By Wojbor A. Woyczynski
This article serves as an exceptional advent to stats for sign research. bear in mind that it emphasizes conception over numerical tools - and that it truly is dense. If one isn't really searching for long motives yet as an alternative desires to get to the purpose fast this ebook will be for them.
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Additional info for A First Course in Statistics for Signal Analysis
Woyczy´ nski, Introductory Statistics and Random Phenomena: Uncertainty, Complexity, and Chaotic Behavior in Engineering and Science, Birkhäuser Boston, Cambridge, MA, 1998. 1 Discrete, continuous, and singular random quantities 53 Fig. 3. f. fX (x) takes values in the interval [−1, 2]. d FX (x) = fX (x). 4 (uniform distribution). The density of a uniformly distributed random quantity X is deﬁned to be a positive constant within a certain interval, say [c, d], and zero outside this interval. 9), fX (x) = (d − c)−1 for c ≤ x ≤ d; 0 elsewhere.
Thus we can approximate the integrals involved in calculation of the Fourier transforms in the same way as one does in numerical integration in calculus, using left-handed rectangles, trapezoids, Simpson’s rule, etc. We use the simplest approximation, which is equivalent to assuming that the signal is constant between the times at which we sample (and rectangular approximations of the area under the function). Therefore, suppose that the sampling period is Ts , with the sampling frequency fs = T1s , so that the signal’s sample is given in the form of a sequence xk = x(kTs ), k = 0, 1, 2, .
Objects 3D vectors Signals with ﬁnite power x = (x1 , x2 , x3 ) y = (y1 , y2 , y3 ) x(t) = y(t) = ∞ m=−∞ zm em (t), PWx < ∞ ∞ m=−∞ wm em (t), PWy < ∞ Bases Unit coordinate vectors Complex exponentials .. e1 (t) = ej2π f0 t e2 (t) = ej2π (2f0 )t e3 (t) = ej2π (3f0 )t .. e1 = (1, 0, 0) e2 = (0, 1, 0) e3 = (0, 0, 1) Scalar products x, y = 3 i=1 xi y i x(t), y(t) 1 P = P 0 x(t)y ∗ (t)dt Orthonormality em , em = δmn em (t), en (t) = δmn Expansions Basis x= Fourier 3 m=1 x(t) = x, em em ∞ i=−∞ x, em em (t) Formulas Pythagoras’ ||x||2 = 3 m=1 Parseval’s 2 xm Scalar product x, y = 3 m=1 P PWx = P1 0 |x(t)|2 dt ∞ = m=−∞ |zm |2 Extended Parseval’s xm y m 1 P = P 0 x(t)y ∗ (t)dt ∗ z m wm ∞ m=−∞ the negative timeline to form either an odd or an even signal, and thus obtain either its pure sine or its pure cosine expansion.