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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">invariant</i> 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.
Highlights
T HE discrete Fourier transform (DFT) as a Fourier representation of finite-length sequences plays a central role in a wide variety of signal-processing applications, including filtering and spectral analysis, since it can be explicitly computed by efficient algorithms, collectively called the fast Fourier transform (FFT)
The principle results of this paper are based on the normalized version of the DFT, which they can be extended to other versions of the DFT
Since every eigenvector of F † gives an invariant function for the DFT, we have the same number of invariant functions
Summary
T HE discrete Fourier transform (DFT) as a Fourier representation of finite-length sequences plays a central role in a wide variety of signal-processing applications, including filtering and spectral analysis, since it can be explicitly computed by efficient algorithms, collectively called the fast Fourier transform (FFT). We answer to the following questions: What are all the linear functions which are invariant under the DFT and which information can be derived from them? The main contributions of this paper are summarized as follows: 1) We derive the linear invariant functions of the DFT, which are invariant relations between time signals and their transformed frequency signals under the DFT. These identities can be considered as linear versions of Parseval’s theorem. A novel invariant non-linear relation between average circular auto-correlation for time and frequency signals is introduced. Explicit examples of the linear Parseval identities are given for 4 and 8-point DFTs
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