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1 1 Fourier 2 : Fourier s(t) Fourier S(!) = s(t) = 1 s(t)e j!t dt (1) S(!)e j!t d! (2) 1 1 s(t) S(!) S(!) =S Λ (!) Λ js T (!)j 2 P (!) = lim T!1 T S T (!) = T=2 T=2 (3) s(t)e j!t dt (4) T P (!) Fourier P (!) =js(!)j 2 (5) F 1 fp (!)g = 1 S(!)S(!)e j!t d! = s( t) Λ s(t) = s( x)s(t x)dx 1

2 Λ R(fi) F 1 [P (!)] = s(t)s(t + fi)dt (6) R(fi) s(t) R(fi) = s(t) Λ s(t + fi)dt (7) 3 cos(! t) e j! t cos(! t)=(e j! t + e j! t )=2 Fourier 2 z(t) = 2 = 1 ß 1 1 S(!)e j!t d! s(t )e j!(t t ) dt d! (8) 1 e j!t d! = ßffi(t)+ j t z(t) =s(t)+ j ß (9) s(t ) t t dt (1) z(t) [1, 2] 2 s(t) Hilbelt Hilbelt FFT Fourier 2 Fourier d dt 6 z(t) jz(t)j 1 2

3 1 s(t) t 1: ( ) 4 Fourier (STFT) Fourier (Short- Fourier Transform; STFT ) S(t;!) = s(t )g Λ (t t)e j!t dt g(t) STFT t 2 spectrogram P (t;!) =js(t;!)j 2 (12) T (11) S(t;!) = t+t=2 t T=2 s(t )e j!t dt (13) Fourier FFT(FastFourier Transform)! ==T T 2 2 ( two-tone ) T 4 FFT (= T ) FFT 2 [3] 3

4 Spectrogram with STFT T=.1T Spectrogram with STFT T=.2T Spectrogram with STFT T=.5T Spectrogram with STFT T=1.T 2: two-tone STFT spectrogram 1 3 chirp ( ) T T T = T FM 5 Wigner STFT 2 Wigner Fourier [1] P (t;!) = s(t + fi 2 )sλ (t fi 2 )e j!fi dfi (14) 4

5 Spectrogram with STFT T=.1T Spectrogram with STFT T=.2T Spectrogram with STFT T=.5T Spectrogram with STFT T=1.T 3: chirp STFT spectrogram Wigner (6) P (!) P (!) = s(t + fi 2 )sλ (t fi 2 )e j!fi dfidt (15) T P (t;!) = t+t=2 t T=2 s(t + fi 2 )sλ (t fi 2 )e j!fi dfidt (16) (14) T! Wigner spectrogram Wigner 4 chirp Wigner STFT 5

6 Wigner distribution Wigner distribution (without analytic function) 4: chirp Wigner 5: Wigner fi=2 P (t;!) =2 s(t + fi)s Λ (t fi)e j2!fi dfi (17) FFT 1 Nyquist (14) Fourier 5 4 s(t) z(t) Wigner P (t;!)d! = js(t)j 2 (18) P (t;!)dt = js(!)j 2 (19) Wigner two-tone Wigner 2 6 ambiguity Wigner 6

7 Wigner distribution 6: Wigner (3) (6) Fourier ambiguity [4] A( ; fi) = s(t + fi 2 )sλ (t fi 2 )e j t dt (2) Wigner ambiguity fi t (2) = A(;fi) = = s(t + fi 2 )sλ (t fi 2 )dt s Λ (t)s(t + fi)dt = R(fi) (21) fi fi = A( ; ) = = js(t)j 2 e j t dt S Λ (!)S(! + )d! (22) fi A( ; fi) s(t) 2 (2) A( ; fi) =A Λ ( ; fi) 7 two-tone chirp ja( ; fi)j fi two-tone 7

8 Ambiguity function two tone signal 2 lag τ 1 1 lag θ Ambiguity function chirp signal 2 lag τ 1 1 lag θ 7: two-tone chirp ambiguity fi chirp Wigner ambiguity A( ; fi) = 1 P (t;!)e j( t fi!) dtd! (23) [5] ambiguity Wigner Fourier 6.1 ambiguity S/N S/N H(!) S/N (S=N) = fi fi fi 1 R 1 1 S(!)H(!)d! fi fi fi 2 N R 1 1 jh(!)j 2 d! 8 (24)

