By Wojbor A. Woyczynski
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Extra resources for A First Course in Statistics for Signal Analysis
3. 4) exist, and the ﬁnite Fourier sums sM (x) of x(t) converge, as M → ∞, to the average value of the signal at the jump: lim sM (t) = M→∞ x(t− ) + x(t+ ) . 1. 1, the ﬁrst three nonzero terms of its cosine expansion were x(t) = a a + 2 π 2 cos 2π t P − 3t 2 cos 2π 3 P + ··· . Hence, in the case of period P = 1 and amplitude a = 1, the ﬁrst four approximating sums are as follows: 1 , 2 2 1 s2 (t) = + cos 2π t, 2 π s0 (t) = 2 1 + cos 2π t, 2 π 2 2 1 s3 (t) = + cos 2π t − cos 6π t. 2. 5). Remark.
N, where the binomial coeﬃcient is deﬁned by n k = n! (n − k)! 1 Discrete, continuous, and singular random quantities n (a + b)n = k=0 51 n k n−k , a b k familiar in the special cases: (a + b)2 = a2 + 2ab + b2 , (a + b)3 = a3 + 3a2 b + 3ab2 + b3 , and so on. Fig. 2. 5 and n = 5. Probabilities pk = pk (n, p) in the binomial probability distribution are probabilities that exactly k “successes” occur in n independent12 Bernoulli experiments in each of which the probability of “success” is p. 5) is here satisﬁed because, in view of the above-mentioned binomial formula, n 1 = (p + q)n = k=0 n k p (1 − p)n−k .
Fourier transform properties. Signal Fourier Transform Linearity αx(t) + βy(t) −→ αX(f ) + βY (f ) Convolution [x ∗ y](t) −→ X(f ) · Y (f ) Diﬀerentiation −→ x (n) (t) (j2π f )n X(f ) Time reversal −→ x(−t) X(−f ) Time delay x(t − t0 ) X(f ) · e−j2π t0 f −→ Frequency translation x(t) · ej2π f0 t −→ X(f − f0 ) Frequency diﬀerentiation (2π )−1 X (n) (f ) −→ (−j)n t n x(t) Frequency convolution −→ x(t)y(t) ∞ −∞ [X ∗ Y ](f ) [x ∗ y](t)e−j2π f t dt = = = ∞ ∞ −∞ ∞ −∞ ∞ −∞ ∞ −∞ −∞ x(s)y(t − s)ds e−j2π f t dt y(t − s)e−j2π f (t−s) x(s)e−j2π f s dsdt y(u)e−j2π f u du · ∞ −∞ x(s)e−j2π f s ds = X(f ) · Y (f ), where the penultimate equality resulted from the substitution t −s = u.