Basic Definitions and The Spectral Estimation Problem Lecture 1 - PDF document

Periodogram Bias Properties Summary of Periodogram Bias Properties: � For “small” N , severe bias � As N ! 1 , W ( ! ) ! � ( ! ) , B ^ � ( ! ) is asymptotically unbiased. so Lecture notes to accompany Introduction to Spectral Analysis Slide L2–10 by P. Stoica and R. Moses, Prentice Hall, 1997

Periodogram Variance N ! 1 As nh i h 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 ( ! ) = y ( t ) e Finite DTFT: N t =1 2 � � i 2 � = = N . ! k and W e Let N 2 � Y ( k ) is the Discrete Fourier Transform (DFT): Then N N N X tk Y ( k ) = y ( t ) W ; k = 0 ; : : : ; N � 1 t =1 N � 1 N f Y ( k ) g f y ( t ) g Direct computation of =0 from t =1 : k 2 O ( N ) flops Lecture notes to accompany Introduction to Spectral Analysis Slide L2–12 by P. Stoica and R. Moses, Prentice Hall, 1997

Radix–2 Fast Fourier Transform (FFT) m N = 2 Assume: N = 2 N X X tk tk Y ( k ) = y ( t ) W + y ( t ) W t =1 t = N = 2+1 N = 2 N k X tk = [ y ( t ) + y ( t + N = 2) W ] W 2 t =1 � +1 ; k for even N k W = with 2 � 1 ; k for odd ~ 2 � i 2 � = N . ~ ~ N = N = 2 and W = W = e Let 4 k = 0 ; 2 ; 4 ; : : : ; N � 2 = 2 p : For ~ N X tp ~ ~ (2 p ) = [ y ( t ) + ( t + )] Y y N W t =1 = 1 ; 3 ; 5 ; 1 = 2 p + 1 : k : : : ; N � For ~ N X t tp ~ ~ (2 p + 1) = f [ y ( t ) ( t + )] W Y � y N g W t =1 ~ N = N = 2 -point DFT computation. Each is a Lecture notes to accompany Introduction to Spectral Analysis Slide L2–13 by P. Stoica and R. Moses, Prentice Hall, 1997

FFT Computation Count k point FFT. c = number of flops for N = 2 Let k Then k 2 c = + 2 c � 1 k k 2 k 2 k = ) c k 2 Thus, 1 = log c N N 2 k 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L2–14 by P. Stoica and R. Moses, Prentice Hall, 1997

Zero Padding Append the given data by zeros prior to computing DFT (or FFT): (1) ; ( N ) ; 0 ; 0 f y : : : ; y : : : g | {z } N Goals: � Apply a radix-2 FFT (so N = power of 2) ^ � Finer sampling of � ( ! ) : � 1 � 1 N N � � � � 2 � 2 � k ! k N N =0 =0 k k ^ φ(ω) continuous curve sampled, N=8 ω Lecture notes to accompany Introduction to Spectral Analysis Slide L2–15 by P. Stoica and R. Moses, Prentice Hall, 1997

Improved Periodogram-Based Methods Lecture 3 Lecture notes to accompany Introduction to Spectral Analysis Slide L3–1 by P. Stoica and R. Moses, Prentice Hall, 1997

Blackman-Tukey Method Basic Idea: Weighted correlogram, with small weight ^ r ( k ) with “large” j k j . applied to covariances M � 1 X � i! k ^ � ( ! ) = w ( k )^ r ( k ) e B T k = � ( M � 1) f w ( k ) g = Lag Window w(k) 1 -M M k Lecture notes to accompany Introduction to Spectral Analysis Slide L3–2 by P. Stoica and R. Moses, Prentice Hall, 1997

Blackman-Tukey Method, con't Z � 1 ^ ^ � ( ! ) = � ( � ) W ( ! � � ) d� p B T 2 � � � W ( ! ) = DTFT f w ( k ) g = Spectral Window ^ � ( ! ) = “locally” smoothed periodogram Conclusion: B T Effect: � Variance decreases substantially � Bias increases slightly M : By proper choice of 2 = var + bias ! 0 as N ! 1 MSE Lecture notes to accompany Introduction to Spectral Analysis Slide L3–3 by P. Stoica and R. Moses, Prentice Hall, 1997

Window Design Considerations Nonnegativeness: Z � 1 ^ ^ � ( ! ) = � ( � ) W ( ! � � ) d� p B T 2 � � � | {z } � 0 W ( ! ) � 0 ( , w ( k ) is a psd sequence) If ^ � ( ! ) � 0 Then: (which is desirable) B T Time-Bandwidth Product � 1 M X w ( k ) k = � ( M � 1) N = = equiv time width e w (0) Z � 1 W ( ! ) d! 2 � � � � = = equiv bandwidth e W (0) N � = 1 e e Lecture notes to accompany Introduction to Spectral Analysis Slide L3–4 by P. Stoica and R. Moses, Prentice Hall, 1997

Window Design, con't � � = 1 = N = 0(1 = M ) e e is the BT resolution threshold. � As M increases, bias decreases and variance increases. ) Choose M as a tradeoff between variance and bias . � Once M is given, N � e (and hence e ) is essentially fixed. ) Choose window shape to compromise between smearing (main lobe width) and leakage (sidelobe level). The energy in the main lobe and in the sidelobes M is given. cannot be reduced simultaneously , once Lecture notes to accompany Introduction to Spectral Analysis Slide L3–5 by P. Stoica and R. Moses, Prentice Hall, 1997

Window Examples M = 25 Triangular Window, 0 −10 −20 dB −30 −40 −50 −60 −3 −2 −1 0 1 2 3 ANGULAR FREQUENCY M = 25 Rectangular Window, 0 −10 −20 dB −30 −40 −50 −60 −3 −2 −1 0 1 2 3 ANGULAR FREQUENCY Lecture notes to accompany Introduction to Spectral Analysis Slide L3–6 by P. Stoica and R. Moses, Prentice Hall, 1997

Bartlett Method Basic Idea: . . . Ν 1 2Μ Μ ^ ^ ^ φ (ω) φ (ω) φ (ω) 1 2 L . . . average ^ φ (ω) B Mathematically: y ( t ) = y ( ( j � 1) M + t ) t = 1 ; : : : ; M j = the j th subsequence 4 ( j = 1 ; : : : ; L = [ N = M ]) � � 2 M � � 1 X � � i! t � ^ � ( ! ) = y ( t ) e � � j j � � M � t =1 � L 1 X ^ ^ � ( ! ) = � ( ! ) B j L j =1 Lecture notes to accompany Introduction to Spectral Analysis Slide L3–7 by P. Stoica and R. Moses, Prentice Hall, 1997

