Stationary Stochastic Processes Table of Formulas, 2017

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Ποδαργη Τομαραίοι
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1 Stationary Stochastic Processes, 07 Stationary Stochastic Processes Table of Formulas, 07 Basics of probability theory The following is valid for probabilities: P(Ω), where Ω is all possible outcomes 0 P(A), where A is some event P(A c ) P(A), where A c is the complement of A P(A B) P(A) + P(B), if the events A and B are mutually exclusive The addition law of probability: P(A B) P(A) + P(B) P(A B) The conditional probability: P(B A) P(A B) P(A) A and B are independent P(A B) P(A) P(B) Stochastic variables p X (k) k x Distribution functions: F X (x 0 ) P(X x 0 ) 0 x0 f X (x) dx k p X (k) k Expected value: E[X] m X x f X (x) dx (X discrete) (X continuous) (X discrete) (X continuous) (k m X ) p X (k) (X discrete) Variance: V[X] E[X ] m X k (x m X ) f X (x) dx (X continuous) Rules for expected value and variance (a and b constants): E[aX + b] ae[x] + b V[aX] a V[X] V[X + b] V[X] E[X + Y] E[X] + E[Y] V[X + Y] V[X] + V[Y] + C[X, Y]

2 Stationary Stochastic Processes, 07 Covariance: C[X, Y] E[(X m X )(Y m Y )] E[XY] m X m Y Correlation coefficient: ρ[x, Y] C[X, Y] V[X] V[Y] Taylor series expansions ("Gauss approximations"): E[g(X,..., X n )] g(e[x ],..., E[X n ]) V[g(X,..., X n )] c i V[X i ] + c i c j C[X i, X j ] i i<j where c i g (x,..., x n ) x i xk E[X k ], k The two-dimensional probability density function of a jointly Gaussian random variable, (X, Y), with E[X] E[Y] 0, V[X] V[Y] and C[X, Y] ρ is { } f X,Y (x, y) π ρ exp ( ρ ) (x ρxy + y ) Trigonometric relations cos(α + β) cos α cos β sin α sin β cos(α β) cos α cos β + sin α sin β sin(α + β) sin α cos β + cos α sin β sin(α β) sin α cos β cos α sin β Stationary stochastic processes Estimation of expected value: ˆm n n t X t V [ ˆm n ] n n V [ ˆm n ] n τ n+ τ (n τ )r X (τ) r X (τ) for large n

3 Stationary Stochastic Processes, 07 3 If ˆm n N(m, V[ ˆm n ]), the confidence interval for m is I m : { ˆm n λ α/ V[ ˆmn ], ˆm n + λ α/ V[ ˆmn ]} with confidence level α. For confidence level 0.95, α 0.05 and λ α/ λ Estimation of covariance function: ˆr n (τ) n τ (X t m X )(X t+τ m X ) for τ 0 n t where m X is replaced by ˆm n if m X is unknown. The Poisson process and the Wiener process A simply increasing process {X(t), t 0} is a homogeneous Poisson process, if X(0) 0 and X(t) has stationary, independent increments. If the intensity is λ, E[X(t)] λt V[X(t)] λt r X (s, t) λ min(s, t) The interarrival times are independent and exponentially distributed with mean value /λ. A Gaussian process {X(t), t 0} is a Wiener process, if X(0) 0, and X(t) has independent increments, where X(t) X(t + h) N(0, σ h), E[X(t)] 0 V[X(t)] σ t r X (s, t) σ min(s, t) Spectral representations Relations between covariance function r X (τ) and spectral density R X (f): Continuous time Discrete time r X (τ) R X(f)e iπfτ df r X (τ) / / R X(f)e iπfτ df R X (f) r X(τ)e iπfτ dτ R X (f) τ r X(τ)e iπfτ

4 4 Stationary Stochastic Processes, 07 Folding (aliasing): Let {Z t, t 0, ±d, ±d,... } be the continuous time process Y(t) sampled with time interval d and sampling frequency f s /d: R Z (f) R Y (f + kf s ) for f s / < f f s / k Sum of harmonic components with random phase and amplitude: X(t) A 0 + A k cos(πf k t + φ k ) where φ k Rect(0, π), A k, k 0,..., n, are independent and E[A 0 ] 0. Covariance function: r X (τ) σ 0 + σ k cos πf k τ k where σ 0 E [A 0] and σ k E [A k ] /. Spectral density: R X (f) b k δ fk (f), k k n where b 0 σ 0 E [A 0], and b k σ k / E [A k ] /4. Linear filters - general theory Impulse response h(u): Y(t) h(u)x(t u) du u h(u)x(t u) (continuous time) (discrete time) Relation between covariance functions: h(u)h(v) r X(τ + u v) du dv r Y (τ) v h(u)h(v) r X(τ + u v) u (continuous time) (discrete time) Relation between spectral densities: R Y (f) H(f) R X (f) where H(f) is the frequency function corresponding to the impulse response h(n).

