Ch. 4 — Capacity of Wireless Channels

Chapter 4 of Wireless Communications (2nd ed. Draft) — Andrea Goldsmith

Abstract

This chapter establishes the theoretical capacity limits for single-user wireless communication channels under various fading conditions and assumptions regarding Channel State Information (CSI). It derives Shannon capacity expressions for time-invariant AWGN, time-varying flat fading, and frequency-selective fading channels, emphasizing the critical role of power adaptation and CSI availability at the transmitter. The central argument is that while fading generally reduces capacity relative to AWGN channels under fixed rate constraints, adaptive power allocation strategies like water-filling can recover or exceed AWGN capacity in specific regimes by exploiting favorable channel states. These results provide the mathematical foundation for designing rate-adaptive and power-control systems in subsequent chapters on modulation and coding.

Key Concepts

• Ergodic Capacity: Defined as the maximum constant data rate achievable with asymptotically small error probability over a time-varying channel, calculated as the expectation of the instantaneous AWGN capacity over the fading distribution $p(\gamma )$. This metric assumes the transmitter does not know the instantaneous channel gain $g[i]$, only the receiver does, forcing the code to span all fading states. It represents the long-term average throughput for delay-tolerant applications like file downloads where the encoder must handle the worst average conditions.
• Outage Capacity: A performance metric used for slowly varying channels where the transmitter fixes a data rate based on a minimum acceptable SNR ${\gamma }_{min}$. If the instantaneous SNR falls below ${\gamma }_{min}$, an outage occurs, and data is lost, allowing for higher average rates than ergodic capacity at the cost of reliability. This is formally defined as $C_{out}=(1-P_{out})B{\log }_2(1+{\gamma }_{min})$, where $P_{out}$ is the probability $p(\gamma <{\gamma }_{min})$.
• Channel Distribution Information (CDI): The scenario where the transmitter and receiver know the statistical distribution of the fading gain $p(g)$ but not the instantaneous realization. Finding the capacity under CDI is generally an open problem for Rayleigh fading, requiring complex optimization over input distributions, though closed-form solutions exist for finite-state Markov channels.
• Channel Side Information (CSI): The knowledge of the instantaneous channel gain $g[i]$ (or SNR $\gamma [i]$) at the receiver (Rx CSI) or both transmitter and receiver (Tx+Rx CSI). Rx CSI allows optimal decoding but fixed transmission rates, whereas Tx+Rx CSI enables the transmitter to adapt power and rate dynamically to maximize capacity.
• Water-Filling Power Adaptation: The optimal power allocation strategy for a transmitter with instantaneous CSI that maximizes ergodic capacity under an average power constraint. The allocation policy $P(\gamma )$ increases power for high SNR states and sets power to zero for states below a cutoff ${\gamma }_0$, mathematically analogous to filling a bowl with a sloped bottom up to a constant water level.
• Channel Inversion: A power adaptation technique where the transmitter increases power inversely proportional to the channel gain ($P\propto 1/\gamma$) to create a time-invariant equivalent channel. While this achieves zero-outage capacity with simple fixed-rate coding, it can reduce capacity to zero in severe fading environments like Rayleigh fading where $E[1/\gamma ]$ is infinite.
• Frequency-Selective Fading Capacity: The capacity of channels where delay spread causes signal distortion across the bandwidth, approximated by dividing the bandwidth into parallel flat fading subchannels of width $B_c$. The total capacity is the sum of capacities of these subchannels, utilizing water-filling in both time and frequency to optimize power distribution.
• Coherence Bandwidth Approximation: Used to convert a time-varying frequency-selective channel into parallel independent flat fading channels. This allows the application of flat fading capacity results to wideband systems by assuming subchannels separated by more than $B_c$ experience independent fading statistics.

