Basic Definitions and The Spectral Estimation Problem

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1 Informal Definition of Spectral Estimation Given: A finite record of a signal Basic Definitions and The Spectral Estimation Problem Determine: The distribution of signal power over frequency signal t=1, 2, t spectral density ω ω+ ω π ω Lecture 1 (angular) frequency in radians/(sampling interval) = frequency in cycles/(sampling interval) Lecture notes to accompany Introduction to Spectral Analysis Slide L1 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L1 2

2 Applications Deterministic Signals Temporal Spectral Analysis Vibration monitoring and fault detection Hidden periodicity finding Speech processing and audio devices discrete-time deterministic data sequence Medical diagnosis If: Seismology and ground movement study Control systems design Then: = Radar, Sonar exists and is called the Discrete-Time Fourier Transform (DTFT) Spatial Spectral Analysis Source location using sensor arrays Lecture notes to accompany Introduction to Spectral Analysis Slide L1 3 Lecture notes to accompany Introduction to Spectral Analysis Slide L1 4

3 Energy Spectral Density Random Signals Parseval's Equality: where = Random Signal random signal probabilistic statements about future variations current observation time t Energy Spectral Density Here: We can write But: = Expectation over the ensemble of realizations where = Average power in PSD = (Average) power spectral density Lecture notes to accompany Introduction to Spectral Analysis Slide L1 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L1 6

4 Second Definition of PSD = Note that where is the finite DTFT of Lecture notes to accompany Introduction to Spectral Analysis Slide L1 8 First Definition of PSD = is the autocovariance sequence (ACS) where Note that (Inverse DTFT) Interpretation: so infinitesimal signal power in the band Lecture notes to accompany Introduction to Spectral Analysis Slide L1 7

5 Transfer of PSD Through Linear Systems System Function: unit delay operator: where Then = = = Lecture notes to accompany Introduction to Spectral Analysis Slide L1 10 Properties of the PSD for all P1: Thus, we can restrict attention to P2: is real, P3: If = Then: Otherwise: Lecture notes to accompany Introduction to Spectral Analysis Slide L1 9

6 The Spectral Estimation Problem The Problem: From a sample Find an estimate of Two Main Approaches : Nonparametric: : Periodogram and Correlogram Methods Derived from the PSD definitions Parametric: Lecture 2 Assumes a parameterized functional form of the PSD Lecture notes to accompany Introduction to Spectral Analysis Slide L1 11 Lecture notes to accompany Introduction to Spectral Analysis Slide L2 1

7 Correlogram : Recall 1st definition of : by and replace Truncate the = Lecture notes to accompany Introduction to Spectral Analysis Slide L2 3 Periodogram : Recall 2nd definition of Given : to get and Drop = Natural estimator Used by Schuster ( 1900) to determine hidden periodicities (hence the name) Lecture notes to accompany Introduction to Spectral Analysis Slide L2 2

8 and Relationship Between, is used in If: the biased ACS estimator Then: can be analyzed simultaneously Consequence: Both and Lecture notes to accompany Introduction to Spectral Analysis Slide L2 5 Covariance Estimators (or Sample Covariances) Standard unbiased estimate: Standard biased estimate: For both estimators: Lecture notes to accompany Introduction to Spectral Analysis Slide L2 4

9 Bias Analysis of the Periodogram Bartlett, or triangular, window Thus, Dirac impulse Ideally: Lecture notes to accompany Introduction to Spectral Analysis Slide L2 7 and Statistical Performance of Summary: ) unbiased: Both are asymptotically (for large as Both have large variance, even for large have poor performance and Thus, Intuitive explanation: may be large for large Even if the errors there are so many that when summed in, the PSD error is large are small, Lecture notes to accompany Introduction to Spectral Analysis Slide L2 6

10 Bartlett Window Smearing and Leakage 0, for Main Lobe Width: smearing or smoothing Details in resolvable φ(ω) separated in smearing by less than ^φ(ω) are not <1/Ν ω ω db 30 Thus: Periodogram resolution limit = ANGULAR FREQUENCY Main lobe 3dB width Sidelobe Level: leakage φ(ω) δ(ω) leakage ^ φ(ω) W B(ω) For small, may differ quite a bit from ω ω Lecture notes to accompany Introduction to Spectral Analysis Slide L2 8 Lecture notes to accompany Introduction to Spectral Analysis Slide L2 9