Comparison of Bartlett and Blackman-Tukey Estimates 8 9 L M � 1 > > 1 < = X X � i! k ^ � ( ! ) = ^ r ( k ) e B j L > > j =1 : ; = � ( M � 1) k 8 9 M � 1 L 1 < = X X � i! k = ^ r ( k ) e j L : ; j =1 = � ( M � 1) k M � 1 X � i! k ^ ( k ) e ' r = � ( M � 1) k Thus: ^ ^ � ( ! ) ' � ( ! ) with a rectangular B B T w ( k ) lag window R ^ ( ! ) implicitly uses ( k ) g , the Bartlett � f w Since B R method has � High resolution (little smearing) � Large leakage and relatively large variance Lecture notes to accompany Introduction to Spectral Analysis Slide L3–8 by P. Stoica and R. Moses, Prentice Hall, 1997

Welch Method Similar to Bartlett method, but � allow overlap of subsequences (gives more subsequences, and thus “better” averaging) � use data window for each periodogram; gives mainlobe-sidelobe tradeoff capability 1 2 N . . . subseq subseq subseq #1 #S #2 S = # of subsequences of length M . Let S > [ N = M ] ) “better (Overlapping means averaging”.) Additional flexibility: The data in each subsequence are weighted by a temporal window ^ � ( ! ) with a Welch is approximately equal to B T non-rectangular lag window. Lecture notes to accompany Introduction to Spectral Analysis Slide L3–9 by P. Stoica and R. Moses, Prentice Hall, 1997

Daniell Method N � 1 , By a previous result, for ^ f � ( ! ) g are (nearly) uncorrelated random variables for p j N � 1 � � 2 � = ! j j N j =0 (2 J + 1) samples in the Idea: “Local averaging” of frequency domain should reduce the variance by about (2 J + 1) . k + J 1 X ^ ^ � ( ! ) = � ( ! ) p D j k 2 J + 1 j = k � J Lecture notes to accompany Introduction to Spectral Analysis Slide L3–10 by P. Stoica and R. Moses, Prentice Hall, 1997

Daniell Method, con't J increases: As � Bias increases (more smoothing) � Variance decreases (more averaging) = 2 J 1 , � = N . Then, for N � Let Z � � 1 ^ ^ � ( ! ) ' � ( ! ) d! p D 2 � � � � � ^ ^ ( ! ) ( ! ) with a rectangular spectral � ' � Hence: D B T window. Lecture notes to accompany Introduction to Spectral Analysis Slide L3–11 by P. Stoica and R. Moses, Prentice Hall, 1997

Summary of Periodogram Methods � Unwindowed periodogram – reasonable bias – unacceptable variance � Modified periodograms – Attempt to reduce the variance at the expense of (slightly) increasing the bias. � BT periodogram ^ � ( ! ) by a suitably selected – Local smoothing/averaging of p spectral window. ^ r ( k ) using a lag – Implemented by truncating and weighting ^ � ( ! ) window in c � Bartlett, Welch periodograms ^ � ( ! ) with a suitable lag – Approximate interpretation: B T window (rectangular for Bartlett; more general for Welch). – Implemented by averaging subsample periodograms. � Daniell Periodogram ^ ( ! ) with a rectangular � – Approximate interpretation: B T spectral window. – Implemented by local averaging of periodogram values. Lecture notes to accompany Introduction to Spectral Analysis Slide L3–12 by P. Stoica and R. Moses, Prentice Hall, 1997

Parametric Methods for Rational Spectra Lecture 4 Lecture notes to accompany Introduction to Spectral Analysis Slide L4–1 by P. Stoica and R. Moses, Prentice Hall, 1997

Basic Idea of Parametric Spectral Estimation Observed Data ^ ^ Estimate ^ φ(ω) =φ(ω,θ) Estimate θ parameters PSD Assumed in φ(ω,θ) functional form of φ(ω,θ) possibly revise assumption on φ(ω) Rational Spectra P � i! k � e k j k j� m � ( ! ) = P � i! k � e k j k j� n � i! . � ( ! ) is a rational function in e � ( ! ) can approximate arbitrarily By Weierstrass theorem , m and n are chosen well any continuous PSD , provided sufficiently large. Note, however: � choice of m and n is not simple � some PSDs are not continuous Lecture notes to accompany Introduction to Spectral Analysis Slide L4–2 by P. Stoica and R. Moses, Prentice Hall, 1997

AR, MA, and ARMA Models � ( ! ) can be By Spectral Factorization theorem, a rational factored as � � 2 B ( ! ) � � 2 � ( ! ) = � � � � � A ( ! ) � � � 1 � n A ( z ) = 1 + + + a z � � � a z n 1 � 1 � m B ( z ) = 1 + b z + � � � + b z m 1 A ( ! ) = A ( z ) j and, e.g. , i! z = e Signal Modeling Interpretation: e ( t ) ( t ) y B ( q ) 2 - - � B ( ! ) � A ( q ) 2 2 � ( ! ) = � � ( ! ) = � � � e y � � A ( ! ) � � white noise �ltered white noise A ( q ) y ( t ) = B ( q ) e ( t ) ARMA: A ( q ) y ( t ) = e ( t ) AR: y ( t ) = B ( q ) e ( t ) MA: Lecture notes to accompany Introduction to Spectral Analysis Slide L4–3 by P. Stoica and R. Moses, Prentice Hall, 1997

ARMA Covariance Structure ARMA signal model: n m X X y ( t ) + a y ( t � i ) = b e ( t � j ) ; ( b = 1) i j 0 i =1 =0 j � y ( t � k ) and take E f�g to give: Multiply by n m X X � r ( k ) + a r ( k � i ) = b E f e ( t � j ) y ( t � k ) g i j i =1 j =0 m X 2 � = � b h j j � k =0 j = 0 for k > m 1 X B ( q ) � k H ( q ) = = h q ; ( h = 1) where 0 k A ( q ) =0 k Lecture notes to accompany Introduction to Spectral Analysis Slide L4–4 by P. Stoica and R. Moses, Prentice Hall, 1997

