Periodogram and Correlogram Methods. Lecture 2

Transcription

1 Periodogram and Correlogram Methods Lecture 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L2 1

2 Periodogram Recall 2nd definition of (!): (!) = lim N!1 E 8 >< >: 1 N NX t=1 y(t)e ;i!t 2 9 >= > Given : fy(t)g N t=1 Drop lim and E fg to get N!1 ^ p (!) = 1 N NX t=1 y(t)e ;i!t 2 Natural estimator Used by Schuster (1900) to determine hidden periodicities (hence the name). Lecture notes to accompany Introduction to Spectral Analysis Slide L2 2

3 Correlogram Recall 1st definition of (!): (!) = 1X k=;1 r(k)e ;i!k Truncate the P and replace r(k) by ^r(k) : ^ c (!) = N;1 X k=;(n;1) ^r(k)e ;i!k Lecture notes to accompany Introduction to Spectral Analysis Slide L2 3

4 Covariance Estimators (or Sample Covariances) Standard unbiased estimate: ^r(k) = 1 N ; k NX t=k+1 y(t)y (t ; k) k 0 Standard biased estimate: ^r(k) = 1 N NX t=k+1 y(t)y (t ; k) k 0 For both estimators: ^r(k) = ^r (;k) k < 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L2 4

5 Relationship Between ^ p (!) and ^ c (!) If: the biased ACS estimator ^r(k) is used in ^ c (!), Then: ^ p (!) = 1 N = NX t=1 N;1 X k=;(n;1) y(t)e ;i!t 2 ^r(k)e ;i!k = ^ c (!) ^ p (!) = ^ c (!) Consequence: Both ^ p (!) and ^ c (!) can be analyzed simultaneously. Lecture notes to accompany Introduction to Spectral Analysis Slide L2 5

6 Statistical Performance of ^ p (!) and ^ c (!) Summary: Both are asymptotically (for large N) unbiased: E n^ p (!)o! (!) as N! 1 Both have large variance, even for large N. Thus, ^ p (!) and ^ c (!) have poor performance. Intuitive explanation: ^r(k) ; r(k) may be large for large jkj Even if the errors f^r(k) ; r(k)g N;1 are small, jkj=0 there are so many that when summed in [^ p (!) ; (!)], the PSD error is large. Lecture notes to accompany Introduction to Spectral Analysis Slide L2 6

7 Bias Analysis of the Periodogram E n^ o p (!) = E n^ o c (!) = = w B (k) = N;1 X k=;(n;1) 1X k=;1 8 < : 0 = N;1 X k=;(n;1) 1 ; jkj N! w B (k)r(k)e ;i!k 1 ; jkj N E f^r(k)g e ;i!k r(k)e ;i!k jkj N ; 1 jkj N = Bartlett, or triangular, window Thus, o E n^ p (!) = 1 2 Z ; ()W B(! ; ) d Ideally: W B (!) = Dirac impulse (!). Lecture notes to accompany Introduction to Spectral Analysis Slide L2 7

8 Bartlett Window W B (!) " sin(!n=2) # 2 W B (!) = 1 N sin(!=2) 0 W B (!)=W B (0), forn = db ANGULAR FREQUENCY Main lobe 3dB width 1=N. For small N, W B (!) may differ quite a bit from (!). Lecture notes to accompany Introduction to Spectral Analysis Slide L2 8

9 Smearing and Leakage Main Lobe Width: smearing or smoothing Details in (!) separated in f by less than 1=N are not resolvable. φ(ω) smearing ^φ(ω) <1/Ν ω ω Thus: Periodogram resolution limit = 1=N. Sidelobe Level: leakage φ(ω) δ(ω) leakage ^ φ(ω) W (ω) B ω ω Lecture notes to accompany Introduction to Spectral Analysis Slide L2 9

10 Periodogram Bias Properties Summary of Periodogram Bias Properties: For small N,severebias As N! 1, W B (!)! (!), so ^(!) is asymptotically unbiased. Lecture notes to accompany Introduction to Spectral Analysis Slide L2 10

11 Periodogram Variance As N! 1 E nh^ io p (! 1 ) ; (! 1 )ih^ p (! 2 ) ; (! 2 ) = ( 2 (! 1 )! 1 =! 2 0! 1 6=! 2 Inconsistent estimate Erratic behavior ^φ(ω) ^ φ(ω) asymptotic mean = φ(ω) ω +- 1 st. dev = φ(ω) too Resolvability properties depend on both bias and variance. Lecture notes to accompany Introduction to Spectral Analysis Slide L2 11

