Fourier transform analysis

The paper is a study of the properties of Fourier transforms and examples of applications to various radio problems. Exponential and trigonometrical expressions of the Fourier transforms are given and the conditions of Validity indicated. The exponential form is a short-hand very useful in nearly all applications provided that the meaning of the symbols is clearly understood. A new notation to designate Fourier transforms is used throughout the paper. The applications of Fourier transforms to radio problems are classified into two main categories: direct application and more advanced applications. In the first category enter all problems where it is required to know the Fourier transform of a function or, inversely, a function whose Fourier transform is already known. Since the Fourier transform of a function represents its frequency and phase characteristics, it can be applied to the study of the frequency spectrum and phase characteristics of electric signals. Frequency spectra of many typical signals such as square, triangular and other types of pulses, sinusoidal signals and modulated carrier, are given. The relation between the shape of the signal and its frequency spectrum is studied, and it is shown that a convenient shape avoids a large radiated band-width. The inverse problem: to calculate a function whose Fourier transform is known, comes naturally in the study of transmission of signals through networks. It is first shown that the response of a network to a sharp impulse applied at the input is a signal whose Fourier transform represents the frequency and phase characteristics of the network. It is also a characteristic of the network itself and can be used to determine the distortion produced on a signal by transmission through the network. The impulse responses of typical networks are given. The networks are divided into three classes: networks with sharp cut-off, with progressive linear cut-off, and with any frequency and phase characteristics. In the - first two classes the impulse response of linear phase-shift and 90 ? out of phase networks is calculated. In the third class it is shown how divergence from linearity in the frequency and in the phase characteristics produces echoes of the ideal response. The transmission of signals through networks is then considered, and it is shown that the distortion produced by networks can be calculated either directly by means of Fourier Transforms, or with the help of the impulse response of the network. The category of more advanced applications is too broad to be investigated in detail in this paper: only a summarised review of the methods used to tackle some typical applications is given. Three applications are mentioned : In the first, a short account of Brillouin´s study on the propagation of signals through dispensive media is given. In the second, som? fundamental properties of the author´s ?Theory of selective transforms? are explained and some applications indicated. Finally, it is shown bow Fourier transforms can be used to study pulse modulation and demodulation in pulse broadcasting.