11 Periodogram Bias Properties Periodogram Variance As Summary of Periodogram Bias Properties: Inconsistent estimate For small As so, severe bias,, is asymptotically unbiased 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 10 Lecture notes to accompany Introduction to Spectral Analysis Slide L2 11

12 Radix 2 Fast Fourier Transform (FFT) Assume: for even for odd with and Let : For : For -point DFT computation Each is a Lecture notes to accompany Introduction to Spectral Analysis Slide L2 13 Discrete Fourier Transform (DFT) Finite DTFT: and Let is the Discrete Fourier Transform (DFT): Then from : Direct computation of flops Lecture notes to accompany Introduction to Spectral Analysis Slide L2 12

13 FFT Computation Count Zero Padding Append the given data by zeros prior to computing DFT (or FFT): Let number of flops for point FFT Goals: Then Apply a radix-2 FFT (so power of 2) Finer sampling of : Thus, ^φ(ω) continuous curve sampled, N=8 ω Lecture notes to accompany Introduction to Spectral Analysis Slide L2 14 Lecture notes to accompany Introduction to Spectral Analysis Slide L2 15

14 Blackman-Tukey Method Basic Idea: Weighted correlogram, with small weight applied to covariances with large Improved Periodogram-Based Methods Lecture 3 Lag Window w(k) 1 -M M k Lecture notes to accompany Introduction to Spectral Analysis Slide L3 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 2

15 Blackman-Tukey Method, con't Window Design Considerations Nonnegativeness: DTFT Spectral Window If is a psd sequence) Conclusion: locally smoothed periodogram Then: (which is desirable) Effect: Time-Bandwidth Product Variance decreases substantially Bias increases slightly equiv time width By proper choice of : equiv bandwidth MSE var bias as Lecture notes to accompany Introduction to Spectral Analysis Slide L3 3 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 4

16 Window Design, con't Window Examples is the BT resolution threshold 0 10 Triangular Window, As increases, bias decreases and variance increases Choose as a tradeoff between variance and bias db ANGULAR FREQUENCY Once fixed is given, (and hence ) is essentially Rectangular Window, 0 Choose window shape to compromise between smearing (main lobe width) and leakage (sidelobe level) The energy in the main lobe and in the sidelobes cannot be reduced simultaneously, once is given db ANGULAR FREQUENCY Lecture notes to accompany Introduction to Spectral Analysis Slide L3 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 6

17 Comparison of Bartlett and Blackman-Tukey Estimates Thus: with a rectangular lag window, the Bartlett implicitly uses Since method has High resolution (little smearing) Large leakage and relatively large variance Lecture notes to accompany Introduction to Spectral Analysis Slide L3 8 Bartlett Method Basic Idea: 1 Μ 2Μ Ν φ (ω) ^L φ (ω) ^2 φ (ω) ^1 average φ (ω) ^B Mathematically: th subsequence the Lecture notes to accompany Introduction to Spectral Analysis Slide L3 7

18 Welch Method Daniell 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 Let # of subsequences of length (Overlapping means better averaging ) subseq #S By a previous result, for, are (nearly) uncorrelated random variables for Idea: Local averaging of samples in the frequency domain should reduce the variance by about Additional flexibility: The data in each subsequence are weighted by a temporal window Welch is approximately equal to non-rectangular lag window with a Lecture notes to accompany Introduction to Spectral Analysis Slide L3 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 10

19 Daniell Method, con't Summary of Periodogram Methods Unwindowed periodogram reasonable bias unacceptable variance As Let Hence: window increases: Bias increases (more smoothing) Variance decreases (more averaging) Then, for, with a rectangular spectral Modified periodograms Attempt to reduce the variance at the expense of (slightly) increasing the bias BT periodogram Local smoothing/averaging of spectral window Implemented by truncating and weighting window in Bartlett, Welch periodograms by a suitably selected using a lag Approximate interpretation: with a suitable lag window (rectangular for Bartlett; more general for Welch) Implemented by averaging subsample periodograms Daniell Periodogram Approximate interpretation: spectral window with a rectangular Implemented by local averaging of periodogram values Lecture notes to accompany Introduction to Spectral Analysis Slide L3 11 Lecture notes to accompany Introduction to Spectral Analysis Slide L3 12

