Theory and applications of time-frequency analysis

Transcription

1 Theory and applications of time-frequency analysis Ville Turunen Abstract: When and how often something happens in a signal? By properly quantizing these questions, we obtain the Born-Jordan time-frequency transform, defining a sharp phase-space energy density. We study properties of different time-frequency transforms, and also present computed examples from acoustic signal processing, quantum mechanics and medical sciences. Mathematical optics, image modelling and algorithms , Hannover 1 / 26

2 Waveform of speech... 1 Signal (word Why ), sampling rate 4000 Hz / 26

3 Wigner for speech (compare to next Born Jordan...) 700 "Why?": Discrete time Wigner distribution / 26

4 ... Born Jordan for speech (compare to next spectrogram...) 700 "Why?": Discrete time Born Jordan distribution / 26

5 ... a spectrogram (compare to previous Born Jordan) 700 Spectrogram with a medium wide Gaussian window / 26

6 ... a spectrogram with wide Gaussian window Spectrogram with a wide Gaussian window / 26

7 ... a spectrogram with narrow Gaussian window 700 Spectrogram with a narrow Gaussian window / 26

8 Some history Tools: Fourier analysis, FFT! 1920s foundations of quantum mechanics (Heisenberg, Born, Jordan, Schrödinger; Dirac, Wigner, Weyl, von Neumann) Spectrograms (early 1940s Bell Labs, Gabor) 1966 Leon Cohen s class of time-frequency distributions (physics, signal processing) 1960s pseudodifferential operators (Hörmander et al) 2010 Born-Jordan L 2 -continuity (Boggiatto De Donno Oliaro) 2011 Born-Jordan uncertainty (Boggiatto Oliaro Carypis) 8 / 26

9 Order on R characterizes Born Jordan [V.T.] Studying a quantum particle on the real line R, we ask: (A) Is particle on the right? (B) Is particle moving right? So, we just ask for the directions of location and of movement (Separate right from left, up from down, future from past...) In other words, the order relation on R is essential here. The uncertainty in (A, B) characterizes the Born Jordan transform, leading to sharp time-frequency (phase-space) analysis. Other time-frequency transforms do not properly answer to (A, B). This has important consequences in signal processing and quantum mechanics. 9 / 26

10 Direction of position Wavefunction ψ H = L 2 (R) describing a quantum particle on R. By Max Born, probability of finding position right of x R is x ψ(y) 2 dy [0, 1]. (1) Localization to [x, ) R is given by projection A x : H H, { u(y) when y x, A x u(y) := (2) 0 otherwise. Observable A x : Is particle having position right of x? Yes (eigenvalue 1) with probability x ψ(y) 2 dy, and we find the updated wavefunction u/ u with u = A x ψ. No (eigenvalue 0) with probability x ψ(y) 2 dy, and the wavefunction becomes v/ v with v = ψ A x ψ. 10 / 26

11 Direction of momentum Change from position x to momentum η by Fourier transform. By Max Born, probability of finding momentum above η R is η ψ(ξ) 2 dξ [0, 1]. (3) Localization to [η, ) R is given by projection B η : H H, B η u(ξ) := {û(ξ) when ξ η, 0 otherwise. (4) Observable B η : Is particle having momentum above η? Yes (eigenvalue 1) with probability η ψ(ξ) 2 dξ, and we find the updated wavefunction u/ u with u = B η ψ. No (eigenvalue 0) with probability η ψ(ξ) 2 dξ. and the wavefunction becomes v/ v with v = ψ B η ψ. 11 / 26

