Fading & Diversity Reception

A deterministic link budget predicts a local-average received power. A real receiver sees that average move as buildings and terrain block the path, then sees the complex voltage fluctuate much faster as delayed replicas add constructively and destructively. Fading is the statistical description of those variations; diversity is the engineering response of observing the same information through multiple, imperfectly correlated channel realizations.

The useful mental model is therefore not “fading adds a few decibels of loss.” It is a random, time-varying filter. Sometimes the whole waveform is scaled, sometimes different frequencies are changed differently, and sometimes the filter moves before the receiver can estimate it. Those are separate questions, and the answer determines whether the remedy is margin, equalization, coding, interleaving, diversity, or a combination.

flowchart LR
  P["Distance-dependent path loss"] --> S["Shadowed local mean"]
  S --> F["Small-scale multipath fading"]
  F --> O["Instantaneous SNR and outage"]
  D["Partly independent observations"] --> C["Selection or coherent combining"]
  C --> O

The upper path says how one branch acquires random signal strength. The lower path says what diversity can change: it does not remove propagation loss; it changes the probability that all usable observations are bad at the same time.

Three axes that must not be collapsed #

ITU-R P.1407 separates rapid variation over distances on the order of a wavelength from slower variation over larger distances caused mainly by shadowing. Two other classifications describe the channel relative to the transmitted waveform. They are independent axes:

This matters because a channel can be flat and fast: one complex gain multiplies the whole narrowband waveform but changes quickly. It can also be frequency-selective and slow: the delayed-path pattern is stable for many symbols but different subcarriers experience different gains. “Slow fading” is sometimes used loosely for shadowing, but that hides the distinction between a large-scale local mean and the time selectivity of the small-scale channel.

From paths to a time-varying filter #

For a complex-baseband transmitted signal $x(t)$, a resolved multipath model is

$$y(t)=\sum_{n=1}^{N} a_n(t)e^{j\phi_n(t)}x\bigl(t-\tau_n(t)\bigr)+n(t).$$

Each path has an amplitude $a_n$, phase $\phi_n$, excess delay $\tau_n$, and possibly a different angle, polarization, and Doppler shift. The equivalent impulse response and frequency response are

$$h(\tau,t)=\sum_n a_n(t)e^{j\phi_n(t)}\delta\bigl(\tau-\tau_n(t)\bigr), \qquad H(f,t)=\int h(\tau,t)e^{-j2\pi f\tau}\,d\tau.$$

The phases are decisive. Moving only a fraction of a wavelength changes relative path phase enough to turn reinforcement into cancellation. At 3.5 GHz the wavelength is about $8.6\ \text{cm}$, so centimetre-scale motion can materially change the received voltage even while distance-dependent path loss is essentially unchanged.

Large-scale shadowing and the local mean #

Path loss captures the median trend with distance. Shadowing is the residual change in local-average power caused by terrain, buildings, foliage, vehicles, and other obstructions. It is commonly modelled as a zero-mean Gaussian deviation in decibels, which means a log-normal multiplier in linear power. 3GPP TR 38.901 uses this form and gives scenario-specific standard deviations and spatial correlation distances rather than one universal shadowing number.

The local mean is itself correlated in space. If a user moves a metre, the blocking geometry usually has not been replaced by an independent one. In the 3GPP model, shadowing autocorrelation decays exponentially with displacement,

$$R_{\mathrm{SF}}(\Delta x)=\exp\left(-\frac{|\Delta x|}{d_{\mathrm{cor}}}\right),$$

where the correlation distance $d_{\mathrm{cor}}$ depends on the scenario. This is why collecting many samples along one short route does not necessarily produce many independent shadowing observations.

Small-scale fading #

After removing the local mean, the remaining complex fluctuation is small-scale fading. It is driven chiefly by relative phases, delays, and Doppler shifts of the multipath components. A stationary terminal can still fade if vehicles, foliage, machinery, or other scatterers move. Conversely, mobility does not guarantee a rich fade: a highly directional link dominated by one component may change far more smoothly.

