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Basic Definitions and the Spectral Estimation Problem Basic Definitions and The Spectral Estimation Problem Lecture 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L1–1 by P. Stoica and R. Moses, Prentice Hall, 1997 Informal Definition of Spectral Estimation Given: A finite record of a signal. Determine: The distribution of signal power over frequency. signal spectral density t ω ω ω+∆ω π t=1, 2, ... = ! (angular) frequency in radians/(sampling interval) = !=2 f =frequencyincycles/(samplinginterval) Lecture notes to accompany Introduction to Spectral Analysis Slide L1–2 by P. Stoica and R. Moses, Prentice Hall, 1997 Applications Temporal Spectral Analysis Vibration monitoring and fault detection Hidden periodicity finding Speech processing and audio devices Medical diagnosis Seismology and ground movement study Control systems design Radar, Sonar Spatial Spectral Analysis Source location using sensor arrays Lecture notes to accompany Introduction to Spectral Analysis Slide L1–3 by P. Stoica and R. Moses, Prentice Hall, 1997 Deterministic Signals 1 y (t)g = f discrete-time deterministic data t=1 sequence 1 X 2 y (t)j < 1 If: j t=1 1 X i! t Y (! ) (t)e Then: = y t=1 exists and is called the Discrete-Time Fourier Transform (DTFT) Lecture notes to accompany Introduction to Spectral Analysis Slide L1–4 by P. Stoica and R. Moses, Prentice Hall, 1997 Energy Spectral Density Parseval's Equality: Z 1 X 1 2 jy (t)j (! )d! = S 2 t=1 where 4 2 S (! ) = jY (! )j = Energy Spectral Density We can write 1 X i! k (k )e S (! ) = k =1 where 1 X (k ) = y (t)y (t k ) t=1 Lecture notes to accompany Introduction to Spectral Analysis Slide L1–5 by P. Stoica and R. Moses, Prentice Hall, 1997 Random Signals Random Signal random signal probabilistic statements about future variations t current observation time 1 X 2 y (t)j = 1 Here: j t=1 n o 2 jy (t)j < 1 But: E fg E = Expectation over the ensemble of realizations n o 2 jy (t)j y (t) E =Averagepowerin PSD = (Average) power spectral density Lecture notes to accompany Introduction to Spectral Analysis Slide L1–6 by P. Stoica and R. Moses, Prentice Hall, 1997 First Definition of PSD 1 X i! k (! ) (k )e = r k =1 (k ) where r is the autocovariance sequence (ACS) r (k ) = E fy (t)y (t k )g r (k ) = r (k ); r (0) jr (k )j Note that Z 1 i! k (! )e d! (k ) = r (Inverse DTFT) 2 Interpretation: Z n o 1 2 r (0) = E jy (t)j = (! )d! 2 so (! )d! = infinitesimal signal power in the band d! ! 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L1–7 by P. Stoica and R. Moses, Prentice Hall, 1997 Second Definition of PSD 8 9 2 > > N < = X 1 i! t (! ) lim E (t)e = y > > N !1 N : ; t=1 Note that 1 2 jY (! )j (! ) = lim E N N !1 N where N X i! t Y (! ) = y (t)e N t=1 y (t)g is the finite DTFT of f . Lecture notes to accompany Introduction to Spectral Analysis Slide L1–8 by P. Stoica and R. Moses, Prentice Hall, 1997 Properties of the PSD (! ) = (! + 2 ) ! P1: for all . Thus, we can restrict attention to ! 2 [; ] () f 2 [1=2; 1=2] (! ) 0 P2: (t) P3: If y is real, (! ) (! ) Then: = (! ) 6= (! ) Otherwise: Lecture notes to accompany Introduction to Spectral Analysis Slide L1–9 by P. Stoica and R. Moses, Prentice Hall, 1997 Transfer of PSD Through Linear Systems 1 X k h q (q ) = System Function: H k k =0 1 1 = q y (t) = y (t 1) where q unit delay operator: e(t) y (t) - - H (q ) 2 (! ) (! ) = jH (! )j (! ) e y e Then 1 X y (t) e(t k ) = h k k =0 1 X i! k h e (! ) H = k k =0 2 (! ) jH (! )j (! ) e y = Lecture notes to accompany Introduction to Spectral Analysis Slide L1–10 by P. Stoica and R. Moses, Prentice Hall, 1997 The Spectral Estimation Problem The Problem: y (1); :::;y(N )g From a sample f ^ (! ) f(! ); ! 