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This article continues the exposition in Parts I [1] and II [2] of certain concepts in the theory of tempered distributions which are useful for a thorough understanding of signal theory. In this series of articles, we strive to present a mathematically rigorous exposition of these ideas to supplement the imprecise treatment of distribution theory encountered in the signal processing literature. Part I contains an overview of the subject and discusses some well-known applications of tempered distributions in signals analysis. Subsequent parts delve more deeply into specialized topics. Thus, Part II focuses on the convolution of tempered distributions, and here in Part III we consider periodic tempered distributions and generalized Fourier series. Part III upholds the style of Parts I and II by maintaining a high level of mathematical rigor, while resisting the temptation to become unnecessarily abstract. Thus, for example, it is not necessary to invoke the notion of topological vector spaces from general distribution theory when the simpler notion of normed linear space will suffice. We also provide very detailed proofs which will hopefully make the material more accessible to physicists, engineers, computer scientists and other non-mathematicians with an interest in this subject. [ABSTRACT FROM AUTHOR]

Abstract: We derive the correct formulation of the Doppler problem in which the received signal is scaled in time by a scale factor dependent on the relative radial velocity of the transmitter and the receiver, normalized by the propagation speed in the transmission medium. For sine waves, the conventional Doppler model in which the received signal is translated in frequency is equivalent to the scale model. However, for signals with non zero bandwidth, the difference can be quite significant, as we demonstrate. We further demonstrate simple methods, based on the discrete Fourier transform (DFT) and phase differentiation of the scale CAF, by which delay and scale may be accurately estimated. [Copyright &y& Elsevier]

Abstract: We describe a new linear time–frequency model in which the instantaneous value of each signal component is mapped to the curve functionally representing its instantaneous frequency. This transform is linear, uniquely defined by the signal decomposition and satisfies linear marginal-like distribution properties. We further demonstrate that the transform generated surface may be estimated from the short time Fourier transform (STFT) by a concentration process based on the phase of the STFT, differentiated with respect to time. Interference may be identified on the concentrated STFT surface, and the signal with the interference removed may be estimated by applying the linear time marginal to the concentrated STFT surface from which the interference components have been removed. [Copyright &y& Elsevier]

Abstract: We propose a new method for adaptively removing noise and interference from a signal. In this method unwanted components are removed from the short time Fourier transform (STFT) surface, and the clean signal is estimated by integrating the modified STFT with respect to frequency. Isolation of the signal and interference components is facilitated by a concentration process based on the phase of the STFT differentiated with respect to time. The concentrated STFT is a linear representation, free of cross terms and having the property that signal and interference components are easily recognized because their distributions are more concentrated in frequency. Interference removal may be accomplished by removing unwanted components from the concentrated STFT, and the clean signal may be estimated by integration of the modified concentrated STFT. We demonstrate the advantages of the proposed method over conventional methods. [Copyright &y& Elsevier]

Abstract: This article continues the exposition in Part I [D.C. Smith, An introduction to distribution theory for signals analysis, Digital Signal Process. 13 (2003) 201–232] of certain key concepts from distribution theory which are essential for understanding signal theory. In this second part, we demonstrate the power of distribution theory by focusing on distributional convolution and some of its applications to signals analysis. The well-known classical convolution theorem (CCT) states that the Fourier transform of a convolution of integrable functions is the product of their Fourier transforms, and is essential in signals processing, by providing a method for removing noise and undesirable spectral artifacts from signals. Unfortunately, signal processing texts routinely apply this theorem nonrigorously, with questionable results, e.g., in attempts to apply it to singular functions in derivations of the Hilbert conjugate and the analytic signal, in attempts to recover an unknown causal impulse response modeling a data channel, and in derivations of the Sampling Theorem [D.C. Smith, An introduction to distribution theory for signals analysis, Digital Signal Process. 13 (2003) 201–232]. Fortunately, many inadequacies of the CCT are overcome by its generalizations to the tempered distributions. In this article we discuss three distributional convolution theorems (DCTs), each of which has important theoretical and practical consequences for signal theory. We demonstrate that when the Cauchy principal value distribution is used to properly define the Hilbert conjugate, the first DCT may be applied to rigorously derive the analytic signal. The second DCT is particularly useful for applications involving compactly supported distributions, including the Dirac delta. The third DCT concerns the lesser-known distributional Laplace transform, which is shown by example to be superior to either the classical or distributional Fourier transform for recovering an unknown impulse response modeling a causal linear time-invariant system. This article upholds the style of Part I, by avoiding unnecessary abstraction and supplying very detailed proofs in hopes of appealing to a wider audience. [Copyright &y& Elsevier]

The theory of tempered distributions provides the mathematical foundation for much of signal theory, especially DSP. Yet signals processing texts do not contain careful expositions of the theory, preferring instead to treat the subject as a collection of rules to be memorized and formally manipulated. Unfortunately, the manipulations are often performed in situations which cannot be justified by the theory, for example, in derivations of the sampling theorem. This article contains a concise, though mathematically rigorous introduction to the theory of tempered distributions. We subsequently apply distribution theory to give rigorous proofs of many of the basic results of signals analysis, including Whittaker''s sampling theorem and the remarkable though lesser-known result that every periodic tempered distribution has a (generalized) Fourier series. The level of presentation is intentionally less abstract and the proofs contain more detail than one normally finds in purely mathematical treatments of distribution theory and will hopefully appeal to a wider audience. [Copyright &y& Elsevier]