12 Expectation of directional uncertainty Uncertainty observable of the observable pair (A, B) is i2π [A, B] = i2π (AB BA). (5) An application of the Cauchy Schwarz inequality yields i2π [A, B]u, u = 4π Im Au, Bu 4π Au Bu. (6) This gives the Heisenberg uncertainty inequality i2π [A, B] 4π ( A) ( B), (7) where in state ψ observable M has expectation M := Mψ, ψ and uncertainty M := Mψ M ψ. For u, v H, define Q(u, v)(x, η) := i2π [A x, B η ]u, v. (8) We call Q(u, v) : R R C the time-frequency transform, and Q[u] = Q(u, u) : R R R the time-frequency distribution. Q[ψ](x, η) is the expectation of uncertainty of (A x, B η ) in state ψ. Especially: Q[ψ](0, 0) is the expectation of uncertainty in directional location and movement, 12 / 26

13 Born Jordan transform... and a brief calculation yields the Born Jordan transform 1/2 Q(u, v)(x, η) = e i2πy η u(x+(τ+1/2)y) v(x+(τ 1/2)y) dτdy. R 1/2 Q[ψ] = Q(ψ, ψ) is a quasi-probability distribution of ψ, or Q[u] = Q(u, u) is an energy density of u. Alternatively, the Born Jordan transform is given by (9) FQ(u, v)(ξ, y) = sinc(ξ y) FW (u, v)(ξ, y), (10) where F is the symplectic Fourier transform, and the Wigner transform W (u, v) is defined by W (u, v)(x, η) = e i2πy η u(x + y/2) v(x y/2) dy. (11) R Unfortunately, Wigner has a very bad property: it is extremely sensitive to noise (unlike the non-unitary Q. Also, Q is not causal nor positive.) 13 / 26

14 Wigner distribution is sensitive to noise: speech example Wigner distribution / 26

15 ... and corresponding Born Jordan distribution Speech: Discrete time Born Jordan distribution / 26

16 Bound for Born Jordan energy density [V.T. 2016] There is the optimal Born Jordan bound Q(u, v)(x, η) π u v (12) for all (x, η) R R. Especially, Q[u](x, η) π u 2. Proof. Point localization at the origin : Lu, v = Q(u, v)(0, 0) 1 z/2 = u(t + z/2) v(t z/2) dt dz z z/2 = K L (x, y) u(y) v(x) dx dy, with the Schwartz kernel K L (x, y) = { x y 1 if xy < 0, 0 if xy / 26

17 Proof of the bound for energy density From Lu, v = = u(y) v(x) y x u(y)v( x) y + x dy dx + dy dx u(y) v(x) dy dx x y u( y)v(x) dy dx, x + y denoting u = Pu + Nu, where Pu(x) = 1 R +(x) u(x), we get Lu, v Hilbert π Pu Nv + π Nu Pv = π ( Pu, Nu ) ( Nv, Pv ) Cauchy Schwarz π = π u v. Pu 2 + Nu 2 Nv 2 + Pv 2 Especially, Lu, u π u / 26

18 EKG (Electro-KardioGram) 0.6 Channel 1: Waveform. Press any key to continue / 26

19 EKG: Born Jordan distribution (absolute value) Channel 1: Time index interval [56001:72000]. Frequency unit Hz. Comparison un / 26

20 Another characterization for Born Jordan [V.T. 2016] Cohen class transforms: C = k W for some k S (R R). Necessary and sufficient (i,ii,iii) for C = Q: (i) C is scale-invariant. This means that if v(x) = λ 1/2 u(λx) for λ > 0 then C[v](x, η) = C[u](λx, η/λ). (ii) C is position-local. This means that if u(x) = 0 whenever x [a, b] R then C[u](x, η) = 0 whenever x [a, b] R. (iii) C maps Dirac delta comb to Dirac delta grid. This means C[δ Z ](x, η) = δ Z (x) + δ Z (η) 1, where the Dirac delta comb is δ Z (x) = k Z δ k (x) = κ Z e i2πx κ, with δ k being the Dirac delta distribution at k Z. Think δ Z as a ticking-of-a-clock. Notice that F (δ Z ) = δ Z. Wigner distribution has properties (i,ii) but not (iii). Spectrograms satisfy none of the properties (i,ii,iii). 20 / 26