AR Signals: Yule-Walker Equations m = 0 . AR: Writing covariance equation in matrix form for k = 1 : : : n : 2 3 2 3 2 3 2 (0) ( � 1) ( � n ) 1 r r : : : r � . . r (1) r (0) a 6 7 6 7 6 0 7 . 1 6 7 6 7 6 7 = . . . ... 6 . 7 6 . 7 6 . 7 r ( � 1) . . . 4 5 4 5 4 5 r ( n ) : : : r (0) a 0 n " # " # 2 1 � R = � 0 These are the Yule–Walker (YW) Equations . Lecture notes to accompany Introduction to Spectral Analysis Slide L4–5 by P. Stoica and R. Moses, Prentice Hall, 1997

AR Spectral Estimation: YW Method Yule-Walker Method: 2 : r ( k ) by ^ r ( k ) and solve for f ^ a g and � ^ Replace i 2 3 2 3 2 3 2 ^ r (0) ^ r ( � 1) : : : ^ r ( � n ) 1 � ^ . . r ^ (1) ^ r (0) ^ a 6 7 6 7 6 0 7 . 1 6 7 6 7 6 7 = . . . ... 6 . 7 6 . 7 6 . 7 ^ r ( � 1) . . . 4 5 4 5 4 5 r ^ ( n ) : : : r ^ (0) ^ a 0 n Then the PSD estimate is 2 ^ � ^ � ( ! ) = ^ 2 j A ( ! ) j Lecture notes to accompany Introduction to Spectral Analysis Slide L4–6 by P. Stoica and R. Moses, Prentice Hall, 1997

AR Spectral Estimation: LS Method Least Squares Method: n X T e ( t ) = y ( t ) + a y ( t � i ) = y ( t ) + ' ( t ) � i i =1 4 = y ( t ) + ^ y ( t ) T . ' ( t ) = [ y ( t 1) ; ( t n )] � : : : ; y � where T to minimize = [ a ] � : : : a Find n 1 N X 2 f ( � ) = j e ( t ) j t = n +1 � � 1 � ^ � = � ( Y Y ) ( Y y ) where This gives 2 3 2 3 y ( n + 1) y ( n ) y ( n � 1) � � � y (1) ( n + 2) ( n + 1) ( n ) (2) y y y � � � y 6 7 6 7 = = y ; Y . . . 4 . 5 4 . . 5 . . . ( N ) ( N 1) ( N 2) ( N n ) y y � y � � � � y � Lecture notes to accompany Introduction to Spectral Analysis Slide L4–7 by P. Stoica and R. Moses, Prentice Hall, 1997

Levinson–Durbin Algorithm Fast, order-recursive solution to YW equations 2 3 2 3 2 3 2 � � � � � � 1 � 0 � 1 � n n . ... . � � 0 6 . 7 6 7 6 7 1 0 6 7 6 7 6 7 = . ... ... . 6 . 7 6 7 6 7 . � � . . n � 1 4 5 4 5 4 5 � � � � � � 0 n 1 0 | {z } R n +1 = either ( k ) or ^ ( k ) . � r r k Direct Solution: 3 � For one given value of n : O ( n ) flops 4 � For k = 1 ; : : : ; n : O ( n ) flops Levinson–Durbin Algorithm: R Exploits the Toeplitz form of n +1 to obtain the solutions 2 = 1 ; ( n ) flops! k : : : ; n in O for Lecture notes to accompany Introduction to Spectral Analysis Slide L4–8 by P. Stoica and R. Moses, Prentice Hall, 1997

Levinson-Durbin Alg, con't Relevant Properties of R : � � T � R x = y $ R ~ x = ~ y , where x ~ = [ x : : : x ] n 1 � Nested structure 2 3 � 2 3 � � � n n +1 R . n +1 . 6 7 R = ; r ~ = . ~ r 4 5 n n +2 n 4 5 � � � r ~ � � n +1 0 n 1 Thus, 2 3 2 3 2 3 2 3 � 2 � 1 1 � n +1 n R n +1 R � = � = 0 r ~ 4 5 4 5 4 5 4 5 n n n +2 n � 0 0 � � ~ r � n 0 n +1 n � = + ~ � � r � where n n n +1 n Lecture notes to accompany Introduction to Spectral Analysis Slide L4–9 by P. Stoica and R. Moses, Prentice Hall, 1997

Levinson-Durbin Alg, con't 2 3 2 3 2 3 2 3 � 2 1 0 � � n n ~ 0 � = = R 0 ; R � 4 5 4 5 4 5 4 5 n n +2 n +2 n 2 0 � 1 � n n Combining these gives: 8 2 3 2 3 9 2 3 2 3 2 2 � 0 � 1 � + k � n < = n n n +1 ~ R � + k = 0 = � 0 4 5 4 5 4 5 4 5 n n n n +2 2 : ; 0 1 + � k � 0 n n n 2 k = � � =� ) Thus, n n n " # " # ~ � � n n � = + k n n +1 0 1 2 2 � 2 2 � = � + k � = � (1 � j k j ) n n n n n n +1 Computation count: � 2 k flops for the step k ! k + 1 2 flops to determine 2 n ) � n f � ; � g k k k =1 2 O ( n ) times faster than the direct solution. This is Lecture notes to accompany Introduction to Spectral Analysis Slide L4–10 by P. Stoica and R. Moses, Prentice Hall, 1997

MA Signals = 0 n MA: y ( t ) = B ( q ) e ( t ) = e ( t ) + b e ( t � 1) + � � � + b e ( t � m ) m 1 Thus, r ( k ) = 0 for j k j > m and m X 2 2 � i! k � ( ! ) = j B ( ! ) j � = r ( k ) e k = � m Lecture notes to accompany Introduction to Spectral Analysis Slide L4–11 by P. Stoica and R. Moses, Prentice Hall, 1997

MA Spectrum Estimation � ( ! ) : Two main ways to Estimate 2 and insert them in f b g and � 1. Estimate k 2 2 � ( ! ) = j B ( ! ) j � � nonlinear estimation problem ^ � � ( ! ) is guaranteed to be � 0 f ^ r ( k ) g in: 2. Insert sample covariances m X � i! k � ( ! ) = r ( k ) e k = � m ^ ( ! ) with a rectangular lag window of � This is � B T 2 m + 1 . length ^ � � ( ! ) is not guaranteed to be � 0 Both methods are special cases of ARMA methods = 0 . n described below, with AR model order Lecture notes to accompany Introduction to Spectral Analysis Slide L4–12 by P. Stoica and R. Moses, Prentice Hall, 1997

