Short-Time Fourier Transform and Gabor Expansion (Introduction)

Overview

In conventional Fourier transform, the signal is compared to complex sinusoidal functions. Because sinusoidal basis functions spread into the entire time domain and are not concentrated in time, the Fourier transform does not explicitly indicate how a signal's frequency contents evolve in time.

Based on the expansion and inner product concepts, a natural way of characterizing a signal in time and frequency simultaneously is to compare the signal with elementary functions that are concentrated in both time and frequency domains, such as the frequency modulated Gaussian function. Because Gaussian-type functions are optimally concentrated in the joint time and frequency domains, the resulting comparisons reflect a signal's behavior in local time and frequency.

In Section 3.1, we briefly introduce the methodology of the short-time Fourier transform (STFT). For the continuous-time STFT, the analysis function and synthesis function have the same form. We can easily recover the original time functions based on the STFT. However, the representation based on the continuous-time STFT is highly redundant (or oversampled). For a compact presentation, we often prefer to use the sampled STFT. In this case, the inverse problem is no longer as straightforward as the case of the continuous-time STFT. The inverse of sampled STFT can be accomplished by the Gabor expansion. Although it is also known as windowed Fourier transform.

Gabor was apparently not motivated to investigate the inverse problem of sampled STFT, the Gabor expansion turns out to be the most elegant algorithm of computing the inverse of the sampled STFT.

Section 3.2 is devoted to the general introduction of the Gabor expansion. Although the idea of the Gabor expansion was rather straightforward, its implementation has been a hot research topic. The continuous-time Gabor expansion in fact has a wider scope for deeper mathematical issues, which has been thoroughly studied by Janssen (see [86], [91], [94], and [95]) as well as many other researches. The discrete-time Gabor expansion, on the other hand, is relatively simple and can be realized with the help of elementary linear algebra. In Section 3.3, we introduce the discrete Gabor expansion for periodic sequences. Then, we develop the discrete Gabor expansion for infinite samples in Section 3.4. In general, given the synthesis function, the dual function is not unique. The natural question is how to choose the dual functions. In Section 3.5, we introduce the orthogonal-like Gabor expansion. The concept of the orthogonal-like has been found very important from both a theoretical and application point of view. In Section 3.6, we present a fast algorithm of computing dual functions. In appendix A, we discuss the existence of the biorthogonal function at critical sampling. In appendix B, we investigate the general optimal algorithm that allows the dual function g[k] to be optimally close to an arbitrary desired function d[k].

Over the years, many techniques have been successfully developed to implement the Gabor expansion, such as Zak transform-based algorithms (see [198] and [199]), filter bank methods [179], as well as the pseudo-frame approach [115]. In this book, we have limited our discussions to the method that was first introduced by Bastiaans (see [8], [9], and [10]) and recently extended by Wexler and Raz [185]. The reader who is interested in methods other than those presented in this chapter may consult the related literature.

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Related Links:
Short-Time Fourier Transform
Gabor Expansion - Inverse Sampled STFT
Gabor Expansion for Discrete Periodic Samples
Discrete Gabor Expansion
Orthogonal-Like Gabor Expansion
Fast Algorithm of Computing Dual Functions

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