This chapter is dedicated to exploring a form of the Fourier transform that can be applied to digital waveforms, the discrete Fourier transform (DFT). The theory is introduced and discussed as a modification to the continuous-time transform, alongside the concept of windowing in the time domain. The fast Fourier transform is explored as an efficient algorithm for the computation of the DFT. The operation of discrete-time convolution is presented as a straight application of the DFT in musical signal processing. The chapter closes with a detailed look at time-varying convolution, which extends the principles developed earlier. The conclusion expands the definition of spectrum once more.
The Hilbert transform is tightly associated with the Fourier transform. As the offset linear canonical transform (OLCT) has been shown to be useful and powerful in signal processing and optics, the concept of generalized Hilbert transform associated with the OLCT has been proposed in the literature. However, some basic results for the generalized Hilbert transform still remain unknown. Therefore, in this paper, theories and properties of the generalized Hilbert transform have been considered. First, we introduce some basic properties of the generalized Hilbert transform. Then, an important theorem for the generalized analytic signal is presented. Subsequently, the generalized Bedrosian theorem for the generalized Hilbert transform is formulated. In addition, a generalized secure single-sideband (SSB) modulation system is also presented. Finally, the simulations are carried out to verify the validity and correctness of the proposed results.
Abstract. We demonstrate the usefulness of some signal-processing tools for the EISCAT data analysis. These tools are somewhat less classical than the familiar periodogram, squared modulus of the Fourier transform, and therefore not as commonly used in our community. The first is a stationary analysis, "Thomson's estimate'' of the power spectrum. The other two belong to time-frequency analysis: the short-time Fourier transform with the spectrogram, and the wavelet analysis via the scalogram. Because of the highly non-stationary character of our geophysical signals, the latter two tools are better suited for this analysis. Their results are compared with both a synthetic signal and EISCAT ion-velocity measurements. We show that they help to discriminate patterns such as gravity waves from noise.
This paper presents a defect detection algorithm for rail health monitoring that could potentially be used with limited bogie. Current wheel and track monitoring requires expensive track instrumentation and/or time consuming operation of railway monitoring vehicles. The proposed health monitoring algorithm can potentially be used with a portable data acquisition system that can be relocated from train to train to monitor and diagnose the conditions of the track as a train is driven during typical day-to-day operation. The algorithm processes the data using wavelets and is able to locate defects and provide information that may help to distinguish between various types of rail defects. In recent years, wavelets have been used extensively in signal processing because of their ability to analyze a signal simultaneously in the time and frequency domains. The Fourier transform has been used traditionally in signal processing to locate dominant frequencies in a signal, but it is unable to provide time localization of those frequencies. Unlike the Fourier transform, the wavelet transform uses a set of basis functions with finite energy, which is advantageous for detecting the irregular events that may show up in a transient signal. The wavelets used in the proposed signal processing routine were chosen for optimal signal decomposition through consideration of the signals that are likely to be generated from common rail and wheel defects, including rail cracks, squats, corrugation, and, wheel out-of-rounds. A sample accelerometer signal was generated from information found in existing literature and was then processed using the proposed defect detection algorithm. Results show the potential of this algorithm to locate and diagnose defects from limited bogie vertical acceleration data. This study is intended to present a proof-of-concept for the proposed defect detection algorithm, providing a basis for which a more comprehensive defect detection and diagnosis algorithm can be developed.
<div>Fourier theory is the backbone of the area of Signal Processing (SP) and Communication Engineering. However, Fourier series (FS) or Fourier transform (FT) do not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). We note a subtle gap in the explanation of these conditions as available in the popular signal processing literature. They lack a certain degree of explanation essential for the proper understanding of the same. For example, </div><div>the original second Dirichlet condition is the requirement of bounded variations over one time period for the convergence of Fourier Series, where there can be at most infinite but countable number of maxima and minima, and at most infinite but countable number of discontinuities of finite magnitude. However, a large body of the literature replaces this statement with the requirements of finite number of maxima and minima over one time period, and finite number of discontinuities. The latter incorrectly disqualifies some signals from having valid FS representation. Similar problem holds in the description of DCs for the Fourier transform. Likewise, while it is easy to relate the first DC with the finite value of FS or FT coefficients, it is hard to relate the second and third DCs as specified in the signal processing literature with the Fourier representation as to how the failure to satisfy these conditions disqualifies those signals from having valid FS or FT representation. <br></div><div><br></div>
In recent years the application of the Hartley transform, originally introduced by Hartley (1942), has gained importance in seismic signal processing and interpretation (Saatcilar et al., 1990, 1992). The Hartley transform is similar to the Fourier transform but is computationally much faster than even the fast Fourier transform (Bracewell, 1983; Bracewell et al., 1986; Sorensen et al., 1985; Pei and Wu, 1985; Duhamel and Vetterli, 1987; Zhou, 1992). Surprisingly, we have not seen a clear definition of the 2-D Hartley transform in the published literature.
<div>Fourier theory is the backbone of the area of Signal Processing (SP) and Communication Engineering. However, Fourier series (FS) or Fourier transform (FT) do not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). We note a subtle gap in the explanation of these conditions as available in the popular signal processing literature. They lack a certain degree of explanation essential for the proper understanding of the same. For example, </div><div>the original second Dirichlet condition is the requirement of bounded variations over one time period for the convergence of Fourier Series, where there can be at most infinite but countable number of maxima and minima, and at most infinite but countable number of discontinuities of finite magnitude. However, a large body of the literature replaces this statement with the requirements of finite number of maxima and minima over one time period, and finite number of discontinuities. The latter incorrectly disqualifies some signals from having valid FS representation. Similar problem holds in the description of DCs for the Fourier transform. Likewise, while it is easy to relate the first DC with the finite value of FS or FT coefficients, it is hard to relate the second and third DCs as specified in the signal processing literature with the Fourier representation as to how the failure to satisfy these conditions disqualifies those signals from having valid FS or FT representation. <br></div><div><br></div>
The linear canonical transform, which can be looked at the generalization of the fractional Fourier transform and the Fourier transform, has received much interest and proved to be one of the most powerful tools in fractional signal processing community. A novel watermarking method associated with the linear canonical transform is proposed in this paper. Firstly, the watermark embedding and detecting techniques are proposed and discussed based on the discrete linear canonical transform. Then the Lena image has been used to test this watermarking technique. The simulation results demonstrate that the proposed schemes are robust to several signal processing methods, including addition of Gaussian noise and resizing. Furthermore, the sensitivity of the single and double parameters of the linear canonical transform is also discussed, and the results show that the watermark cannot be detected when the parameters of the linear canonical transform used in the detection are not all the same as the parameters used in the embedding progress.
Two different versions of kernels associated with the 2-D Hartley transforms are investigated in relation to their Fourier counterparts. This newly emerging tool for digital signal processing is an alternate means of analyzing a given function in terms of sinusoids and is an offshoot of Fourier transform. Being a real‐valued function and fully equivalent to the Fourier transform, the Hartley transform is more efficient and economical than its progenitor. Hartley and Fourier pairs of complete orthogonal transforms comprise mathematical twins having definite physical significance. The direct and inverse Hartley transforms possess the same kernel, unlike the Fourier transform, and hence have the dual distinction of being both self reciprocal and having the convenient property of occupying the real domain. Some of the properties of the Hartley transform differ marginally from those of the Fourier transform.
Time-frequency tools of signal processing for EISCAT data analysis