ARMA Signals ARMA models can represent spectra with both peaks (AR part) and valleys (MA part). A ( q ) y ( t ) = B ( q ) e ( t ) 2 � � m P � i! k � e ( ! ) B � � k 2 k = � m � � � ( ! ) = � = � � 2 A ( ! ) j A ( ! ) j � � where � = f [ B ( q ) e ( t )][ B ( q ) e ( t )] � E � k g k � = E f [ A ( q ) y ( t )][ A ( q ) y ( t � k )] g n n X X � = a a r ( k + p � j ) j p j =0 p =0 Lecture notes to accompany Introduction to Spectral Analysis Slide L4–13 by P. Stoica and R. Moses, Prentice Hall, 1997

ARMA Spectrum Estimation Two Methods: 2 � � B ( ! ) 2 2 � ( ! ) = � � f a ; b ; � g in � 1. Estimate i j � � A ( ! ) � nonlinear estimation problem; can use an approximate linear two-stage least squares method ^ � � ( ! ) is guaranteed to be � 0 P m � i! k � e = � m k k ( k ) g in � ( ! ) = f a ; r 2. Estimate i 2 j A ( ! ) j � linear estimation problem (the Modified Yule-Walker method). ^ � � ( ! ) is not guaranteed to be � 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L4–14 by P. Stoica and R. Moses, Prentice Hall, 1997

Two-Stage Least-Squares Method Assumption: The ARMA model is invertible: A ( q ) e ( t ) = y ( t ) ( q ) B = y ( t ) + � y ( t � 1) + � y ( t � 2) + � � � 1 2 = AR ( 1 ) with j � j ! 0 as k ! 1 k K Step 1: Approximate, for some large e ( t ) ( t ) + ( t 1) + + ( t ) ' y � y � � � � � y � K 1 K K f � g 1a) Estimate the coefficients =1 by using AR k k modelling techniques. 1b) Estimate the noise sequence ^ e ( t ) = y ( t ) + � ^ y ( t � 1) + � � � + � ^ y ( t � K ) 1 K and its variance N 1 X 2 2 � ^ = j e ( t ) j ^ N � K t = K +1 Lecture notes to accompany Introduction to Spectral Analysis Slide L4–15 by P. Stoica and R. Moses, Prentice Hall, 1997

Two-Stage Least-Squares Method, con't f e ( t ) g by ^ e ( t ) in the ARMA equation, Step 2: Replace A ( q ) y ( t ) ' B ( q ) ^ e ( t ) f a ; b g by applying least squares and obtain estimates of i j techniques. a b Note that the i and j coefficients enter linearly in the above equation: ( t ) ^ ( t ) [ � y ( t 1) ( t n ) ; y � e ' � : : : � y � ^ e ( t 1) ^ e ( t m )] � � : : : � T � = [ a : : : a b : : : b ] n m 1 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L4–16 by P. Stoica and R. Moses, Prentice Hall, 1997

Modified Yule-Walker Method ARMA Covariance Equation: n X r ( k ) + a r ( k � i ) = 0 ; k > m i i =1 k = m + 1 ; : : : ; m + M In matrix form for 2 3 2 3 ( m ) ( m + 1) ( m + 1) r : : : r � n r " # a 1 ( m + 1) ( m � + 2) ( m + 2) r r n r . 6 7 6 7 . = � . . . . ... 4 . . 5 4 . 5 . . . a ( m + 1) ( m + ) n ( m + ) r M � : : : r � n M r M f r ( k ) g by f ^ r ( k ) g and solve for f a g . Replace i M = n , fast Levinson-type algorithms exist for If f ^ a g . obtaining i M > n overdetermined Y W system of equations; least If ^ f a g . squares solution for i Note: For narrowband ARMA signals, the accuracy of f ^ a g is often better for M > n i Lecture notes to accompany Introduction to Spectral Analysis Slide L4–17 by P. Stoica and R. Moses, Prentice Hall, 1997

Summary of Parametric Methods for Rational Spectra Computational Guarantee ^ � ( ! ) � 0 ? Method Burden Accuracy Use for AR: YW or LS low medium Yes Spectra with (narrow) peaks but no valley MA: BT low low-medium No Broadband spectra possibly with valleys but no peaks ARMA: MYW low-medium medium No Spectra with both peaks and (not too deep) valleys ARMA: 2-Stage LS medium-high medium-high Yes As above Lecture notes to accompany Introduction to Spectral Analysis Slide L4–18 by P. Stoica and R. Moses, Prentice Hall, 1997

Parametric Methods for Line Spectra — Part 1 Lecture 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5–1 by P. Stoica and R. Moses, Prentice Hall, 1997

Line Spectra Many applications have signals with (near) sinusoidal components. Examples: � communications � radar, sonar � geophysical seismology ARMA model is a poor approximation Better approximation by Discrete/Line Spectrum Models � ( ! ) 6 2 (2 � ) 6 � 2 3 (2 � � ) 1 6 2 (2 � � ) 6 2 2 � - ! � � ! ! ! � 1 2 3 An “Ideal” line spectrum Lecture notes to accompany Introduction to Spectral Analysis Slide L5–2 by P. Stoica and R. Moses, Prentice Hall, 1997

Line Spectral Signal Model Signal Model: Sinusoidal components of frequencies 2 f ! g and powers f � g , superimposed in white noise of k k 2 . � power y ( t ) = x ( t ) + e ( t ) t = 1 ; 2 ; : : : n X i ( ! t + � ) x ( t ) = � e k k k | {z } k =1 x ( t ) k Assumptions: � > 0 ! 2 [ � � ; � ] A1: k k (prevents model ambiguities) f ' g = independent rv's, uniformly A2: k [ � � ; � ] distributed on (realistic and mathematically convenient) 2 e ( t ) = circular white noise with variance � A3: 2 � E f e ( t ) e ( s ) g = � � E f e ( t ) e ( s ) g = 0 t;s (can be achieved by “slow” sampling) Lecture notes to accompany Introduction to Spectral Analysis Slide L5–3 by P. Stoica and R. Moses, Prentice Hall, 1997

Covariance Function and PSD Note that: n o � i' i' = 1 , for = � E e p e j p j n o n o n o � i' � i' i' i' = � E e p e j E e p E e j 2 � � Z � 1 � � i' = e d' = 0 , for p 6 = j � � � 2 � � � � Hence, n o 2 � i! k p E x ( t ) x ( t � k ) = � e � p p;j j p � r ( k ) = E f y ( t ) y ( t � k ) g P n 2 i! k 2 = + � e p � � ; 0 p k p =1 and n X 2 2 � ( ! ) = 2 � � � ( ! � ! ) + � p p p =1 Lecture notes to accompany Introduction to Spectral Analysis Slide L5–4 by P. Stoica and R. Moses, Prentice Hall, 1997