12 Improved Periodogram-Based Methods Lecture 3 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 1

13 Blackman-Tukey Method Basic Idea: Weighted correlogram, with small weight applied to covariances ^r(k) with large jkj. ^ BT (!) = M;1 X k=;(m;1) w(k)^r(k)e ;i!k fw(k)g = Lag Window w(k) 1 -M M k Lecture notes to accompany Introduction to Spectral Analysis Slide L3 2

14 Blackman-Tukey Method, con't ^ BT (!) = 1 2 Z ; ^ p ()W (! ; )d W (!) = DTFTfw(k)g = Spectral Window Conclusion: ^ BT (!) = locally smoothed periodogram Effect: Variance decreases substantially Bias increases slightly By proper choice of M: MSE = var + bias 2! 0 as N! 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 3

15 Window Design Considerations Nonnegativeness: ^ BT (!) = 1 2 Z ; ^ p () {z } 0 W (! ; )d If W (!) 0 (, w(k) is a psd sequence) Then: ^ BT (!) 0 (which is desirable) Time-Bandwidth Product N e = e = M;1 X k=;(m;1) 1 2 Z w(0) w(k) ; W (!)d! W (0) = equiv time width = equiv bandwidth N e e = 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 4

16 Window Design, con't e = 1=N e = 0(1=M) 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). Theenergyinthemainlobeandinthesidelobes cannot be reduced simultaneously,oncem is given. Lecture notes to accompany Introduction to Spectral Analysis Slide L3 5

17 Window Examples 0 Triangular Window, M = db ANGULAR FREQUENCY Rectangular Window, M = db ANGULAR FREQUENCY Lecture notes to accompany Introduction to Spectral Analysis Slide L3 6

18 Bartlett Method Basic Idea: 1 Μ 2Μ... Ν φ (ω) ^1 φ (ω) ^2... average φ (ω) ^L φ (ω) ^B Mathematically: y j (t) = y((j ; 1)M + t) t = 1 ::: M = the jth subsequence (j = 1 ::: L 4 = [N=M]) ^ j (!) = 1 M MX 2 y j (t)e ;i!t t=1 ^ B (!) = 1 L LX j=1 ^ j (!) Lecture notes to accompany Introduction to Spectral Analysis Slide L3 7

19 Thus: Comparison of Bartlett and Blackman-Tukey Estimates ^ B (!) = 1 L = ' LX j=1 M;1 X 8 >< >: k=;(m;1) M;1 X k=;(m;1) M;1 X k=;(m;1) 8 < : 1 L LX j=1 ^r(k)e ;i!k 9 > = ^r j (k)e ;i!k > 9 = ^r j (k) ^ B (!) ' ^ BT (!) with a rectangular lag window w R (k) e;i!k Since ^ B (!) implicitly uses fw R (k)g, the Bartlett method has High resolution (little smearing) Large leakage and relatively large variance Lecture notes to accompany Introduction to Spectral Analysis Slide L3 8

20 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 #1 subseq #2... subseq #S Let S =# of subsequences of length M. (Overlapping means S > [N=M] ) better averaging.) Additional flexibility: The data in each subsequence are weighted by a temporal window Welch is approximately equal to ^ BT (!) with a non-rectangular lag window. Lecture notes to accompany Introduction to Spectral Analysis Slide L3 9

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

22 Daniell Method, con't As J increases: Bias increases (more smoothing) Variance decreases (more averaging) Let = 2J=N. Then, for N 1, ^ D (!) ' 1 2 Z ; ^ p (!)d! Hence: ^ D (!) ' ^ BT (!) with a rectangular spectral window. Lecture notes to accompany Introduction to Spectral Analysis Slide L3 11

23 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 Local smoothing/averaging of ^ p (!) by a suitably selected spectral window. Implemented by truncating and weighting ^r(k) using a lag window in ^ c (!) Bartlett, Welch periodograms Approximate interpretation: ^ BT (!) with a suitable lag window (rectangular for Bartlett; more general for Welch). Implemented by averaging subsample periodograms. Daniell Periodogram Approximate interpretation: ^ BT (!) with a rectangular spectral window. Implemented by local averaging of periodogram values. Lecture notes to accompany Introduction to Spectral Analysis Slide L3 12

