new to Analytica 4.5

FFTInv( x, Freq, T )

The Inverse Fast Fourier Transform (FFT) converts a frequency spectrum equally spaced values, «x», from the discrete frequency domain to the discrete time (or spatial) domain. «T» is the time index and «Freq» is the Frequency index, and these two indexes should have the same length. The frequency spectrum, «x», is indexed by «Freq» and contains complex numbers, encoding both the amplitude and phase at each frequency. The result is indexed by «T» and contains complex numbers in general. The units of «Freq» are cycles per «units of T».

The FFT function does the forwart transformation from the time (or spatial) domain to the frequency domain.

The Discrete Fourier Transform

FFTInv computes the inverse discrete Fourier transform (DFT) in an efficient manner. The inverse DFT is defined given by

[math]\displaystyle{ h_k = {1\over n} \sum_{i=0}^{n-1} x_i e^{-2j\pi i k/n} }[/math]

where [math]\displaystyle{ j }[/math] is the imaginary number [math]\displaystyle{ \sqrt{-1} }[/math], and n is the number of points in «T» and «Freq». This is a discrete approximation to the continuous Fourier Transform given by

[math]\displaystyle{ h(t) = \int_{-\infty}^\infty x(f) e^{-2j \pi f t} df }[/math]