Linear Version of Parseval’s Theorem

Abstract

Parseval’s theorem states that the energy of a signal is preserved by the discrete Fourier transform (DFT). Parseval’s formula shows that there is a nonlinear invariant function for the DFT, so the total energy of a signal can be computed from the signal or its DFT using the same nonlinear function. In this paper, we try to answer the question of whether there are linear invariant functions for the DFT, and how they can be found, along with their potential applications in digital signal processing. In order to answer this question, we first prove that the only linear equations that are preserved by the DFT are its orthogonal projections. Then, using Hilbert spaces and adjoint operators, we propose an algorithm that computes all linear invariant functions for the DFT. These linear invariant functions are also shown to be useful and important in a variety of signal-processing applications, particularly for finding some boundaries for transformed signals without explicitly evaluating the DFT, and vice versa. Additionally, using the proposed identities, we demonstrate that the average of a circular auto-correlation function for a large class of signals is preserved by the DFT. Finally, the results reported in this paper are verified for several short-length and long-length DFTs, including a 256-point DFT.

Keywords

• Adjoint operators
• discrete Fourier transform (DFT)
• Hilbert spaces
• invariant functions
• Parseval’s theorem