Understanding Ergodic Capacity: Testing Your Wireless Speed Calculator

Introduction

We already met ergodic capacity as an expectation. Now treat it as something you compute: how do you estimate it stably, and is there a closed form to check against? Call this a short lab before MIMO.

Motive

Monte Carlo is easy to get slightly wrong—too few samples, the wrong distribution for \(|h|^2\), or \(\log\) where you meant \(\log_2\). A closed form keeps you honest. Once the scalar case matches, you can use the same Monte Carlo pattern on \(\log\det\) for MIMO.

Technical development

\[C_{\mathrm{erg}}(\rho)=\int_0^\infty \log_2(1+\rho u)\,e^{-u}\,du = \frac{e^{1/\rho}}{\ln 2}\,E_1\!\left(\frac{1}{\rho}\right),\]

where \(E_1\) is the exponential integral. The integral is exactly the expectation of \(\log_2(1+\rho|h|^2)\) for unit-mean exponential \(|h|^2\).

How this achieves the motive

Agreement between Monte Carlo and the exponential-integral expression tells you your random-number generation and your units are correct. Disagreement usually means someone used \(\log\) instead of \(\log_2\), or drew complex Gaussians without normalising variance.

Sample simulation

import numpy as np
from scipy.special import expi

def erg_mc(rho, n=100000, rng=None):
    rng = rng or np.random.default_rng()
    return np.mean(np.log2(1 + rho * rng.exponential(1.0, n)))

def erg_ei(rho):
    # exp(1/rho) * E_1(1/rho) / ln(2);  E_1(x) = -expi(-x)
    return np.exp(1 / rho) * (-expi(-1 / rho)) / np.log(2)

for sdb in (0, 10, 20):
    rho = 10 ** (sdb / 10)
    print("%2d dB   MC=%.4f   EI=%.4f" % (sdb, erg_mc(rho), erg_ei(rho)))

Sample results

The printed MC and EI columns should match to about two decimal places with \(10^5\) samples. If they disagree, fix the scalar case before touching MIMO. Figure 1 places those numbers on the full SNR axis.

Next topic

Replace the scalar \(|h|^2\) by the eigenvalues of \(\mathbf{HH}^H\) and the same averaging philosophy yields MIMO ergodic capacity.

References

1. A. Goldsmith, Wireless Communications, Cambridge University Press, Cambridge, UK, 2005. Chapter 4.
2. M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels, 2nd ed., John Wiley & Sons, Hoboken, NJ, 2005.
3. D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, Cambridge, UK, 2005. Chapter 5.

Frequently asked questions

What closed-form expression checks a Rayleigh ergodic-capacity Monte Carlo? For unit-mean exponential \(|h|^2\), \(C_{\mathrm{erg}}(\rho)=e^{1/\rho}E_1(1/\rho)/\ln 2\), where \(E_1\) is the exponential integral. Agreement with Monte Carlo to about two decimals at \(10^5\) samples is a good sanity check before moving to MIMO.

What are common numerical mistakes in ergodic-capacity simulations? Too few channel draws, drawing complex Gaussians without unit variance per dimension, and using natural logarithm where capacity is defined with \(\log_2\). Any of these shifts the curve away from the exponential-integral reference.

Why verify the scalar case before simulating MIMO capacity? MIMO capacity replaces \(|h|^2\) by the eigenvalues of \(\mathbf{HH}^H\) inside a log-determinant, but the Monte Carlo pattern is the same. If the scalar loop is wrong, the matrix loop will be wrong in the same way and harder to debug.