Short-time Fourier transform for quaternionic signals

Transcription

1 Short-time Fourier transform for quaternionic signals Joint work with Y. Fu and U. Kähler P. Cerejeiras Departamento de Matemática Universidade de Aveiro New Trends and Directions in Harmonic Analysis, Fractional Operator Theory, and Image Analysis - Inzell /28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

2 Gabor systems Analytic signals Definition A discrete system {ψ j, j J} is a frame for a Hilbert space H if there exist 0 < A B < such that A f 2 j J f, ψ j 2 B f 2 for all f H. Definition We define the analysis operator as the linear operator F : H l 2 (J), where Ff = c, with c j = f, ψ j. Its synthesis operator F is given by F c = j J c jψ j ; The frame operator is given by FF : H H. 2/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

3 Gabor systems Analytic signals Definition A discrete system {ψ j, j J} is a frame for a Hilbert space H if there exist 0 < A B < such that A f 2 j J f, ψ j 2 B f 2 for all f H. Definition We define the analysis operator as the linear operator F : H l 2 (J), where Ff = c, with c j = f, ψ j. Its synthesis operator F is given by F c = j J c jψ j ; The frame operator is given by FF : H H. 2/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

4 Gabor systems Analytic signals Then one has a good characterization of f : f, ψ j = 0, j J f = 0; a good reconstruction scheme for f : f = f, ψ j ψ j = f, ψ j ψ j. j J j J 1 The frame is said to be tight if A = B; 2 If the frame is tight and A = 1 then the frame is orthogonal; 3 The frame is said to be exact if after redrawing an arbitrary ψ j0 from the system it is no longer a frame. 3/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

5 Gabor systems Analytic signals Then one has a good characterization of f : f, ψ j = 0, j J f = 0; a good reconstruction scheme for f : f = f, ψ j ψ j = f, ψ j ψ j. j J j J 1 The frame is said to be tight if A = B; 2 If the frame is tight and A = 1 then the frame is orthogonal; 3 The frame is said to be exact if after redrawing an arbitrary ψ j0 from the system it is no longer a frame. 3/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

6 Gabor systems Analytic signals Then one has a good characterization of f : f, ψ j = 0, j J f = 0; a good reconstruction scheme for f : f = f, ψ j ψ j = f, ψ j ψ j. j J j J 1 The frame is said to be tight if A = B; 2 If the frame is tight and A = 1 then the frame is orthogonal; 3 The frame is said to be exact if after redrawing an arbitrary ψ j0 from the system it is no longer a frame. 3/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

7 Windowed Fourier transform Gabor systems Analytic signals Definition For a given window g L 2 (R) we define the windowed Fourier transform of f L 2 (R) as F win f (t, ω) = + f (x)e 2πωix g(x t)dx. Problem For discretization of the WFT what choice of ω 0, t 0 such that the signal is characterized by its coefficients? one has a numerically stable reconstruction of the signal? 4/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

8 Windowed Fourier transform Gabor systems Analytic signals Definition For a given window g L 2 (R) we define the windowed Fourier transform of f L 2 (R) as F win f (t, ω) = + f (x)e 2πωix g(x t)dx. Problem For discretization of the WFT what choice of ω 0, t 0 such that the signal is characterized by its coefficients? one has a numerically stable reconstruction of the signal? 4/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

9 Gabor systems Analytic signals Discrete windowed Fourier transform Lemma For a given window g and its dual window g in M 1 we have that the Gabor frame operator converges in L 2 (R). 5/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

10 Gabor systems Analytic signals Discrete windowed Fourier transform Lemma For a given window g and its dual window g in M 1 we have that the Gabor frame operator converges in L 2 (R). 5/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

11 Gabor systems Analytic signals The Balian-Low theorem expresses the fact that time-frequency concentration and non-redundancy are incompatible properties for Gabor systems that is to say, either g m,n = e i2πm g( n), n, m Z, + x 2 g(x) 2 dx = or + ξ 2 ĝ(ξ) 2 dξ =. 6/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