20 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) Parametric Methods for Rational Spectra Lecture 4 Rational Spectra is a rational function in By Weierstrass theorem, can approximate arbitrarily well any continuous PSD, provided and are chosen sufficiently large Note, however: choice of and is not simple some PSDs are not continuous Lecture notes to accompany Introduction to Spectral Analysis Slide L4 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 2

21 * ' ARMA Covariance Structure ARMA signal model: 4 to give: and take Multiply by 4 for where Lecture notes to accompany Introduction to Spectral Analysis Slide L4 4 AR, MA, and ARMA Models can be By Spectral Factorization theorem, a rational factored as and, eg, Signal Modeling Interpretation: " -, * $%$%$&$ + (!) (!) $%$%$&$ # -! /0! ARMA: AR: MA: Lecture notes to accompany Introduction to Spectral Analysis Slide L4 3

22 AR Spectral Estimation: YW Method Yule-Walker Method: : and and solve for by Replace Then the PSD estimate is Lecture notes to accompany Introduction to Spectral Analysis Slide L4 6 AR Signals: Yule-Walker Equations AR: Writing covariance equation in matrix form for : These are the Yule Walker (YW) Equations Lecture notes to accompany Introduction to Spectral Analysis Slide L4 5

23 Levinson Durbin Algorithm Fast, order-recursive solution to YW equations or either Direct Solution: flops : For one given value of flops : For to obtain the solutions Levinson Durbin Algorithm: Exploits the Toeplitz form of flops! in for Lecture notes to accompany Introduction to Spectral Analysis Slide L4 8 AR Spectral Estimation: LS Method Least Squares Method: 4 where to minimize Find where This gives Lecture notes to accompany Introduction to Spectral Analysis Slide L4 7

24 Levinson-Durbin Alg, con't Combining these gives: Thus, Computation count: flops for the step flops to determine times faster than the direct solution This is Lecture notes to accompany Introduction to Spectral Analysis Slide L4 10 Levinson-Durbin Alg, con't : Relevant Properties of, where Nested structure Thus, where Lecture notes to accompany Introduction to Spectral Analysis Slide L4 9

25 MA Signals MA Spectrum Estimation Two main ways to Estimate : 1 Estimate and and insert them in MA: nonlinear estimation problem is guaranteed to be Thus, and for 2 Insert sample covariances in: This is length with a rectangular lag window of is not guaranteed to be Both methods are special cases of ARMA methods described below, with AR model order Lecture notes to accompany Introduction to Spectral Analysis Slide L4 11 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 12

26 6 6 ARMA Signals ARMA Spectrum Estimation Two Methods: ARMA models can represent spectra with both peaks (AR part) and valleys (MA part) where 1 Estimate in nonlinear estimation problem; can use an approximate linear two-stage least squares method is guaranteed to be 2 Estimate in linear estimation problem (the Modified Yule-Walker method) is not guaranteed to be Lecture notes to accompany Introduction to Spectral Analysis Slide L4 13 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 14

27 Two-Stage Least-Squares Method Two-Stage Least-Squares Method, con't Assumption: The ARMA model is invertible: AR with as Step 2: Replace by in the ARMA equation, Step 1: Approximate, for some large and obtain estimates of techniques by applying least squares 1a) Estimate the coefficients modelling techniques by using AR Note that the and above equation: coefficients enter linearly in the 1b) Estimate the noise sequence and its variance Lecture notes to accompany Introduction to Spectral Analysis Slide L4 15 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 16

28 Lecture notes to accompany Introduction to Spectral Analysis Slide L4 18 Summary of Parametric Methods for Rational Spectra Computational 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 Modified Yule-Walker Method ARMA Covariance Equation: 4 In matrix form for and solve for by Replace, fast Levinson-type algorithms exist for If obtaining system of equations; least If overdetermined squares solution for Note: For narrowband ARMA signals, the accuracy of is often better for Lecture notes to accompany Introduction to Spectral Analysis Slide L4 17

29 Line Spectra Many applications have signals with (near) sinusoidal components Examples: communications radar, sonar Parametric Methods for Line Spectra Part 1 geophysical seismology ARMA model is a poor approximation Better approximation by Discrete/Line Spectrum Models Lecture 5 An Ideal line spectrum Lecture notes to accompany Introduction to Spectral Analysis Slide L5 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 2

