Deterministic Discrete-Time Signal Transforms

Chapter 15 presents a detailed analysis of discrete-time signals and systems in the frequency domain, including the theory of the discrete Fourier series, the discrete-time Fourier transform, and the discrete Fourier transform, and key examples relevant for the analysis and synthesis of signals processed in the discrete transceiver blocks of a communication system. Amplitude spectra, magnitude spectra, phase spectra, and power spectra are defined and calculated for typical signals. Using a unique notation that distinguishes between energy signals and power signals, the correlation function and power or energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of linear-time-invariant systems, using the notions of impulse responses, correlation functions, and power spectral densities for both power and energy signals, is presented. The basic theory of the z-transform is also presented.

Transforms of Deterministic Continuous-Time Signals

Chapter 12 presents a detailed analysis of continuous-time signals and systems in the frequency domain, including the theory of Fourier series and Fourier transforms, and key examples relevant for the analysis and synthesis of signals processed in the digital transceiver blocks of a communication system. The amplitude, magnitude, phase, and power spectra are defined and calculated for typical signals. In particular, the Fourier transform of periodic signals is presented, due to its importance in communication systems theory and practice. Using a unique notation that distinguishes energy and power signals, the correlation, power, and energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of a linear-time-invariant system, using the concepts of impulse response, system correlation function, and power spectral density, both for power signals and energy signals, is presented. In addition, Parseval’s theorem and the Rayleigh theorem are proven.

The Energy Spectral Density of the Mertens Function

The Mertens function is the summatory Mobius function but the Mertens function can be generated recursively without using this definition. This recursive definition is the basis of autocorrelations that can be done on sequences of Mertens function values. Fourier transforms of the autocorrelations result in the energy spectral density. A likely upper bound of the absolute value of the Mertens function is determined.

Analysis of wave spectrum by Blackman-Tukey method and fast fourier method

Energy of surface waves is almost dissipated as propagating from shallow water to muddy flat and in mangrove forest. The study aims to analyze energy spectral density by using Blackman-Tukey (BT) and Fast Fourier Transform (FFT) methods in order to analyze the wave energy in mangrove areas in Cu Lao Dung (Soc Trang province). BT method is easy to use especially in short time series. Selection of lag number m is very important to determine the energy amount and number of peaks. Whereas, FFT method helps us to analyze the shift of spectral energy as waves propagate into shallower water. The results show that wave energy is dissipated from shallow water to muddy flat and into mangrove forests. The spectral energy shifts from low frequency to higher frequency as propagating into mangrove forests. This can prove the non-linear characteristics of waves in mangrove forests and the complicated hydrodynamic processes in mangrove forests.

White noise wave generation method controlled by a rotary valve

To meet the demands of white noise waves with different and higher energy spectral density, a new white noise wave generation method was proposed and a corresponding wave-making system controlled by a rotary valve was developed. The theoretical model of the new method was established, and the hydraulic transmission function of the wave-making system was solved. After the control parameters of the white noise waves in the wave-making system were obtained, the experiment tests were carried out to generate the white noise waves, and its energy spectral density was analyzed by Fourier analysis. The results show that white noise waves with different amplitudes have different energy spectral densities. And the energy spectral density of the generated waves is approximately a constant. Therefore, the present white noise wave generation method is feasible in marine engineering tests, and the wave with variable amplitude and higher energy spectral density can be generated.

Research on an Optimization Method for a Partially Responsive Continuous Phase Modulated (CPM) Signal Based on an Optimal Generic Function

This paper establishes an optimal generic function model in order to obtain a continuous phase modulated (CPM) signal with a smoother phase modulation function. This is achieved by finding the solution to the symbol signals at different lengths of the CPM function. In the solution process, the unknown amount that needs to be solved is reduced by using the even function symmetry characteristic of the signal to be solved. For each different form of the signal, the time domain form of the CPM function and the corresponding normalized energy spectral density are compared under the influence of the phase modulation signal length and the generic function parameter n. The data transmission rate is improved by introducing inter-symbol interference, and the modulation process is realized using six-way parallel transmission when the CPM function is 6T. The simulation results show that the CPM function obtained by establishing an optimal generic function model has high-quality time-frequency characteristics. The real-time phase trajectory and the high-order derivative are both continuous, and the modulated signal has constant envelope characteristics. The CPM function has a fast rolling-off in the frequency domain and small out-of-band radiation, which greatly improves the characteristics of the frequency band utilization.