Ferkans — Interactive Telecom Tutor

Beyond Flat Fading

Sections 6.3–6.4 assumed the channel was a single complex coefficient (narrowband/flat fading). Modern wideband systems (OFDM, spread-spectrum) transmit signals whose bandwidth $W$ may exceed the coherence bandwidth $B_c$ . Different frequency components then experience different fading — the channel is frequency-selective. We now characterise this through the power delay profile, RMS delay spread, and coherence bandwidth.

Definition:
Power Delay Profile

The power delay profile (PDP) is the average power as a function of excess delay:

$P(\tau) = E[|h(\tau; t)|^2]$

For a discrete multipath channel with $L$ paths:

$P(\tau) = \sum_{l=0}^{L-1} P_l\, \delta(\tau - \tau_l)$

where $P_l = E[|\alpha_l|^2]$ is the average power of the $l$ -th path. The PDP is obtained from measurements by averaging $|h(\tau; t)|^2$ over time or spatial locations.

Definition:
Mean Excess Delay and RMS Delay Spread

From the PDP, the key delay statistics are:

Mean excess delay:

$\bar{\tau} = \frac{\sum_l P_l \tau_l}{\sum_l P_l}$

RMS delay spread:

$\sigma_\tau = \sqrt{\overline{\tau^2} - \bar{\tau}^2}$

where $\overline{\tau^2} = \sum_l P_l \tau_l^2 / \sum_l P_l$ .

Environment	Typical $\sigma_\tau$
Indoor (office)	10–50 ns
Urban micro	100–300 ns
Urban macro	1–3 $\mu$ s
Hilly terrain	3–10 $\mu$ s

Theorem: Coherence Bandwidth

The coherence bandwidth $B_c$ is the frequency separation over which the channel's frequency response remains correlated. It is inversely proportional to the RMS delay spread:

$B_c \approx \frac{1}{5\sigma_\tau} \quad \text{(correlation } \geq 0.9\text{)}$

$B_c \approx \frac{1}{2\pi\sigma_\tau} \quad \text{(correlation } \geq 0.5\text{)}$

Channel classification:

$W \ll B_c$ ( $T_s \gg \sigma_\tau$ ): flat fading — all frequencies fade together
$W \gg B_c$ ( $T_s \ll \sigma_\tau$ ): frequency-selective fading — different subcarriers fade independently

If two frequencies are separated by more than $B_c$ , their fading is essentially independent. A wideband signal spanning many coherence bandwidths experiences frequency diversity — some subcarriers may be in a deep fade while others are strong. This is the basis for OFDM and frequency-domain scheduling.

Proof

Frequency correlation function

The frequency correlation is $R_H(\Delta f) = \int P(\tau)\, e^{-j2\pi \Delta f \tau}\, d\tau$ .

This is the Fourier transform of the PDP. For an exponentially decaying PDP $P(\tau) = (1/\sigma_\tau)e^{-\tau/\sigma_\tau}$ :

$R_H(\Delta f) = \frac{1}{1 + j2\pi \Delta f \sigma_\tau}$ ,

$|R_H(\Delta f)| = \frac{1}{\sqrt{1 + (2\pi \Delta f \sigma_\tau)^2}}$ .

Setting $|R_H| = 0.5$ gives $\Delta f = \sqrt{3}/(2\pi\sigma_\tau) \approx 1/(2\pi\sigma_\tau)$ . $\blacksquare$

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Power Delay Profile and Delay Spread

Adjust the number of clusters and exponential decay rate to see how the PDP shape affects the RMS delay spread and coherence bandwidth.

Parameters

Number of clusters3

Decay constant (ns)200

Intra-cluster spread (ns)50

Coherence Bandwidth vs Delay Spread

Visualise the frequency correlation function for different delay spreads. See how larger delay spread narrows the coherence bandwidth.

Parameters

RMS delay spread

\sigma_\tau

(ns)200

Flat vs Frequency-Selective Fading

Compare what happens to a narrowband signal ( $W \ll B_c$ ) versus a wideband signal ( $W \gg B_c$ ) passing through a multipath channel.

Parameters

Signal bandwidth

W

(MHz)5

RMS delay spread

\sigma_\tau

(ns)200

Definition:
WSSUS — Wide-Sense Stationary Uncorrelated Scattering

The WSSUS assumption simplifies channel statistics:

Wide-sense stationary (WSS): the fading statistics do not change over the time interval of interest — i.e., $E[h(\tau_1; t)\, h^*(\tau_2; t + \Delta t)]$ depends only on $\Delta t$ , not on $t$ .
Uncorrelated scattering (US): scatterers at different delays are uncorrelated — i.e., $E[h(\tau_1; t)\, h^*(\tau_2; t)] = P(\tau_1)\,\delta(\tau_1 - \tau_2)$ .

Under WSSUS, the channel is fully characterised by the power delay profile $P(\tau)$ and the Doppler spectrum $S_H(f)$ . This is the standard assumption for system design.

WSSUS is an approximation. It breaks down for non-stationary scenarios (e.g., high-speed trains, drones) and for ultra-wideband channels where scatterer correlations matter.