12 The Zak transform - 1 Gabor systems Analytic signals The Zak transform (J. Zak, 1967) is an important tool for studying the frame given by Gabor systems. Definition For f L 2 (R), we have (Zf )(t, ω) = k Z e 2πi kω f (t k), (t, ω) [0, 1) 2. It defines a unitary operator from L 2 (R) to L 2 ([0, 1) 2 ). In abstract harmonic analysis the Zak transform is called the Weil-Brezin map; The harmonic waves e 2πi nω, n Z in the Zak transform have constant frequencies, which can be seen as the derivative of the linear phase φ(ω) = 2πnω; 7/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

13 The Zak transform - 1 Gabor systems Analytic signals The Zak transform (J. Zak, 1967) is an important tool for studying the frame given by Gabor systems. Definition For f L 2 (R), we have (Zf )(t, ω) = k Z e 2πi kω f (t k), (t, ω) [0, 1) 2. It defines a unitary operator from L 2 (R) to L 2 ([0, 1) 2 ). In abstract harmonic analysis the Zak transform is called the Weil-Brezin map; The harmonic waves e 2πi nω, n Z in the Zak transform have constant frequencies, which can be seen as the derivative of the linear phase φ(ω) = 2πnω; 7/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

14 The Zak transform - 1 Gabor systems Analytic signals The Zak transform (J. Zak, 1967) is an important tool for studying the frame given by Gabor systems. Definition For f L 2 (R), we have (Zf )(t, ω) = k Z e 2πi kω f (t k), (t, ω) [0, 1) 2. It defines a unitary operator from L 2 (R) to L 2 ([0, 1) 2 ). In abstract harmonic analysis the Zak transform is called the Weil-Brezin map; The harmonic waves e 2πi nω, n Z in the Zak transform have constant frequencies, which can be seen as the derivative of the linear phase φ(ω) = 2πnω; 7/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

15 The Zak transform -2 Gabor systems Analytic signals The nontrivial harmonic waves e iθa(2πω) have positive time-varying frequencies and are expected to be better suitable and adaptable, along with different choices of a, to nonlinear and non-stationary time-frequency analysis. 8/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

16 Analytic signals Gabor systems Analytic signals Applying to a given signal f (x) the Hilbert transform Hf (x) = 1 + f (y) dy, y R. 2π y x we obtain the complex-valued function F (x) = f (x) + ihf (x) = a(x)e iθ(x) F(x) is the analytic signal, a(x) the amplitude and θ(x) the phase; ω(x) = θ (x) is called instantaneous frequency; The pair (a, θ) is called canonical modulation pair of f (x). 9/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

17 Analytic signals Gabor systems Analytic signals Applying to a given signal f (x) the Hilbert transform Hf (x) = 1 + f (y) dy, y R. 2π y x we obtain the complex-valued function F (x) = f (x) + ihf (x) = a(x)e iθ(x) F(x) is the analytic signal, a(x) the amplitude and θ(x) the phase; ω(x) = θ (x) is called instantaneous frequency; The pair (a, θ) is called canonical modulation pair of f (x). 9/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

18 Gabor systems Analytic signals Analytic signals - Mathematical background Riemann-Hilbert problem: F z = 0 in Im(z) > 0 ReF(z) = f (x) in Im(z) = 0 ImF (z 0 ) = c in Im(z 0 ) > 0 Cauchy integral transform: F (z) := Cf (z) = 1 2πi Plemelj-Sokhotzki formula: tr ± Cf (z 0 ) = lim z z0 Cf (z) = 1 2 Γ 1 ζ z f (ζ)dζ [ f (z 0 ) ± 1 πi Γ ] 1 f (ζ)dζ ζ z 0 10/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

