perhaps most importantly, careless use of asymptotic arguments may yield erroneous conclusions. For example in the effective SINRs in multi-cell Massive MIMO, the coherent interference scales with M (number of antennas) – which yields the commonly held misconception that coherent interference is the main impairment caused by pilot contamination. But in fact, in many relevant circumstances it is not (see case studies here): the main impairment for “reasonable” values of M is the reduction in coherent beamforming gain due to reduced estimation quality, which in turn is independent of M.

In addition, the number of antennas beyond which the far-field assumption is violated is actually smaller than what one might first think (problem 3.14).

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