Quaternion domain Laplace transform of the pulse vector

Abstract

The Laplace transform (LT) is widely used in radio engineering for signal and circuit analysis. The PL greatly facilitates the solution of differential equations, the calculation of transfer functions, the finding of impulse responses, etc. Multiple-Input Multiple-Output (MIMO) systems are becoming more common today. With input influences on such systems, at the output signals are obtained, the elements of which are closely related to each other, and changes in some influencing elements of the input vector change the values of others. Such changes are usually associated with the preservation of the vector norm during transformation. Obviously, this completely changes the shape of the output response and, accordingly, its spectrum. To calculate such changes, it is possible to use the usual PL of real signals and the corresponding theorems. However, this approach requires a significant investment of time and computational resources. If you change the amplitude, shape, time shift of at least one pulse, you will have to repeat all the calculations again. Quaternion transformations, including the Laplace transform, have been studied in many works. However, these studies are often of a general theoretical nature or are used only to obtain the Fractional Quaternion Laplace Transform of 2D images. To calculate the LT of the impulse vector when using the MIMO scheme, it is proposed to use hypercomplex numbers, in the particular case, quaternions. Quaternion is a hypercomplex number with one scalar and three imaginary numbers i, j, k. To get rid of operations with imaginary numbers, the quaternion is represented as an orthogonal 4×4 matrix. The matrix, in turn, is decomposed into 4 basis matrices. Moreover, operations with matrices correspond to operations with imaginary units and the quaternion as a whole. It is shown that the quaternionic Laplace transform (QLT) of the vector is represented as a one-dimensional integral from 0 to ∞ of the vector. In this case, the matrix exponent in the power of the quaternion frequency matrix S = Eσ + 1/√3(I + J + K)ω is used as the transformation kernel, where E, I, J, K are basis matrices. The main properties of the QLT are considered. It is shown that in terms of the notation form, the properties of the QLT correspond to the properties of the LT of real functions, taking into account the non-commutativity of matrix multiplication. Therefore, to calculate the QLT, it is possible to use the well-known expressions for the LT of real pulses with the replacement of the complex frequency s by the matrix of quaternion frequencies S. Expressions for the QLT are obtained for the pulse vectors, which are often used to solve radio engineering problems. It is shown that for σ = 0 these expressions correspond to the quaternionic Fourier transform of the vector pulses. In general, vector pulses can have different delays, amplitudes and shapes. Expressions are obtained for finding the QLT of such vectors.