Properties of the PSD P1:(w)=(w+2T)for all w. Thus,we can restrict attention to w∈[-π，π]→f∈[-1/2,1/2] P2:p(w)≥0 P3:If y(t)is real, Then:(w)=o(-w) Otherwise:(w)(-w) 2020-01-1 16
2020-01-18 16 Properties of the PSD
Two Main Approaches o The Problem From a sample sequence Find an estimate of PSD o Nonparametric approaches Derived from the PSD definitions o Parametric approaches ● Assumes a parameterized functional form of the PSD 2020-01-18 17
2020-01-18 17 Two Main Approaches The Problem From a sample sequence Find an estimate of PSD Nonparametric approaches Derived from the PSD definitions Parametric approaches Assumes a parameterized functional form of the PSD
S2.Periodogram and correlogram o Periodogram o Correlogram o Statistical Performance Bias variance o From Finite DTFT to DFT,FFT 2020-01-18 18
2020-01-18 18 S2. Periodogram and correlogram Periodogram Correlogram Statistical Performance Bias variance From Finite DTFT to DFT, FFT
(1)Periodogram o From the second definition Drop“im”and“E{-}”oget N+o∞ 12 B(w) 1-N ∑y(t)e-iwt t=1 o Natural estimator o Used by Schuster (1900)to determine “hidden periodicities”(hence the name). 2020-01-18 19
2020-01-18 19 (1) Periodogram Natural estimator Used by Schuster (1900) to determine “hidden periodicities” (hence the name). From the second definition
(2)Correlogram o From the first definition Truncate the“∑”and replace“r(k)”by“r(k)”: N-1 币c(w)= ∑(k)eiak k=-(N-1) 2020-01-18 20
2020-01-18 20 (2) Correlogram From the first definition