UDFT

UDFT, or Unitary Discrete Fourier Transform, is a variant of the discrete Fourier transform that uses a symmetric normalization to make the transform unitary. For an N-point sequence x[n], the UDFT is defined by X[k] = (1/√N) ∑_{n=0}^{N-1} x[n] e^{-j 2π k n / N}, and the inverse transform is x[n] = (1/√N) ∑_{k=0}^{N-1} X[k] e^{j 2π k n / N}. With this normalization, the transform matrix is unitary, meaning its conjugate transpose serves as the inverse.

Key properties of the UDFT include energy preservation and numerical stability. Because the forward and inverse transforms are each other’s conjugate transpose, the L2 norm of the signal is preserved: ∑|x[n]|² = ∑|X[k]|². This unitary nature also implies that the UDFT is an orthogonal basis transformation in the complex vector space of finite sequences.

Relation to the standard DFT is primarily a matter of normalization. The conventional DFT uses a 1/N scaling for the inverse transform, which does not render the transform unitary. The UDFT can be computed using the same fast algorithms as the DFT by applying appropriate scaling factors, making it compatible with established FFT techniques.

Applications of the UDFT appear in areas where energy conservation and unitary operations are desirable. In signal processing, it provides a unitary framework for spectral analysis and filtering. In quantum information and certain numerical methods, unitary transforms are favored for their stability properties and compatibility with linear algebraic operations. The UDFT is often discussed in contrast to the standard DFT to emphasize normalization and unitary structure.