Sample Covariances Standard unbiased estimate: 1 N (k)= , ∑y(t)g*(t-k),k≥0 =k+1 Standard biased estimate: ()= 1 ∑(t)g*(t-k),k≥0 t=k+1 2020-01-18 21
2020-01-18 21 Sample Covariances
Relationship between two estimations o If:the biased ACS estimator is used o Then: bn(w）= HZ e tr t=1 N-1 三 r(k)e-iwk k=-(N-1) =Φc(w) 2020-01-18 22
2020-01-18 22 Relationship between two estimations If: the biased ACS estimator is used Then:
(3)Statistical Performance Both are asymptotically (for large N)unbiased: E{⊙p(w)}→(w)asN→∞ ●Both have"large”variance,even for large N. o Thus,Both have poor performance. 2020-01-18 23
2020-01-18 23 (3) Statistical Performance Thus, Both have poor performance
Intuitive explanation (k)-r(k)may be large for large ·Even ifheo(F(k)-r(k)}☆二a are small. there are "so many"that when summed in [p(w)-(w)],the PSD error is large. 2020-01-18 24
2020-01-18 24 Intuitive explanation
Bias Analysis N-1 E{⊙(u)}=E{(ω)}= Etr(k)e-iwk k=-(N-1) r(k)e-iwk 三 ∑wB(k)r(k)e-iwk k=-0 wp(k) ={-别 |k≤N-1 Ik≥N 202 Bartlett,or triangular,window 25
2020-01-18 25 Bias Analysis