Key Equations and Algorithms

• AWGN Channel Capacity: $C=B{\log }_2(1+\gamma )$, where $B$ is the bandwidth and $\gamma$ is the received Signal-to-Noise Ratio (SNR). This fundamental formula establishes the upper bound on reliable communication rates for a constant channel, serving as the baseline for comparing fading channel performance.
• Ergodic Capacity with Receiver CSI: $C=\int B{\log }_2(1+\gamma )p(\gamma )d\gamma$. This integral represents the average capacity over all fading realizations when the transmitter is unaware of instantaneous fading, effectively averaging the capacity of the underlying AWGN channel weighted by the SNR distribution.
• Optimal Power Allocation (Water-Filling): $P(\gamma )=P_{avg}\left(\frac{1}{{\gamma }_0}-\frac{1}{\gamma }\right)$ for $\gamma \ge {\gamma }_0$, and 0 otherwise. This Lagrangian solution maximizes the capacity expression $\int B{\log }_2(1+P(\gamma )\gamma /P_{avg})p(\gamma )d\gamma$ subject to the average power constraint $\int P(\gamma )p(\gamma )d\gamma =P_{avg}$.
• Cutoff Value Determination: ${\int }_{{\gamma }_0}^{\infty }\left(\frac{1}{{\gamma }_0}-\frac{1}{\gamma }\right)p(\gamma )d\gamma =1$. This equation determines the threshold ${\gamma }_0$ required to satisfy the average power constraint, ensuring that power is only allocated when the channel gain is sufficiently high to justify transmission.
• Zero-Outage Capacity (Channel Inversion): $C_{inv}=B{\log }_2(1+\sigma )$ where $\sigma =1/E[1/\gamma ]$. This capacity is achieved by maintaining a constant received power $\sigma$ regardless of fading, effectively inverting the channel gain to eliminate amplitude variations at the cost of significant power fluctuations.
• Frequency-Selective Water-Filling: $P(f)={\left(\frac{1}{{\gamma }_0}-\frac{1}{|H(f){|}^{2}P/N_0}\right)}^{+}$. This extends the time-domain water-filling concept to the frequency domain for time-invariant frequency-selective channels, allocating more power to frequency subchannels with stronger gains $|H(f)|$.
• Parallel Flat Fading Subchannel Model: Capacity is approximated as $\sum C_j({\bar{P}}_j)$ where subchannels $j$ have bandwidth $B_c$. This algorithm partitions the wideband signal bandwidth into independent narrowband blocks, allowing the summation of individual flat-fading capacities under a total power constraint.

Key Claims and Findings

• Fading Penalty with Receiver CSI Only: Under ergodic capacity conditions with receiver CSI only, fading always reduces capacity compared to an AWGN channel with the same average SNR due to Jensen’s inequality, meaning variability in the channel is detrimental without adaptation.
• Low SNR Advantage: In the low SNR regime (below 0 dB), a fading channel with transmitter and receiver CSI can actually exceed the capacity of an AWGN channel with the same average power because the transmitter exploits rare high-SNR states to transmit data at higher rates.
• Adaptation Gains: Transmitter adaptation (power and rate) significantly improves capacity over receiver-only CSI in severe fading, but the gain diminishes as the number of diversity branches increases and the channel approaches AWGN characteristics.
• Trade-off Between Complexity and Capacity: Channel inversion achieves zero-outage capacity with low implementation complexity (fixed-rate coding) but suffers large capacity losses compared to ergodic capacity in Rayleigh fading, whereas water-filling maximizes capacity but requires dynamic rate adaptation and high complexity.
• Frequency Selectivity as Parallel Channels: Time-varying frequency-selective channels can be effectively modeled as a set of parallel time-varying flat fading channels, enabling the use of parallel channel capacity formulas by respecting the coherence bandwidth $B_c$ for independence.
• Outage Probability vs. Rate: Higher transmission rates can be sustained by accepting a non-zero outage probability, as demonstrated by the capacity-outage curve where $C$ increases rapidly as $P_{out}$ is allowed to increase beyond zero.
• MIMO Capacity Scaling: The introduction of multiple antennas (MIMO) increases capacity linearly with $M=\min (M_t,M_r)$ in rich scattering environments, though the specific capacity formulas for MIMO are reserved for later chapters.

Terminology

• Channel Side Information (CSI): Knowledge of the instantaneous channel gain $g[i]$ or SNR $\gamma [i]$ available at the receiver (Rx CSI) or both ends (Tx+Rx CSI), used to adapt transmission parameters.
• Ergodic Capacity: The maximum achievable rate averaged over fading states for delay-insensitive applications, representing the long-term spectral efficiency of the channel $\int C(\gamma )p(\gamma )d\gamma$.
• Water-Filling Cutoff (${\gamma }_0$): The SNR threshold below which the transmitter allocates zero power to the channel in the optimal power allocation strategy, determined by the average power constraint.
• Outage Probability ($P_{out}$): The probability that the instantaneous SNR $\gamma$ falls below a target threshold ${\gamma }_{min}$, resulting in a failure to decode the fixed-rate transmission correctly.
• Coherence Bandwidth ($B_c$): The frequency separation over which channel responses are highly correlated; used in frequency-selective analysis to define the bandwidth of independent parallel subchannels ($B_c\approx 1/{\sigma }_{Tm}$).
• Channel Inversion: A power control technique where transmit power is adjusted as $P\propto 1/\gamma$ to compensate for fading, resulting in a constant received SNR and zero outage probability at the cost of average power.
• Channel Distribution Information (CDI): Knowledge of the fading statistics (PDF $p(\gamma )$) without knowledge of instantaneous realizations, which is insufficient for optimal power adaptation without further assumptions.
• Frequency-Selective Fading: A channel condition where the signal bandwidth exceeds the coherence bandwidth, causing distortion and inter-symbol interference (ISI) due to multipath delay spread, requiring equalization or multicarrier modulation.