9 N Parseval js h ()j 2 Schwarz fi fi fi fi 1 fififi fi S(!)H(!)d! 2 1» 1 (S=N)» 1 js(!)j2 d! 1 (25) 1 jh(!)j2 d! (25) R 1 1 js(!)j 2 d! N (26) H(!) =S Λ (!) (27) (matched filter) S/N h(t) = SΛ (!)e j!t d! = s Λ ( t) (28) s(t) h(t) g(t) = 1 1 s(t x)h(x)dx = 1 1 sλ (t )s(t + t)dt = ρ(t) (29) v d! d g(t) = 1 1 s Λ (t )s(t + t)e j! dt dt = A(! d ;t)e j! t d 2 (3) ambiguity ambiguity 7 Wigner Wigner Φ(t;!) P (t;!) = Φ(t t ;!! )P (t ;! )dt d! (31) Wigner P (t;!) 2 Fourier ambiguity [6] A( ; fi) =ffi( ; fi)a( ; fi) (32) 9

10 ffi( ; fi) = 1 Φ(t;!)e j( t fi!) dtd! (33) 2 Fourier (kernel) [1, 5] (32) ambiguity ambiguity Wigner ffi( ; fi) =1 P (t;!) ffi( ; fi) P (t;!) = 1 P (t;!) = R(t; fi) = 1 e j (t t ) j!fi ffi( ; fi)s(t + fi 2 )sλ (t fi 2 )dt dfid (34) R(t; fi) 1 P (t;!)e j!fi d! (35) R(t; fi)e j!fi dfi (36) e j (t t ) ffi( ; fi)s(t + fi 2 )sλ (t fi 2 )dt d (37) [1] (36) FFT R(t; fi) 7.1 Wigner (SPWD) Φ(t;!) =Φ 1 (t)φ 2 (!) (38) Wigner (Smoothed Pseudo Wigner Distribution SPWD ) [5]! Φ(t;!) = exp ψ t2 ff!2 (39) 2 fi 2 Gauss R(t; fi) R(t; fi) = 1 ffi( ; fi) =ßfffi exp ψ ff2 2 pßfi exp ψ (t t ) 2 fi2 fi 2 ff ! fi2 fi 2 4 (4)! s(t + fi 2 )sλ (t fi 2 )dt (41)

11 Smoothed Wigner: α= 4., β= 1. Smoothed Wigner: α= 8., β= 2. 8: cross-chirp SPWD chirp ff fi Wigner 8 2 chirp (cross-chirp ) ff fi 7.2 Choi-Williams SPWD Wigner Wigner ((18) (19) ) Choi-Williams [6]! ffi( ; fi) =exp ψ 2 fi 2 (42) ff R(t; fi) R(t; fi) = 1 ψ 1 q 4ßfi 2 =ff exp (t! t ) 2 s(t + fi 4fi 2 =ff 2 )sλ (t fi 2 )dt (43) ff 9 cross-chirp ff Wigner [5] 1 2 Choi-Williams SPWD 11

12 Choi Williams: σ= 2. Choi Williams: σ= 1. 9: cross-chirp Choi-Williams Choi Williams: σ=.5 Smoothed Wigner: α= 5., β= 3. 1: Choi-Williams SPWD 2 SPWD [7] 7.3 Wigner spectrogram STFT spectrogram Wigner spectrogram Wigner js(t;!)j 2 = P g (t t;!!)p (t ;! )dt d! (44) 12

13 [5] P g (t;!) g(t) Wigner Φ(t;!) =P g ( t;!) = Gauss g( t + fi 2 )gλ ( t fi 2 )ej!fi dfi (45) g(t) =e t2 =fl 2 Φ(t;!) = p fl exp! ψ 2t2 fl fl2! (39) ff = fl= p 2 fi = p 2=fl fffi =1 STFT t! ==2 fffi ' 32 SPWD 2 8 (46) 7.4 Polynomial Wigner-Ville Wigner (Wigner-Ville Distribution; WVD) ffl ffl PWVD(Polynomial WVD) [8] z(t) =e jffi(t) WVD W z (t; f) =F[z t + fi 2 z Λ t fi 2 f i (t; fi) ]=F[e jfffi(t+fi=2) ffi(t fi=2)g ] (47) f i (t; fi) =fffi(t + fi=2) ffi(t fi=2)g=fi (48) FM ffi(t) = px i= a i t i (49) 13

14 f i (t; fi) = 1 fi Polynomial WVD q=2 X k= q=2 b k ffi(t + c k fi) (5) W g z (t; f) = F[K g z (t; fi)] (51) K g z (t; fi) = q=2 Y k= q=2 [z(t + c k fi)] b k (52) Peak Filtering - Peak Filtering[9] 1 WVD x(t) n(t) s(t) =x(t)+n(t) (53) y s (t) = e jr t 1 s( )d = y x (t)y n (t) (54) y x (t) = e jr t 1 x( )d (55) y n (t) = e jr t 1 n( )d (56) y s (t) WVD x(t) 8 wavelet wavelet 8.1 wavelet wavelet W (t; a) = s(t ) p 1 ψ! t h Λ t dt (57) a a 14