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Example: OFDM Subcarrier Spacing and Coherence Bandwidth

An OFDM system has subcarrier spacing $\Delta f = 15$ kHz (LTE standard). The urban channel has $\sigma_\tau = 1\,\mu$ s.

(a) Compute the coherence bandwidth $B_c$ (at 0.5 correlation).

(b) How many subcarriers fit within one coherence bandwidth?

(c) Is this flat or frequency-selective fading per subcarrier?

Solution

Coherence bandwidth

$B_c \approx 1/(2\pi\sigma_\tau) = 1/(2\pi \times 10^{-6}) = 159$ kHz.

Subcarriers per coherence bandwidth

$N = B_c / \Delta f = 159 / 15 \approx 10.6$ subcarriers.

Classification

Each individual subcarrier has bandwidth $\Delta f = 15$ kHz $\ll B_c = 159$ kHz, so each subcarrier experiences flat fading. However, the total system bandwidth (e.g., 10 MHz = 667 subcarriers) is much larger than $B_c$ , so the channel is frequency-selective across the full band.

This is exactly why OFDM works: it converts a frequency-selective channel into many flat-fading subchannels. $\blacksquare$

Common Mistake: Mean Delay vs RMS Delay Spread

Mistake:

Using the maximum excess delay instead of the RMS delay spread for coherence bandwidth calculations.

Correction:

The maximum excess delay (the delay of the last arriving path above a threshold) can be much larger than $\sigma_\tau$ and gives a pessimistic estimate of frequency selectivity. The RMS delay spread is the second central moment of the PDP and is the correct parameter for coherence bandwidth formulas.

Quick Check

A channel has $\sigma_\tau = 100$ ns. A signal has bandwidth $W = 200$ kHz. Is this flat or frequency-selective fading?

Flat fading

Frequency-selective fading

Cannot determine without knowing the carrier frequency

It depends on the mobile speed

Correction:

Flat fading

$B_c \approx 1/(5 \times 100 \times 10^{-9}) = 2$ MHz $\gg W = 0.2$ MHz. Since $W \ll B_c$ , this is flat fading.

⚠️Engineering Note

Cyclic Prefix Design Trade-offs

The cyclic prefix (CP) in OFDM must absorb the channel's maximum excess delay to prevent inter-symbol interference. Practical design constraints:

CP length $\geq \tau_{\max}$ : In LTE, the normal CP is 4.7 $\mu$ s (sufficient for urban macro with $\sigma_\tau \leq 1\,\mu$ s, since $\tau_{\max} \approx 4\sigma_\tau$ ). The extended CP is 16.7 $\mu$ s for extreme delay spread scenarios.
Overhead penalty: CP duration is wasted energy and bandwidth. For LTE normal CP: $4.7/(66.7 + 4.7) = 6.6$ % overhead. For 5G NR at 120 kHz spacing: $0.59/(8.33 + 0.59) = 6.6$ % (same ratio by design).
Numerology selection: 5G NR offers subcarrier spacings of 15, 30, 60, 120, 240 kHz. Larger spacing means shorter symbols and shorter CP — suitable only for low-delay-spread environments (e.g., mmWave indoor). Choosing the wrong numerology for the deployment scenario causes ISI.
Timing advance: Even with sufficient CP, the base station must command each UE to advance its transmission timing to compensate for propagation delay. Timing advance errors effectively reduce the usable CP length.

Practical Constraints

•
CP must exceed maximum excess delay of the deployment scenario
•
CP overhead is proportional to delay spread / symbol duration ratio
•
Timing advance errors reduce effective CP length

📋 Ref: 3GPP TS 38.211, §5.3 (OFDM baseband signal generation)

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RMS Delay Spread

$\sigma_\tau = \sqrt{\overline{\tau^2} - \bar{\tau}^2}$ : the second central moment of the power delay profile. Determines the coherence bandwidth $B_c \approx 1/(5\sigma_\tau)$ .

Coherence Bandwidth

$B_c \approx 1/(5\sigma_\tau)$ : the frequency range over which the channel response is correlated. Determines flat vs frequency-selective classification.

WSSUS

Wide-Sense Stationary Uncorrelated Scattering: the standard channel assumption combining temporal stationarity with uncorrelated scattering at different delays.

Wideband Channel Characterization

Beyond Flat Fading

Definition: Power Delay Profile

Definition: Mean Excess Delay and RMS Delay Spread

Theorem: Coherence Bandwidth

Frequency correlation function

Power Delay Profile and Delay Spread

Parameters

Coherence Bandwidth vs Delay Spread

Parameters

Flat vs Frequency-Selective Fading

Parameters

Definition: WSSUS — Wide-Sense Stationary Uncorrelated Scattering

Example: OFDM Subcarrier Spacing and Coherence Bandwidth

Coherence bandwidth

Subcarriers per coherence bandwidth

Classification

Common Mistake: Mean Delay vs RMS Delay Spread

Quick Check

Cyclic Prefix Design Trade-offs

RMS Delay Spread

Coherence Bandwidth

WSSUS

Definition:
Power Delay Profile

Definition:
Mean Excess Delay and RMS Delay Spread

Definition:
WSSUS — Wide-Sense Stationary Uncorrelated Scattering