1、Gabor Order Tracking SAE Noise & Vibration Conference May 1, 2001A Comparison of the newly Proposed Gabor Order TrackingTechnique vs. Other Order Tracking MethodsMichael F. AlbrightGeneral ManagerSignal.X Technologies LLCShie QianNational Instruments CorporationRoush Industries Inc., Roush Anatrol D
2、ivisionSAE Paper 2001-01-1471Gabor Order Tracking SAE Noise & Vibration Conference May 1, 2001What is an Order An Orderis the periodic response to periodic forcing modified by path dynamics Orderis expressed in multiples of the underlying periodic process such as shaft rotation speed, gear mesh, fir
3、ing frequency, etc. Traditionally, Order magnitude is shown against a coarse RPM or Time axisGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001What is Order Tracking A method to extract the instantaneous frequency and amplitude modulation of harmonic components from a complex signal M
4、ethods are: Frequency domain, Power spectra against a coarse RPM axis Time Domain, Vold-Kalman Time/Frequency domain, Gabor Order Tracking (GOT)Gabor Order Tracking SAE Noise & Vibration Conference May 1, 2001The Discrete Gabor Expansion Represents a signal as a weighted summation of time shifted an
5、d frequency modulated elementary functions (Gabor Atoms)Some Gabor AtomsGabor SamplingLatticeNnkjNnnmMmeMmkhCkS/21,1)()(=Gabor CoefficientsGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001The Gabor Transform Also known as the Short Time Fourier Transform (STFT) or equivalently, the i
6、nverse Gabor Expansion1 The relationship between the elementary function and what is known as its Dual, is the key to understanding the Gabor Transform and its inverse, the Gabor expansion=10/2*,)()(LkNnkjnmeMmkkSCNnkjNnnmMmeMmkhCkS/21,1)()(=)(k)(kh1 First published in 1980 by B. BastiaansGabor Orde
7、r Tracking SAE Noise & Vibration Conference May 1, 2001The Dual Function Relationship Orthogonal Transformations A signal is expressed in terms of its similarity to a pre-selected family of orthogonal basis functions Therefore , reconstruction is in terms of the identical family of basis functions E
8、xamples : Fourier Transform, Wavelet Transform =212121211001CCCCSS|=212121211001SSSSCCSimple 2D matrix analogyGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001The Dual Function RelationshipNon-Orthogonal Transformations Projection (Analysis) and re-construction (Synthesis) are with r
9、espect to non-orthogonal basis A simple matrix inverse links Analysis and Synthesis functions Dual function relationship implies that and can be reversed( )0,11=( )2/3,2/12=( )13/3,1 =( )3/32,02=22C 11C SS()+=8/31C2/2C)2/2,2/2(S)B)A21)A)B)C)D( )2/3,2/12=( )0,11=SS( )3/32,02=( )13/3,1 =( )()=23/32C6/
10、332C)2/2,2/2(S)D)C2122C 11C 1C2C2C1CSSSSSCC=1 implies over-sampling in time/frequency Analysis and synthesis operations are with respect to over-determined function sets CCCCST=|321321 SSSSSC =321321,SCoptTT=1+=+=82/313C8/312C,2/21C)2/2,2/2(S)B)AA)B)C)D)0,1(1= 5.0,2/331C2C3CS= 2/3,25.02)0,5.0(1=35.0
11、,4/3 =33C 22C 11C 123S1C2C3C12311C 22C 33C S=+=8/3223C8/32122C,4/21C)D)C= 2/3,5.02SSolution is byPseudo-inverseGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001 Orthogonal-Like Transformation Special cases L Can be rank deficient, undersampling L can be an Identity matrix, Critical S
12、ampling L can be diagonal, Tight frameThe Dual Function Relationship +=TT1 LT= TTL =1 TTL =Dual FunctionOperator=+=4/1323C4/3122C,2/21C)2/2,2/2(S)B)AA)B)C)D)1,0(1=( )5.,2/33=1C2C3CS23/1,3/3 =1)3/2,0( =33/1,3/3 =33C 22C 11C 123S1C2C3C12311C 33C S=+=6/1323C6/1322C,3/21C)D)C= 5.,2/32S22C Gabor Order Tr
13、acking SAE Noise & Vibration Conference May 1, 2001The Wexler-Raz Identity Consider the Problem Statement are the Gabor Coefficients and are Bi-Orthogonal Wexler-Raz Identity2solves for given (or vice versa)nmnmnmhSkS,)(= NnkjnmeMmk/2,)(=NnkjnmeMmkhh/2,)(=)()()()(10*/2qpNMkeqNkhLkMpkj=+ =HnmS,nm,nmh
14、,)(kh )(kOr in matrix form2J. Wexler S. Raz published in 1990Gabor Order Tracking SAE Noise & Vibration Conference May 1, 2001The Wexler-Raz Identity Solution is by Pseudo Inverse and will converge for increasing Gabor Sample Rate When they are equivalent, a “near” Tight-Frameis approximated A typic
15、al example:)(k )(khCritical SamplingX2 Over-Sampling X4 Over-SamplingGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001The Orthogonal-Like Gabor Transform Achieving a Tight Frame is subject to: Window selection Window variance Frequency Domain Sampling Step Time Domain Sampling StepNn
16、kjMmNnnmeMmkhckS/21010,)()(=10/2*,)()(LkNnkjnmeMmkhkScGabor Sample RateThen the Gabor Transform and Gabor Expansion Exhibit The Orthogonal-Like Property That Analysis And Synthesis Operations Utilize the Identical Window FunctionGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001Gabor
17、Order Tracking Order masking and re-construction Extract 60 bin wide mask at center frequency of a signal with two sidebands and a suppressed carrier frequencyGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001Gabor Order Tracking Filter shape approximations Analyze a long sequence of
18、broaband random dataVold-Kalman Gabor Order TrackingGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001Gabor Order Tracking WOT Vehicle run-up Extract 4th order interior noise with Gabor Order TrackingGabor Order Tracking SAE Noise & Vibration Conference May 1, 2001Gabor Order Tracking
19、 WOT Vehicle run-up Comparison of GOT result to Typical Vold-Kalman extraction GOT produces the smoother result due to superior stop-band selectivity (note that further optimization of Vold-Kalman should permit eventual equivalence to GOT result)Gabor Order Tracking SAE Noise & Vibration Conference
20、May 1, 2001Summary Gabor Order Tracking is just one of many possible masking strategies made possible by the Gabor methods of Time/Frequency Analysis It is now possible to perform high performance order tracking without a tachometer signal Crossing orders are not yet handled by the method The redundancy of the over-sampled Gabor transform permits excellent performance in the presence of noise Performance is limited only by ones ability to isolate desired Gabor coefficients from the undesired ones