30 Covariance Function and PSD Note that:, for, for Hence, and Lecture notes to accompany Introduction to Spectral Analysis Slide L5 4 Line Spectral Signal Model Signal Model: Sinusoidal components of frequencies and powers, superimposed in white noise of power Assumptions: A1: (prevents model ambiguities) independent rv's, uniformly distributed on (realistic and mathematically convenient) A2: circular white noise with variance A3: (can be achieved by slow sampling) Lecture notes to accompany Introduction to Spectral Analysis Slide L5 3

31 Nonlinear Least Squares (NLS) Method Let: Lecture notes to accompany Introduction to Spectral Analysis Slide L5 6 Parameter Estimation Estimate either: (Signal Model) (PSD Model) Major Estimation Problem: are determined: Once can be obtained by a least squares method from + residuals OR: can be derived by a least Both and squares method from + residuals with Lecture notes to accompany Introduction to Spectral Analysis Slide L5 5

32 Nonlinear Least Squares (NLS) Method, con't NLS Properties Then: Excellent Accuracy: var for Example: SNR db This gives: Then var and Difficult Implementation: The NLS cost function is multimodal; it is difficult to avoid convergence to local minima Lecture notes to accompany Introduction to Spectral Analysis Slide L5 7 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 8

33 Unwindowed Periodogram as an Approximate NLS Method Unwindowed Periodogram as an Approximate NLS Method, con't For a single (complex) sinusoid, the maximum of the unwindowed periodogram is the NLS frequency estimate: Assume: Assume: Then: Then: (finite DTFT) the locations of the peaks of largest provided that (Unwindowed Periodogram) which is the periodogram resolution limit So, with no approximation, If better resolution desired then use a High/Super Resolution method Lecture notes to accompany Introduction to Spectral Analysis Slide L5 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 10

34 High-Order Yule-Walker Method, con't Estimation Procedure: using an ARMA MYW technique Estimate, along with Roots of give spurious roots Lecture notes to accompany Introduction to Spectral Analysis Slide L5 12 High-Order Yule-Walker Method Recall: : Degenerate ARMA equation for Let arbitrary Then Lecture notes to accompany Introduction to Spectral Analysis Slide L5 11

35 4 High-Order and Overdetermined YW Equations High-Order and Overdetermined YW Equations, con't Fact: rank ARMA covariance: SVD of : with In matrix form for with Thus,, diagonal and nonsingular This is a high-order (if ) and overdetermined (if ) system of YW equations The Minimum-Norm solution is Important property: The additional spurious zeros of are located strictly inside the unit circle, if the Minimum-Norm solution is used Lecture notes to accompany Introduction to Spectral Analysis Slide L5 13 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 14

36 HOYW Equations, Practical Solution Let but made from instead of Let, SVD of Compute, be defined similarly to Then are closest to the unit circle are found from the,, zeroes of from the that When the SNR is low, this approach may give spurious frequency estimates when ; this is the price paid for increased accuracy when Parametric Methods for Line Spectra Part 2 Lecture 6 Lecture notes to accompany Introduction to Spectral Analysis Slide L5 15 Lecture notes to accompany Introduction to Spectral Analysis Slide L6 1

37 and Its Properties Eigendecomposition of Let: : eigenvalues of : orthonormal eigenvectors associated with : orthonormal eigenvectors associated with Thus, Lecture notes to accompany Introduction to Spectral Analysis Slide L6 3 The Covariance Matrix Equation Let: ) (for Note: rank Define where Then with Lecture notes to accompany Introduction to Spectral Analysis Slide L6 2

38 MUSIC Method Thus, are the unique solutions of Let: made from the eigenvectors of Lecture notes to accompany Introduction to Spectral Analysis Slide L6 5 and Its Properties, Eigendecomposition of con't : As rank nonsingular Note: ) rank (since rank with Therefore, since, Lecture notes to accompany Introduction to Spectral Analysis Slide L6 4

