A First Course in Statistics for Signal Analysis by Wojbor A. Woyczynski

By Wojbor A. Woyczynski

This article serves as a great creation to statistical data for sign research. remember that it emphasizes concept over numerical tools - and that it's dense. If one isn't really trying to find long factors yet as an alternative desires to get to the purpose quick this booklet should be for them.

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1. 2. 3 in the case of arbitrary period P and amplitude a. 3. Graph them and compare the graphs with the graph of the original signal. 8. cos(2π mf0 t), a for 0 ≤ t < −a for P 2 P 2; ≤ t < P. 5 produce graphs comparing the signal x(t) and its finite Fourier sums of order 1, 3, and 6. Find the complex and real Fourier series for the periodic signal with period P = 1 defined by the formula x(t) = 1− 0 t 2 for 0 ≤ t < 12 ; for 1 2 ≤ t < 1. Produce graphs comparing the signal x(t) and its finite Fourier sums of order 1, 3, and 6.

The histogram of daily voltage readings on an electrical outlet. 2. In this chapter, we will discuss analytical tools for the study of such random quantities. The discrete and continuous random quantities are introduced, but we also show that, in the presence of fractal phenomena, the above classification is not exhaustive. , will symbolize measurements of experiments with uncertain outcomes. 1 Discrete, continuous, and singular random quantities 49 PX (a, b] = P(a < X ≤ b) = P(X ∈ (a, b]) that X takes values in the interval (a, b].

Fig. 3. 3. The period P = 1, and the amplitude a = 1. For m = 0, 1/2 z0 = 1/4 = 1 . 4 For m ≠ 0, zm = |zm |eiθm = = 1/2 1/4 1 −j3π m/4 e πm e−j2π mt dt = 1 [e−j2π m/2 − e−j2π m/4 ] −j2π m ejπ m/4 − e−jπ m/4 2j = π 1 sin m e−j3π m/4 . πm 4 Thus πm 1 j3π m sin , and θm = − πm 4 4 and the complex Fourier series for x(t) is |zm | = ∞ x(t) = πm 1 1 + sin e−j3π m/4 ej2π mt . 4 m=−∞,m≠0 π m 4 26 2 Spectral Representation of Deterministic Signals Again, pairing up the complex exponentials with opposite signs in the exponents, we obtain the real expansion in terms of the cosines, but this time with phase shifts that depend on m: ∞ x(t) = πm 1 2 3π m + sin cos 2π mt − , 4 m=1 π m 4 4 which, using the trigonometric formula cos(α + β) = cos α cos β − sin α sin β, can be written as a general real Fourier series ∞ x(t) = a0 + am cos(2π mt) + bm sin(2π mt), m=1 with a0 = 1 , 4 πm 3π m 2 sin cos , πm 4 4 πm 3π m 2 sin sin .

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