19 Gabor systems Analytic signals Analytic signals - Mathematical background Riemann-Hilbert problem: F z = 0 in Im(z) > 0 ReF(z) = f (x) in Im(z) = 0 ImF (z 0 ) = c in Im(z 0 ) > 0 Cauchy integral transform: F (z) := Cf (z) = 1 2πi Plemelj-Sokhotzki formula: tr ± Cf (z 0 ) = lim z z0 Cf (z) = 1 2 Γ 1 ζ z f (ζ)dζ [ f (z 0 ) ± 1 πi Γ ] 1 f (ζ)dζ ζ z 0 10/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

20 Gabor systems Analytic signals Why does it work for signal analysis? Hardy decomposition: L p (Γ) = H + p (Γ) H p (Γ); Poisson kernel 1 P(z, ζ) = Re. ζ z 0 Basic idea: Poisson kernel is also low-pass filter! Scale-space analysis by Gaussian kernel can be replaced by scale-space analysis using Poisson kernel; f (x) = a(x)e iθ(x) ; Weyl-relation [x, D] = i with D = i x < ω >= ω Ff (ω) 2 dω = θ (x)a(x) 2 dx 11/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

21 Extension to C 2 Gabor systems Analytic signals Riemann-Hilbert problem F = 0 z 1 (z 1, z 2 ) C 2, z 1 < 1 & z 2 < 1, F = 0 z 2 (z 1, z 2 ) C 2, z 1 < 1 & z 2 < 1, ReF (ξ 1, ξ 2 ) = f (ξ 1, ξ 2 ) ξ 1 = 1, ξ 2 = 1. Solution: F(z 1, z 2 ) := Cf (z 1, z 2 ) = 1 4π 2 1 T (ξ 2 1 z 1 )(ξ 2 z 2 ) f (ξ 1, ξ 2 )dξ 1 dξ 2. 12/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

22 Extension to C 2 Gabor systems Analytic signals Riemann-Hilbert problem F = 0 z 1 (z 1, z 2 ) C 2, z 1 < 1 & z 2 < 1, F = 0 z 2 (z 1, z 2 ) C 2, z 1 < 1 & z 2 < 1, ReF (ξ 1, ξ 2 ) = f (ξ 1, ξ 2 ) ξ 1 = 1, ξ 2 = 1. Solution: F(z 1, z 2 ) := Cf (z 1, z 2 ) = 1 4π 2 1 T (ξ 2 1 z 1 )(ξ 2 z 2 ) f (ξ 1, ξ 2 )dξ 1 dξ 2. 12/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

23 Gabor systems Analytic signals Hypercomplex signal by T. Bülow Two different imaginary units: z 1 = x 1 + iy 1 and z 2 = x 2 + jy 2 Solution of Riemann-Hilbert problem: Cf (z 1, z 2 ) = 1 1 4π 2 T (ξ 2 1 z 1 )(ξ 2 z 2 ) f (ξ 1, ξ 2 )dξ 1 dξ 2. Plemelj-Sokhotzki formula (just one side): trcf (x 1, x 2 ) = 1 4 (I + ih 1)(I + jh 2 )f (x 1, x 2 ) = 1 4 (f + ih 1f + jh 2 f + khf )(x 1, x 2 ). 13/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

24 Gabor systems Analytic signals Hypercomplex signal by T. Bülow Two different imaginary units: z 1 = x 1 + iy 1 and z 2 = x 2 + jy 2 Solution of Riemann-Hilbert problem: Cf (z 1, z 2 ) = 1 1 4π 2 T (ξ 2 1 z 1 )(ξ 2 z 2 ) f (ξ 1, ξ 2 )dξ 1 dξ 2. Plemelj-Sokhotzki formula (just one side): trcf (x 1, x 2 ) = 1 4 (I + ih 1)(I + jh 2 )f (x 1, x 2 ) = 1 4 (f + ih 1f + jh 2 f + khf )(x 1, x 2 ). 13/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