24 Parametric Methods for Rational Spectra Lecture 4 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 1

25 Basic Idea of Parametric Spectral Estimation Observed Data Assumed functional form of φ(ω,θ) Estimate parameters in φ(ω,θ) ^θ Estimate PSD ^φ(ω) =φ(ω,θ) ^ possibly revise assumption on φ(ω) Rational Spectra (!) = Pjkjm ke ;i!k Pjkjn ke ;i!k (!) is a rational function in e ;i!. By Weierstrass theorem, (!) can approximate arbitrarily well any continuous PSD, providedm and n are chosen 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

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

27 ARMA Covariance Structure ARMA signal model: y(t)+ nx i=1 a i y(t;i) = mx j=0 b j e(t;j) (b 0 = 1) Multiply by y (t ; k) and take E fg to give: r(k) + nx i=1 a i r(k ; i) = where H(q) = B(q) A(q) = mx j=0 = 2 m X b j E fe(t ; j)y (t ; k)g j=0 b j h j;k = 0 for k > m 1X k=0 h k q ;k (h 0 = 1) Lecture notes to accompany Introduction to Spectral Analysis Slide L4 4

28 AR Signals: Yule-Walker Equations AR: m = 0. Writing covariance equation in matrix form for k = 1 ::: n: r(0) r(;1) ::: r(;n) r(1) r(0) r(;1) 5 r(n) ::: r(0) " # " 1 2 R = 0 # 1 a 1. a n = These are the Yule Walker (YW) Equations. Lecture notes to accompany Introduction to Spectral Analysis Slide L4 5

29 MA Signals MA: n = 0 y(t) = B(q)e(t) = e(t) + b 1 e(t ; 1) + + b m e(t ; m) Thus, r(k) = 0 for jkj > m and (!) = jb(!)j 2 2 = mx k=;m r(k)e ;i!k Lecture notes to accompany Introduction to Spectral Analysis Slide L4 11

30 MA Spectrum Estimation Two main ways to Estimate (!): 1. Estimate fb k g and 2 and insert them in (!) = jb(!)j 2 2 nonlinear estimation problem ^(!) is guaranteed to be 0 2. Insert sample covariances f^r(k)g in: (!) = mx k=;m r(k)e ;i!k This is ^ BT (!) with a rectangular lag window of length 2m + 1. ^(!) is not guaranteed to be 0 Both methods are special cases of ARMA methods described below, with AR model order n = 0. Lecture notes to accompany Introduction to Spectral Analysis Slide L4 12

31 ARMA Signals ARMA models can represent spectra with both peaks (AR part) and valleys (MA part). where A(q)y(t) = B(q)e(t) 2 P mk=;m k e ;i!k = A(!) ja(!)j 2 (!) = 2 B(!) k = E f[b(q)e(t)][b(q)e(t ; k)] g = E f[a(q)y(t)][a(q)y(t ; k)] g = nx nx j=0 p=0 a j a p r(k + p ; j) Lecture notes to accompany Introduction to Spectral Analysis Slide L4 13

32 ARMA Spectrum Estimation Two Methods: 1. Estimate fa i b j 2 g in (!) = 2 B(!) 2 A(!) nonlinear estimation problem; can use an approximate linear two-stage least squares method ^(!) is guaranteed to be 0 2. Estimate fa i r(k)g in (!) = P m k=;m ke ;i!k ja(!)j 2 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

33 Modified Yule-Walker Method ARMA Covariance Equation: r(k) + nx i=1 a i r(k ; i) = 0 In matrix form for k = m + 1 ::: m+ M r(m) ::: r(m ; n +1) r(m +1) r(m ; n +2)..... r(m + M ; 1) ::: r(m ; n + M) " a1 #. a n k > m = ; r(m +1) r(m +2). r(m + M) Replace fr(k)g by f^r(k)g and solve for fa i g. If M = n, fast Levinson-type algorithms exist for obtaining f^a i g. If M > n overdetermined Y W system of equations; least squares solution for f^a i g. Note: For narrowband ARMA signals, the accuracy of f^a i g is often better for M > n Lecture notes to accompany Introduction to Spectral Analysis Slide L4 17

34 Summary of Parametric Methods for Rational Spectra Computational ^(!) 0 Guarantee 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