Parameter Estimation Estimate either: n 2 � f ! ; � ; ' g ; � (Signal Model) k k k k =1 2 n 2 � f ! ; � g ; � (PSD Model) k k k =1 f ^ g Major Estimation Problem: ! k f ! ^ g are determined: Once k 2 ^ � f � g can be obtained by a least squares method from k n X k + residuals 2 i ! ^ p ^ r ( k ) = � e p p =1 OR: � Both f � ^ g and f ' ^ g can be derived by a least k k squares method from n X t + residuals i ! ^ y ( t ) = � e k k k =1 i' � = � e with k . k k Lecture notes to accompany Introduction to Spectral Analysis Slide L5–5 by P. Stoica and R. Moses, Prentice Hall, 1997

Nonlinear Least Squares (NLS) Method � � 2 N n � � X X � � i ( ! t + ' ) min ( t ) y � � e k k � � k � � f ! ;� ;' g k k k t =1 � � k =1 | {z } ( ! ;�;' ) F Let: i' = � � e k k k T � = [ � : : : � ] n 1 T Y = [ y (1) : : : y ( N )] 2 3 i! i! e 1 � � � e n . . . . = 6 7 B . . 4 5 iN ! iN ! e � � � e n 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L5–6 by P. Stoica and R. Moses, Prentice Hall, 1997

Nonlinear Least Squares (NLS) Method, con't Then: � 2 F = ( Y � B � ) ( Y � B � ) = k Y � B � k � � 1 � � � = [ � � ( B B ) B Y ] [ B B ] � � 1 � [ � � ( B B ) B Y ] � � � � 1 � + Y ( B ) Y � Y B B B Y This gives: � � � 1 � ^ � � = ( B B ) B Y � ! = ! ^ and � 1 � � � ! ^ = a rg max Y B ( B B ) B Y ! Lecture notes to accompany Introduction to Spectral Analysis Slide L5–7 by P. Stoica and R. Moses, Prentice Hall, 1997

NLS Properties Excellent Accuracy: 2 6 � ( ! ^ ) = ( for N � 1) var k 2 3 N � k N = 300 Example: 2 2 = � =� = 30 dB SNR k k q � 5 . ( ^ ) 10 ! � Then var k Difficult Implementation: F is multimodal; it is difficult to The NLS cost function avoid convergence to local minima. Lecture notes to accompany Introduction to Spectral Analysis Slide L5–8 by P. Stoica and R. Moses, Prentice Hall, 1997

Unwindowed Periodogram as an Approximate NLS Method For a single (complex) sinusoid, the maximum of the unwindowed periodogram is the NLS frequency estimate: n = 1 Assume: � B B = N Then: N X � � i! t B Y = y ( t ) e = Y ( ! ) (finite DTFT) t =1 1 � 1 2 � � � Y B ( B B ) B Y = j Y ( ! ) j N ^ = ( ! ) � p = (Unwindowed Periodogram) So, with no approximation , ^ ^ ! = a rg max � ( ! ) p ! Lecture notes to accompany Introduction to Spectral Analysis Slide L5–9 by P. Stoica and R. Moses, Prentice Hall, 1997

Unwindowed Periodogram as an Approximate NLS Method, con't 1 n > Assume: Then: n f ! ^ g ' the locations of the n largest k k =1 ^ � ( ! ) peaks of p provided that inf 2 � j ! � ! j > = N p k which is the periodogram resolution limit. If better resolution desired then use a High/Super Resolution method. Lecture notes to accompany Introduction to Spectral Analysis Slide L5–10 by P. Stoica and R. Moses, Prentice Hall, 1997

High-Order Yule-Walker Method Recall: n X i ( ! t + ' ) y ( t ) = x ( t ) + e ( t ) = � e + e ( t ) k k k | {z } k =1 x ( t ) k ( t ) : y “Degenerate” ARMA equation for i! � 1 (1 � e k q ) x ( t ) k n o i ( ! t + ' ) i! i [ ! ( t � 1)+ ' ] = = 0 � e k k � e k e k k k Let L 4 X � k � ( q ) = 1 + = A ( q ) A ( q ) B b q k k =1 i! � 1 i! � 1 A ( q ) = (1 1 ) (1 n ) � e q � � � � e q � ( q ) = A arbitrary B ( q ) x ( t ) � 0 ) Then B ( q ) y ( t ) = B ( q ) e ( t ) Lecture notes to accompany Introduction to Spectral Analysis Slide L5–11 by P. Stoica and R. Moses, Prentice Hall, 1997

High-Order Yule-Walker Method, con't Estimation Procedure: ^ L � Estimate f b g i =1 using an ARMA MYW technique i n ^ � Roots of B ( q ) give f ! ^ g L � n =1 , along with k k “spurious” roots. Lecture notes to accompany Introduction to Spectral Analysis Slide L5–12 by P. Stoica and R. Moses, Prentice Hall, 1997

High-Order and Overdetermined YW Equations ARMA covariance: L X r ( k ) + b r ( k � i ) = 0 ; k > L i i =1 k = L + 1 ; : : : ; L + M In matrix form for 2 3 2 3 r ( L ) : : : r (1) r ( L + 1) 6 r ( L + 1) : : : r (2) 7 6 r ( L + 2) 7 6 7 6 7 b = � . . . 6 . . 7 6 . 7 . . . 4 5 4 5 ( L + 1) ( M ) ( L + ) r M � : : : r r M | {z } | {z } 4 4 =� = � L > n ) and overdetermined This is a high-order (if M > L ) system of YW equations. (if Lecture notes to accompany Introduction to Spectral Analysis Slide L5–13 by P. Stoica and R. Moses, Prentice Hall, 1997

High-Order and Overdetermined YW Equations, con't (�) = n Fact: rank � � : � = U � V SVD of � � U = ( M � n ) with U U = I n � � � V = ( n � L ) with V V = I n � = ( n n ) , diagonal and nonsingular � � Thus, � ( U � V ) b = � � The Minimum-Norm solution is y � 1 � b = � � � = � V � U � ( L n ) spurious � Important property: The additional ( q ) are located strictly inside the unit circle, if B zeros of b is used. the Minimum-Norm solution Lecture notes to accompany Introduction to Spectral Analysis Slide L5–14 by P. Stoica and R. Moses, Prentice Hall, 1997

