Any finite sequence of numbers can be represented exactly as a combination of complex oscillations, even when the original data looks nothing like a wave. This is the mathematics behind the Discrete Fourier Transform (DFT), one of the fundamental tools of digital signal processing. When engineers record sound, measure vibrations, or collect data from sensors, they obtain a sequence of samples. The DFT provides a different way to represent those same measurements, using complex exponentials at discrete frequencies. The mathematics relies on a remarkable property called orthogonality. For N samples, the DFT uses N orthogonal complex exponentials. Each frequency component has a corresponding complex coefficient, and the complete set of coefficients contains enough information to reconstruct every original sample. When a complex sinusoid aligns exactly with a DFT frequency bin, its contributions reinforce at that bin and cancel at the others. When its frequency falls between bins, the finite observation window causes its contribution to spread across multiple frequencies. This is known as spectral leakage. These properties have enormous practical value in engineering. In audio processing, Fourier analysis helps identify frequency components and design digital filters. In mechanical engineering, it helps detect periodic vibrations in rotating machinery. In wireless communications, OFDM systems use Fourier transforms to generate and recover signals carried by orthogonal subcarriers. The mathematics also simplifies certain computations. Circular convolution in the sample domain becomes pointwise multiplication in the frequency domain. The Fast Fourier Transform (FFT) makes these calculations efficient by exploiting mathematical symmetries, reducing the computational work from the quadratic cost of direct evaluation to roughly N log N for standard FFT algorithms. There is one important distinction. The DFT preserves all the information in the original sequence of samples, but it cannot recover information already lost during sampling. It is an exact change of mathematical representation, not a way to recover frequencies that sampling has made indistinguishable.