By Wojbor A. Woyczynski
This article serves as a very good advent to stats for sign research. bear in mind that it emphasizes thought over numerical equipment - and that it truly is dense. If one isn't searching for long reasons yet as an alternative desires to get to the purpose fast this booklet can 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.
0 < pi < 1, pi = 1. f. 6) i=1 where u(x) is the unit step function. f. has jumps of size pi at locations xi , and is constant at other points of the real line. , A ∩ B = ∅. 1 (Bernoulli distribution). In this case the values of X, that is the outcomes of the experiment, are assumed to be either 1 or 0 (think about it as a model of an experiment in which “success” or “failure” are the only possible outcomes), with P(X = 1) = p > 0, P(X = 0) = q > 0, with p, q satisfying condition p + q = 1.
1. Common Fourier transforms. Signal Fourier Transform e−a|t| −→ 2a , a2 + (2π f )2 2 −→ e−π f −→ sin π f πf −→ sin2 π f π 2f 2 ej2π f0 t −→ δ(f − f0 ) δ(t) −→ 1 cos 2π f0 t −→ δ(f + f0 ) + δ(f − f0 ) 2 sin 2π f0 t −→ jδ(f + f0 ) − δ(f − f0 ) 2 −→ 1 1 δ(f ) + 2 j2π f t · u(t) −→ j 1 δ (f ) − 4π 4π 2 f 2 e−at · u(t) −→ 1 , a + j2π f e−π t ⎧ ⎨1 for |t| ≤ 1 ; 2 ⎩0 for |t| > 1 . 2 ⎧ ⎨1 − |t| for |t| ≤ 1; ⎩0 for |t| > 1. ⎧ ⎨0 u(t) = ⎩1 for t < 0; for t ≥ 0. a>0 2 a>0 tools that help carry out operations such as distributional diﬀerentiation, distributional multiplication, etc.