Ch. 7 — Diversity

Chapter 7 of Wireless Communications (2nd ed. Draft) — Andrea Goldsmith

Abstract

This chapter presents a comprehensive analysis of diversity techniques employed to mitigate the effects of wireless channel fading on system performance. It establishes the theoretical framework for various receiver and transmitter combining schemes, including Selection Combining (SC), Maximal-Ratio Combining (MRC), and the Alamouti space-time block code. A central technical contribution is the introduction of the Moment Generating Function (MGF) method as a unified approach for evaluating average error probability across independent fading branches. The chapter argues that while MRC achieves optimal diversity order and array gain, simpler techniques like SC and Equal-Gain Combining (EGC) offer viable performance-complexity trade-offs. Additionally, it distinguishes between transmit diversity scenarios where Channel State Information (CSI) is known versus unknown at the transmitter.

Key Concepts

• Selection Combining (SC): A receiver diversity technique that selects the single branch with the highest instantaneous signal-to-noise ratio (SNR) for processing. This method requires monitoring all branches but co-phasing is not required since only one signal is used. SC offers a simpler implementation than MRC but suffers from diminishing diversity gain as the number of branches increases.
• Switch-and-Stay Combining (SSC): A threshold-based combining scheme designed to reduce receiver complexity by sequentially scanning branches until one exceeds a threshold ${\gamma }_T$. The system remains on a selected branch until its SNR drops below ${\gamma }_T$, at which point it switches to another branch. Performance is optimized when the threshold equals the target outage SNR ${\gamma }_0$, offering performance between no diversity and ideal SC.
• Equal-Gain Combining (EGC): A combining technique that co-phases signals from all branches and sums them with equal weighting factors. Unlike MRC, EGC does not require knowledge of instantaneous SNR amplitudes, only phases, reducing complexity while maintaining performance within approximately 1 dB of MRC.
• Maximal-Ratio Combining (MRC): The optimal linear receiver diversity technique where branch signals are weighted proportional to their respective amplitudes and co-phased before summation. MRC maximizes the output SNR by ensuring the weights $a_i$ satisfy $a_i\propto r_i$, resulting in an output SNR equal to the sum of individual branch SNRs.
• Generalized Combining (GC): A hybrid scheme that selects the $L$ branches with the highest SNRs from a total of $M$ available branches, where $1<L<M$, and combines them using MRC or EGC. GC provides performance superior to selecting fewer branches but inferior to using all $M$ branches, balanced against hardware constraints.
• Transmit Diversity with CSIT: A system configuration where multiple transmit antennas utilize Channel Side Information at the Transmitter (CSIT) to pre-code signals for coherent combining at the receiver. When CSI is known, weights are chosen to maximize received SNR, achieving performance identical to receiver-side MRC with linear array gain.
• Alamouti Space-Time Block Code: A specific transmit diversity scheme designed for two-antenna systems where channel knowledge is unavailable at the transmitter. It exploits both spatial and temporal diversity to achieve a diversity order of 2 without requiring channel feedback or complex processing at the receiver.
• Moment Generating Function (MGF) Analysis: A mathematical technique derived in Section 7.4 that simplifies the calculation of average error probability by converting the convolution of branch SNR distributions into a product of MGFs. This method is particularly effective for independent fading paths, yielding closed-form expressions or single finite-range integrals.
• Diversity Order: A metric representing the slope of the average probability of error curve versus SNR in the high-SNR limit. It quantifies the number of independent fading paths effectively combined, with MRC achieving the full diversity order equal to the number of branches $M$.
• Array Gain: The SNR improvement achieved by combining multiple diversity branches relative to a single branch system at moderate SNR levels. MRC provides an array gain proportional to the number of branches $M$, whereas SC exhibits diminishing returns as $M$ increases.

