Bridging the gap between the short-time Fourier transform (STFT), wavelets, the constant-Q transform and multi-resolutio
- PDF / 1,094,066 Bytes
- 9 Pages / 595.276 x 790.866 pts Page_size
- 86 Downloads / 151 Views
ORIGINAL PAPER
Bridging the gap between the short-time Fourier transform (STFT), wavelets, the constant-Q transform and multi-resolution STFT Carlos Mateo1
· Juan Antonio Talavera2
Received: 16 October 2019 / Revised: 24 February 2020 / Accepted: 23 April 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract The short-time Fourier transform (STFT) is extensively used to convert signals from the time-domain into the time–frequency domain. However, the standard STFT has the drawback of having a fixed window size. Recently, we proposed a variant of that transform which fixes the window size in the frequency domain (STFT-FD). In this paper, we revisit that formulation, showing its similarity to existing techniques. Firstly, the formulation is revisited from the point of view of the STFT and some improvements are proposed. Secondly, the continuous wavelet transform (CWT) equation is used to formulate the transform in the continuous time using wavelet theory and to discretize it. Thirdly, the constant-Q transform (CQT) is analyzed showing the similarities in the equations of both transforms, and the differences in terms of how the sweep is carried out are discussed. Fourthly, the analogies with multi-resolution STFT are analyzed. Finally, the representations of a period chirp and an electrocardiogram signal in the time–frequency domain and the time-scale domain are obtained and used to compare the different techniques. The analysis in this paper shows that the proposed transform can be expressed as a variant of STFT, and as an alternative discretization of the CWT. It could also be considered a variant of the CQT and a special case of multi-resolution STFT. Keywords Wavelets · Short-time Fourier transform · Constant-Q transform · Time–frequency · Time-scale · Electrocardiogram
1 Introduction The short-time Fourier transform (STFT) can be applied to convert a signal from the time-domain into the time–frequency domain. It has been used to process signals in many research areas, for example in image processing [1], speech [2], engineering [3, 4], biology and medicine [5]. The STFT can be used to analyze non-stationary signals, determining how the spectral content of signals changes over time. This transform localizes the signal in time using a window. However, the standard STFT transform has the disadvantage of using a fixed window size. On one hand, long windows have better frequency resolution but poor time resolution. On the other hand, short windows provide better time resolution but lower frequency resolution [6]. Several alternatives can be
B
Carlos Mateo [email protected]
1
The Institute for Research in Technology, School of Engineering (ICAI), Universidad Pontificia Comillas, Madrid 28015, Spain
2
Esdras A., S.L., Las Rozas de Madrid, Spain
used to improve this transform, such as adaptive STFT [7, 8] and multi-resolution STFT [9–11]. Adaptive techniques adjust the window size depending on local signal characteristics. Therefore, with adaptive STFT, different window sizes are used fo
Data Loading...
