Performance comparison of FFT vs. Wavelets

Hello everyone, I am new to the Wavelets topic, but I understtod some of the basic comparison between Fast Fourier Transform (Frequency domain) and Wavelets (Time and Frequency domain). I'd like to understand the theoritical computational duration when comparing FFT and Wavelets. Can anyone help me understand this subject better?

Thank you!

Answer 1

[Harsh Kumar]

Hi Prashanth,

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.

2. Resolution:

• 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 :

% Generate a sample signal
fs = 1000; % Sampling frequency
t = 0:1/fs:1; % Time vector
f1 = 10; % Frequency of sinusoid 1
f2 = 50; % Frequency of sinusoid 2
x = sin(2*pi*f1*t) + sin(2*pi*f2*t);

% FFT
tic; % Start timer
X = fft(x);
fftTime = toc; % End timer

% Wavelet Transform
tic; % Start timer
wname = 'db4'; % Wavelet name (Daubechies 4)
level = 5; % Decomposition level
[C, L] = wavedec(x, level, wname);
waveletTime = toc; % End timer

% Display computational duration
fprintf('FFT duration: %.4f seconds\n', fftTime);

FFT duration: 0.0056 seconds

fprintf('Wavelet transform duration: %.4f seconds\n', waveletTime);

Wavelet transform duration: 1.0509 seconds

Documentation link for details : mathworks.com/help/wavelet/gs/from-fourier-analysis-to-wavelet-analysis.html