25 Gabor systems Analytic signals Hypercomplex signal by T. Bülow Two different imaginary units: z 1 = x 1 + iy 1 and z 2 = x 2 + jy 2 Solution of Riemann-Hilbert problem: Cf (z 1, z 2 ) = 1 1 4π 2 T (ξ 2 1 z 1 )(ξ 2 z 2 ) f (ξ 1, ξ 2 )dξ 1 dξ 2. Plemelj-Sokhotzki formula (just one side): trcf (x 1, x 2 ) = 1 4 (I + ih 1)(I + jh 2 )f (x 1, x 2 ) = 1 4 (f + ih 1f + jh 2 f + khf )(x 1, x 2 ). 13/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

26 Quaternionic algebra Quaternionic Fourier transform Carriers Quaternion algebra H The quaternion algebra is an extension of complex numbers to a 4D algebra. Every element of H is a linear combination of a real scalar and three orthogonal imaginary units (denoted by i, j, k) with real coefficients H = {q : q = q 0 + iq 1 + jq 2 + kq 3, q 0, q 1, q 2, q 3 R}, where the elements i, j, k obey the Hamilton s multiplication rules ij = ji = k, jk = kj = i, ki = ik = j, i 2 = j 2 = k 2 = ijk = 1. The scalar part is denoted as Sc(q) = q 0 and has the cyclic multiplication symmetry Sc(qrs) = Sc(rsq), q, r, s H. 14/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

27 H-valued function space Quaternionic Fourier transform Carriers H-conjugation of a given q = q 0 + iq 1 + jq 2 + kq 3 = Sc(q) + Vec(q) is q = Sc(q) Vec(q). Consider the quaternion-valued function space L 2 (R 2 ; H) equipped with the quaternionic-valued inner product (f, g) := f (x)g(x)dx. R 2 Additionally, consider also the complex-valued inner product f, g := Sc(f, g) = Sc[f (x)g(x)]dx. R 2 15/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

28 H-valued function space Quaternionic Fourier transform Carriers H-conjugation of a given q = q 0 + iq 1 + jq 2 + kq 3 = Sc(q) + Vec(q) is q = Sc(q) Vec(q). Consider the quaternion-valued function space L 2 (R 2 ; H) equipped with the quaternionic-valued inner product (f, g) := f (x)g(x)dx. R 2 Additionally, consider also the complex-valued inner product f, g := Sc(f, g) = Sc[f (x)g(x)]dx. R 2 15/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

29 H-valued function space Quaternionic Fourier transform Carriers H-conjugation of a given q = q 0 + iq 1 + jq 2 + kq 3 = Sc(q) + Vec(q) is q = Sc(q) Vec(q). Consider the quaternion-valued function space L 2 (R 2 ; H) equipped with the quaternionic-valued inner product (f, g) := f (x)g(x)dx. R 2 Additionally, consider also the complex-valued inner product f, g := Sc(f, g) = Sc[f (x)g(x)]dx. R 2 15/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

30 Quaternionic Fourier transform Carriers Definition Given a 2D quaternion-valued signal f L 2 (R 2, H), we defined its quaternionic Fourier transform as F H [f ](w) = e 2πix 1ω 1 f (x)e 2πjx 2ω 2 dx, w = (ω 1, ω 2 ), x = (x 1, x 2 ) R 2. R 2 Definition Given a windowed function g L 2 (R 2, R), we set the windowed quaternionic Fourier transform (WQFT) of a 2D quaternion-valued signal f L 2 (R 2, H) as Q g [f ](w, b) = e 2πix 1ω 1 f (x)g(x b)e 2πjx 2ω 2 dx. R 2 16/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

31 Quaternionic Fourier transform Carriers Definition Given a 2D quaternion-valued signal f L 2 (R 2, H), we defined its quaternionic Fourier transform as F H [f ](w) = e 2πix 1ω 1 f (x)e 2πjx 2ω 2 dx, w = (ω 1, ω 2 ), x = (x 1, x 2 ) R 2. R 2 Definition Given a windowed function g L 2 (R 2, R), we set the windowed quaternionic Fourier transform (WQFT) of a 2D quaternion-valued signal f L 2 (R 2, H) as Q g [f ](w, b) = e 2πix 1ω 1 f (x)g(x b)e 2πjx 2ω 2 dx. R 2 16/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