5 Stationary Stochastic Processes, 07 5 Differentiation: X (t) exists (in quadratic mean) if r X (t) exists. This is equivalent to (πf) R(f)df <. If X (t) exists, the following relations hold: r X (τ) r X(τ) R X (f) (πf) R X (f) V [X (t)] (πf) R X (f) df r X,X (τ) r X(τ) r X (j),x (k)(τ) ( )j r (j+k) X (τ) Integration: [ E ] g(s)x(s) ds g(s)e[x(s)] ds [ C g(s)x(s) ds, ] h(t)y(t) dt g(s)h(t) C[X(s), Y(t)] ds dt Cross-covariance and cross-spectrum: r X,Y (τ) C[X(t), Y(t + τ)] e iπfτ R X,Y (f) df R X,Y (f) H(f)R X (f) A X,Y (f)e iφ X,Y(f) where A X,Y (f) is the amplitude spectrum and Φ X,Y (f) the phase spectrum. The squared coherence spectrum is κ X,Y(f) A X,Y (f) R X (f)r Y (f) AR- MA- and ARMA-models White noise in discrete time: {e t, t 0, ±,...}, E[e t ] 0 and V[e t ] σ : R e (f) σ for / f / AR(p)-process: (a 0 ) X t + a X t + a X t a p X t p e t

6 6 Stationary Stochastic Processes, 07 Yule-Walker equations for covariance function: r X (k) + a r X (k ) a p r X (k p) { σ for k 0 0 for k,,... Spectral density: MA(q)-process: (b 0 ) R X (f) p k0 a ke iπfk σ Covariance function: X t e t + b e t + b e t b q e t q r X (τ) { σ j kτ b jb k for τ q 0 for τ > q Spectral density: q R X (f) b k e iπfk σ k0 Matched filter and Wiener filter Matched filter: with white noise: with colored noise: s(t u) c h(u) c s(t u) SNR N 0 s(t u) du SNR c h(v)r N (u v) dv h(u)h(v)r N (u v) du dv Wiener filter: H(f) SNR R S (f) R S (f) + R N (f) RS (f) df RS (f)r N (f) R S (f)+r N (f) df

7 Stationary Stochastic Processes, 07 7 Spectral estimation Periodogram of the sequence {x(t), t 0,,,... n }, where X (f) n t0 x(t)e iπft. ] E [ˆRx (f) ˆR x (f) X (f) n τ / / k n (τ)r X (τ)e iπfτ K n (f u)r X (u)du where k n (τ) τ for n + τ n and K n n(f) n τ n+ k n(τ)e iπfτ. ] { R V [ˆRx (f) X (f) for 0 < f < / R X (f) for f 0, ±/ The distribution of the periodogram estimate is ˆR x (f) R X (f) χ () for 0 < f < / Modified periodogram ˆR w (f) n x(t)w(t)e iπft n t0 / X (ν)w(f ν)dν n Lag-windowing Averaging of spectrum ˆR lw (f) / τ / / ˆR av (f) K k Ln (τ)ˆr x (τ)e iπfτ K Ln (f ν)ˆr x (ν)dν K ˆR x,j (f) where K different spectrum estimates, ˆR x,j (f), j... K, are used. The distribution is ˆR av (f) R X (f) χ (K) for 0 < f < / K j

8 8 Stationary Stochastic Processes, 07 Fourier transforms g(τ) (α > 0) G(f) e iπfτ g(τ) dτ e α τ α α +(πf) α +τ π α e πα f τ e α τ (α (πf) ) (α +(πf) ) τ k e α τ k! {(α + iπf) k+ + (α iπf) k+ } (α +(πf) ) k+ e ατ π/α exp( (πf) 4α ) e α τ cos(πf 0 τ) e α τ sin(πf 0 τ) { α if τ 0 sin(πατ) πτ if τ 0 { α τ if τ α 0 if τ > α g(τ)h(τ) g(τ) h(τ) g(t)h(τ t)dt g (τ) α α α +(πf 0 + πf) α +(πf 0 +πf) πf 0 πf α +(πf 0 + πf 0+πf πf) α +(πf 0 +πf) { / if f α 0 if f > α { α ( ( if f 0 α (πf) cos πf )) α if f 0 G(f) H(f) G(ν)H(f ν)dν G(f)H(f) iπf G(f) g(ατ) α G( f α ) α g( τ α ) G(αf) g(τ τ 0 ) G(f)e iπfτ 0 g(τ)e iπf 0τ G(f f 0 ) Parseval s theorem: n x(t) t0 / / X(f) df, where X(f) n t0 x(t)e iπft.

9 Stationary Stochastic Processes, 07 9 Gaussian distribution table F(x) Φ(x) x

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Stationary Stochastic Processes, 06 Stationary Stochastic Processes Table of Formulas, 06 Basics of probability theory The following is valid for probabilities: P(Ω), where Ω is all possible outcomes 0