A composite model may use log-normal variation for the local mean and Rayleigh or Rician variation around that mean. Mixing these scales before fitting a distribution produces parameters that have no clean physical meaning.

Delay spread, coherence bandwidth, and selectivity #

The power delay profile (PDP) is the power in $h(\tau,t)$ as a function of excess delay, averaged over an explicitly chosen short time or local spatial region. If resolved paths have linear powers $P_n$ and delays $\tau_n$, define

$$\bar{\tau}=\frac{\sum_n P_n\tau_n}{\sum_n P_n}, \qquad \tau_{\mathrm{rms}}= \sqrt{\frac{\sum_n P_n(\tau_n-\bar{\tau})^2}{\sum_n P_n}}.$$

The RMS delay spread $\tau_{\mathrm{rms}}$ describes the time dispersion of the channel. The frequency correlation follows from the Fourier transform of the PDP: frequencies separated by a small amount see nearly the same complex channel; larger separations progressively decorrelate.

Coherence bandwidth $B_c$ must always carry a correlation criterion. ITU-R P.1407 defines $B_x$ as the frequency offset where the normalized channel correlation falls to $x\%$ of its zero-offset value, with 0.5 and 0.9 given as typical thresholds. There is therefore no source-independent identity such as “$B_c=1/(5\tau_{\mathrm{rms}})$.” Such formulas are rough proxies whose constants encode a PDP shape and a chosen threshold.

• If $B_s\ll B_c$, $H(f,t)$ is nearly constant across the occupied signal band: the channel is approximately flat fading.
• If $B_s$ is comparable to or larger than $B_c$, different parts of the signal see different gains and phases: the channel is frequency-selective.

Frequency selectivity produces pulse spreading and potentially inter-symbol interference in a single-carrier waveform. An equalizer or an OFDM cyclic prefix addresses the delayed structure; those mechanisms are covered in symbol timing, carrier recovery, and ISI. Diversity addresses the probability of bad observations. They can complement each other, but one is not a substitute for the other.

3GPP's standardized CDL and TDL profiles make the separation concrete. TR 38.901 allows the normalized tap delays to be scaled to example RMS delay spreads of 10, 30, 100, 300, or 1,000 ns. Those are evaluation profiles, not universal measurements. The same occupied bandwidth can therefore be flat under one profile and strongly selective under another.

Doppler, coherence time, and the Jakes model #

Motion changes each path length and therefore its phase. For a path arriving at angle $\theta$ relative to the velocity vector, the classical Doppler shift is

$$f_D(\theta)=\frac{v}{\lambda}\cos\theta, \qquad f_m=\frac{v}{\lambda}=\frac{v f_c}{c},$$

where $v$ is speed, $\lambda$ is wavelength, $f_c$ is carrier frequency, and $f_m$ is the maximum magnitude for this geometry. A spread of arrival angles creates a spread of Doppler frequencies, hence a time-varying complex channel.

Coherence time $T_c$ is the time offset at which normalized channel correlation falls to a declared threshold. Like coherence bandwidth, its numerical value depends on the threshold and the channel spectrum. The scale is inversely related to Doppler spread, but $T_c=1/f_m$, $1/(2f_m)$, and other rules are not interchangeable definitions.

At $f_c=3.5\ \text{GHz}$ and $v=30\ \text{m/s}$,

$$\lambda=\frac{c}{f_c}\approx 0.0857\ \text{m}, \qquad f_m\approx 350\ \text{Hz}, \qquad \frac{1}{f_m}\approx 2.86\ \text{ms}.$$

The last number is a variation scale, not a threshold-defined coherence time. A receiver must estimate the channel often enough for the weights, equalizer, and detector to remain valid over the operation they support.