Key Equations and Algorithms

• MRC Output SNR: ${\gamma }_{\Sigma }={\sum }_{i=1}^M{\gamma }_i$. This expression indicates that the combined SNR is the sum of the individual branch SNRs ${\gamma }_i$, maximizing the total energy collected from all diversity paths.
• MGF of Branch SNR (Rayleigh): $M_{{\gamma }_i}(s)=(1-s{\bar{\gamma }}_i{)}^{-1}$. This defines the moment generating function for the exponential distribution of SNR in Rayleigh fading, where ${\bar{\gamma }}_i$ is the average branch SNR.
• Average Error Probability via MGF (MRC): ${\bar{P}}_b=\frac{1}{\pi }{\int }_0^{\pi /2}{\prod }_{i=1}^M{M}_{{\gamma }_i}\left(-\frac{g}{{\sin }^2\varphi }\right)d\varphi$. This algorithm computes the average bit error probability for MRC by integrating the product of individual branch MGFs, applicable to modulations approximated by $P_s=\alpha Q(\sqrt{\beta \gamma })$.
• Alamouti Received Signal Model: $\mathbf{y}={\mathbf{H}}_A\mathbf{s}+\mathbf{n}$. This matrix equation represents the received symbol vector over two time periods, where ${\mathbf{H}}_A$ is the effective channel matrix constructed from channel gains $h_1,h_2$.
• SSC CDF (Two-Branch): $F_{{\gamma }_{\Sigma }}(\gamma )=[1-e^{-\gamma /\bar{\gamma }}][1-e^{-\max (0,\gamma -{\gamma }_T)/\bar{\gamma }}]$. This cumulative distribution function describes the combined SNR statistics for two-branch SSC in i.i.d. Rayleigh fading, dependent on the threshold ${\gamma }_T$.
• Diversity Order Definition: $d={\lim }_{\bar{\gamma }\to \infty }\frac{-\log (P_b)}{\log (\bar{\gamma })}$. This limit defines the diversity order based on the asymptotic slope of the error probability curve, characterizing the system’s resilience to deep fades at high SNR.
• Optimal SSC Threshold: ${\gamma }_T={\gamma }_0$. This rule states that to minimize the outage probability for a given target threshold ${\gamma }_0$ in SSC, the switching threshold must be set equal to the outage target.
• EGC CDF (Two-Branch): $P_{{\gamma }_{\Sigma }}(\gamma )=1-Q\left(2\sqrt{\gamma \bar{\gamma }}\right)+\sqrt{\frac{\gamma }{\bar{\gamma }}}{e}^{-(\gamma /\bar{\gamma })}{I}_1\left(2\sqrt{\gamma \bar{\gamma }}\right)$. This expression (referenced via text description) defines the CDF for EGC output SNR involving the Q-function and modified Bessel function $I_1$.

Key Claims and Findings

• Maximal-Ratio Combining is the only diversity-combining technique that achieves full diversity order equal to the number of antennas $M$ simultaneously with optimal array gain.
• Equal-Gain Combining typically exhibits a power penalty of less than 1 dB compared to Maximal-Ratio Combining, justifying its use in systems with reduced complexity requirements.
• For Switch-and-Stay Combining, the switching threshold ${\gamma }_T$ that minimizes outage probability is exactly equal to the target SNR threshold ${\gamma }_0$ required for the specific modulation.
• Transmit diversity with perfect Channel State Information at the Transmitter (CSIT) achieves the same performance as receiver-side Maximal-Ratio Combining, providing a diversity order of $M$.
• The Alamouti scheme achieves full transmit diversity (order 2) for two antennas without requiring channel knowledge at the transmitter, though it provides only an array gain of 1.
• Using Moment Generating Functions simplifies the analysis of diversity systems with independent but not necessarily identically distributed branches, avoiding complex convolution integrals.
• Selection Combining suffers from diminishing returns in array gain as the number of branches $M$ increases, unlike MRC where array gain increases linearly with $M$.

Terminology

• Branch: A specific signal path between a transmitter and receiver antenna pair experiencing independent fading statistics.
• Outage Probability: The probability $P_{out}$ that the instantaneous SNR of the combiner output falls below a predetermined threshold ${\gamma }_0$ required for reliable communication.
• Array Gain: The shift in the average SNR curve resulting from combining multiple branches, providing an SNR improvement independent of the diversity slope at moderate SNRs.
• Co-phasing: The process of adjusting the phase of signals on different branches so that they add constructively at the combiner input.
• Diversity Loss: The performance degradation in a specific combining scheme relative to optimal MRC, often due to lack of optimal weight selection or correlation between branches.
• Channel Side Information at Transmitter (CSIT): Knowledge of the instantaneous complex channel gains available at the transmitter, enabling beamforming and optimal weight assignment.
• Fading Correlation: A statistical dependence between the fading amplitudes of different diversity branches; performance degradation is negligible for correlations below 0.5.
• Space-Time Block Code (STBC): A coding scheme that encodes data across multiple transmit antennas and time slots to provide diversity gains without requiring CSI at the transmitter.
• Hard Decision Decoding: A decoding method where received bits are strictly decided as 0 or 1 prior to channel decoding, though this chapter focuses primarily on modulation error probability analysis.
• Switching Mechanism: The logic used in SSC to determine when to transition from the active branch to a backup branch based on threshold comparisons.