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Ebook A Student’s guide to fourier transforms (3rd edition): Part 2

(BQ) Part 2 book "A Student’s guide to fourier transforms" has contents: Applications 2 - signal analysis and communication theory; Applications 3 - interference spectroscopy and spectral line shapes; two dimensional fourier transforms, multi dimensional fourier transforms; the formal complex fourier transform; discrete and digital fourier transforms. | 4 Applications 2: signal analysis and communication theory 4.1 Communication channels Although the concepts involved in communication theory are general enough to include bush-telegraph drums, alpine yodelling or a ship’s semaphore flags, by ‘communication channel’ is usually meant a single electrical conductor, a waveguide, a fibre-optic cable or a radio-frequency carrier wave. Communication theory covers the same general ground as information theory, which discusses the ‘coding’ of messages (such as Morse code, not to be confused with encryption, which is what spies do) so that they can be transmitted efficiently. Here we are concerned with the physical transmission, by electric currents or radio waves, of the signal or message that has already been encoded. The distinction is that communication is essentially an analogue process, whereas information coding is essentially digital. For the sake of argument, consider an electrical conductor along which is sent a varying current, sufficient to produce a potential difference V (t) across a terminating impedance of one ohm (1 ). The mean level or time-average of this potential is denoted by the symbol hV (t)i defined by the equation: hV (t)i D 1 2T T V (t)dt. T The power delivered by the signal varies from moment to moment, and it too has a mean value: hV 2 (t)i D 1 2T T V 2 (t)dt. T For convenience, signals are represented by functions like sinusoids which, in general, disobey one of the Dirichlet conditions described at the beginning of 66 4.1 Communication channels 67 Chapter 2: they are not square-integrable: T V 2 (t)dt ! 1. lim T !1 T However, in practice, the signal begins and ends at finite times and we regard the signal as the product of V (t) with a very broad top-hat function. Its Fourier transform – which tells us about its frequency content – is then the convolution of the true frequency content with a sinc-function so narrow that it can for most purposes be ignored. We thus assume that V (t)