Topics in Time-Frequency Analysis
Total Page:16
File Type:pdf, Size:1020Kb
Kernels Multitaper basics MT spectrograms Concentration Topics in time-frequency analysis Maria Sandsten Lecture 7 Stationary and non-stationary spectral analysis February 10 2020 1 / 45 Kernels Multitaper basics MT spectrograms Concentration Wigner distribution The Wigner distribution is defined as, Z 1 τ ∗ τ −i2πf τ Wz (t; f ) = z(t + )z (t − )e dτ; −∞ 2 2 with t is time, f is frequency and z(t) is the analytic signal. The ambiguity function is defined as Z 1 τ ∗ τ −i2πνt Az (ν; τ) = z(t + )z (t − )e dt; −∞ 2 2 where ν is the doppler frequency and τ is the lag variable. 2 / 45 Kernels Multitaper basics MT spectrograms Concentration Ambiguity function Facts about the ambiguity function: I The auto-terms are always centered at ν = τ = 0. I The phase of the ambiguity function oscillations are related to centre t0 and f0 of the signal. I jAz1 (ν; τ)j = jAz0 (ν; τ)j when Wz1 (t; f ) = Wz0 (t − t0; f − f0). I The cross-terms are always located away from the centre. I The placement of the cross-terms are related to the interconnection of the auto-terms. 3 / 45 Kernels Multitaper basics MT spectrograms Concentration The quadratic class The quadratic or Cohen's class is usually defined as Z 1 Z 1 Z 1 Q τ ∗ τ i2π(νt−f τ−νu) Wz (t; f ) = z(u+ )z (u− )φ(ν; τ)e du dτ dν: −∞ −∞ −∞ 2 2 The filtered ambiguity function can also be formulated as Q Az (ν; τ) = Az (ν; τ) · φ(ν; τ); and the corresponding smoothed Wigner distribution as Q Wz (t; f ) = Wz (t; f ) ∗ ∗Φ(t; f ); using the corresponding time-frequency kernel. 4 / 45 Kernels Multitaper basics MT spectrograms Concentration Properties of the ambiguity kernel In order to preserve the marginals and the total energy condition of the time-frequency distribution, the ambiguity kernel must fulfill φ(0; τ) = φ(ν; 0) = 1; φ(0; 0) = 1: For a real-valued distribution, φ∗(−ν; −τ) = φ(ν; τ): 5 / 45 Kernels Multitaper basics MT spectrograms Concentration The Choi-Williams distribution The Choi-Williams distribution or the Exponential distribution (ED), is defined from the ambiguity kernel 2 2 − ν τ φED (ν; τ) = e σ ; where σ is a design parameter. The distribution satisfies the marginals and total energy condition, φED (0; τ) = φED (ν; 0) = 1; φED (0; 0) = 1: The Choi-Williams distribution is real-valued, 2 2 ∗ − ν τ φED (−ν; −τ) = e σ = φED (ν; τ): The Choi-Williams distribution works well for suppression of cross-terms found on a diagonal line between the auto-terms. 6 / 45 Kernels Multitaper basics MT spectrograms Concentration Example 7 / 45 Kernels Multitaper basics MT spectrograms Concentration Separable kernels Separable kernels is defined by φ(ν; τ) = G1(ν) · g2(τ); The smoothed Wigner-Ville distribution becomes F Wz (t; f ) = Wz (t; f ) ∗ ∗Φ(t; f ) = g1(t) ∗ Wz (t; f ) ∗ G2(f ): I If G1(ν) = 1 a doppler-independent kernel is given as φ(ν; τ) = g2(τ), resulting in the pseudo-Wigner distribution, with smoothing only in frequency. I If g2(τ) = 1 a lag-independent kernel is φ(ν; τ) = G1(ν) where the time-frequency formulation gives smoothing only in time. 8 / 45 Kernels Multitaper basics MT spectrograms Concentration Doppler-independent kernel 9 / 45 Kernels Multitaper basics MT spectrograms Concentration Lag-independent kernel 10 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation The Rihaczek distribution The energy of a complex-valued signal is Z 1 Z 1 E = jz(t)j2dt = z(t)z∗(t)dt −∞ −∞ Z 1 Z 1 = z(t)Z ∗(f )e−i2πft dfdt: −∞ −∞ The Rihaczek distribution is defined as ∗ −i2πft Rz (t; f ) = z(t)Z (f )e : 11 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation The Rihaczek distribution The marginals are satisfied as Z 1 Z 1 ∗ −i2πft 2 Rz (t; f )df = z(t) Z (f )e df = jz(t)j ; −∞ −∞ and Z 1 Z 1 ∗ −i2πft 2 Rz (t; f )dt = Z (f ) z(t)e dt = jZ(f )j : −∞ −∞ The ambiguity kernel is found as φ(ν; τ) = e−iπντ : (Assignment!) From this expression we also easily verify that the marginals are satisfied as φ(ν; 0) = φ(0; τ) = 1. Is the Rihaczek distribution real-valued? (φ∗(−ν; −τ) = φ(ν; τ)?): 12 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Example 13 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Example 14 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation The spectrogram again We can also formulate the quadratic class as Z 1 Z 1 Q −i2πf τ Wz (t; f ) = rz (u; τ)ρ(t − u; τ)e du dτ; −∞ −∞ where the time-lag kernel is defined as Z 1 ρ(t; τ) = φ(ν; τ)ei2πνt dν; −∞ and the IAF as τ τ r (t; τ) = z(t + )z∗(t − ): z 2 2 (Assignment!) 