The Discrete Fourier Transform, Part 1

This paper describes an implementation of the Discrete Fourier Transform (DFT) and the Inverse Discrete Fourier Transform (IDFT). We show how the computation of the DFT and IDFT may be performed in Java and show why such operations are typically considered slow. This is a multi-part paper, in part 2, we discuss a speed up of the DFT and IDFT using a class of algorithms known as the FFT (Fast Fourier Transform) and the IFFT (Inverse Fast Fourier Transform). Part 3 demonstrates the computation of the PSD (Power Spectral Density) and applications of the DFT and IDFT. The applications include filtering, windowing, pitch shifting and the spectral analysis of re-sampling. 1 THE DISCRETE FOURIER TRANSFORM Let vj , j ∈ 0...N −1 [ ] (1) be the sampled version of the waveform, v(t) where N is the number of samples. Equation (1) numbers from zero, rather than one, to reflect the start point of arrays in Java. Fourier transform of v(t) is given by V ( f ) = F[v(t)] = v(t)e ift dt −∞ ∞ ∫ (2). V(f) can only exists if v(t) is absolutely integrable, i.e.