32 Multiplication operator Quaternionic Fourier transform Carriers Aim: to express the kernel of the WQFT! Solution: to introduce the multiplication operator. For f, g L 2 (R 2, H), we define their pointwise product as C[f, g](x) = f (x)g(x). Based on this, we define the left and right multiplicative operators, C[λ, ], resp. C[, λ], as C[λ, ]f (x) := f (x)λ, resp. f (x)c[, λ] := λf (x). 17/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

33 Multiplication operator Quaternionic Fourier transform Carriers Aim: to express the kernel of the WQFT! Solution: to introduce the multiplication operator. For f, g L 2 (R 2, H), we define their pointwise product as C[f, g](x) = f (x)g(x). Based on this, we define the left and right multiplicative operators, C[λ, ], resp. C[, λ], as C[λ, ]f (x) := f (x)λ, resp. f (x)c[, λ] := λf (x). 17/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

34 Multiplication operator - 2 Quaternionic Fourier transform Carriers Then C[λ, ] := C[, λ] and C[, λ] := C[λ, ]. Definition For a windowed function g L 2 (R 2, R), then g w,b (x) = e 2πjx 2ω 2 g(x b)c[e 2πix 1ω 1, ] is the kernel of the WQFT. 18/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

35 Multiplication operator - 2 Quaternionic Fourier transform Carriers Then C[λ, ] := C[, λ] and C[, λ] := C[λ, ]. Definition For a windowed function g L 2 (R 2, R), then g w,b (x) = e 2πjx 2ω 2 g(x b)c[e 2πix 1ω 1, ] is the kernel of the WQFT. 18/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

36 Consequences Quaternionic Fourier transform Carriers Then f (x)g w,b = g w,b f (x); so that (f, g w,b ) := f (x)g w,b d 2 x R 2 the WQFT can be write as Q g [f ](w, b) = (f, g w,b ). Theorem [Reconstruction formula] For g L 2 (R 2 ; R) non-zero real-valued window function, then every f L 2 (R 2 ; H) can be reconstructed via f (x) = 1 e 2πix 1ω 1 Q g [f ](w, b)g(x b)e 2πjx 2ω 2 d 2 wd 2 b. g L 2 R 2 R 2 19/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

37 Consequences Quaternionic Fourier transform Carriers Then f (x)g w,b = g w,b f (x); so that (f, g w,b ) := f (x)g w,b d 2 x R 2 the WQFT can be write as Q g [f ](w, b) = (f, g w,b ). Theorem [Reconstruction formula] For g L 2 (R 2 ; R) non-zero real-valued window function, then every f L 2 (R 2 ; H) can be reconstructed via f (x) = 1 e 2πix 1ω 1 Q g [f ](w, b)g(x b)e 2πjx 2ω 2 d 2 wd 2 b. g L 2 R 2 R 2 19/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

38 Consequences Quaternionic Fourier transform Carriers Then f (x)g w,b = g w,b f (x); so that (f, g w,b ) := f (x)g w,b d 2 x R 2 the WQFT can be write as Q g [f ](w, b) = (f, g w,b ). Theorem [Reconstruction formula] For g L 2 (R 2 ; R) non-zero real-valued window function, then every f L 2 (R 2 ; H) can be reconstructed via f (x) = 1 e 2πix 1ω 1 Q g [f ](w, b)g(x b)e 2πjx 2ω 2 d 2 wd 2 b. g L 2 R 2 R 2 19/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

39 Expansion Quaternionic Fourier transform Carriers Moreover, if the system {1g m,n : m, n Z 2 } is a Gabor orthonormal basis, a function f L 2 (R 2 ; H) admits the expansion f (x) = c m,n (1g m,n )(x) m,n Z 2 with c m,n = f, 1g m,n. This expansion can be regarded as the discrete form of the previous continuous reconstruction formula. 20/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