39 Spectral and Root MUSIC Methods Pisarenko Method Spectral MUSIC Method: the locations of the highest peaks of the pseudo-spectrum function: Pisarenko is a special case of MUSIC with (the minimum possible value) Root MUSIC Method: If: Then:, can be found from the roots of the angular positions of the roots of: no problem with spurious frequency estimates that are closest to the unit circle Here, computationally simple (much) less accurate than MUSIC with Note: Both variants of MUSIC may produce spurious frequency estimates Lecture notes to accompany Introduction to Spectral Analysis Slide L6 6 Lecture notes to accompany Introduction to Spectral Analysis Slide L6 7

40 Min-Norm Method Min-Norm Method, con't Spectral Min-Norm Goals: Reduce computational burden, and reduce risk of false frequency estimates Uses Let (as in MUSIC), but only one vector in (as in Pisarenko) the vector in, with first element equal to one, that has minimum Euclidean norm Root Min-Norm the locations of the pseudo-spectrum the angular positions of the polynomial highest peaks in the roots of the that are closest to the unit circle Lecture notes to accompany Introduction to Spectral Analysis Slide L6 8 Lecture notes to accompany Introduction to Spectral Analysis Slide L6 9

41 ESPRIT Method Let, where Then Also, let Then with Recall is uniquely So has the same eigenvalues as determined as Lecture notes to accompany Introduction to Spectral Analysis Slide L6 11 Min-Norm Method: Determining Let Then: Min-Norm solution: exists iff, As: ) (This holds, at least, for gives: Multiplying the above equation by Lecture notes to accompany Introduction to Spectral Analysis Slide L6 10

42 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 ESPRIT Implementation From the eigendecomposition of, find, then and The frequency estimates are found by: are the eigenvalues of where 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

43 Basic Ideas Two main PSD estimation approaches: 1 Parametric Approach: Parameterize finite-dimensional model by a Filter Bank Methods 2 Nonparametric Approach: Implicitly smooth by assuming that is nearly constant over the bands Lecture 7 2 is more general than 1, but 2 requires to ensure that the number of estimated values ( ) is leads to the variability / resolution compromise associated with all nonparametric methods Lecture notes to accompany Introduction to Spectral Analysis Slide L7 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L7 2

44 Filter Bank Approach to Spectral Estimation! " # $ % & ' ( ) $ * + ', - & ( $ % $, / & 0 ' : : ; ; < 7 : : 3 4 = >?@ A B C D E ( F G H I J K LNM L O P C Q E O C Q E S Q T U ( V G H I J K W L X M L C Q E S Q T U ( Y C D E (a) consistent power calculation (b) Ideal passband filter with bandwidth U C Q E constant on Q Z [ D \ I J U ] D ^ I J U _ 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 Filter Bank Interpretation of the Periodogram where otherwise center frequency of 3dB bandwidth of Lecture notes to accompany Introduction to Spectral Analysis Slide L7 4

45 Filter Bank Interpretation of the Periodogram, con't Possible Improvements to the Filter Bank Approach as a function of, for 1 Split the available sample, and bandpass filter each subsample more data points for the power calculation stage db This approach leads to Bartlett and Welch methods Use several bandpass filters on the whole sample Each filter covers a small band centered on ANGULAR FREQUENCY provides several samples for power calculation Conclusion: The periodogram is a filter bank PSD estimator with bandpass filter as given above, and: narrow filter passband, power calculation from only 1 sample of filter output 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 5 Lecture notes to accompany Introduction to Spectral Analysis Slide L7 6

46 Capon Method, con't Capon Filter Design Problem: subject to Solution: The power at the filter output is: in a passband centered which should be the power of on (filter length) The Bandwidth Conclusion Estimate PSD as: with Lecture notes to accompany Introduction to Spectral Analysis Slide L7 8 Capon Method Idea: Data-dependent bandpass filter design where Lecture notes to accompany Introduction to Spectral Analysis Slide L7 7

47 Capon Properties Relation between Capon and Blackman-Tukey Methods Consider with Bartlett window: is the user parameter that controls the compromise between bias and variance: as increases, bias decreases and variance increases 4 Capon uses one bandpass filter only, but it splits the -data point sample into ( ) subsequences of length with maximum overlap Then we have Lecture notes to accompany Introduction to Spectral Analysis Slide L7 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L7 10

48 Relation between Capon and AR Methods Let AR be the th order AR PSD estimate of Then AR Spatial Methods Part 1 Consequences: Lecture 8 Due to the average over, generally has less statistical variability than the AR PSD estimator Due to the low-order AR terms in the average, generally has worse resolution and bias properties than the AR method Lecture notes to accompany Introduction to Spectral Analysis Slide L7 11 Lecture notes to accompany Introduction to Spectral Analysis Slide L8 1

