Bounded-Magnitude Discrete Fourier Transform

Analyzing the magnitude response of a finite-length sequence is a ubiquitous task in signal processing. However, the discrete Fourier transform (DFT) provides only discrete sampling points of the response characteristic. This work introduces bounds on the magnitude response, which can be efficiently computed without additional zero padding. The proposed bounds can be used for more informative visualization and inform whether additional frequency resolution or zero padding is required.

The DFT is one of the prime analysis methods for spectral content of a finite-length signal [1]. The DFT, usually computed by applying the fast Fourier transform algorithm, is used for analysis and processing in many applications. Primarily, the magnitude of the DFT is commonly used for signal analysis and visualization purposes, as it describes the strength of the individual frequency components.

Although a DFT of sufficient length carries the complete information of the sequence, direct visualization of the DFT magnitude points can be hard to interpret; see Figure 1 for an example. Many pieces of scientific computing software readily provide plotting features connecting the individual DFT data with line segments. However, such a linear interpolation can be misleading, as the actual shape of the magnitude response is described by the discrete-time Fourier transform (DTFT); see Figure 1. Numerically, the DTFT is usually approximated by a zero-padded higher-order DFT. While sufficient zero padding can lead to higher accuracy, it is often not clear how much zero padding is required.