35 Parametric Methods for Line Spectra Part 1 Lecture 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 1

36 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 6 (!) (2 2 1 ) (2 2 2 ) (2 2 3 ) ;! 1! 2! 3 An Ideal line spectrum -! Lecture notes to accompany Introduction to Spectral Analysis Slide L5 2

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

38 Covariance Function and PSD Note that: E E n o e i' p e ;i' j n = e i' pe ;i' j 1 2 Z o = 1, forp = j = E n ; ei' d' e i' p 2 o E n o e ;i' j = 0, forp 6= j Hence, E nx p (t)x j (t ; k) o = 2 p ei! pk p j r(k) = E fy(t)y (t ; k)g = P n p=1 2 pe i! pk + 2 k 0 and (!) = 2 nx p=1 2 p(! ;! p ) + 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 4

39 Parameter Estimation Estimate either: f! k k ' k g n k=1 2 (Signal Model) f! k 2 k gn k=1 2 (PSD Model) Major Estimation Problem: f^! k g Once f^! k g are determined: f^ 2 kg can be obtained by a least squares method from ^r(k) = OR: nx p=1 2 pe i^! pk + residuals Both f^ k g and f^' k g can be derived by a least squares method from y(t) = nx k=1 with k = k e i' k. k e i^! kt + residuals Lecture notes to accompany Introduction to Spectral Analysis Slide L5 5

40 Nonlinear Least Squares (NLS) Method Let: min f! k k ' k g NX nx 2 t=1 y(t) ; k e i(! kt+' k ) k=1 {z } k = k e i' k F (! ') = [ 1 ::: n ] T Y = [y(1) :::y(n)] T B = e i! 1. e i! n. e in! 1 e in! n Lecture notes to accompany Introduction to Spectral Analysis Slide L5 6

41 Nonlinear Least Squares (NLS) Method, con't Then: F = (Y ; B) (Y ; B) = ky ; Bk 2 = [ ; (B B) ;1 B Y ] [B B] [ ; (B B) ;1 B Y ] +Y Y ; Y B(B B) ;1 B Y This gives: ^ = (B B) ;1 B Y!=^! and ^! = arg max! Y B(B B) ;1 B Y Lecture notes to accompany Introduction to Spectral Analysis Slide L5 7

42 NLS Properties Excellent Accuracy: 62 var (^! k ) = N 3 2 (for N 1) k Example: N = 300 SNR k = 2 k =2 = 30 db Then q var(^! k ) 10 ;5. Difficult Implementation: The NLS cost function F is multimodal; it is difficult to avoid convergence to local minima. Lecture notes to accompany Introduction to Spectral Analysis Slide L5 8

43 Parametric Methods for Line Spectra Part 2 Lecture 6 Lecture notes to accompany Introduction to Spectral Analysis Slide L6 1

44 The Covariance Matrix Equation Let: a(!) = [1 e ;i! ::: e ;i(m;1)! ] T A = [a(! 1 ) ::: a(! n )] (m n) Note: rank(a) = n (for m n ) Define where Then ~y(t) 4 = y(t) y(t ; 1). y(t ; m + 1) = A~x(t) + ~e(t) ~x(t) = [x 1 (t) :::x n (t)] T ~e(t) = [e(t) :::e(t ; m + 1)] T with R 4 = E f~y(t)~y (t)g = AP A + 2 I P = E f~x(t)~x (t)g = n Lecture notes to accompany Introduction to Spectral Analysis Slide L6 2

45 Eigendecomposition of R and Its Properties Let: R = AP A + 2 I (m > n) 1 2 ::: m : eigenvalues of R fs 1 :::s n g: orthonormal eigenvectors associated with f 1 ::: n g fg 1 ::: g m;n g: orthonormal eigenvectors associated with f n+1 ::: m g S = [s 1 :::s n ] (m n) G = [g 1 :::g m;n ] (m (m ; n)) Thus, R = [S G] m 3 " # 7 S 5 G Lecture notes to accompany Introduction to Spectral Analysis Slide L6 3

46 Eigendecomposition of R and Its Properties, con't As rank(ap A ) = n: = 2 6 k > 2 k = 2 k = 1 ::: n k = n + 1 ::: m 4 1 ; n ; = nonsingular Note: RS = AP A S + 2 S = S n S = A(PA S ;1 ) 4 = AC with jcj 6= 0 (since rank(s) = rank(a) = n). Therefore, since S G = 0, A G = 0 Lecture notes to accompany Introduction to Spectral Analysis Slide L6 4