49 The Spatial Spectral Estimation Problem Simplifying Assumptions Far-field sources in the same plane as the array of sensors Non-dispersive wave propagation Problem: Detect and locate radiating sources by using an array of 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 ) Hence: The waves are planar and the only location parameter is direction of arrival (DOA) (or angle of arrival, AOA) The number of sources the detection problem) is known (We do not treat 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 2 Lecture notes to accompany Introduction to Spectral Analysis Slide L8 3

50 Array Model Single Emitter Case Narrowband Assumption the signal waveform as measured at a reference point (eg, at the first sensor) the delay between the reference point and the th sensor Assume: The emitted signals are narrowband with known carrier frequency Then: the impulse response (weighting function) of sensor where vary slowly enough so that noise at the th sensor (eg, thermal noise in sensor electronics; background noise, etc) Time delay is now to a phase shift : Note: (continuous-time signals) Then the output of sensor ( is convolution operator) Basic Problem: Estimate the time delays known but unknown with where function Hence, the th sensor output is is the th sensor's transfer This is a time-delay estimation problem in the unknown input case Lecture notes to accompany Introduction to Spectral Analysis Slide L8 4 Lecture notes to accompany Introduction to Spectral Analysis Slide L8 5

51 Vector Representation for a Narrowband Source Let the emitter DOA the number of sensors Then and do not depend NOTE: enters via both For omnidirectional sensors the on Lecture notes to accompany Introduction to Spectral Analysis Slide L8 7 Complex Signal Representation The noise-free output has the form: (translate to baseband): Demodulate highpass lowpass to obtain Lowpass filter Hence, by low-pass filtering and sampling the signal 4 ) we get the complex representation: (for or is the complex envelope of where Lecture notes to accompany Introduction to Spectral Analysis Slide L8 6

52 Analog Processing Block Diagram Analog processing for each receiving array element! " # $ % & # ' ( ) * ) +, - & /10 2 / , 5 6 $, 5 6 $ 7 % 8 9 : 9 : ; ( < ; ( < ; = ( < ; = ( < - - > ) - > ) Lecture notes to accompany Introduction to Spectral Analysis Slide L8 8 Multiple Emitter Case Given emitters with received signals: DOAs: Linear sensors Let Then, the array equation is: Use the planar wave assumption to find the dependence of on Lecture notes to accompany Introduction to Spectral Analysis Slide L8 9

53 Uniform Linear Arrays Spatial Frequency Let: signal wavelength By direct analogy with the vector made from uniform samples of a sinusoidal time series, ULA Geometry Sensor #1 = time delay reference The function spatial frequency is one-to-one for Time Delay for sensor : As spatial sampling period where = wave propagation speed is a spatial Shannon sampling theorem Lecture notes to accompany Introduction to Spectral Analysis Slide L8 10 Lecture notes to accompany Introduction to Spectral Analysis Slide L8 11

54 Spatial Filtering Spatial filtering useful for Spatial Methods Part 2 DOA discrimination (similar to frequency discrimination of time-series filtering) Lecture 9 Nonparametric DOA estimation There is a strong analogy between temporal filtering and spatial filtering Lecture notes to accompany Introduction to Spectral Analysis Slide L9 1 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 2

55 Analogy between Temporal and Spatial Filtering Analogy between Temporal and Spatial Filtering Spatial Filter: Temporal FIR Filter: the spatial samples obtained with a sensor array Spatial FIR Filter output: If then filter transfer function Narrowband Wavefront: The array's (noise-free) response to a narrowband ( sinusoidal) wavefront with complex envelope is: We can select to enhance or attenuate signals with different frequencies The corresponding filter output is filter transfer function We can select to enhance or attenuate signals coming from different DOAs Lecture notes to accompany Introduction to Spectral Analysis Slide L9 3 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 4