15 Scalogram: γ= 5. Scalogram: γ= 2. 11: two-tone scalogram a! [1, 11] h(t) t = analysis wavelet a h(t)! a =! =! t wavelet STFT h(t) =g(t)e j! t (58) (57) W (t;!) = s!! e j!t! s(t )g Λ (t t) e j!t dt! (59) (11) g(t) STFT spectrogram jw (t; a)j 2 scalogram 11 g(t) =exp( t 2 =fl 2 ) scalogram scalogram spectrogram 11 2 scalogram 8.2 Wigner scalogram spectrogram scalogram Wigner [5, 11] jw (t; a)j 2 = P h ψ t t a ;a!! P (t ;! )dt d! (6) P h (t;!) analysis wavelet h(t) Wigner a! ψ!! jw (t;!)j 2 = P h (t! t);! P (t ;! )dt d! (61)!! 15

16 12: Wigner spectrogram scalogram affine affine Wigner! ~P a (t;!) = Ψ ψ!(t t );! P (t ;! )dt d! (62)! [5] wavelet Ψ a (ff; fi) = P h ff ;! fi! = h ff + fi! 2 (63) h Λ ff fi e j! fifi dfi (64)! 2 7 affine Ψ a scalogram spectrogram 12 (BT) ((39) fffi ) Wigner (BT=) affine spectrogram scalogram (BT=1) [12] 16

17 8.3 wavelet (57) a (!) t wavelet N N (18) (19) wavelet analysis wavelet Meyer [13] (57) W (t; a) = s(t )h Λ a;t(t )dt (65) h a;t (t ) = p 1 ψ! t t h (66) a a w i;k = h i;k s(t)h Λ i;k(t)dt (67) h i;k = 2 i=2 h(2 i t k) (68) s(t) = X i X Meyer analysis wavelet k w i;k h i;k (t) (69) H(!) =e j!=2 qj(!=2) 2 J(!) 2 (7) J(!) = K(!) = L(!) = q K(!)K(!) L(4ß=3!) L(! =3) + L(4ß=3!) ( exp( 1=!2 ) (! >) (!» ) h(t) Fourier 13 h(t) H(!) H(!) =3»j!j»8ß=3 h(t) t = t =1=2 analysis wavelet (; 1) 17

18 h(t) H(ω) t ω 13: analysis wavelet Fourier wavelet w i;k (67) s(t) h(t) s(t) FFT H(!) FFT Meyer wavelet H(!) 9 STFT spectrogram Wigner Wigner SPWD ambiguity wavelet scalogram spectrogram Wigner wavelet affine 18

19 [1, 5, 11] 3 [5] [1] L. Cohen, -frequency distributions A review, Proc. IEEE, Vol.77, pp , [2],,,, pp , Jan [3] R. C. Singleton, An algorithm for computing the mixed radix fast Fourier transform, IEEE AU-17, pp.93, [4] M.I. Skolnik, Introduction to radar systems, McGraw Hill, New York, 198. [5] F. Hlawatsch and G.F. Boudreaux-Bartels, Linear and quadratic time-frequency signal representations, IEEE Sig. Proc. Mag., Vol.9, pp.21 67, 1992, [6] H-I Choi and W.J. Williams, Improved time-frequency representation of multicomponent signals using exponential kernels, IEEE Trans. Acoust. Speech Sig. Proc., Vol.37, pp , [7] D.L. Jones and T.W. Parks, A resolution comparison of several time-frequency representations, IEEE Trans. Sig. Proc., Vol.4, pp , [8] B. Boashash and P. O'Shea, Polynomial Wigner-Ville distributions and their relationship to time-varying higher order spectra, IEEE Trans. Signal Processing, Vol. 42, No. 1, pp , [9] M.J. Arnold and B. Boashash, - Peak Filtering: A non-model based signal enhancementscheme, Proc. Int. Symp. Information Theory and its Applications, pp , [1] C.K. Chui (, ),,, [11] O. Rioul and M. Vetterli, Wavelets and Signal processing, IEEE Sig. Proc. Mag., Vol.8, pp.14 38, [12] O. Rioul and P. Flandrin, -scale energy distributions: A general class extending wavelet transforms, IEEE Trans. Sig. Proc., Vol.4, pp , [13] M. Yamada and K. Ohkitani, Orthonormal waveletanalysis of turbulence, Fluid Dyna. Res., Vol.8, pp ,

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