40 Quaternionic Zak transform Quaternionic Zak transform Definition Given a function f L 2 (R 2, H), we set its quaternionic Zak transform as Z H [f ](x, w) = k Z 2 e 2πjk 2ω 2 f (x k)e 2πik 1ω 1, x, w [0, 1) 2. Properties 1 Z H [f ] is a well defined function in L 2 ([0, 1) 2 [0, 1) 2, H); 2 Z H [f ](x + n, w) = e 2πjn 2ω 2ZH [f ](x, w)e 2πin 1ω 1, and Z H [f ](x, w + n) = Z H [f ](x, w), where n Z 2 ; 3 Z H is a unitary operator from L 2 (R 2, H) to L 2 ([0, 1) 2 [0, 1) 2, H), i.e., Zf, Zg = f, g ; 21/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

41 Quaternionic Zak transform Quaternionic Zak transform Definition Given a function f L 2 (R 2, H), we set its quaternionic Zak transform as Z H [f ](x, w) = k Z 2 e 2πjk 2ω 2 f (x k)e 2πik 1ω 1, x, w [0, 1) 2. Properties 1 Z H [f ] is a well defined function in L 2 ([0, 1) 2 [0, 1) 2, H); 2 Z H [f ](x + n, w) = e 2πjn 2ω 2ZH [f ](x, w)e 2πin 1ω 1, and Z H [f ](x, w + n) = Z H [f ](x, w), where n Z 2 ; 3 Z H is a unitary operator from L 2 (R 2, H) to L 2 ([0, 1) 2 [0, 1) 2, H), i.e., Zf, Zg = f, g ; 21/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

42 Quaternionic Zak transform Quaternionic Zak transform Definition Given a function f L 2 (R 2, H), we set its quaternionic Zak transform as Z H [f ](x, w) = k Z 2 e 2πjk 2ω 2 f (x k)e 2πik 1ω 1, x, w [0, 1) 2. Properties 1 Z H [f ] is a well defined function in L 2 ([0, 1) 2 [0, 1) 2, H); 2 Z H [f ](x + n, w) = e 2πjn 2ω 2ZH [f ](x, w)e 2πin 1ω 1, and Z H [f ](x, w + n) = Z H [f ](x, w), where n Z 2 ; 3 Z H is a unitary operator from L 2 (R 2, H) to L 2 ([0, 1) 2 [0, 1) 2, H), i.e., Zf, Zg = f, g ; 21/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

43 Quaternionic Zak transform Quaternionic Zak transform Definition Given a function f L 2 (R 2, H), we set its quaternionic Zak transform as Z H [f ](x, w) = k Z 2 e 2πjk 2ω 2 f (x k)e 2πik 1ω 1, x, w [0, 1) 2. Properties 1 Z H [f ] is a well defined function in L 2 ([0, 1) 2 [0, 1) 2, H); 2 Z H [f ](x + n, w) = e 2πjn 2ω 2ZH [f ](x, w)e 2πin 1ω 1, and Z H [f ](x, w + n) = Z H [f ](x, w), where n Z 2 ; 3 Z H is a unitary operator from L 2 (R 2, H) to L 2 ([0, 1) 2 [0, 1) 2, H), i.e., Zf, Zg = f, g ; 21/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

44 Quaternionic Zak transform Quaternionic Zak transform Properties (cont.) 4 f (x) = Z H [f ](x, w)dw, x R 2 ; [0,1) 2 5 F H [f ]( w) = e 2πin 1ω 1 Z H [f ](x, w)e 2πjn 2ω 2 dx, x R 2 ; [0,1) 2 Space Z of all φ : R 2 H such that φ(x + n, w) = e 2πjn 2ω 2 φ(x, w)e 2πin 1ω 1, φ(x, w + n) = φ(x, w), n Z 2, φ 2 = φ(x, w) 2 dxdw <. [0,1) 2 [0,1) 2 22/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

