Ferkans — Interactive Telecom Tutor

Why Fading Changes Everything

In Chapters 10 and 12 we studied channels with fixed, known gains. The AWGN channel $Y = X + Z$ has a deterministic relationship between input power and capacity. But in wireless communications, the channel gain fluctuates randomly due to multipath propagation, shadowing, and mobility. The receiver may experience a strong signal in one time slot and a weak one in the next.

The central question of this chapter is: how does this randomness affect the fundamental limits of reliable communication? The answer depends critically on two factors: (1) the time scale of fading relative to the codeword length, and (2) who knows the channel realization — the receiver, the transmitter, or both. These distinctions lead to qualitatively different capacity formulas and engineering strategies.

Definition:
Scalar Fading Channel

The scalar fading channel is described by

$Y = HX + Z,$

where:

$X$ is the transmitted signal with average power constraint $\mathbb{E}[|X|^2] \leq P$ ,
$H$ is the fading gain, a random variable drawn from a known distribution $f_H(h)$ ,
$Z \sim \mathcal{N}(0, N)$ is additive Gaussian noise independent of $H$ and $X$ ,
$Y$ is the received signal.

The instantaneous received SNR given fading realization $H = h$ is

$\text{SNR}(h) = \frac{|h|^2 P}{N}.$

We write $\text{SNR} = P/N$ for the average SNR (without fading), so $\text{SNR}(h) = |h|^2 \cdot \text{SNR}$ . When the fading distribution has $\mathbb{E}[|H|^2] = 1$ , the average received SNR equals $\text{SNR}$ .

Fading gain

The random multiplicative coefficient $H$ in the channel model $Y = HX + Z$ . It captures the combined effect of multipath propagation, scattering, and possibly shadowing on the received signal amplitude and phase.

Related: CSIR, CSIT

Rayleigh fading

A fading model where $H \sim \mathcal{CN}(0, 1)$ , so $|H|^2$ is exponentially distributed with mean 1. This arises when there is no dominant line-of-sight path and many scattered components contribute to the received signal. It is the most common model for non-line-of-sight wireless channels.

Related: Fading gain

Definition:
Channel State Information at the Receiver (CSIR)

We say the system has CSIR when the decoder knows the fading realization $H$ perfectly. Formally, the decoder has access to $(Y^n, H^n)$ where $H^n = (H_1, \ldots, H_n)$ are the fading gains for the $n$ channel uses.

In practice, CSIR is obtained through pilot-aided channel estimation. As long as the estimation error is small relative to the noise, the perfect-CSIR model is a good approximation.

CSIR is a reasonable assumption in most modern wireless systems. For example, in LTE and 5G NR, reference signals (pilots) are embedded in every resource block, enabling the receiver to track the channel.

Definition:
Channel State Information at the Transmitter (CSIT)

We say the system has CSIT when the encoder also knows the fading realization $H$ causally — that is, at time $i$ the encoder knows $H_1, \ldots, H_i$ (and possibly all future realizations in the non-causal case). Full CSI means both CSIR and CSIT hold.

With CSIT, the encoder can adapt its transmission strategy to the current channel state, for example by allocating more power when the channel is strong and less (or none) when the channel is weak.

CSIR

Channel state information at the receiver. The decoder knows the fading realization $H$ for each channel use.

Related: CSIT

CSIT

Channel state information at the transmitter. The encoder knows the fading realization $H$ and can adapt its transmission strategy accordingly.

Related: CSIR

Definition:
Ergodic (Fast) Fading

A fading channel is ergodic when the codeword spans many independent fading realizations. Formally, the fading process $\{H_i\}$ is stationary and ergodic, and the codeword length $n$ is large enough that the empirical distribution of $(H_1, \ldots, H_n)$ converges to the true distribution of $H$ .

In this regime, the time-averaged mutual information $\frac{1}{n}\sum_{i=1}^n I(X_i; Y_i | H_i = h_i)$ concentrates around its expectation $\mathbb{E}_H[I(X; Y | H)]$ , and an ergodic capacity exists.

Ergodic fading is a good model when the coherence time of the channel is much shorter than the codeword duration — for example, a mobile user at vehicular speed with a latency-tolerant application.

Definition:
Quasi-Static (Slow) Fading

A fading channel is quasi-static when the fading realization $H$ is constant for the entire duration of a codeword and changes independently between codewords. The channel is drawn once as $H \sim f_H$ and remains fixed for all $n$ channel uses:

$Y_i = H \cdot X_i + Z_i, \quad i = 1, \ldots, n.$

In this regime, no ergodic averaging is possible. The effective channel for a given codeword is an AWGN channel with SNR $|H|^2 \cdot \text{SNR}$ , but the transmitter (without CSIT) cannot know which SNR it will get.

Quasi-static fading arises when the coherence time exceeds the codeword duration — for example, a pedestrian user transmitting a short packet. In this regime, the concept of outage capacity replaces ergodic capacity.

