Estimate the psd of a signal

In the following code I calculate the PSD of my signal, a delta, using the FFT. Now, I want to estimate this PSD using the CWT coefficients, using the Global Wavelet Spectrum. But the graphics that remain are not comparable with the original PSD. I need the decomposition to be in the first 16 scales, with the db6 wavelet:

Fs = 1000; % Hz

duracion = 1;

N = Fs * duracion;

t = linspace(-duracion/2, duracion/2, N);

delta = zeros(1, N);

delta(round(N/2)) = sqrt(N);

cantidad_muestras = length(delta);

N = length(delta);

X = fft(delta);

Pxx1_delta = 2*abs(X(1:N/2)).^2/(N*Fs); % PSD

frecuencias = linspace(0, Fs, cantidad_muestras);

frecuencias=frecuencias(1:500);

delta_time=1/Fs;

scales=1:16;

freqs=scal2frq(scales,'db6',delta_time); % computes pseudo frequencies from the scales ..

% Wavelet transform and power spectrum

coefs_base=cwt(delta,scales,'db6'); % computes the Wavelet Transform for dataset at scales

pow_base=(abs(coefs_base)).^2; % computes the Wavelet Time-Frequency Power Spectrum for dataset

% bias correction

bias_corr_pow_base=ones(length(scales),length(t));

for k=1:length(scales)

bias_corr_pow_base(k,:)=pow_base(k,:)/(scales(k));

% after the work of C. Torrence and G. Compo

end

% Time averaged wavelet power spectrum

time_avg_cor_pow_base_spec=mean(bias_corr_pow_base,2);

figure;

plot(freqs,time_avg_cor_pow_base_spec)

title('Bias Corrected Time Averaged Power Spectrum Dataset');

hold on

plot(frecuencias, Pxx1_delta)

return

Answer 1

[Hassaan]

% Constants and Signal Definition
Fs = 1000; % Sampling frequency in Hz
duracion = 1; % Duration in seconds
N = Fs * duracion; % Total number of samples
delta = zeros(1, N); % Initialize delta function
delta(round(N/2)) = sqrt(N); % Impulse in the center

% FFT-based PSD
X = fft(delta); 
Pxx1_delta = 2*abs(X(1:N/2)).^2/(N*Fs); % PSD computation
frecuencias = linspace(0, Fs/2, N/2); % Frequency vector for FFT

% CWT-based PSD
delta_time = 1/Fs;
scales = 1:16; % Define scales
freqs = scal2frq(scales, 'db6', delta_time); % Convert scales to frequencies
coefs_base = cwt(delta, scales, 'db6'); % CWT
pow_base = (abs(coefs_base)).^2; % Power spectrum from CWT coefficients
bias_corr_pow_base = pow_base ./ scales'; % Correcting for scale bias
time_avg_pow_base_spec = mean(bias_corr_pow_base, 2); % Global Wavelet Spectrum

% Interpolate Wavelet Spectrum to match FFT frequencies
interp_wavelet_spectrum = interp1(freqs, time_avg_pow_base_spec, frecuencias, 'linear', 'extrap');

% Plotting
figure;
loglog(frecuencias, interp_wavelet_spectrum, 'LineWidth', 2, 'DisplayName', 'Interpolated Wavelet PSD');
hold on;
loglog(frecuencias, Pxx1_delta, 'LineWidth', 2, 'DisplayName', 'FFT PSD');
xlabel('Frequency (Hz)');
ylabel('Power');
title('Comparison of FFT and Interpolated Wavelet-Based PSD');
legend('show');
grid on;

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[Gisela Clemente]

Thanks for the reply! I don't understand why PSD calculated from the CWT does not converge to the traditional PSD. How do you understand these results?