HOYW Equations, Practical Solution ^ � = � but made from f ^ r ( k ) g instead of f r ( k ) g . Let ^ ^ ^ U , � , V be defined similarly to U , � , V from the Let ^ � . SVD of � 1 � ^ ^ ^ ^ b = � V � U � ^ Compute n ^ f ! ^ g n zeroes of B ( q ) that Then =1 are found from the k k are closest to the unit circle. When the SNR is low, this approach may give spurious L > n ; this is the price paid for frequency estimates when L > n . increased accuracy when Lecture notes to accompany Introduction to Spectral Analysis Slide L5–15 by P. Stoica and R. Moses, Prentice Hall, 1997

Parametric Methods for Line Spectra — Part 2 Lecture 6 Lecture notes to accompany Introduction to Spectral Analysis Slide L6–1 by P. Stoica and R. Moses, Prentice Hall, 1997

The Covariance Matrix Equation Let: � i! � i ( m � 1) ! T a ( ! ) = [1 e : : : e ] A = [ a ( ! ) : : : a ( ! )] ( m � n ) n 1 ( A ) = n m � n ) Note: rank (for Define 2 3 ( t ) y ( t 1) 4 6 y � 7 6 7 ~ ( t ) = = x ( t ) ~ + ~ ( t ) y A e . 6 . 7 . 4 5 y ( t � m + 1) where T ~ x ( t ) = [ x ( t ) : : : x ( t )] n 1 T e ~ ( t ) = [ e ( t ) : : : e ( t � m + 1)] Then 4 2 � � R = E f y ~ ( t ) ~ y ( t ) g = AP A + � I with 2 3 2 � 0 1 ... � 6 7 P = E f x ( t ) ~ x ~ ( t ) g = 4 5 2 0 � n Lecture notes to accompany Introduction to Spectral Analysis Slide L6–2 by P. Stoica and R. Moses, Prentice Hall, 1997

Eigendecomposition of R and Its Properties � 2 R = AP A + � I ( m > n ) Let: � � � � : : : � � R m : eigenvalues of 1 2 f s ; : : : s g : orthonormal eigenvectors associated n 1 f � ; : : : ; � g with n 1 f g ; : : : ; g g : orthonormal eigenvectors associated m � n 1 f � ; : : : ; � g with m n +1 S = [ s : : : s ] ( m � n ) n 1 G = [ g : : : g ] ( m � ( m � n )) 1 m � n Thus, 2 3 � " # 1 � S ... 6 7 R = [ S G ] � 4 5 G � m Lecture notes to accompany Introduction to Spectral Analysis Slide L6–3 by P. Stoica and R. Moses, Prentice Hall, 1997

Eigendecomposition of R and Its Properties, con't � As rank ( AP ) = A n : 2 = 1 ; � > � k : : : ; n k 2 � = � k = n + 1 ; : : : ; m k 2 3 2 � � � 0 1 � ... 6 7 � = = nonsingular 4 5 2 0 � � � n Note: 2 3 0 � 1 � 2 ... = + = 6 7 R S AP A S � S S 4 5 0 � n � 4 � 1 � S = A ( P A S � ) = AC j C j 6 = 0 (since rank ( S ) = rank ( A ) = n ). with � S G = 0 , Therefore, since � A G = 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L6–4 by P. Stoica and R. Moses, Prentice Hall, 1997

MUSIC Method 2 3 � a ( ! ) 1 . � . = 6 7 = 0 A G G . 4 5 � ( ! ) a n n ) f a ( ! ) g ? R ( G ) k =1 k Thus, n f ! g =1 are the unique solutions of k k � � ( ! ) GG a ( ! ) = 0 . a Let: N 1 X � ^ R = y ~ ( t ) ~ y ( t ) N t = m ^ ^ S ; G = S ; G made from the ^ R eigenvectors of Lecture notes to accompany Introduction to Spectral Analysis Slide L6–5 by P. Stoica and R. Moses, Prentice Hall, 1997

Spectral and Root MUSIC Methods Spectral MUSIC Method: n ^ = the locations of the f ! g n highest peaks of the k =1 k “pseudo-spectrum” function: 1 ; ! 2 [ � � ; � ] � ^ ^ � a ( ! ) G G a ( ! ) Root MUSIC Method: n f ! ^ g = the angular positions of the n roots of: k k =1 T � 1 � ^ ^ a ( z ) G G a ( z ) = 0 that are closest to the unit circle. Here, � 1 � ( m � 1) T a ( z ) = [1 ; ] z ; : : : ; z Note: Both variants of MUSIC may produce spurious frequency estimates. Lecture notes to accompany Introduction to Spectral Analysis Slide L6–6 by P. Stoica and R. Moses, Prentice Hall, 1997

Pisarenko Method m = n + 1 Pisarenko is a special case of MUSIC with (the minimum possible value). = + 1 m n If: ^ G = g ^ Then: 1 , n ) f ^ ! g =1 can be found from the roots of k k T � 1 a ( z ) ^ g = 0 1 � no problem with spurious frequency estimates � computationally simple + 1 � (much) less accurate than MUSIC with m � n Lecture notes to accompany Introduction to Spectral Analysis Slide L6–7 by P. Stoica and R. Moses, Prentice Hall, 1997

Min-Norm Method Goals: Reduce computational burden, and reduce risk of false frequency estimates. m � n (as in MUSIC), but only one vector in Uses R ( G ) (as in Pisarenko). Let " # 1 ^ R ( G ) , with first element equal = the vector in ^ g to one, that has minimum Euclidean norm. Lecture notes to accompany Introduction to Spectral Analysis Slide L6–8 by P. Stoica and R. Moses, Prentice Hall, 1997

Min-Norm Method, con't Spectral Min-Norm n f ! ^ g = the locations of the n highest peaks in the =1 k “pseudo-spectrum” � � 2 " # � 1 � � 1 � ( ! ) � = a � � ^ g � � Root Min-Norm n f ! ^ g = the angular positions of the n roots of the k =1 polynomial " # 1 T � 1 a ( z ) ^ g that are closest to the unit circle. Lecture notes to accompany Introduction to Spectral Analysis Slide L6–9 by P. Stoica and R. Moses, Prentice Hall, 1997

Min-Norm Method: Determining ^ g " # � � g 1 ^ S = Let � S g m � 1 Then: " # " # 1 1 � ^ ^ 2 R ( G ) ) S = 0 g ^ g ^ � � ) S ^ g = � � � 1 � � � � ^ g = � S ( S S ) � Min-Norm solution: � 1 exists iff � � � � ^ ^ � � � � I = S S = �� + S S , ( S S ) As: � 2 = 6 = 1 � � k � k 1 .) N � (This holds, at least, for � gives: Multiplying the above equation by 2 � � � � (1 � k � k ) = ( S S ) � � � 1 2 � � ) ( S S ) � = �= (1 � k � k ) 2 � ^ = �= (1 ) ) g � S � k � k Lecture notes to accompany Introduction to Spectral Analysis Slide L6–10 by P. Stoica and R. Moses, Prentice Hall, 1997

