SIMO receive-diversity channel models — from one transmit antenna to several receivers

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MIMO — Diversity path

1. Intro → 2. Diversity vs mux → 3. SIMO → 4. Selection combining → 5. MRC

Tools: Diversity & multiplexing demo

Introduction

Turn off every transmit antenna but one, and keep several receive antennas. That SIMO setup is the cleanest way into spatial diversity in class: no space-time code to design, no precoder to argue about—just a vector of observations of one symbol, and the question how to combine them.

Motive

Hardware often grows receive antennas before transmit antennas (phones listen with multiple chains while still transmitting on one). Selection combining and MRC are usually the first algorithms you try on that hardware. Their SNR gains—and their costs in channel knowledge—start from the SIMO model.

Technical development

\[\mathbf{r}=\mathbf{h}s+\mathbf{n}, \qquad \mathbf{h}\in\mathbb{C}^{N}.\]

A linear combiner produces the scalar \(y=\mathbf{w}^H\mathbf{r}\). Three classical recipes:

  • Selection combining (SC): set \(w\) to pick the branch with largest \(|h_i|\). Simple, one RF chain after the switch, but discards energy on the other antennas.
  • Maximum ratio combining (MRC): \(\mathbf{w}=\mathbf{h}\). Co-phases and weights by amplitude; maximises instantaneous SNR among linear combiners.
  • Equal-gain combining (EGC): co-phase only, equal amplitudes. Nearly as good as MRC when branch SNRs are similar, with simpler AGC.
SER SC MRC
Figure 1: Preview of QPSK symbol-error rate under SC and MRC for several branch counts (full discussion in the next two articles).

How this achieves the motive

If the entries of \(\mathbf{h}\) fade independently, the combiner output SNR is far less likely to be small than any single branch SNR. That is diversity in one sentence. SC realises it with a hard choice; MRC realises it with a soft weighted sum. Figure 1 already shows MRC pulling ahead of SC at the same \(N\)—we will explain why in the MRC article.

Sample simulation

import numpy as np

N, snr_db = 4, 10
snr = 10 ** (snr_db / 10)
rng = np.random.default_rng(0)
h = (rng.standard_normal(N) + 1j * rng.standard_normal(N)) / np.sqrt(2)
gamma = np.abs(h) ** 2 * snr
print("branch SNRs:", np.round(gamma, 3))
print("SC chooses branch", int(np.argmax(gamma)), "with SNR", round(gamma.max(), 3))
print("MRC output SNR (sum):", round(gamma.sum(), 3))

Sample results

For this seed the MRC sum exceeds the best single branch—the power that selection combining discards. Re-run with other seeds; the inequality should hold for almost every draw.

Next topic

We treat selection combining in full: outage formula, curves, and a QPSK SER experiment.

References

  1. A. Goldsmith, Wireless Communications, Cambridge University Press, Cambridge, UK, 2005.
  2. D. G. Brennan, “Linear diversity combining techniques,” Proceedings of the IRE, vol. 47, no. 6, pp. 1075–1102, June 1959.
  3. M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels, 2nd ed., John Wiley & Sons, Hoboken, NJ, 2005.

Frequently asked questions

What is a SIMO receive-diversity model? One transmit antenna and \(N\) receive antennas yield the vector observation \(\mathbf{r}=\mathbf{h}s+\mathbf{n}\). A linear combiner \(y=\mathbf{w}^H\mathbf{r}\) turns that vector into a single decision statistic. The choice of \(\mathbf{w}\) defines selection combining, MRC, or equal-gain combining.

Why do handsets often grow receive chains before transmit chains? Receive diversity needs only additional RF front-ends and baseband processing on the handset, whereas transmit diversity or uplink MIMO also needs coordinated multi-antenna transmission and usually more uplink power amplifiers. Selection combining and MRC are therefore natural first algorithms on multi-antenna receivers.

How do selection combining, MRC, and equal-gain combining differ? Selection combining keeps only the strongest branch. MRC co-phases and amplitude-weights every branch and maximises instantaneous SNR among linear combiners. Equal-gain combining co-phases with equal amplitudes and is nearly as good as MRC when branch SNRs are similar, with simpler automatic gain control.

Does receive diversity require channel state information? Selection combining needs only per-branch SNR (or envelope) measurements. MRC and equal-gain combining need per-branch channel estimates to co-phase correctly. Poor CSI erodes the combining gain.

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