i would like to compare each other wavelet and Fourier transform on given signal,let us consider following signal
function [ x ] = generate1(N,m,A3)
f1 = 100;
f2 = 200;
T = 1./f1;
t = (0:(N*T/m):(N*T))'; %'
wn = randn(length(t),1); %zero mean variance 1
x = 20.*sin(2.*pi.*f1.*t) + 30.*cos(2.*pi.*f2.*t) + A3.*wn;
%[pks,locs] = findpeaks(x);
plot(x)
end
which can be called by $B=generate1$(3,500,1) for example,when i have done separately FFT and wavelet transform,i have got following pictures
for Fourier and
for wavelet,please help me to compare each other these two graph,does they express same picture,but drawn in different basis?
Fourier and wavelet are inner product transforms. There is a basis fucntion which will be multiplied to the signal and then the integral of the calculated value is the transform. The basis function determine the transform features. The basis function in fourier transform is $b\omega(t):=e^{j\omega t}$
$\operatorname{Fourier}[s(t)](\omega)=\langle\operatorname{basis}(\omega,t),s(t)\rangle=\int e^{*j\omega \tau}\cdot s(\tau)\,\mathrm{d}\tau$ note: star means conjugate
The Fourier transform will map a time domain signal to a frequency domain plot and only will tell you what frequencies a signal has. But you may want to know when these frequencies have happened. The Wavelet transform uses many basis functions, for example Morlet functions, which are defined as $\psi(t)=(\pi f_b)^{-0.5}e^{j2\pi f_c}e^{-t^{2}f_b^{-1}}$. We calculate a window function based on $\psi((t-\tau)s^{-1})$, so we slide this window over the signal with different scale values $S$.
$\operatorname{wavelet}[s(t)]=\int \psi((t-\tau)s^{-1})sig(\tau)\,\mathrm{d}\tau$ .
This transform will give us both time and frequency information. But comparing wavelet and Fourier transforms is pointless, you may like to compare STFT and Wavelet which both are time frequency transforms. If you are interested in matlab you can use the spectrogram function to calculate STFT, and then compare those results.