What Jakes does and does not say #

For many equal-strength diffuse paths arriving uniformly in azimuth in a two-dimensional scattering geometry, the classical/Jakes Doppler power spectrum has the idealized shape

$$S(f)= \begin{cases} \displaystyle \frac{K_0}{\pi f_m\sqrt{1-(f/f_m)^2}}, & |f|<f_m,\\[6pt] 0, & |f|\ge f_m, \end{cases}$$

where $K_0$ normalizes total power. The high density near $\pm f_m$ corresponds to paths close to the direction of travel. ITU-R P.1407 presents Jakes as a specific narrowband terrestrial model; 3GPP uses the classical spectrum for its TDL taps and adds a discrete Doppler component to selected LOS taps.

Real angular spectra may be directional, blocked, three-dimensional, or dominated by moving scatterers. Then the measured Doppler spectrum can be asymmetric, multi-peaked, or shifted. Jakes is a reproducible benchmark, not evidence that a measured channel must have its U-shaped spectrum.

Envelope models are marginals, not complete channels #

The following models describe the distribution of envelope or power at a point after a particular averaging and normalization choice. They do not specify delay taps, Doppler correlation, spatial correlation, or shadowing. A simulator needs those structures separately.

For a Rayleigh envelope with mean received power $\Omega$, the normalized instantaneous power $G=R^2/\Omega$ is exponential:

$$p_G(g)=e^{-g},\qquad F_G(g)=1-e^{-g},\qquad g\ge 0.$$

Rician $K$ is a power ratio, normally reported in decibels as $K_{\mathrm{dB}}=10\log_{10}K$. A large $K$ makes the envelope concentrate around the dominant component; $K=0$ leaves only the Rayleigh diffuse term. For wideband channels, a different tap or cluster can have a different $K$; calling the entire channel “Rician” can conceal that structure.

The Nakagami-$m$ envelope density is

$$p_R(r)=\frac{2m^m}{\Gamma(m)\Omega^m}r^{2m-1} \exp\left(-\frac{mr^2}{\Omega}\right),\qquad r\ge 0.$$

Its flexibility is useful, but $m$ is not a count of physical paths and should not be interpreted as a Rician $K$ factor. ITU-R P.1057 explicitly distinguishes Nakagami-$m$ from Nakagami-Rice.

Outage probability and fade margin #

An outage event must name the metric and observation interval: for example, post-combiner SNR below a decoder threshold for one codeword, block error rate above a limit, or mutual information below the transmitted rate. Define

$$P_{\mathrm{out}}=\Pr\{\gamma<\gamma_{\mathrm{th}}\}.$$

For one flat Rayleigh branch with average SNR $\bar\gamma$, instantaneous SNR is exponential and

$$P_{\mathrm{out},1}=1-\exp\left(-\frac{\gamma_{\mathrm{th}}}{\bar\gamma}\right).$$

Solving this equation for the average-to-threshold ratio gives the ideal Rayleigh fade margin needed for target outage $\epsilon$:

$$M_{\mathrm{fade}}=10\log_{10}\frac{\bar\gamma}{\gamma_{\mathrm{th}}} =-10\log_{10}\left[-\ln(1-\epsilon)\right].$$

For $\epsilon=0.01$, this is about $20.0\ \text{dB}$. That striking number explains why a single uncoded Rayleigh branch is expensive to make highly reliable by power alone. It is not a complete link-budget margin: shadowing, interference, implementation loss, estimation error, coding, traffic definitions, and correlated outages require their own treatment.

For frequency-selective channels, one narrow notch does not necessarily cause an outage. The result depends on waveform, interleaving, code rate, resource allocation, and receiver processing. For slow shadowing, availability may be dominated by the distribution of the local mean rather than the instantaneous Rayleigh tail. Always compute outage at the same abstraction level as the requirement.

Diversity: change the tail, not just the mean #

Diversity sends or receives the same information through observations that are unlikely to fade together. The important resource is decorrelation, not a nominal branch count.