47 MUSIC Method A G = a (! 1 ). a (! n ) G = 0 ) fa(! k )g n k=1? R(G) Thus, f! k g n k=1 are the unique solutions of a (!)GG a(!) = 0. Let: ^R = 1 N NX t=m ~y(t)~y (t) ^S ^G = S Gmade from the eigenvectors of ^R Lecture notes to accompany Introduction to Spectral Analysis Slide L6 5

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

49 Spatial Methods Part 1 Lecture 8 Lecture notes to accompany Introduction to Spectral Analysis Slide L8 1

50 The Spatial Spectral Estimation Problem Source vsource 2 B BB B BN ccc cc cc cc c c v Source ; ; ; Sensor ; ; Sensor m Sensor 2 Problem: Detect and locate n radiating sources by using an array of m passive sensors. 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

51 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

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

53 Narrowband Assumption Assume: The emitted signals are narrowband with known carrier frequency! c. Then: x(t) = (t) cos[! c t + '(t)] where (t) '(t) vary slowly enough so that (t ; k ) ' (t) '(t ; k ) ' '(t) Time delay is now ' to a phase shift! c k : x(t ; k ) ' (t)cos[! c t + '(t) ;! c k ] h k (t) x(t ; k ) ' jh k (! c )j(t) cos[! c t + '(t) ;! c k +argfh k (! c )g] where H k (!) = Ffh k (t)g is the kth sensor's transfer function Hence, the kth sensor output is y k (t) = jh k (! c )j(t) cos[! c t + '(t) ;! c k + arg H k (! c )] + e k (t) Lecture notes to accompany Introduction to Spectral Analysis Slide L8 5

54 Complex Signal Representation The noise-free output has the form: z(t) = (t)cos[! c t + (t)] = n e i[! c t+ (t)] + e ;i[! c t+ (t)] o = (t) 2 Demodulate z(t) (translate to baseband): 2z(t)e ;! ct = (t)f e i (t) {z } lowpass + e ;i[2! ct+ (t)] {z } highpass Lowpass filter 2z(t)e ;i! ct to obtain (t)e i (t) g Hence,bylow-passfiltering and sampling the signal ~y k (t)=2 = y k (t)e ;i! ct = y k (t)cos(! c t) ; iy k (t) sin(! c t) we get the complex representation: (for t 2 Z) or y k (t) = (t) e {z i'(t) } s(t) jh k (! c )j e i arg[h k (! c)] {z } H k (! c ) y k (t) = s(t)h k (! c ) e ;i! c k + e k (t) where s(t) is the complex envelope of x(t). e ;i! c k + e k (t) Lecture notes to accompany Introduction to Spectral Analysis Slide L8 6

55 Vector Representation for a Narrowband Source Let Then y(t) = = the emitter DOA m = the number of sensors a() = y 1(t). y m (t) 4 H 1(! c ) e ;i! c 1. H m (! c ) e ;i! c m e(t) = 2 6 y(t) = a()s(t) + e(t) e 1(t). e m (t) NOTE: enters a() via both f k g and fh k (! c )g. For omnidirectional sensors the fh k (! c )g do not depend on. Lecture notes to accompany Introduction to Spectral Analysis Slide L8 7

56 Given n emitters with Multiple Emitter Case received signals: fs k (t)g n k=1 DOAs: k Linear sensors ) y(t) = a( 1 )s 1 (t) + + a( n )s n (t) + e(t) Let A = [a( 1 ) :::a( n )] (m n) s(t) = [s 1 (t) :::s n (t)] T (n 1) Then, the array equation is: y(t) = As(t) + e(t) Use the planar wave assumption to find the dependence of k on. Lecture notes to accompany Introduction to Spectral Analysis Slide L8 9

57 Uniform Linear Arrays Source h J JJ J J^ JJ J JJ - J+ JJ J] sin J J^d J JJ ] J v? 1 v JJ v v v p p p m J JJ d - ULA Geometry Sensor Sensor #1 = time delay reference Time Delay for sensor k: k = (k ; 1) d sin c where c = wave propagation speed Lecture notes to accompany Introduction to Spectral Analysis Slide L8 10

58 Spatial Frequency Let:! s 4 =!c d sin c = 2 d sin c=f c = c=f c = signal wavelength a() = [1 e ;i! s ::: e ;i(m;1)! s ] T = 2 d sin By direct analogy with the vector a(!) made from uniform samples of a sinusoidal time series,! s = spatial frequency The function! s 7! a() is one-to-one for As j! s j $ dj sin j =2 1 d =2 d = spatial sampling period d =2 is a spatial Shannon sampling theorem. Lecture notes to accompany Introduction to Spectral Analysis Slide L8 11

59 Spatial Filtering Spatial filtering useful for DOA discrimination (similar to frequency discrimination of time-series filtering) Nonparametric DOA estimation There is a strong analogy between temporal filtering and spatial filtering. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 2

60 Analogy between Temporal and Spatial Filtering u(t) =e i!t z 1 m ; 1 s? q ;1 s? q? q ; e ;i!. e ;i(m;1)! 1 {z } a(!) u(t) C - - h 0 h 1 q h - P - [h a(!)]u(t) (Temporal sampling) (a) Temporal filter narrowband source with DOA= z Z ZZ Z~ a! -! a 1 reference point 0 1 a!! a a!! a q q 2 - m - 1 e ;i! 2(). e ;i!m() C A {z } a() s(t) h 0? h 1? q @@R - P [h a()]s(t) - (Spatial sampling) (b) Spatial filter Lecture notes to accompany Introduction to Spectral Analysis Slide L9 5

61 Spatial Filtering, con't Example: The response magnitude jh a()j of a spatial filter (or beamformer) for a 10-element ULA. Here, h = a( 0 ),where 0 = Theta (deg) Magnitude Lecture notes to accompany Introduction to Spectral Analysis Slide L9 6

62 Spatial Filtering Uses Spatial Filters can be used To pass the signal of interest only, hence filtering out interferences located outside the filter's beam (but possibly having the same temporal characteristics as the signal). To locate an emitter in the field of view, by sweeping the filter through the DOA range of interest ( goniometer ). Lecture notes to accompany Introduction to Spectral Analysis Slide L9 7

63 Nonparametric Spatial Methods A Filter Bank Approach to DOA estimation. Basic Ideas Design a filter h() such that for each It passes undistorted the signal with DOA = It attenuates all DOAs 6= Sweep the filter through the DOA range of interest, and evaluate the powers of the filtered signals: E n jy F (t)j 2o = with R = E fy(t)y (t)g. E n jh ()y(t)j 2o = h ()Rh() The (dominant) peaks of h ()Rh() give the DOAs of the sources. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 8

64 Beamforming Method Assume the array output is spatially white: R = E fy(t)y (t)g = I Then: E n jyf (t)j 2o = h h Hence: In direct analogy with the temporally white assumption for filter bank methods, y(t) can be considered as impinging on the array from all DOAs. Filter Design: min h (h h) subject to h a() = 1 Solution: h = a()=a ()a() = a()=m E n jyf (t)j 2o = a ()Ra()=m 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 9

65 Implementation of Beamforming ^R = 1 N NX t=1 y(t)y (t) The beamforming DOA estimates are: f^ k g = the locations of the n largest peaks of a () ^Ra(). This is the direct spatial analog of the Blackman-Tukey periodogram. Resolution Threshold: inf j k ; p j > wavelength array length = array beamwidth Inconsistency problem: Beamforming DOA estimates are consistent if n = 1, but inconsistent if n > 1. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 10

66 Capon Method Filter design: min h (h Rh) subject to h a() = 1 Solution: h = R ;1 a()=a ()R ;1 a() E n jy F (t)j 2o = 1=a ()R ;1 a() Implementation: f^ k g = the locations of the n largest peaks of 1=a () ^R ;1 a(): Performance: Slightly superior to Beamforming. Both Beamforming and Capon are nonparametric approaches. They do not make assumptions on the covariance properties of the data (and hence do not depend on them). Lecture notes to accompany Introduction to Spectral Analysis Slide L9 11

67 Parametric Methods Assumptions: The array is described by the equation: y(t) = As(t) + e(t) The noise is spatially white and has the same power in all sensors: E fe(t)e (t)g = 2 I The signal covariance matrix is nonsingular. P = E fs(t)s (t)g Then: R = E fy(t)y (t)g = AP A + 2 I Thus: The NLS, YW, MUSIC, MIN-NORM and ESPRIT methods of frequency estimation can be used, almost without modification, for DOA estimation. Lecture notes to accompany Introduction to Spectral Analysis Slide L9 12