56 - +, +, + & B A@? a` Z_ Y \ X W V V V V V T V T s T y y y v ~ q } t b c W Z ayt v ~ r } q } t t Œ t wx w w w w w u u u u u u Œ w Œ w v w~ q t š œ t p o W mb b c X Spatial Filtering, con't Example: The response magnitude of a spatial filter (or beamformer) for a 10-element ULA Here,, where Theta (deg) Magnitude Lecture notes to accompany Introduction to Spectral Analysis Slide L9 6 Analogy between Temporal and Spatial Filtering FHG F, #%$! " P M S PQOR MON L IJK F : #! " <2=3=3=4=4=3=4=5=4=4=3=4=3=3=4> ' ( & ' ), ( * * & - - / ;, ( + 9 ' : + #!ED C (a) Temporal filter i g h f de bc [ ] ^ Z[ XY W Y TT U V V V V V V y y{z t 8 8 ~ } `al W a ay ³ ²± ª«Ÿ ˆ H ƒ Ž Ž ˆ H Š ƒ Ÿ ž Xnl m ^X jk l (b) Spatial filter Lecture notes to accompany Introduction to Spectral Analysis Slide L9 5

57 Spatial Filtering Uses Nonparametric Spatial Methods A Filter Bank Approach to DOA estimation Basic Ideas Spatial Filters can be used Design a filter such that for each 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 ) It passes undistorted the signal with DOA = It attenuates all DOAs Sweep the filter through the DOA range of interest, and evaluate the powers of the filtered signals: with The (dominant) peaks of DOAs of the sources give the Lecture notes to accompany Introduction to Spectral Analysis Slide L9 7 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 8

58 Beamforming Method Implementation of Beamforming Assume the array output is spatially white: Then: The beamforming DOA estimates are: Hence: In direct analogy with the temporally white assumption for filter bank methods, can be considered as impinging on the array from all DOAs Filter Design: Solution: subject to the locations of the largest peaks of This is the direct spatial analog of the Blackman-Tukey periodogram Resolution Threshold: wavelength array length array beamwidth Inconsistency problem: Beamforming DOA estimates are consistent if inconsistent if, but Lecture notes to accompany Introduction to Spectral Analysis Slide L9 9 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 10

59 Capon Method Parametric Methods Filter design: Assumptions: subject to The array is described by the equation: Solution: The noise is spatially white and has the same power in all sensors: Implementation: The signal covariance matrix the locations of the largest peaks of is nonsingular Performance: Slightly superior to Beamforming Then: 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 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

60 Nonlinear Least Squares Method Nonlinear Least Squares Method Minimizing over gives Properties of NLS: Then tr Performance: high Computational complexity: high Main drawback: need for multidimensional search Thus, tr For, this is precisely the form of the NLS method of frequency estimation Lecture notes to accompany Introduction to Spectral Analysis Slide L9 13 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 14

61 YW-MUSIC DOA Estimator largest peaks of the where, is the array transfer vector for, at DOA, using is defined similarly to Properties: Computational complexity: medium Performance: satisfactory if Main advantages: weak assumption on need not be calibrated the subarray Lecture notes to accompany Introduction to Spectral Analysis Slide L9 16 Yule-Walker Method Assume: Then: Also assume: rank rank, and the SVD of is Then: rank Properties: Lecture notes to accompany Introduction to Spectral Analysis Slide L9 15

62 MUSIC and Min-Norm Methods ESPRIT Method Assumption: The array is made from two identical subarrays separated by a known displacement vector Both MUSIC and Min-Norm methods for frequency estimation apply with only minor modifications to the DOA estimation problem Spectral forms of MUSIC and Min-Norm can be used for arbitrary arrays Let # sensors in each subarray (transfer matrix of subarray 1) (transfer matrix of subarray 2) Root forms can be used only with ULAs MUSIC and Min-Norm break down if the source signals are coherent; that is, if rank rank Modifications that apply in the coherent case exist Then, where the time delay from subarray 1 to subarray 2 for a signal with DOA = : where is the subarray separation and is measured from the perpendicular to the subarray displacement vector Lecture notes to accompany Introduction to Spectral Analysis Slide L9 17 Lecture notes to accompany Introduction to Spectral Analysis Slide L9 18

63 ESPRIT Method, con't ESPRIT Scenario source θ subarray 1 known displacement vector subarray 2 Properties: Requires special array geometry Computationally efficient No risk of spurious DOA estimates Does not require array calibration Note: For a ULA, the two subarrays are often the first and last array elements, so and Lecture notes to accompany Introduction to Spectral Analysis Slide L9 19

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