2 [; ]g Find an estimate of : Two Main Approaches : Nonparametric: – Derived from the PSD definitions. Parametric: – Assumes a parameterized functional form of the PSD Lecture notes to accompany Introduction to Spectral Analysis Slide L1–11 by P. Stoica and R. Moses, Prentice Hall, 1997 Periodogram and Correlogram Methods Lecture 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L2–1 by P. Stoica and R. Moses, Prentice Hall, 1997 Periodogram (! ) Recall 2nd definition of : 8 9 2 > > N < = X 1 i! t (! ) = lim E y (t)e > > N !1 N : ; t=1 N y (t)g Given : f t=1 E fg Drop “ lim ” and “ ” to get N !1 2 N X 1 i! t ^ (! ) (t)e = y p N t=1 Natural estimator Used by Schuster ( 1900) to determine “hidden periodicities” (hence the name). Lecture notes to accompany Introduction to Spectral Analysis Slide L2–2 by P. Stoica and R. Moses, Prentice Hall, 1997 Correlogram (! ) Recall 1st definition of : 1 X i! k (! ) = r (k )e k =1 P (k ) ^r (k ) Truncate the “ ” and replace “r ” by “ ”: N 1 X i! k ^ ^r (k )e (! ) = c k =(N 1) Lecture notes to accompany Introduction to Spectral Analysis Slide L2–3 by P. Stoica and R. Moses, Prentice Hall, 1997 Covariance Estimators (or Sample Covariances) Standard unbiased estimate: N X 1 y (t)y (t k ); k 0 ^r (k ) = N k t=k +1 Standard biased estimate: N X 1 r^(k ) = y (t)y (t k ); k 0 N t=k +1 For both estimators: ^r (k ) = ^r (k ); k < 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L2–4 by P. Stoica and R. Moses, Prentice Hall, 1997 ^ ^ (! ) (! ) Relationship Between and p c ^ ^r (k ) (! ) If: the biased ACS estimator is used in c , Then: 2 N X 1 i! t ^ y (t)e (! ) = p N t=1 N 1 X i! k ^r (k )e = k =(N 1) ^ = (! ) c ^ ^ (! ) = (! ) p c Consequence: ^ ^ (! ) (! ) Both and can be analyzed simultaneously. p c Lecture notes to accompany Introduction to Spectral Analysis Slide L2–5 by P. Stoica and R. Moses, Prentice Hall, 1997 ^ ^ (! ) (! ) Statistical Performance of and p c Summary: N Both are asymptotically (for large ) unbiased: n o ^ E N ! 1 (! ) ! (! ) as p N Both have “large” variance, even for large . ^ ^ (! ) (! ) Thus, and have poor performance. p c Intuitive explanation: r^(k ) r (k ) jk j may be large for large N 1 ^r (k ) r (k )g Even if the errors f are small, jk j=0 there are “so many” that when summed in ^ (! ) (! )] [ , the PSD error is large. p Lecture notes to accompany Introduction to Spectral Analysis Slide L2–6 by P. Stoica and R. Moses, Prentice Hall, 1997 Bias Analysis of the Periodogram N 1 n o n o X i! k ^ ^ E (! ) = E (! ) = E f^r (k )g e p c k =(N 1) ! N 1 X jk j i! k r (k )e 1 = N k =(N 1) 1 X i! k = w (k )r (k )e B k =1 8 < jk j 1 ; jk j N 1 N w (k ) = B : 0; jk j N = Bartlett, or triangular, window Thus, Z o n 1 ^ ( )W (! ) d (! ) = E B p 2 (! ) = (! ) Ideally: W Dirac impulse . B Lecture notes to accompany Introduction to Spectral Analysis Slide L2–7 by P. Stoica and R. Moses, Prentice Hall, 1997 (! ) Bartlett Window W B " # 2 1 sin(!N=2) W (! ) = B N sin(!=2) W (! )=W (0) = 25 ,forN B B 0 −10 −20 −30 dB −40 −50 −60 −3 −2 −1 0 1 2 3 ANGULAR FREQUENCY 1=N Main lobe 3dB width . W (! ) (! ) For “small” N , may differ quite a bit from . B Lecture notes to accompany Introduction to Spectral Analysis Slide L2–8 by P. Stoica and R. Moses, Prentice Hall, 1997 Smearing and Leakage Main Lobe Width: smearing or smoothing (! ) f 1=N Details in separated in by less than are not resolvable. φ(ω) ^φ(ω) smearing <1/Ν ω ω =N Thus: Periodogram resolution limit = 1 . Sidelobe Level: leakage φ(ω)≅δ(ω) ^φ(ω)≅ (ω) WB leakage ω ω Lecture notes to accompany Introduction to Spectral Analysis Slide L2–9 by P. Stoica and R. Moses, Prentice Hall, 1997 Periodogram Bias Properties Summary of Periodogram Bias Properties: N For “small” ,severebias N ! 1 W (! ) ! (! ) As , , B ^ (! ) so is asymptotically unbiased. Lecture notes to accompany Introduction to Spectral Analysis Slide L2–10 by P. Stoica and R. Moses, Prentice Hall, 1997 Periodogram Variance ! 1 As N nh ih io ^ ^ E (! ) (! ) (! ) (! ) 1 1 2 2 p p ( 2 (! ); ! = ! 1 1 2 = 0; ! 6= ! 1 2 Inconsistent estimate Erratic behavior ^φ(ω) ^φ(ω) asymptotic mean = φ(ω) -+ 1 st. dev = φ(ω) too ω Resolvability properties depend on both bias and variance. Lecture notes to accompany Introduction to Spectral Analysis Slide L2–11 by P. Stoica and R. Moses, Prentice Hall, 1997 Discrete Fourier Transform (DFT) N X i! t y (t)e (! ) = Finite DTFT: Y N t=1 2 i 2 N k = W = e Let ! and . N 2 k ) ( Then Y is the Discrete Fourier Transform (DFT): N N N X tk y (t)W ; k = 0;:::;N 1 Y (k ) = t=1 N 1 N Y (k )g fy (t)g Direct computation of f from : t=1 k =0 2 (N ) O flops Lecture notes to accompany Introduction to Spectral Analysis Slide L2–12 by P.
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