Mathematics is the mother of all the sciences, engineering and technology, and a normed division algebra of all finite dimensions is the mathematical holy grail. In search of a real three-dimensional, normed, associative, division algebra, Hamilton discovered quaternions that form a non-commutative division algebra of quadruples. Later works showed that there are only four real division algebras with 1, 2, 4, and 8 dimensions. This work overcomes this limitation and introduces generalized hypercomplex numbers of all dimensions that are extensions of the traditional complex numbers. The space of these numbers forms non-distributive normed division algebra that is extendable to all finite dimensions. To obtain these extensions, we defined a unified multiplication, designated as scaling and rotative multiplication, fully compatible with the existing multiplication. Therefore, these numbers and the corresponding algebras reduce to distributive normed algebras for dimensions 1 and 2. Thus, this work presents a generalization of $\mathbb{C}$ in higher dimensions along with interesting insights into the geometry of the vectors in the corresponding spaces.

In search of a real three-dimensional, normed, associative, division algebra, Hamilton discovered quaternions that form a non-commutative division algebra of quadruples. Later works showed that there are only four real division algebras with 1, 2, 4, or 8 dimensions. This work overcomes this limitation and introduces generalized hypercomplex numbers of all dimensions that are extensions of the traditional complex numbers. The space of these numbers forms non-distributive normed division algebra that is extendable to all finite dimensions. To obtain these extensions, we defined a unified multiplication, designated as scaling and rotative multiplication, fully compatible with the existing multiplication. Therefore, these numbers and the corresponding algebras reduce to distributive normed algebras for dimensions 1 and 2. Thus, this work presents a generalization of $\mathbb{C}$ in higher dimensions along with interesting insights into the geometry of the vectors in the corresponding spaces.

Fourier theory is one of the most important tools used ubiquitously for understanding the spectral content of a signal, extracting and interpreting information from signals, and transmitting, processing, and analyzing the signals and systems. Undergraduate engineering students are exposed to these concepts, usually in their second year, to build their foundation in the areas of signal processing and communication engineering. However, the popular signal processing literature \cite{book1,book2,book3,book4} does not offer a clear explanation regarding the convergence or the existence of Fourier representations for certain well-known signals. Because of this subtle gap, it becomes hard for young students to assimilate the Fourier theory with clarity, and they are forced to be familiar with some of these concepts without understanding them. To bring clarity to the existence and the convergence of Fourier representation, including Fourier series and transform, lecture notes were published recently in IEEE Signal Processing Magazine’s September 2022 issue \cite{PSDCs}. This lecture note is in the continuation with technical details added from yet another mathematical topic of distribution theory that connects delta Dirac functions with the Fourier theory. The distribution theory by Schwartz in 1945 is one of the great revolutions in mathematical function analysis. It is considered as a completion of differential calculus, similar to how the revolutionary measure theory or Lebesgue integration theory proposed in 1903, is considered as a completion of integral calculus. Both these theories unlocked new paradigms of mathematical development. Although distribution theory is a powerful tool for understanding Fourier theory, it is ignored in engineering textbooks. In this lecture note, we utilize the concepts of this theory to show how some signals that fail to exhibit FT in the conventional sense can have FT in the distributional sense.

This work presents a novel perfect reconstruction filter bank decomposition (PRFBD) for nonlinear and nonstationary time series and image data representation and analysis. The Fourier decomposition method (FDM), an adaptive approach wholly based on the Fourier representation, is shown to be a special case of the proposed PRFBD. The adaptive Fourier–Gauss decomposition (FGD) proposed in this work is a variation of the FDM, which is based on the FR and Gaussian filtering. Similarly, we also consider Butterworth filtering to develop adaptive Fourier–Butterworth decomposition (FBD). The proposed theory can decompose any signal (time series, image, or other data) into a set of the desired number of Fourier intrinsic band functions (FIBFs), which follow the amplitude-modulation and frequency-modulation (AM-FM) representations. A generic filterbank representation is also provided, where perfect reconstruction can be ensured for any given set of lowpass or highpass filters. We have performed an extensive analysis of both simulated and real-life data (COVID-19 pandemic, Earthquake and Gravitational waves) to demonstrate the efficacy of the proposed method. The resolution results in the time-frequency representation demonstrate that the proposed method is more promising than the state-of-the-art approaches.

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, 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.

This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed.