The Averaged Periodogram The averaged periodogram is an improved periodogram method,also called Welch's method. How to improve? .()-I(@) varlPl-var[P/wl
The Averaged Periodogram • The averaged periodogram is an improved periodogram method, also called Welch’s method. How to improve? ( ) ( ) 1 1 av L per peri i P S L ω ω Λ Λ = = ∑
Each segment length M segment number L 50%overlapping segments Each segment length M 2N/(L 1) L=2N/M-1
50% overlapping segments
The Averaged Periodogram Compared with periodogram method: (1)The variance of the averaged periodogram method is decreased by 1/L; varIP()]-Ivar[P,()] (2)The frequency resolution is poorer; (3)By padding zero,it can increase the observation points of frequency,but no frequency resolution increases. (4)Adding window to reduce the frequency leakage,Hamming window
The Averaged Periodogram Compared with periodogram method: (1) The variance of the averaged periodogram method is decreased by 1/L; (2) The frequency resolution is poorer; (3) By padding zero, it can increase the observation points of frequency, but no frequency resolution increases. (4) Adding window to reduce the frequency leakage, Hamming window
Blackman-Tukey Spectral Estimation BT Method (Indirect method): To improve the performance of the periodogram method. Sr(f)=∑w[k]r(k)exp(-j2π)
Blackman-Tukey Spectral Estimation BT Method (Indirect method): To improve the performance of the periodogram method. ( ) [ ] ( ) ( ) ( ) 1 BT 1 exp 2 N k N S f w k r k j fk π Λ Λ − =− − = − ∑
Blackman-Tukey method BT Method: St()=∑w[k]F(k)exp(-j2π) k=-(N-1) 1.0≤w[k≤w[0=1,k<M,M≤N-1 2.w(k)=0,k>M 3.w(-k)=w(k)
Blackman-Tukey method BT Method: ( ) [ ] ( ) ( ) ( ) 1 BT 1 exp 2 N k N S f w k r k j fk π Λ Λ − =− − = − ∑ [ ] [ ] ( ) ( ) ( ) 1. 0 0 1, , 1 2. 0, 3. w k w k M M N w k k M w k w k ≤ ≤ = < ≤ − = > − =