Ergodic vs. Quasi-Static Fading

Property	Ergodic (Fast) Fading	Quasi-Static (Slow) Fading
Fading dynamics	$H$ changes many times per codeword	$H$ is constant per codeword
Capacity concept	Ergodic capacity $C_{\text{erg}} = \mathbb{E}[\log(1 + \|H\|^2 \text{SNR})]$	Outage capacity $C_\epsilon$
Reliable rate	Any $R < C_{\text{erg}}$ is achievable with vanishing error	Rate $R$ incurs outage probability $P_{\text{out}}(R) > 0$
Coding strategy	Code across many fading states	Cannot average out the fading; must accept outage
Practical scenario	High mobility, long codewords	Low mobility, short packets (URLLC)

Common Fading Distributions

The results in this chapter hold for any fading distribution, but the following models are most commonly used:

Rayleigh fading: $H \sim \mathcal{CN}(0, 1)$ , so $|H|^2 \sim \text{Exp}(1)$ . Models rich scattering with no line-of-sight.
Rician fading: $H = \sqrt{K/(K+1)} + \sqrt{1/(K+1)} \cdot W$ where $W \sim \mathcal{CN}(0,1)$ and $K$ is the Rician factor. Includes a deterministic line-of-sight component.
Nakagami- $m$ fading: $|H|^2 \sim \text{Gamma}(m, 1/m)$ where $m \geq 1/2$ . Includes Rayleigh ( $m=1$ ) as a special case and can approximate Rician for appropriate $m$ .
Deterministic (AWGN): $H = 1$ with probability 1. The AWGN channel is a degenerate special case of the fading model.

Notice that as the fading distribution concentrates around its mean (less randomness), the capacity approaches that of the AWGN channel.

Historical Note: From Telephone Lines to Wireless: The Information Theory of Fading

1960s-2000s

Shannon's 1948 paper focused on the AWGN channel, but he was well aware that real channels fluctuate. The systematic information-theoretic treatment of fading channels began with the work of Wolowitz (1959) and Gallager (1968), who studied the compound channel and ergodic capacity. The modern framework for fading channels with CSI was crystallized by Goldsmith and Varaiya (1997), who derived the water-filling capacity with CSIT, and by Biglieri, Proakis, and Shamai (1998), who provided a comprehensive treatment.

The quasi-static fading model and outage capacity were formalized by Ozarow, Shamai, and Wyner (1994), leading to the influential diversity-multiplexing tradeoff framework of Zheng and Tse (2003). These results fundamentally shaped how engineers think about wireless system design — moving from the AWGN mindset of "one capacity number" to a probabilistic framework where reliability and rate are traded off.

Quick Check

In the scalar fading channel $Y = HX + Z$ with $H \sim \mathcal{CN}(0,1)$ and $Z \sim \mathcal{N}(0, N)$ , what is the distribution of the instantaneous received SNR $|H|^2 \cdot \text{SNR}$ ?

Gaussian with mean $\text{SNR}$ and variance $\text{SNR}^{2}$

Exponential with mean $\text{SNR}$

Chi-squared with 2 degrees of freedom

Rayleigh distributed

Correction:

Exponential with mean

\text{SNR}

Since $H \\sim \\mathcal{CN}(0,1)$ , we have $|H|^2 \\sim \\text{Exp}(1)$ , so $|H|^2 \\cdot \\text{SNR} \\sim \\text{Exp}(1/\\text{SNR})$ with mean $\\text{SNR}$ .

Common Mistake: CSI Is Not Free

Mistake:

Assuming perfect CSIR in capacity calculations and forgetting that channel estimation consumes pilot resources (time, frequency, power) that reduce the effective throughput.

Correction:

In practice, a fraction of the available resources must be dedicated to pilot symbols for channel estimation. The net throughput is $(1 - \alpha) \cdot C$ where $\alpha$ is the pilot overhead. For rapidly varying channels (high Doppler), $\alpha$ can be substantial. The Hassibi-Hochwald (2003) training-based capacity framework quantifies this tradeoff rigorously.

Why This Matters: Fading Models in 5G NR

In 5G New Radio (NR), the channel is modeled as a fading channel at every level of the protocol stack. The 3GPP TR 38.901 channel model specifies statistical fading parameters (delay spreads, angular spreads, Rician factors) for various deployment scenarios. The system is designed assuming CSIR is available at the UE via demodulation reference signals (DMRS), and partial CSIT is obtained at the gNB through uplink sounding (SRS) in TDD mode or through limited feedback (PMI) in FDD mode.

The ergodic capacity formula governs the long-term average throughput in mobile scenarios, while the outage capacity framework applies to ultra-reliable low-latency communication (URLLC) where a single transmission must succeed with high probability.

Key Takeaway

The scalar fading channel $Y = HX + Z$ is the fundamental building block for wireless information theory. The capacity depends on two key distinctions: (1) whether fading is ergodic or quasi-static, and (2) whether the transmitter has CSI. Each combination leads to a different capacity formula with different engineering implications.

The Fading Channel Model