Coherence bandwidth of S21

How do I code this formula in Matlab to find the coherence bandwidth of an S21 data set (a vector of complex numbers vs. frequency)?

Coherence bandwidth is defined according to the correlation function of the frequency response (R(χ)).

R(χ) = ∫ H (f )H∗ (f + χ)df

integrated from fmin to fmax, where χ is the frequency shift, B = [fmin ,fmax ] is the frequency bandwidth and {.}∗ denotes the complex conjugate.

Then, the coherence bandwidth of the channel at the level ρ ∈ [0,1], is defined as the frequency Bρ for which the absolute value of the correlation function falls to a value equal to ρ times its maximum corresponding to χ = 0.

Bρ = χ such that |R(χ)| = ρ|R(0)|

Answer 1

[Arnav]

Hi @Derek,

We can go about doing the task in MATLAB as you described but first, we need to assume some data. Here, I am considering the frequency response of a band pass filter:

%% Sample frequency data
f = linspace(1e9, 2e9, 1000); % Frequency range = 1GHz to 2 GHz
%% Sample data for a band-pass filter
f0 = 1.5e9; % Center frequency at 1.5 GHz
BW = 0.2e9; % Bandwidth of 200 MHz
S21_mag = exp(-((f - f0) / (BW / 2)).^2);
S21_phase = -2 * pi * (f - f0) / max(f);
S21 = S21_mag .* exp(1i * S21_phase);

You may load your own data into variables f and S21 instead of the generic bandpass filter that I have used. The following steps calculate the Correlation function R:

R = @(chi) arrayfun(@(c) trapz(f, S21 .* conj(interp1(f, S21, f + c, 'linear', 0))), chi);

Here R is a function handle that takes an input vector chi and maps each element of chi, say x, to a numeric integral that evaluates R(x). This is done using a function trapz which calculates the numeric integral.

Next, we find the region of chi that correspond to amplitude greater than or equal to ρ|R(0)|

R0 = R(0); % Max value of R
rho = 0.5; % Example level, adjust as needed

chi_values = linspace(-max(f) + min(f), max(f) - min(f), 2000); % Possible χ values (both negative and positive)
R_chi = abs(R(chi_values)); % Calculate R(χ) for each χ

% This finds the chi indices that fall within the bandwidth 
B_rho_indices = find(R_chi >= rho * abs(R0)); 
B_rho = chi_values([B_rho_indices(1), B_rho_indices(end)]); % Corresponding χ values for bandwidth edges

Now, we print the value of the bandwidth along with a plot that makes it easy to visualize:

%% Plotting the outputs:
% Plot the correlation function |R(χ)|
figure;
subplot(2, 1, 1);
plot(chi_values, R_chi, 'b', 'LineWidth', 2);
hold on;
yline(rho * abs(R0), 'r--', 'LineWidth', 2); % Line for ρ|R(0)|
xline(B_rho(1), 'g--', 'LineWidth', 2); % Line for negative Bρ
xline(B_rho(2), 'g--', 'LineWidth', 2); % Line for positive Bρ
xlabel('Frequency Shift χ (Hz)');
ylabel('|R(χ)|');
title('Correlation Function vs Frequency Shift');
legend('|R(χ)|', 'ρ|R(0)|', 'Bρ');
grid on;

% Display the coherence bandwidth
fprintf('Coherence bandwidth Bρ at level ρ = %.2f is from %.2f Hz to %.2f Hz\n', rho, B_rho(1), B_rho(2));

Coherence bandwidth Bρ at level ρ = 0.50 is from -117558779.39 Hz to 117558779.39 Hz

fprintf('Total coherence bandwidth Bρ is %.2f Hz\n', B_rho(2) - B_rho(1));

Total coherence bandwidth Bρ is 235117558.78 Hz

% Plot the transfer function of the band-pass filter
subplot(2, 1, 2);
plot(f, 20*log10(abs(S21)), 'b', 'LineWidth', 2);
xlabel('Frequency (Hz)');
ylabel('Magnitude (dB)');
title('Transfer Function of the Band-Pass Filter');
grid on;

Along with the output we also get a plot showing the bandwidth between the green dashed lines.

Refer to the following documentation for more information:

Function handles:

https://www.mathworks.com/help/matlab/ref/function_handle.html?searchHighlight=function%20handles&s_tid=srchtitle_support_results_2_function%20handles#responsive_offcanvas

trapz:

https://www.mathworks.com/help/matlab/ref/trapz.html?searchHighlight=trapz&s_tid=srchtitle_support_results_1_trapz#responsive_offcanvas

arrayfun:

https://www.mathworks.com/help/matlab/ref/arrayfun.html?searchHighlight=arrayfun&s_tid=srchtitle_support_results_1_arrayfun