Fourier Transform - Signal Processing

Contrary to the rest of the book where signal and image processing FT methods are introduced from an application-oriented perspective, the first two chapters of this book will introduce several fundamental theorems and important definitions behind the Fourier analysis and other signal processing techniques from a more mathematics-oriented perspective. The FT decomposes functions into a frequency spectra which contain information about the original signals. In signal processing, any periodic signal (with period P) can be represented as summation of an infinite number of instances of an aperiodic function that are shifted by integer multiples of P. Alternatively, the periodic function can be represented as a complex Fourier series where the coefficients are proportional to the sampling of the Continuous Fourier Transform (CFT) at intervals of 1/P. Sampling theorems are fundamental in digital signal processing because it establishes a relationship between continuous-time signals and discrete-time signals. When Fourier coefficients are samples of the periodic function at constant time intervals, it becomes equivalent to a periodic summation of the CFT known as a discrete-time Fourier transform (DFT), and its computation is normally called of Fourier transform (FFT) algorithm. After the mathematics-oriented chapters, the book focuses on FT and DFT spectrum analysis techniques and concepts such as frequency estimation, spectral leakage, Mellin and Scale transforms which are of great usefulness in stationary signal processing. However, FT data processing techniques alone are inadequate methods for tracking changes in signal magnitude, frequency or phase, i.e. for the analysis of non-stationary signals. Short-time Fourier transform (STFT) methods are the most widely used in analyzing the time-frequency properties of non-stationary signals. The remaining chapters focus on the data analysis techniques based on linear transformations generalizing the FT and STFT such as the Fractional Fourier transform (FrFT) and the Local Polynomial Fourier Transform (LPFT), respectively. The last chapters are focused on alternative methods such as the Time Stretch Dispersive Fourier Transform ( a.k.a the warped-stretch transform) used in real-time spectroscopy as well as Wavelet, Radon, Fan-beam, Ridgelet and Curvelet transforms used seismic data analysis.