45 Quaternionic Zak transform Quaternionic Zak transform Properties (cont.) 4 f (x) = Z H [f ](x, w)dw, x R 2 ; [0,1) 2 5 F H [f ]( w) = e 2πin 1ω 1 Z H [f ](x, w)e 2πjn 2ω 2 dx, x R 2 ; [0,1) 2 Space Z of all φ : R 2 H such that φ(x + n, w) = e 2πjn 2ω 2 φ(x, w)e 2πin 1ω 1, φ(x, w + n) = φ(x, w), n Z 2, φ 2 = φ(x, w) 2 dxdw <. [0,1) 2 [0,1) 2 22/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

46 Quaternionic Zak transform Quaternionic Zak transform Properties (cont.) 4 f (x) = Z H [f ](x, w)dw, x R 2 ; [0,1) 2 5 F H [f ]( w) = e 2πin 1ω 1 Z H [f ](x, w)e 2πjn 2ω 2 dx, x R 2 ; [0,1) 2 Space Z of all φ : R 2 H such that φ(x + n, w) = e 2πjn 2ω 2 φ(x, w)e 2πin 1ω 1, φ(x, w + n) = φ(x, w), n Z 2, φ 2 = φ(x, w) 2 dxdw <. [0,1) 2 [0,1) 2 22/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

47 Consequences Quaternionic Zak transform The quaternion Zak transform Z H is an unitary map between L 2 (R 2 ; H) and Z. We know the inverse map as well: for every φ Z, Z 1 H φ(x) = [0,1) 2 φ(x, w)dw. Lemma Given f L 2 (R 2, H), and a Gabor system {1g m,n, m, n Z 2 } we have m,n Z 2 f, g m,n 2 = Z H [f ]Z H [1g m,n ] 2. 23/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

48 Consequences Quaternionic Zak transform The quaternion Zak transform Z H is an unitary map between L 2 (R 2 ; H) and Z. We know the inverse map as well: for every φ Z, Z 1 H φ(x) = [0,1) 2 φ(x, w)dw. Lemma Given f L 2 (R 2, H), and a Gabor system {1g m,n, m, n Z 2 } we have m,n Z 2 f, g m,n 2 = Z H [f ]Z H [1g m,n ] 2. 23/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

49 Consequences Quaternionic Zak transform The quaternion Zak transform Z H is an unitary map between L 2 (R 2 ; H) and Z. We know the inverse map as well: for every φ Z, Z 1 H φ(x) = [0,1) 2 φ(x, w)dw. Lemma Given f L 2 (R 2, H), and a Gabor system {1g m,n, m, n Z 2 } we have m,n Z 2 f, g m,n 2 = Z H [f ]Z H [1g m,n ] 2. 23/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

50 Quaternionic Zak transform Link with Weyl-Heisenberg algebra Annihilation/creation operators: a 1 = 1 2π x 1 +x 1, a 1 = 1 2π x 1 +x 1, a 2 = 1 2π x 2 +x 2, a 2 = 1 2π x 2 +x 2 Link with Zak transform: Z H [a 2 f ](x, w) = A 2 Z H [f ](x, w), Z H [f a 1 ](x, w) = Z H A 2 [f ](x, w) where A 1 = 1 2πi ( ω 1 + i x1 ) + x 1, A 2 = 1 2πj ( ω 2 + j x2 ) + x 2 ; Quaternionic time-frequency shift: Z H [M θ T p f ](x, w) = e 2πj(p 2ω 2 +θ 2 x 2 ) Z H [f ](x, w)e 2πi(p 1ω 1 +θ 1 x 1 ) 24/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

