0 t1 t2
نویسندگان
چکیده
منابع مشابه
Applied Stochastic Processes Problem Set 8
Recall, from Theorem 8.1-2 on page 490 in [4], that a random process X(t) with autocorrelation function RXX(t1, t2) has a m.s. derivative at time t if ∂RXX(t1, t2)/∂t1∂t2 exists at t1 = t2 = t. Furthermore, we recall that the correlation function RXX(t1, t2) is related to the covariance function KXX(t1, t2) as follows. RXX(t1, t2) = KXX(t1, t2) + μX(t1)μX(t2) (1) For this problem we have a cons...
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Q: Consider an uncorrelated (white) Gaussian signal s(t). Calculate its 3rd and 4th order moments, m3 = ⟨s(t1)s(t2)s(t3)⟩ and m4 = ⟨s(t1)s(t2)s(t3)s(t4)⟩. Confirm with simulation. The third moment m3 = ⟨s(t1)s(t2)s(t3)⟩ is zero. Namely, if t1 ̸= t2 ̸= t3 then m3 = ⟨s⟩ = 03 = 0. If t1 = t2 = t3, m3 = ⟨ s3 ⟩ = ́ N (0, σ2)x3dx = 0, as this is the integral over an odd function. For t1 = t2 ̸= t3, m3 = ...
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Let ω(h, k) be a modulus of continuity, that is, ω(h, k) is a continuous function on the square [0, 2π] × [0, 2π], nondecreasing in each variable, and possessing the following properties: ω(0, 0) = 0, ω(t1 + t2, t3) ≤ ω(t1, t3) + ω(t2, t3), ω(t1, t2 + t3) ≤ ω(t1, t2) + ω(t1, t3). Yu ([3]) introduced the following classes of functions: HH := {f(x, y) : ‖f(x, y)− f(x+ h, y)− f(x, y + k) + f(x+ h,...
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Proof (of Lemma 2). Assume that S `C t t0. By definition we get that S `C t. If (CANCEL) was used to derive S `C t t0 then t has the form t1 :t2 :t3 and t0 has the form t1 :t3, and furthermore we know that S t1 ! S0 t2 ! S0. It follows immediately that S `C t1:t3, and hence that S `C t0 and finalC(S, t) = finalC(S, t 0), as required. On the other hand, if (SWAP) was used to derive S `C t t0 the...
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and Applied Analysis 3 Lemma 2.1. If α ∈ 0, 1 , then 1 2α log 2 − logα > 3 log 2. Proof. For α ∈ 0, 1 , let f α 1 2α log 2 − logα , then simple computations lead to f ′ α 2 ( log 2 − 1 − 2 logα − 1 α , 2.1 f ′′ α 1 α2 1 − 2α . 2.2 From 2.2 we clearly see that f ′′ α > 0 for α ∈ 0, 1/2 , and f ′′ α < 0 for α ∈ 1/2, 1 . Then from 2.1 we get f ′ α ≤ f ′ ( 1 2 ) 4 ( log 2 − 1 < 0 2.3 for α ∈ 0, 1 ....
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