Quick question, say I have a complex series of data $f(t)$, so that at each data point $t_i$ have a real and imaginary number, is it correct to calculate the power spectrum of that series (so I want power at some set of frequencies) by saying that at some frequency $w$, the contribution to the signal is given by:
$(a + jb)*(\cos(wt)+i\sin(wt))$
with $j$ complex $j$, and then power is $a^2 + b^2$.
should I be able to decompose any signal like this (i.e.; if i generated a random signal for both the real and imaginary parts seperately, should I still be able to find the power spectrum like this)?
Cheers
Essentially yes. The discrete Fourier transform in its original formulation is for complex inputs and outputs. And since all Fourier transforms are complex linear, you can combine the DFT of the real part and the DFT of the imaginary part to obtain the DFT of the non-separated complex input vector. But that is really not necessary and often leads to a higher computational complexity.
Note that for real signals, the Bruun FFT algorithm computes the DFT using only really real intermediate results, so in theory this FFT applied to real and imaginary parts should be about as fast as the Cooley-Tukey FFT algorithm. But Bruun's FFT has a higher organizational overhead, so handle with care.