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EECS 501 DISCRETE-TIME RANDOM PROCESSES Fall 2001 DEF: A discrete-time random process=random sequence x(n) is mapping x(n, ω) : (Z × Ω) → R where Ω=sample space and Z = {integers}.
- Fix no ∈ Z → x(no, ω)=random variable indexed by no.
- Fix ωo ∈ Ω → x(n, ωo)=sample function=realization.
- Can think of x(n) as a random vector of infinite length.
THM: Kolmogorov Extension Thm.: Discrete-time random process x(n) is completely specified by its joint pdfs fx(i 1 )...x(iN )(X 1... XN ). EX: An iidrp (independent identically distributed random process) has fx(i 1 )...x(iN )(X 1... XN ) = fx(X 1 )fx(X 2 ) · · · fx(XN ) for any i 1... iN.
DEF: x(n) is N th-order stationary if joint pdfs of order N have: fx(i 1 )...x(iN )(X 1... XN ) = fx(i 1 +j)...x(iN +j)(X 1... XN ) for any j. Means: Shifting time origin does not affect marginal pdfs of order N. EX: 1 st-order stationary ⇔ fx(i)(X) = fx(j)(X) ⇔ x(n) idrp (not iidrp).
THM: N th-order stationary→ (N − K)th-order stationary for 0 ≤ K ≤ N − 1. Proof: Integrate marginals of order N K times→marginals of order N − K. DEF: x(n) SSS strict sense stationary ⇔ N th-order stationary for all N. EX: iidrp x(n) is SSS since fx(i 1 )...x(iN )(X 1... XN ) = fx(X 1 ) · · · fx(XN ).
DEF: Mean μ(n) = E[x(n)]. Variance function σ x^2 (n) = Kx(n, n) where: DEF: (Auto)covariance Kx(i, j) = E[x(i)x(j)] − E[x(i)]E[x(j)] = λx(i),x(j). DEF: (Auto)correlation Rx(i, j) = E[x(i)x(j)] = Kx(i, j) if x(n) is 0-mean. DEF: Cross-covariance Kxy (i, j) = E[x(i)y(j)] − E[x(i)]E[y(j)] = Kyx(j, i). DEF: x(n)uncorrelated ⇔ Kx(i, j) = 0 for i 6 = j. Kx(i, i) may vary with i.
- Kx(i, i) = σ^2 x(i) ≥ 0. (2.) Kx(i, j) = Kx(j, i) (symmetry).
- |Kx(i, j)| ≤
Kx(i, i)Kx(j, j) (Schwarz inequality).
∑N
i=
∑N
j=1 aiKx(ni, nj^ )aj^ ≥^ 0 for any^ ni, nj^ , N^ (psd function).
DEF: x(n) WSS wide sense stationary ⇔ μ(n) = μ and Kx(i, j) = Kx(i − j). Props: (1) Kx(0) = σ^2 x(n) ≥ 0; (2) Kx(i) = Kx(−i); (3) |Kx(i)| ≤ Kx(0). Note: iid→SSS→ N th-order→ 2 nd-order→WSS→ 1 st-order↔id. DEF: x(n) Gaussian ↔ {x(i 1 ), x(i 2 )... x(iN )} JGRV for all i 1... iN. Note: For Gaussian rp: (1) Kolmogorov specified; (2) SSS⇔WSS.
EECS 501 DISCRETE-TIME RPs THROUGH LTI SYSTEMS Fall 2001 LTI: A discrete-time system is LTI linear time-invariant if its response to input x(n) is output y(n) =
i=−∞ h(i)x(n−i) =^
i=−∞ h(n−i)x(i) where h(n)=impulse response of system: x(n) = δ(n) → y(n) = h(n). DEF: Random→ y(n, ω) =
h(n − i)x(i, ω) for each ω ∈ Ω=sample space.
Then: E[y(n)] =
i=−∞ h(n^ −^ i)E[x(i)] =^
i=−∞ h(i)E[x(n^ −^ i)].
Ky (m, n) =
h(m − i)h(n − j)Kx(i, j) =
h(i)h(j)Kx(m − i, n − j). and: Kxy (m, n) =
i=−∞ h(i)Kx(m, n^ −^ i) =^
i=−∞ h(n^ −^ i)Kx(m, i).
- System BIBO stable and μ(n), Kx(n, n) < ∞ →these well-defined.
- x(n) Gaussian→ y(n) Gaussian→only need E[y(n)] and Ky (m, n).
Note: x(n) WSS→ E[x(i)] = μ and Kx(i, j) = Kx(i − j). Above simplify to:
i=−∞ h(i)μ^ =^ H(e
j (^0) )μ=constant.
h(i)h(j)Kx((m − i) − (n − j))
h(i)h(j)Kx((m − n) − i + j) = Ky (m − n). y(n) is also WSS.
h(i)Kx(m − n + i) = Kxy (m − n). x, y jointly WSS.
Transfer function: H(ejω^ ) =
n=−∞ h(n)e
−jωn. h(n) = 1 2 π
∫ (^) π −π H(e
jω (^) )ejωndω. PSD: Sx(ejω^ ) =
n=−∞ Kx(n)e
−jωn (^) = Kx(0) + 2 ∑∞ n=1 Kx(n) cos(ωn). Then: Sy (ejω^ ) = H(ejω^ )H(e−jω^ )Sx(ejω^ ) = |H(ejω^ )|^2 Sx(ejω^ ). Useful later!
DEF: A 1-sided discrete-time rp x(n) is defined only for times n = 0, 1... DEF: A 1-sided rp x(n) is II⇔it has (stationary) independent increments ⇔ {x(i 1 ) − x(0), x(i 2 ) − x(i 1 ), x(i 3 ) − x(i 2 ).. .} are independent rvs for all 0 < i 1 < i 2 <... and pdf of x(i 1 ) − x(i 2 ) depends only on i 1 − i 2.
THM: y(n) is II, y(0)=0 ⇔ y(n) =
∑n i=1 x(i) for some iidrp^ x(n).^ Proof: ⇒: x(n) = y(n) − y(n − 1) → x(n) iidrp and y(n) =
∑n i=1 x(i). ⇐: y(n) =
∑n i=1 x(i)^ →^ y(i^2 )^ −^ y(i^1 ) =^
∑i 2 i 1 +1 x(i) are independent rvs.
THM: y(n) II→ E[y(n)] = μn and Ky (i, j) = σ^2 min[i, j] for constants μ, σ^2. Proof: Apply formulae for LTI systems to h(n) = 1 for n ≥ 0; 0 for n < 0: E[y(n)] =
∑n− 1 i=0 1 ·^ E[x(n^ −^ i)] =^ nμ^ where^ μ^ =^ E[x(n)]. Ky (m, n) =
∑m− 1 i=
j=0 n^ −^11 ·^1 ·^ σ
(^2) δ(i − j) = σ (^2) min[m, n]. Note that an II process is not WSS since Ky (m, n) 6 = Ky (m − n).