15 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation The spectrogram kernel The spectrogram also belongs to the quadratic class which can be seen from the following derivation: Z 1 ∗ −i2πft1 2 Sz (t; f ) = j h (t1 − t)z(t1)e dt1j ; −∞ Z 1 Z 1 ∗ −i2πft1 ∗ i2πft2 = ( h (t1 − t)z(t1)e dt1)( h(t2 − t)z (t2)e dt2): −∞ −∞ τ τ With t1 = u + 2 and t2 = u − 2 , we get Sz (t; f ) = Z 1 Z 1 τ τ τ τ = z(u+ )z∗(u− )h∗(u+ −t)h(u− −t)e−i2πf τ dτdu: −∞ −∞ 2 2 2 2 16 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation The spectrogram kernel We can identify the time-lag distribution τ τ r (u; τ) = z(u + )z∗(u − ); z 2 2 and the time-lag kernel τ τ ρ (t − u; τ) = h∗(u − t + )h(u − t − ); h 2 2 giving Z 1 Z 1 −i2πf τ Sz (t; f ) = rz (u; τ)ρh(t − u; τ)e dτdu; −∞ −∞ which we recognize as the quadratic class. 17 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation The spectrogram kernel The window function can be expressed as an ambiguity kernel φh(ν; τ) which is the ambiguity function of a window h(t), i.e., Z 1 ∗ τ τ −i2πνt φh(ν; τ) = h (−t + )h(−t − )e dt: −∞ 2 2 The spectrogram does not fulfill the marginals. Why? 18 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Example Wigner−Ville Spectrogram Choi−Williams 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 20 20 20 0.1 0.1 0.1 0 0.05 0 0.05 0 0.05 0 0 0 −20 −0.05 −20 −0.05 −20 −0.05 τ −0.1 ν τ −0.1 ν τ −0.1 ν 19 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Multitaper spectrogram analysis Replacement with u = (t1 + t2)=2 and τ = t1 − t2 in Z 1 Z 1 Q −i2πf τ Wz (t; f ) = rz (u; τ)ρ(t − u; τ)e du dτ; −∞ −∞ give ZZ t1 + t2 t1 + t2 W Q (t; f ) = r ( ; t −t )ρ(t− ; t −t )e−i2πf (t1−t2)dt dt ; z z 2 1 2 2 1 2 1 2 where the IAF t + t t + t t − t t + t t − t r ( 1 2 ; t −t ) = z( 1 2 + 1 2 )z∗( 1 2 − 1 2 ) = z(t )z∗(t ): z 2 1 2 2 2 2 2 1 2 20 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Multitaper spectrogram analysis Define a rotated time-lag kernel as t + t ρrot (t ; t ) = ρ( 1 2 ; t − t ); 1 2 2 1 2 and we get ZZ Q ∗ rot −i2πft1 i2πft2 Wz (t; f ) = z(t1)z (t2)ρ (t − t1; t2)e e dt1dt2: rot rot ∗ If the kernel ρ (t1; t2) = (ρ (t2; t1)) , then solving the integral Z rot ρ (t1; t2)u(t1)dt1 = λ u(t2); results in eigenvalues λk and eigenfunctions uk (t). 21 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Multitaper spectrogram analysis The quadratic class for any kernel is now rewritten as a weighted sum of spectrograms, 1 Z Q X −i2πft1 ∗ 2 Wz (t; f ) = λk j z(t1)e uk (t − t1)dt1j : k=1 The quadratic class of time-frequency distributions can always be approximated, K Z Q X −i2πft1 ∗ 2 Wz (t; f ) ≈ λk j z(t1)e uk (t − t1)dt1j ; k=1 where K is chosen as the number including all λk > . If K is reasonable small, the computations are efficient. 22 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Classification of time-frequency representations M h X T Sz (t; f ) = Φ (t; f )∗∗ Wz (t; f ) ≈ sj ·uj (t)∗ vj (f ) ∗ Wz (t; f ) ; j=1 Johan Brynolfsson and Maria Sandsten, ”Classification of One-Dimensional Non-Stationary Signals Using the Wigner-Ville Distribution in Convolutional Neural Networks, EUSIPCO, Kos island, Greece, 2017. 23 / 45 Kernels Multitaper basics MT spectrograms Concentration Rihaczek dist. Spectrogram kernel Multitaper interpretation Classification of time-frequency representations Eight different classes of multi-component chirp signals (class 1 and 2), sinusoids (class 3-6) and multi-component short Gaussian signals (class 7 and 8).