I understand that you are willing to understand the computational duration when comparing ‘Fast Fourier Transform’ and ‘Wavelets’ both theoretically and programmatically .
Theoretical comparisons between FFT and Wavelets are as follows:
1. Frequency vs. Time-Frequency Analysis:
- FFT: Primarily for frequency analysis, transforms time-domain signal into frequency representation.
- Wavelets: Provides time and frequency localization, analyzing signal at different time intervals.
- FFT: Uniform frequency resolution, lacks time localization.
- Wavelets: Variable resolution in time and frequency, high for high-frequency components, low for low-frequency components.
3. Multiresolution Analysis:
- Wavelets support multiresolution analysis, analyzing signals at different scales or resolutions for detailed examination.
Refer to the below code snippet for better understanding :
x = sin(2*pi*f1*t) + sin(2*pi*f2*t);
[C, L] = wavedec(x, level, wname);
fprintf('FFT duration: %.4f seconds\n', fftTime);
fprintf('Wavelet transform duration: %.4f seconds\n', waveletTime);