Coherence Bandwidth and Coherence Time

 

Coherence Bandwidth

Coherence bandwidth is a concept in wireless communication and signal processing that relates to the frequency range over which a wireless channel remains approximately constant in terms of its characteristics.

coherence bandwidth is The inverse of Doppler spread delay time, or any spread delay time due to fading in general. The coherence bandwidth is related to the delay spread of the channel, which is a measure of the time it takes for signals to traverse the channel. The two are related by the following formulae:

Coherence bandwidth = 1/(2*pi*delay spread time)

Or, Coherence Bandwidth = 1/(2*pi*root-mean-square delay spread time)

(Coherence bandwidth in Hertz)

For instance, the coherence bandwidth is 2 MHz when the delay spread is {1/(2*pi*2*10^6)} = 80 ns in a household indoor environment.

For narrowband approximation,

Coherence Bandwidth = 1/root-mean-square delay spread time

Coherence bandwidth is a measure of the frequency spread over which a wireless communication channel behaves approximately like a flat fading channel. In other words, it's the bandwidth over which the channel's frequency response remains relatively constant. Coherence bandwidth is a crucial parameter in the design of wireless communication systems, particularly for systems that employ frequency-selective fading channel models.

 

Frequency Auto-correlation Function

The frequency auto-correlation function RH(Δf) of the channel transfer function H(f) is defined as: 

RH(Δf) = E{H(f)H∗(f+Δf)}

E{⋅} denotes the expectation operator, and H*(t) is the complex conjugate of H(t).

Coherence Bandwidth Definition

The coherence bandwidth is often defined as the frequency separation Δf over which the auto-correlation function RH(Δf) drops to a certain fraction of its maximum value, typically 0.5 or 1/e: 

Bc≈1/Tm

Where Tm is the multi-path delay spread time or root-mean-square delay spread time, which characterizes the extent of multi-path propagation.

 

 

Coherence Time

Coherence time, in the context of wireless communication, is a fundamental concept related to the temporal properties of wireless channels. It represents the duration for which the channel conditions remain approximately constant. In other words, it's the amount of time during which the wireless channel's characteristics, including phase, amplitude, and delay, can be considered relatively stable.

The relationship between coherence time and coherence bandwidth depends on the specific characteristics of the wireless channel and can vary from one scenario to another. In some cases, you may find that they are inversely related, meaning that a wider coherence bandwidth corresponds to a shorter coherence time and vice versa. However, this relationship is not a strict rule, and the actual values of coherence time and coherence bandwidth depend on the specific channel conditions and environment.

The coherence time and coherence bandwidth are related but not simply inversely proportional to each other. They are both important parameters for characterizing the time and frequency variations in wireless channels, and their values can vary depending on the specific channel properties and circumstances.

 

Time Auto-correlation Function

The time auto-correlation function RH(Δt) of the channel impulse response h(t) is defined as: 

RH(Δt)=E{h(t)h∗(t+Δt)}

E{⋅} denotes the expectation operator, and h∗(t) is the complex conjugate of h(t).

Coherence Time Definition

The coherence time is often defined as the time lag Δt over which the auto-correlation function RH(Δt) drops to a certain fraction of its maximum value, typically 0.5 or 1/e: 

Tc≈1/(2*pi*fD)

Where fD is the Doppler spread, which characterizes the rate of change of the channel due to relative motion. 

For example, if a vehicle is moving at 30 m/s and the carrier frequency is 2 GHz:

Then, doppler spread, fD = v*f/c= (30 * 2 * 10^9) / (3 * 10^8) = 200 Hz

So, coherence time using the general approximation is 1/(2*pi*200) = 0.8 ms (approx)

 

 

 

MATLAB Code to find Coherence Time and Coherence Bandwidth

% The code is written by SalimWireless.Com clc; clear all; close all; % Define the frequency range and corresponding PSD values f = linspace(0, 1e6, 1000); % Frequency range (adjust as needed) % Example PSD values corresponding to the frequencies in 'f' PSD_values = rand(size(f)); % Replace with your actual PSD values % Plot the PSD for visualization (optional) figure; plot(f, PSD_values); title('Power Spectral Density (PSD)'); xlabel('Frequency (Hz)'); ylabel('Spectral Density'); % Find coherence bandwidth coherence_bandwidth = trapz(f, PSD_values) / max(PSD_values); % Approximate as the area under the PSD divided by its maximum value % Print coherence bandwidth disp(['Coherence Bandwidth: ', num2str(coherence_bandwidth), ' Hz']); % Find coherence time coherence_time = 1 / coherence_bandwidth; % Estimate coherence time as the reciprocal of the coherence bandwidth % Print coherence time disp(['Coherence Time: ', num2str(coherence_time), ' seconds']);  

Output

Coherence Bandwidth: 498867.5012 Hz
Coherence Time: 2.0045e-06 seconds

 

 

 

 

Fig 1: PSD of the generated frequency components

Relationship between Coherence Time and Delay Spread

The coherence time of a wireless channel is related to its delay spread. Delay spread refers to the time difference between the arrival of the first and last significant multipath components of a signal. Coherence time represents the time over which the channel's impulse response remains relatively constant, and it's inversely proportional to the delay spread.

The relationship between coherence time (Tc) and delay spread (Td​) can be approximated using the formula:

Tc≈1/β⋅Td​

where β is a factor depending on the specific characteristics of the wireless environment, typically ranging from 2 to 4.

 

MATLAB Code to find Relationship between Coherence Time and delay Spread 

% The code is written by SalimWireless.Com clc; clear all; close all; % Define delay spread (in seconds) delay_spread = 1e-6; % Example value (adjust as needed) % Define the factor beta (typically between 2 and 4) beta = 3; % Example value (adjust as needed) % Calculate coherence time coherence_time = 1 / (beta * delay_spread); % Display coherence time disp(['Coherence Time: ', num2str(coherence_time), ' seconds']);  

Output

Coherence Time: 333333.3333 seconds