ESPRIT Method A = [ I 0] A Let 1 m � 1 A = [0 I ] A 2 m � 1 = A A D , where Then 2 1 2 3 � i! 0 e 1 ... 6 7 D = 4 5 � i! 0 e n S = [ I 0] S Also, let 1 m � 1 S = [0 I ] S 2 m � 1 S = AC with j C j 6 = 0 . Then Recall � 1 = = = S A C A D C S C D C 2 2 1 1 | {z } � � has the same eigenvalues as D . � is uniquely So determined as � 1 � � � = ( S S ) S S 1 1 1 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L6–11 by P. Stoica and R. Moses, Prentice Hall, 1997

ESPRIT Implementation ^ ^ ^ R , find S , then S From the eigendecomposition of 1 and ^ S 2 . The frequency estimates are found by: n f ^ ! g = � a rg( � ^ ) k k k =1 n f � ^ g where =1 are the eigenvalues of k k � � 1 � ^ ^ ^ ^ ^ � = ( S S ) S S 1 1 1 2 ESPRIT Advantages: � computationally simple � no extraneous frequency estimates (unlike in MUSIC or Min–Norm) � accurate frequency estimates Lecture notes to accompany Introduction to Spectral Analysis Slide L6–12 by P. Stoica and R. Moses, Prentice Hall, 1997

Summary of Frequency Estimation Methods Computational Accuracy / Risk for False Method Burden Resolution Freq Estimates Periodogram small medium-high medium Nonlinear LS very high very high very high Yule-Walker medium high medium Pisarenko small low none MUSIC high high medium Min-Norm medium high small ESPRIT medium very high none Recommendation: � Use Periodogram for medium-resolution applications � Use ESPRIT for high-resolution applications Lecture notes to accompany Introduction to Spectral Analysis Slide L6–13 by P. Stoica and R. Moses, Prentice Hall, 1997

Filter Bank Methods Lecture 7 Lecture notes to accompany Introduction to Spectral Analysis Slide L7–1 by P. Stoica and R. Moses, Prentice Hall, 1997

Basic Ideas Two main PSD estimation approaches: � ( ! ) by a 1. Parametric Approach: Parameterize finite-dimensional model. 2. Nonparametric Approach: Implicitly smooth � f � ( ! ) g � ( ! ) is nearly = � � by assuming that ! constant over the bands [ ! � � � ; ! + � � ] ; � � 1 2 is more general than 1, but 2 requires 1 N � > to ensure that the number of estimated values = 2 � = 2 � � = 1 =� ) is < N . ( N � > 1 leads to the variability / resolution compromise associated with all nonparametric methods. Lecture notes to accompany Introduction to Spectral Analysis Slide L7–2 by P. Stoica and R. Moses, Prentice Hall, 1997

Filter Bank Approach to Spectral Estimation ( ! ) 6 H 1 - ! ! c Filtered P o w er in Bandpass Filter ^ ( t ) ( ! ) y � Division b y Signal P o w er the band c with v arying ! and - - - - c Calculation �lter bandwidth �xed bandwidth 6 � ( ! ) 6 � ( ! ) ~ y y � @ @ @ @ � Q � Q � � � - - ! ! � � ! � � � ! � c c Z Z + � � ! � ) ( a ) 1 =� ( b ) 1 =� ( c ) 2 ^ � ( ! ' j H ( � ) j � ( � ) d� ' � ( � ) d� ' � ( ! ) F B 2 � 2 � � � ! � � � (a) consistent power calculation � (b) Ideal passband filter with bandwidth � ( � ) constant on [ ! 2 � + 2 � ] � 2 � � ; ! � (c) Note that assumptions (a) and (b), as well as (b) and (c), are conflicting. Lecture notes to accompany Introduction to Spectral Analysis Slide L7–3 by P. Stoica and R. Moses, Prentice Hall, 1997

Filter Bank Interpretation of the Periodogram � � 2 N � � 1 4 X � ~ � � i ! t ^ � ( ! ~ ) = y ( t ) e � � p N � � � t =1 � 2 � � N � � 1 X � � i ~ ! ( N � t ) = ( t ) e y � � N � � � t =1 � � � 2 1 � � X � � = N h y ( N � k ) � � k � � � k =0 � where ( 1 i ! ~ k e ; k = 0 ; : : : ; N � 1 N h = k 0 ; otherwise 1 iN ( ! ~ � ! ) 1 1 e � X � i! k H ( ! ) = h e = k i ( ~ ! � ! ) N e � 1 k =0 � center frequency of H ( ! ) = ! ~ � 3dB bandwidth of H ( ! ) ' 1 = N Lecture notes to accompany Introduction to Spectral Analysis Slide L7–4 by P. Stoica and R. Moses, Prentice Hall, 1997

Filter Bank Interpretation of the Periodogram, con't j H ( ! ) j as a function of ( ! ~ � ! ) , for N = 50 . 0 −5 −10 −15 dB −20 −25 −30 −35 −40 −3 −2 −1 0 1 2 3 ANGULAR FREQUENCY ^ � ( ! ) is a filter bank PSD Conclusion: The periodogram p estimator with bandpass filter as given above, and: � narrow filter passband, � power calculation from only 1 sample of filter output. Lecture notes to accompany Introduction to Spectral Analysis Slide L7–5 by P. Stoica and R. Moses, Prentice Hall, 1997

Possible Improvements to the Filter Bank Approach 1. Split the available sample , and bandpass filter each subsample. � more data points for the power calculation stage. This approach leads to Bartlett and Welch methods. 2. Use several bandpass filters on the whole sample . ~ ! . Each filter covers a small band centered on � provides several samples for power calculation. This “multiwindow approach” is similar to the Daniell method. Both approaches compromise bias for variance , and in fact are quite related to each other: splitting the data sample can be interpreted as a special form of windowing or filtering. Lecture notes to accompany Introduction to Spectral Analysis Slide L7–6 by P. Stoica and R. Moses, Prentice Hall, 1997

