The Fourier transform and some properties
We have defined the Fourier transform pairs on an infinite time and frequency scale. In the practical world of computers, the time and frequency domain data are assumed to be at discrete sample locations, and with a finite number of samples. In this realm, the Fourier transform becomes the discrete Fourier series where the input and transformed functions must be considered to be periodic, and attention given to wrap-around and aliasing.
Applying the derivative in the frequency domain becomes a very simple procedure. When the data has been Fourier transformed, the derivative is found by applying a 90 degree phase shift (the “j” part), and multiplying the amplitudes of the frequency values by the ramp type function ω or 2πf. The inverse Fourier transform then gives the exact derivative.
Note that derivatives found by subtracting two samples is only an approximation.
Higher order (n) derivatives are found in a similar manner by applying n90 degrees, and multiplying the amplitudes by ωn.
Integral are found very simply by dividing the spectrum by jω.
The exponential of a complex value may be expressed as eJθ = cos(θ) + j sin(θ) It has a magnitude of one, i.e. |ejθ| = 1
The phase = φ may be found from
The above time shift t0 is a linear phase shift in the frequency domain θ= ωt0.
Non-integer static shifts (say t0) can be made accurately in the frequency domain with a simple linear phase shift.