RMS Delay Spread, Excess Delay Spread and Multi-path ...(with MATLAB + Simulator)

📘 Overview of Delay Spread and Multi-path
🧮 Excess Delay spread
🧮 Power delay Profile
🧮 RMS Delay Spread
📚 Further Reading

Delay Spread, Excess Delay Spread, and Multipath (MPCs)

The fundamental distinction between wireless and wired connections is that in wireless connections signal reaches at receiver thru multipath signal propagation rather than directed transmission like co-axial cable. Wireless Communication has no set communication path between the transmitter and the receiver. The line of sight path, also known as the LOS path, is the shortest and most direct communication link between TX and RX. The other communication pathways are called non-line of sight (NLOS) paths. Reflection and refraction of transmitted signals with building walls, foliage, and other objects create NLOS paths. [ Read More about LOS and NLOS Paths]

Multipath Components or MPCs:

The linear nature of the multipath channel is evident. This signifies that each multipath component signal is a delayed and scaled version of the original transmitted signal.

Let me give you an example to help you understand. Let's assume we're sending an impulse signal from the transmitter. The single impulse response is transmitted to the receiver via LOS and NLOS pathways. The signal is only transmitted via NLOS paths if a LOS path is unavailable. The probability of LOS Communication decreases as the density of the region increases. Because there are numerous obstacles between the transmitter and the receiver, such as buildings, etc.

Excess Delay Spread:

Excess delay spread is the arrival time difference between the first and final multipath components (MPCs) at the receiver side. For example, suppose the first multipath component arrives at the receiver at time t1, and the last multipath component arrives at time t2. The Excess Delay Spread will then be (t2 -t1).

Power Delay Profile:

The Power Delay Profile shows how received power changes with time dispersion or time delay caused by multipath in a wireless communication channel.

In the above equation, Ac denotes the multipath intensity profile. τ denotes time delay, μTm denotes average delay spread.

You can also think of Ac as a power profile that exponentially decreases over time as a multipath delay in time.

RMS Delay Spread:

The RMS Delay Spread is the square root of the power delay profile's second central moment. We all know that we get multipath components at the receiver end of the wireless communication process. As a result, to obtain the necessary data, we must use stronger multipath and then add them. Then we divide the total value by the total weights. In the case of the power delay profile computation, power decreases exponentially with time.

Notation for RMS Delay Spread

σ_{τ_m}: Root Mean Square (RMS) delay spread — measures how much the multipath delays are spread around the mean delay.
τ (tau): Excess delay variable (time delay of a multipath component relative to the first arriving path).
μ_{τ_m}: Mean excess delay — the average delay of the received signal weighted by power.
A_c(τ): Power delay profile (PDP) — represents the received signal power as a function of delay τ.
∫₀^∞: Integration over all possible delays (from 0 to infinity).
dτ: Infinitesimal increment of delay.
(τ − μ_{τ_m})²: Squared deviation of each delay from the mean delay.
Numerator: Weighted variance of delay (spread of delays).
Denominator: Total received power (normalization factor).

Here, in the above equation, σTm denotes rms delay spread. It shows how the RMS delay spread relates to the average delay spread. Apart from the average delay spread, we take the square root value of the square of the difference between the average delay spread and the instantaneous delay spread of the multipath component. [Get MATLAB Code for RMS Delay Spread]

Why is there significant multipath in the case of very high frequencies?

The signal coverage is often limited at higher frequencies compared to lower frequencies. In metropolitan or urban scenarios, the LOS component is often blocked, leaving only NLOS pathways. At high frequencies, many paths are lost due to high penetration loss and lack of diffraction, meaning only a few robust NLOS components reach the receiver. Because path loss is directly proportional to the carrier frequency, these higher frequency signals experience significant attenuation.

Why RMS Delay Spread is essential for wireless Communication:

In today's wireless Communication, RMS delay spread is a critical characteristic. It depends on an area's physical constructions, like buildings, foliage, etc. We will receive many copies of the same sent signal. If we broadcast the following signal immediately after the first, the MPCs of the first symbol cause Inter-symbol interference (ISI) on the receiving side. To eliminate interference, we broadcast signals at intervals significantly greater than the RMS delay spread.

Why the Power Delay profile is essential:

The Power Delay Profile shows how received power varies with the time dispersion of MPCs. For high frequencies, often only a few MPCs carry the majority of total energy.

Now, we are continuously moving to higher frequency bands. You know these bands experience more reflection, scattering, etc. That results in more multipath. And multipath causes fading. And the type of fading tells us whether it is flat fading or frequency selective fading. Different multipaths reaches the receiver at different time. Higher frequency bands experience more Doppler spread as compared to lower frequency bands. You know 5G wireless technology is operating at millimeter wave band, so the Doppler effect will be huge. So, currently, researchers are focusing on the delay-doppler domain to mitigate the effect of Doppler delay spread.

MATLAB Code for calculating different types of delay spread

% The code is written by SalimWireless.Com

clc;
clear;
close all;

% Sample delay and power arrays
delays = [0, 1, 2, 3, 4]; % Delay in ns
powers_dB = [0.5, 0.2, 0.1, 0.1, 0.1]; % Power in dB

% Convert dB powers to linear scale
powers_linear = 10.^(powers_dB / 10);

% Sum the linear powers
total_power_linear = sum(powers_linear);

% Normalize the linear power to get a probability density function (pdf)
pdf = powers_linear / total_power_linear;

% Compute weighted delay
weighted_delay = sum(delays .* pdf);

% Compute RMS delay spread
rms_delay_spread = sqrt(sum((delays - weighted_delay).^2 .* pdf));

% Compute mean delay spread
mean_delay_spread = weighted_delay;

% Compute maximum excess delay
max_excess_delay = max(delays) - min(delays);

% Display results
fprintf('Mean Delay Spread: %.2f ns\n', mean_delay_spread);
fprintf('RMS Delay Spread: %.2f ns\n', rms_delay_spread);
fprintf('Maximum Excess Delay: %.2f ns\n', max_excess_delay);
web('https://www.salimwireless.com/search?q=rms%20delay%20spread', '-browser');

Output

Mean Delay Spread: 1.96 ns
RMS Delay Spread: 1.43 ns
Maximum Excess Delay: 4.00 ns

Also Read:

[1] Read more about RMS Delay Spread

[2] More about Channel Input Response (CIR)

[3] Difference between AWGN and Rayleigh Fading

[4] Saleh Valenzuela Channel Model for high frequencies communication

[5] Impact of Rayleigh Fading and AWGN on Digital Communication Systems

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