Frequency diversity requires information to span independently faded frequencies. Merely widening an uncoded signal into a selective channel can add distortion rather than protection. OFDM gains frequency diversity only when coding/interleaving or repetition lets the decoder combine information across subcarriers. Spread-spectrum systems can do something analogous when the receiver resolves and combines distinct multipath or spectral contributions.

Spatial separation is likewise not a magic distance. Half-wavelength spacing is a convenient array heuristic under particular rich-scattering assumptions, not a guarantee of independent fading. A narrow angular spread, a dominant LOS component, shared blockage, mutual coupling, similar beams, or common RF impairments can keep branches correlated even when the antenna ports are distinct. 3GPP TR 38.901 therefore models angular spreads, large-scale-parameter cross-correlations, spatial consistency, per-scenario correlation distances, and cross-polarization power ratio rather than declaring all antenna branches independent.

Selection, equal-gain, and maximal-ratio combining #

For $L$ flat-fading observations

$$y_\ell=h_\ell x+n_\ell, \qquad \gamma_\ell=\frac{|h_\ell|^2E_s}{N_\ell},$$

the common receive combiners make different demands:

With white equal-variance branch noise, MRC uses weights proportional to $h_\ell^*$. It maximizes instantaneous output SNR and naturally gives a weak branch little influence. With correlated or unequal interference-plus-noise, the optimum linear weight is proportional to

$$\mathbf{w}\propto \mathbf{R}_{n}^{-1}\mathbf{h},$$

not merely $\mathbf{h}^*$. Calling a plain conjugate sum “optimal MRC” in an interference-limited array silently assumes away the covariance that matters.

SC avoids coherent summation and can be cheap, but throws away usable energy from non-selected branches. EGC recovers most coherent combining benefit when amplitudes are difficult to weight accurately. Both EGC and MRC fail gracefully only while their phase estimates remain fresh; fast fading, residual carrier offset, calibration error, or unequal branch delay can turn addition into partial cancellation.

Correlation determines the effective diversity order #

For independent branches, the event “all branches are bad” becomes much less likely. For two independent, identically distributed Rayleigh branches,

$$P_{\mathrm{out,SC}}=\left(1-e^{-x}\right)^2, \qquad P_{\mathrm{out,MRC}}=1-e^{-x}(1+x), \qquad x=\frac{\gamma_{\mathrm{th}}}{\bar\gamma}.$$

At high average SNR, an $L$-branch independent Rayleigh combiner produces an outage tail proportional to $(\gamma_{\mathrm{th}}/\bar\gamma)^L$: this exponent is the diversity order. Correlation reduces the slope and, in the limit of identical branches, extra ports provide no diversity at all.

The correlation to inspect is the joint complex channel over the actual signaling resources. Branches may share:

• the same log-normal shadowing or blockage state;
• the same dominant path and angular cluster;
• frequency samples closer than the required coherence-bandwidth criterion;
• time samples closer than the required coherence-time criterion;
• antenna mutual coupling, common oscillator error, AGC behavior, or interference;
• polarization leakage or a propagation environment with little depolarization.

ITU-R P.530 makes the same point for fixed terrestrial links: diversity improvement depends on branch decorrelation, and wideband digital improvement also depends on in-band distortion. Its empirical spacing procedures are specific to the documented fixed-link geometries and frequency/path-length ranges; they should not be transplanted as cellular antenna-spacing rules.

Worked combining and outage example #

The MRC result is better than SC because both branches contribute energy even when neither is individually selected. The squared-outage improvement for SC comes from independence. If the two fades were identical, both combiners would lose that second-order tail benefit; MRC could still provide array gain from two independent receiver-noise observations, but not two independent fade opportunities.