Capon Method Idea: Data-dependent bandpass filter design. m X y ( t ) = h y ( t � k ) F k k =0 2 3 y ( t ) . . 6 7 = [ h h : : : h ] . m 0 1 4 5 | {z } y ( t � m ) � h | {z } y ~ ( t ) n o 2 � � ( t ) j = = ~ ( t ) ~ ( t ) g E j y h R h; R E f y y F m X � i! k � H ( ! ) = h e = h a ( ! ) k =0 k � i! � im! T a ( ! ) = [1 ; e : : : e ] where Lecture notes to accompany Introduction to Spectral Analysis Slide L7–7 by P. Stoica and R. Moses, Prentice Hall, 1997

Capon Method, con't Capon Filter Design Problem: � � min ( h R h ) h a ( ! ) = 1 subject to h � 1 � � 1 = h R a=a R a Solution: 0 The power at the filter output is: n o 2 � 1 � � E j y ( t ) j = h R h = 1 =a ( ! ) R a ( ! ) F 0 0 y ( t ) in a passband centered which should be the power of ! . on 1 1 ' = The Bandwidth m +1 (filter length) Conclusion Estimate PSD as: + 1 m ^ � ( ! ) = � ^ � 1 a ( ! ) R a ( ! ) with N 1 X � ^ R = y ~ ( t ) y ~ ( t ) N � m t = m +1 Lecture notes to accompany Introduction to Spectral Analysis Slide L7–8 by P. Stoica and R. Moses, Prentice Hall, 1997

Capon Properties � m is the user parameter that controls the compromise between bias and variance: m increases, bias decreases and variance – as increases. � Capon uses one bandpass filter only, but it splits the N -data point sample into ( N � m ) subsequences of m with maximum overlap. length Lecture notes to accompany Introduction to Spectral Analysis Slide L7–9 by P. Stoica and R. Moses, Prentice Hall, 1997

Relation between Capon and Blackman-Tukey Methods ^ � ( ! ) with Bartlett window : Consider B T m m + 1 � j k j X � i! k ^ � ( ! ) = ^ r ( k ) e B T m + 1 = � m k m m 1 X X ( t � s ) � i! = ^ r ( t � s ) e m + 1 t =0 s =0 � ^ ( ! ) a ( ! ) a R ^ = ; R = [^ r ( i � j )] m + 1 Then we have � ^ ( ! ) a ( ! ) a R ^ ( ! ) = � B T m + 1 m + 1 ^ ( ! ) = � C � ^ � 1 ( ! ) a ( ! ) a R Lecture notes to accompany Introduction to Spectral Analysis Slide L7–10 by P. Stoica and R. Moses, Prentice Hall, 1997

Relation between Capon and AR Methods Let 2 ^ � � AR k ^ ( ! ) = k ^ 2 j A ( ! ) j k k th order AR PSD estimate of y ( t ) . be the Then 1 ^ ( ! ) = � C m 1 X � AR ^ 1 = ( ! ) k m + 1 k =0 Consequences: ^ � Due to the average over k , � ( ! ) generally has less C statistical variability than the AR PSD estimator. � Due to the low-order AR terms in the average, ^ � ( ! ) generally has worse resolution and bias C properties than the AR method. Lecture notes to accompany Introduction to Spectral Analysis Slide L7–11 by P. Stoica and R. Moses, Prentice Hall, 1997

Spatial Methods — Part 1 Lecture 8 Lecture notes to accompany Introduction to Spectral Analysis Slide L8–1 by P. Stoica and R. Moses, Prentice Hall, 1997

The Spatial Spectral Estimation Problem Source 2 v B Source 1 B v � Source @ � B n � � , , , � v � � � � @ � , B , � , c � c , � c � @ , N B , c R @ c , c c , � c c , � c c c @ � @ � @ � @ � @ � @ � Sensor m Sensor 1 Sensor 2 n radiating sources by using Problem: Detect and locate m passive sensors. an array of Emitted energy: Acoustic, electromagnetic, mechanical Receiving sensors: Hydrophones, antennas, seismometers Applications: Radar, sonar, communications, seismology, underwater surveillance Basic Approach: Determine energy distribution over space (thus the name “spatial spectral analysis”) Lecture notes to accompany Introduction to Spectral Analysis Slide L8–2 by P. Stoica and R. Moses, Prentice Hall, 1997

Simplifying Assumptions � Far-field sources in the same plane as the array of sensors � Non-dispersive wave propagation Hence: The waves are planar and the only location parameter is direction of arrival (DOA) (or angle of arrival, AOA). � The number of sources n is known. (We do not treat the detection problem) � The sensors are linear dynamic elements with known transfer characteristics and known locations (That is, the array is calibrated .) Lecture notes to accompany Introduction to Spectral Analysis Slide L8–3 by P. Stoica and R. Moses, Prentice Hall, 1997

Array Model — Single Emitter Case x ( t ) = the signal waveform as measured at a reference point ( e.g. , at the “first” sensor) � = the delay between the reference point and the k k th sensor h ( t ) = the impulse response (weighting function) of k k sensor � e ( t ) = k th sensor ( e.g. , thermal noise in “noise” at the k sensor electronics; background noise, etc.) t 2 R (continuous-time signals). Note: k is Then the output of sensor � ( t ) = ( t ) x ( t ) + � ( t ) y h � � � e k k k k � = convolution operator). ( f � g with h ( t ) Basic Problem: Estimate the time delays k k x ( t ) unknown. known but This is a time-delay estimation problem in the unknown input case. Lecture notes to accompany Introduction to Spectral Analysis Slide L8–4 by P. Stoica and R. Moses, Prentice Hall, 1997

Narrowband Assumption Assume: The emitted signals are narrowband with known ! carrier frequency c . x ( t ) = � ( t ) cos[ ! t + ' ( t )] Then: c � ( t ) ; ' ( t ) vary “slowly enough” so that where � ( t ) � ( t ) ; ' ( t ) ' ( t ) � � ' � � ' k k ' to a phase shift ! � Time delay is now k : c x ( t � � ) ' � ( t ) cos[ ! t + ' ( t ) � ! � ] c c k k h ( t ) � x ( t � � ) k k ( ! ) j � ( t ) cos [ ! + ' ( t ) + a rg ( ! ) g ] ' j H t � ! � f H c c c c k k k ( ! ) = ( t ) g is the H F f h k th sensor's transfer where k k function k th sensor output is Hence, the � y ( t ) = j H ( ! ) j � ( t ) c k k � cos[ ! t + ' ( t ) � ! � + a rg H ( ! )] + � e ( t ) c c c k k k Lecture notes to accompany Introduction to Spectral Analysis Slide L8–5 by P. Stoica and R. Moses, Prentice Hall, 1997