51 Quaternionic Zak transform Link with Weyl-Heisenberg algebra Annihilation/creation operators: a 1 = 1 2π x 1 +x 1, a 1 = 1 2π x 1 +x 1, a 2 = 1 2π x 2 +x 2, a 2 = 1 2π x 2 +x 2 Link with Zak transform: Z H [a 2 f ](x, w) = A 2 Z H [f ](x, w), Z H [f a 1 ](x, w) = Z H A 2 [f ](x, w) where A 1 = 1 2πi ( ω 1 + i x1 ) + x 1, A 2 = 1 2πj ( ω 2 + j x2 ) + x 2 ; Quaternionic time-frequency shift: Z H [M θ T p f ](x, w) = e 2πj(p 2ω 2 +θ 2 x 2 ) Z H [f ](x, w)e 2πi(p 1ω 1 +θ 1 x 1 ) 24/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

52 Quaternionic Zak transform Link with Weyl-Heisenberg algebra Annihilation/creation operators: a 1 = 1 2π x 1 +x 1, a 1 = 1 2π x 1 +x 1, a 2 = 1 2π x 2 +x 2, a 2 = 1 2π x 2 +x 2 Link with Zak transform: Z H [a 2 f ](x, w) = A 2 Z H [f ](x, w), Z H [f a 1 ](x, w) = Z H A 2 [f ](x, w) where A 1 = 1 2πi ( ω 1 + i x1 ) + x 1, A 2 = 1 2πj ( ω 2 + j x2 ) + x 2 ; Quaternionic time-frequency shift: Z H [M θ T p f ](x, w) = e 2πj(p 2ω 2 +θ 2 x 2 ) Z H [f ](x, w)e 2πi(p 1ω 1 +θ 1 x 1 ) 24/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

53 Case of Gaussian window Quaternionic Zak transform Gaussian window M θ T p φ(x), φ(x) = 2 1/2 e π x 2 : Z H [M θ T p φ](x, w) = e 2πj(p 2ω 2 +θ 2 x 2 ) e π x 2 Θ(z 2 )Θ(z 1 )e 2πi(p 1ω 1 +θ 1 x 1 ) with z 1 = ω 1 + ix 1, z 2 = ω 2 + jx 2 and Θ(z) = 2 1/2 θ 3 (z; i) - Jacobi elliptic function; Link with annihilation operator: (M θ T p φ)a 1 = (M θ T p φ)(ω 1 + ip 1 ) a 2 (M θ T p φ) = (ω 2 + jp 2 )(M θ T p φ) 25/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

54 Frame condition Quaternionic Zak transform Theorem A Gabor system {1g m,n, m, n Z 2 } is 1 a frame for L 2 (R 2, H) if there exist bounds 0 < A B < such that A Z H [g](x, w) 2 B, a.e. x, w R 2. 2 an orthonormal basis for L 2 (R 2, H) iff additionally Z H [g](x, w) 2 = 1, for a.e. x, w R 2. 26/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

55 Quaternionic Zak transform Balian-Low Theorem Let g L 2 (R 2, R) be such that the associated Gabor system {1g m,n, m, n Z 2 } is a frame for L 2 (R 2, H). Then x k ω k =, k = 1, 2, where x k = x kf 2 f 2, ω k = ω kf H [f ] 2, k = 1, 2. F H [f ] 2 27/28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals

56 References Quaternionic Zak transform [1] I. Daubechies: Ten lectures on wavelets, SIAM, [2] T. Bülow, G. Sommer, Hypercomplex signals: A novel extension of the analytic signal to the multidimensional case, IEEE Transactions on signal processing 49, N.o 11, , [3] P. Cerejeiras, Y. Fu, U. Kähler, Quaternionic Zac transform and the Balian-Low theorem, submitted. [4] P. Cerejeiras, Y. Fu, U. Kähler, The Balian-Low theorem for a new kind of Gabor systems, accepted in Applicable Analysis. [5] V.P. Palamodov, Relaxed Gabor expansion at critical density and a certainty principle, ArXiv, /28 P. Cerejeiras A Short-time Fourier transform for quaternionic signals