How to calculate Complex Fourier Transform ?

I have two orthogonal directional time-displacement signals measured from a rotating shaft. I have combined the x(t) and y(t) to form r(t)= x(t)+ j. y(t) . Now I want to decompose this complex signal into forward rotating and backward rotating. That is to say, positive and negative frequencies. Conventional fft ignores negative frequencies isn't it ? What I like to have is in this form Rf.e ^jwt + Rb.e^-jwt. Essentially separate forward and backward components from the original signal and then take ifft individually and use the time signals separately. How to obtain complex Fourier transform in this fashion ?? I have gone through this tutorial.

http://www.analog.com/static/imported-files/tech_docs/dsp_book_Ch31.pdf

But I am still new to signal processing and I am not very clear how to achieve the algorithm given in this link in a practical way with codes in MATLAB. Kind help will be much appreciated.

Answer 1

[Matt J]

That is to say, positive and negative frequencies. Conventional fft ignores negative frequencies isn't it ?

No, it doesn't ignore them. The negative frequency band is indexed by k=N/2...N. Recall that the complex sinusoid (the discrete Fourier basis function) with frequency index k=N-p is the same as k=-p:

 exp(2j*pi*k*n/N)

   = exp(2j*pi*(N-p)*n/N)
   = exp(2j*pi*(N)*n/N) * exp(2j*pi*(-p)*n/N)
   = exp(2j*pi*(-p)*n/N)