Fourier Transforms

Fourier Transform analysis of sampled sound

The Fourier Transform´s ability to represent time-domain data in the frequency domain and vice-versa has many applications. One of the most frequent applications is analysing the spectral (frequency) energy contained in data that has been sampled at evenly-spaced time intervals. Other applications include fast computation of convolution (linear systems responses, digital filtering, correlation (time-delay estimation, similarity measurements) and time-frequency analysis.

The fast version of this transform, the Fast Fourier Transform (or FFT) was first developed by Cooley and Tukey² and later refined for even greater speed and for use with different data lengths through the "mixed-radix" algorithm. Igor computes the FFT using a fast multidimensional prime factor decomposition Cooley-Tukey algorithm.

While the the Fourier Transform is mathematically complicated, Igor´s Fourier Transforms dialog makes it easy to use:

Fourier Transforms dialog showing output type and graphing options.

Igor´s FFT operation supports advanced calculations, some of which are beyond the scope of the Fourier Transforms dialog:

FFT of 2-Dimensional, 3-D, and 4-D data
Input Data Length Padding
Input Data Windowing
Results as Complex, Real-only, Imaginary-only, Magnitude, Magnitude Squared, or Phase
Hypercomplex sine transform
Hypercomplex cosine transform

In addition to the FFT, Igor provides these other transforms:

Discrete Wavelet Transform (DWT)
Continuous Wavelet Transform (CWT)
Hilbert Transform

References