Discrete Fourier Transform Notes - Signals and Systems

Fourier Transform and Discrete Fourier Transform Notes

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Discrete Fourier Transform

The Discrete Fourier Transform is a numerical variant of the Fourier Transform. Specifically, given a vector of n input amplitudes such as {f₀, f₁, f₂, ... , f_n-2, f_n-1}, the Discrete Fourier Transform yields a set of n frequency magnitudes.

The DFT is defined as such:

here, k is used to denote the frequency domain ordinal, and n is used to represent the time-domain ordinal. The big "N" is the length of the sequence to be transformed.

The Inverse DFT (IDFT) is given by the following equation:

where  is defined as:

 Again, "N" is the length of the transformed sequence.