21 Proof idea of Born Jordan characterization Let FC[u] = φ FW [u]. We must show φ(ξ, y) = sinc(ξ y). (1) φ(ξ, y) = ϕ(ξ y) for some ϕ S (R). (2,1) ϕ(x) = 0 for almost all x > 1/2. So we have analytic ϕ Thm. {: C C by Paley Wiener Schwartz (3,2,1) ϕ(k) = 1 if k = 0, 0 if k Z \ {0}. So û = ϕ/sinc : C C is analytic. Then ϕ = u 1 [ 1/2,1/2] gives u(x + k y) dy [ 1/2,1/2] ϕ(x + k) = k Z k Z = u(x) dx = û(0) = ϕ(0) sinc(0) = 1, so ϕ(x) = 1 for almost all x [ 1/2, 1/2]. Thus we obtain 1/2 ϕ(ξ y) = e i2πxy ξ ϕ(x) dx = e i2πxy ξ dx = sinc(ξ y). 1/2 21 / 26

22 Inversion of Born Jordan distribution [V.T.] Mapping u Q[u] is not invertible: Q[λu] = λ 2 Q[u]. But mapping [u] Q[u] is invertible, where the equivalence class [u] = {u : λ C, λ = 1}. It is easy to see invertibility of Wigner [u] W [u], and by (10) also [u] Q[u] can be inverted. Notice the marginals: Q[u](x, η) dη = u(x) 2, Q[u](x, η) dx = û(η) 2. (13) R R Yet these marginals are not enough to find [u]. Suppose u(x) 0 for Schwartz function u. Then we have inversion u(x + h) = h u(x) 1 R(x kh, h) (14) k=0 for all h 0, where 1 is the partial derivative in the first variable, R(x, y) := e i2πy η Q[u](x, η) dη. (15) R 22 / 26

23 Quantization a a C Cohen class transform C gives the corresponding quantization a a C, a C = a C (X, D) from symbols a : R R C to pseudo-differential operators a C. Such linear operator a C = a C (X, D) is defined by the duality u, a C v = C(u, v), a. (16) Properties of C are reflected in a a C. E.g. marginal conditions C[u](x, η) dη = u(x) 2, C[u](x, η) dx = û(η) 2, (17) R mean that if a(x, η) = f (x) and b(x, η) = ĝ(η) then a C u(x) = f (x) u(x) (multiplication) and b C u(x) = g u(x) = R g(x y) u(y) dy (convolution). Especially, if (17) and a(x, η) = x and b(x, η) = η, then a C = X and b C = D, where Xu(x) = x u(x), Du(x) = 1 i2π u (x). R 23 / 26

24 Quantization examples For example, the Wigner transform C = W from formula (11) gives rise to the Weyl quantization a a W : a W u(x) = e i2π(x y) η a( x + y, η) u(y) dy dη. (18) 2 R R Born Jordan transform Q gives Born Jordan quantization a a Q, 1 a Q u(x) = e i2π(x y) η 2 a((τ+ 1 R R 1 2 )x (τ 1 )y, η) dτ u(y) dy dη. 2 2 (19) What makes the Born Jordan quantization unique among quantizations is that if a(x, η) = f (x) and b(x, η) = ĝ(η) then {a, b} Q = i2π [a Q, b Q ], (20) where the Poisson bracket {a, b} is reduced to ( a b {a, b}(x, η) = x η a ) b (x, η) = f (x) ĝ (η). (21) η x (20)-remark: no-go-theorems [Groenewold 1946, van Hove 1951]. 24 / 26

25 MRI + patient (frequency unit 1 Hz) Restore/Delete/+/ /Zoom/Compute. Frequency unit Hz / 26

26 MRI + patient after noise filtering Restore/Delete/+/ /Zoom/Compute. Frequency unit Hz / 26