Diversity versus MIMO #

Multiple antennas do not specify the objective. Diversity transmission/reception sends redundant views of the same information to reduce outage. Spatial multiplexing sends different streams to increase rate through multiple spatial degrees of freedom. Both use a channel matrix, but they exploit it differently:

• diversity asks whether at least one robust combination survives a fade;
• multiplexing asks how many sufficiently strong, distinct singular modes the channel supports;
• beamforming concentrates energy in one or more directions and can supply array gain without adding diversity order;
• a practical MIMO link can trade among diversity, beamforming gain, and multiplexing rate as rank, SNR, feedback quality, and traffic goals change.

A rank-one LOS channel can support receive combining and array gain while offering little spatial-multiplexing rank. Conversely, a rich high-rank channel can carry several streams, but doing so spends redundancy that could otherwise reduce outage. “$4\times4$ MIMO” names antenna-port counts, not four independent fades and not four guaranteed data streams.

Measuring and diagnosing a fading channel #

A fading study is credible only when the measurement separates propagation from the instrument and declares its averaging scales.

1. Capture calibrated complex response. Measure $H(f,t)$ or $h(\tau,t)$ across the full operating bandwidth, all relevant antenna ports and polarizations, and enough time or distance to cover the intended statistics. Preserve phase; RSSI alone cannot diagnose coherent combining.
2. State resolution and sampling. Bandwidth sets delay resolution; time sampling must resolve the expected Doppler; record antenna patterns, polarization, trajectory, speed, carrier, clock references, and environment.
3. Remove deterministic and large-scale trends deliberately. Correct known path-loss trend and estimate a local mean over a window large enough to suppress wavelength-scale fading but small enough not to erase shadowing changes. Report the window instead of treating it as an invisible preprocessing choice.
4. Check stationarity before pooling. Segment routes when PDP, Doppler spectrum, LOS state, or local mean changes. A long non-stationary trace can manufacture a misleading heavy-tailed “fading distribution.”
5. Measure delay and frequency structure. Estimate PDP, mean excess delay, RMS delay spread, frequency correlation, and $B_x$ at a declared threshold. Apply a documented noise-floor threshold and sensitivity analysis.
6. Measure temporal behavior. Estimate time autocorrelation, Doppler spectrum, $T_x$ at a declared threshold, level-crossing rate (LCR), and average fade duration (AFD). A CDF says how often the signal is low; LCR and AFD say how bursts are organized.
7. Fit distributions after scale separation. Compare empirical CDFs and especially the outage tail against Rayleigh, Rician, or Nakagami-$m$. Estimate Rician $K$ only where a stable dominant component is physically and statistically plausible.
8. Measure branch dependence directly. Compute the complex covariance matrix versus time, frequency, and displacement; inspect joint fade events; then replay the actual combiner. Pairwise power correlation alone can miss phase structure and higher-order joint tails.
9. Validate the end metric. Reconstruct post-equalizer/post-combiner SNR, error rate, or decoder outage on held-out routes. A channel-model parameter fit is not success unless it predicts the system event being designed against.

A practical design sequence #

1. Define the outage event, threshold, duration, and availability target.
2. Use radio propagation and the link budget to establish the local-average SNR before small-scale fading.
3. Measure or select a scenario-valid model for shadowing, PDP/delay spread, Doppler, envelope law, and spatial/polarization correlation.
4. Compare $B_s$ with a threshold-defined $B_c$ and processing duration with a threshold-defined $T_c$.
5. Choose diversity dimensions that the channel actually decorrelates and that the system can afford.
6. Choose SC, EGC, MRC, or an interference-aware spatial combiner according to available channel/noise estimates.
7. Add equalization and channel coding where selectivity and burst errors require them; distribute coded information across independently faded resources.
8. Evaluate the joint outage tail with measured correlation and estimation errors, then verify it with representative channel captures.

The key design question is never merely “how many branches?” It is: how many independently useful observations of the coded information remain after real correlation, channel-estimation error, and receiver constraints are included?