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

By Wojbor A. Woyczynski

ISBN-10: 0817643982

ISBN-13: 9780817643980

This article serves as a great advent to statistical data for sign research. bear in mind that it emphasizes conception over numerical equipment - and that it's dense. If one isn't really searching for long motives yet in its place desires to get to the purpose speedy this ebook could be for them.

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Example text

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.

Here and subsequently, u(t) denotes Heaviside’s unit step function, equal to 0 for t < 0 and 1 for t ≥ 0. 2. The Fourier transform of the signal x(t) = cos 2π t can be found in a similar fashion, as direct integration of ∞ −∞ cos (2π t)e−j2π f t dt is impossible. But one observes that the inverse transform ∞ −∞ ej2π t + e−j2π t 1 (δ(f − 1) + δ(f + 1))ej2π f t df = = cos 2π t, 2 2 so the Fourier transform of cos 2π t is δ(f −1)+δ(f +1) . ” There exists a large theory of Dirac delta “functions,” and of similar mathematical objects called distributions (in the sense of Schwartz),7 which develops 7 For a more complete exposition of the theory and applications of the Dirac delta and related “distributions,” see A.

Is rich enough to make the Fourier representation possible for any finite power signal. This assumption, often called completeness of the above orthonormal system, can actually be rigorously proven. Approximation at each time instant t separately. This type of approximation is often called the pointwise approximation and the goal is to verify that, for each time instant t, lim sM (t) = x(t). 2) Here the situation is delicate, as examples at the end of this section will show, and the assumption that signal x(t) has finite power is not sufficient to guarantee the pointwise approximation.

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