Signalbehandling Formelsamling

5 KorrelationKorrelation, korskorrelation, spektrum, korspektrum och koherens mellan in- ochutsignal.Tidskontinuerlig process:Tidsdiskret process:y(t) = h(t) ∗ x(t)Y (F ) = H(F ) · X(F )r yy (τ) = r hh (τ) ∗ r xx (τ)R yy (F ) = |H(F )| 2 R xx (F )r yx (τ) = h(τ) ∗ r xx (τ)R yx (F ) = H(F ) · R xx (F )r xx (τ) = ∫ t x(t)x(t − τ)dtr yx (τ) = ∫ t y(t)x(t − τ)dtγ xx (τ) = E{x(t)x(t − τ)}γ yx (τ) = E{y(t)x(t − τ)}y(n) = h(n) ∗ x(n)Y (f) = H(f) · X(f)r yy (n) = r hh (n) ∗ r xx (n)R yy (f) = |H(f)| 2 · R xx (f)r yx (n) = h(n) ∗ r xx (n)R yx (f) = H(f) · R xx (f)r xx (n) = ∑ l x(l)x(l − n)r yx (n) = ∑ l y(l)x(l − n)γ xx (n) = E{x(l)x(l − n)}γ yx (n) = E{y(l)x(l − n)}Normalfördelade stok.var. X i ∈ N(m i ,σ i )E{X 1 X 2 X 3 X 4 } = E{X 1 X 2 } E{X 3 X 4 } + E{X 1 X 3 } E{X 2 X 4 } ++ E{X 1 X 4 } E{X 2 X 3 }−2m 1 m 2 m 3 m 436