Characteristic Behaviors of Wavelet and Fourier Spectral Coherences ABSTRACT

Transcription

1 1 2 Characteristic Behaviors of Wavelet and Fourier Spectral Coherences Yueon-Ron Lee and Jin Wu ABSTRACT Here we examine, as well as make comparison of, the behaviors of coherences based upon both an wavelet optimum function basis and the Fourier spectral function basis. The main topics include: effects of data length on convergent tendencies for the two types of coherences; differences in coherences associated with complex and real functions related to the optimum wavelet basis; effects of data quality on the coherence distributions; symptoms arising from non-concurrent or displaced measurements; statistical implications; and physical depictions of analytical aspects or mathematical counterparts. Overall, the present study, on the one hand, serves as an extension of our previous studies concerning the identification of an optimum wavelet function basis for water wave related signals and the relevant data analysis methodology, and it fully vindicates the usefulness and superiority of the identified basis; on the other, it serves as an example for the basis practical applications. coherence function basis Lee and Wu 1996a 1996b 2003a 2003b Hilbert analytic signal procedure Lee and Wu

2 Riesz wavelet bases entropy Asyst cardinal spline semi-orthogonal wavelet m 0 (ξ) Daubechies 1992 linear phase filtering time-frequency windows Phillips 1977 wavelet admissibility condition g(t) h(t) t cross correlation function inner product c(t) = g(t + τ), h(τ) τ r s (t) r s (t) = c(t) g(t) h(t), (1) ĉ(t) g(t) h(t) = G(ω)H(ω) G(ω) H(ω), (2) ω t R 2 s (ω) = E[G(ω)H(ω)] 2 ( E[ G(ω) 2 ]E[ H(ω) 2 ] ) 1/2, (3) E normalization the wavelet resolution of identity g, h = 1 1 c ψ 0 a 2 g, ψ a,b h, ψ a,b dbda. (4) a g a, h a = 1 c ψ 1 a 2 g, ψ a,b h, ψ a,b db. (5) R 2 w (a) = E b [ g, ψ a,b h, ψ a,b ] 2 ( Eb [ g, ψ a,b 2 ]E b [ h, ψ a,b 2 ] ) 1/2, (6) b translation pa-

3 rameter E b unity b order degrees of freedom variance shift non-invarince property phase noises ambiguity effects Liu 1994 Liu ψ ψ Riesz Morlet wavelet ψ(t) = π 1/4 (e iω 0t e ω0 2/2 )e t2 /2, (7) ω 0 Chui 1992a 1992b Heisenberg uncertainty principle 5 m cm 24 cm ASI TSI LDV Stokes cm 6 m/sec

4 [(16,100)WC-C10:12-Exp0-F40-Erf1<11.0,5.0>] [(21,128)SC]pts=512,#.sec=2:18,(f=40,inc=1) [(21,64)SC]pts=256,#.sec=4:37,(f=40,inc=1) 09/20/04-11:35 WC-Wind (F) CW-F0LW6DxR0-L1 1F CS-F0LW6DxR0-L1 1F CSSF0LW6DxR0-L1 1F F F F F F F F F F F F F [(16,100)WC-C10:12-Exp0-F40-Erf1<11.0,5.0>] [(21,128)SC]pts=512,#.sec=2:18,(f=40,inc=1) [(21,64)SC]pts=256,#.sec=4:37,(f=40,inc=1) 09/20/04-11:35 WC-Wind (F) CW-F0LW6DxR0-L2 2F CS-F0LW6DxR0-L2 2F CSSF0LW6DxR0-L2 2F F F F F F F F F F F F F [(16,100)WC-C10:12-Exp0-F40-Erf1<11.0,5.0>] [(21,128)SC]pts=512,#.sec=2:18,(f=40,inc=1) [(21,64)SC]pts=256,#.sec=4:37,(f=40,inc=1) 09/20/04-11:35 WC-Wind (F) CW-F0LW6DxR0-L4 4F CS-F0LW6DxR0-L4 4F CSSF0LW6DxR0-L4 4F F F F F F F F F F F F F cm FFT [(21,64)SC]pts=256,#.sec=4:37,(f=40,inc=1) 09/20/04-11:35 CSSF0LW6DxR0_9 9F F F F F cm FFT sec non-stationary support lengthes

5 regularity Lee and Wu 1996a b cutoff Nyquist rate 1 4 Hz 3 ψ modulated Gaussian function m/sec Fermi-Pasta-Ulam recurrence phenomen Debnath 1994 Lee and Wu LDV LDV 40% 5

6 [(37,90)WC-C10-Exp0-F40-Erf1<11.0,5.0>] 09/10/04-16:10 [(37,90)WC-R10-Exp0-F40-Erf1<11.0,5.0>] 09/10/04-16:10 WC-B_Wind (F) CW-B0QWxD4R0 B RC-B_Wind (F) CR-B0QWxD4R0 B B B Wavelet coherence (R) B B m/sec [(37,90)WC-R10-Exp0-F40-Erf1<11.0,5.0>] 09/10/04-16:10 [(37,64)SC]pts=256,#.sec=4,(f=40,inc=5) 09/10/04-16:10 RC-A_Stokes (F) CR-A0QM3DxR0 A CS-A0QM3DxR0 A Wavelet coherence (R) A A A A A A LDV 7 Time (sec) Time (sec) Time (sec) ( f0 p3 c1 s9 i5 ) 09/21/04**15:45 Case: q0w6030 Chan: cm/sec ( f0 p3 c1 s9 i5 ) 09/21/04**15:45 Case: q0w6030 Chan: cm/sec ( f0 p3 c1 s9 i5 ) 09/21/04**15:45 Case: q0w6030 Chan: LDV cm

7 [(37,90)WC-C10-Exp0-F40-Erf1<11.0,5.0>] 09/10/04-16:13 [(37,64)SC]pts=256,#.sec=4,(f=40,inc=5) 09/10/04-16:14 CW-A0QM1D4R0 A CS-A0QM1D4R0 A A A A A [(19,90)WC-C10-Exp0-F40-Erf1<11.0,5.0>] 09/10/04-16:12 [(19,64)SC]pts=256,#.sec=4,(f=40,inc=5) 09/10/04-16:12 CW-A0PM1-CC A CS-A0PM1-CC A A A A A A A A A A A [(19,90)WC-C10-Exp0-F40-Erf1<11.0,5.0>] 09/10/04-16:12 [(19,64)SC]pts=256,#.sec=4,(f=40,inc=5) 09/10/04-16:12 CW-A0PM3-CC A CS-A0PM3-CC A A A A A A A A A A A [(37,90)WC-C10-Exp0-F40-Erf1<11.0,5.0>] 09/14/04-08:04 [(37,64)SC]pts=256,#.sec=4,(f=40,inc=5) 09/14/04-08:05 CW-A0UM1D3R0 A CS-A0UM1D3R0 A A A A A [(37,90)WC-C10-Exp0-F40-Erf1<11.0,5.0>] 09/14/04-08:04 [(37,64)SC]pts=256,#.sec=4,(f=40,inc=5) 09/14/04-08:05 CW-A0UM3D3R0 A CS-A0UM3D3R0 A A A A A

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