By John R. Pierce
Книга just about all approximately Waves just about all approximately WavesКниги Физика Автор: John R. Pierce Год издания: 2006 Формат: djvu Издат.:Dover guides Страниц: 224 Размер: 1,1 ISBN: 0486453022 Язык: Английский0 (голосов: zero) Оценка:This textual content considers waves the good unifying notion of physics. With minimum arithmetic, it emphasizes the habit universal to phenomena akin to earthquake waves, ocean waves, sound waves, and mechanical waves. subject matters comprise speed, vector and intricate illustration, strength and momentum, coupled modes, polarization, diffraction, and radiation. 1974 version.
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15 Starting from the wide-sense stationary process XðtÞ we deﬁne a new process as YðtÞ ¼ XðtÞ À Xðt þ TÞ. (a) Show that the mean value of YðtÞ is zero, even if the mean value of XðtÞ is not zero. (b) Show that 2Y ¼ 2fRXX ð0Þ À RXX ðTÞg. 9 (c) If YðtÞ ¼ XðtÞ þ Xðt þ TÞ ﬁnd expressions for E½YðtÞ and 2Y . Compare these results with the answers found in (a) and (b). 16 Determine for each of the following functions whether it can be the autocorrelation function of a real wide-sense stationary process XðtÞ.
A complicating factor is that for a random waveform (a sample function of the stochastic process) the Fourier transform generally does not exist. 1 THE POWER SPECTRUM Due to the problems with the Fourier transform, a theoretical description of stochastic processes must basically start in the time domain, as given in Chapter 2. In this chapter we will conﬁne ourselves exclusively to wide-sense stationary processes with the autocorrelation function RXX ðÞ. Let us assume that it is allowed to apply the Fourier transform to RXX ðÞ.
3:32Þ Proofs of the properties: 1. Þ. 2. Þ the cross-power spectrum will in general be a complex-valued function. Þ d ð3:34Þ For real processes the cross-correlation function is real as well and the ﬁrst integral represents the real part of the power spectrum and is obviously even. The second integral represents the imaginary part of the power spectrum which is obviously odd. 3. 30). MODULATION OF STOCHASTIC PROCESSES 47 4. The fourth property is quite straightforward. For orthogonal processes, by deﬁnition, RXY ðÞ ¼ RYX ðÞ 0, and so are the corresponding Fourier transforms.