STFT and DGT phase conventions and phase derivatives interpretation

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1 STFT DGT phase conventions phase derivatives interpretation Zdeněk Průša January 14, 2016 Contents About this document 2 1 STFT phase conventions Phase derivatives DGT Phase derivatives Matlab functions Algorithms Reassignment 6 4 Alternative conventions 6 1

2 About this document The purpose of this document is to sumarize common STFT DGT phase conventions their effect on the phase derivatives. Consequently, the phase derivatives are linked to terms like instantaneous time t(ω, t), instantaneous frequency ω(ω, t), chanelized instantaneous frequency CIF(ω, t), local group delay LGD(ω, t), to derivatives of t(ω, t) ω(ω, t) over time or frequency t, ω, t ω. Numerical approximation of the phase derivatives using DGT is discussed also. 1 STFT phase conventions STFT of signal f(t) using window g(t) phase convention pc will be denoted as The translation modulation operators are defined as V pc (ω, t) M(ω, t)e iφpc(ω,t). (1) T τ f(t) f(t τ) M ω f(t) f(t)e jωt. (2) STFT with frequency-invariant phase is obtained as V fi (ω, t) f, M ω T t g (3) STFT with time-invariant phase is obtained as STFT with symmetric phase is obtained as f(τ)e jωτ g(τ t) dτ. (4) V ti (ω, t) f, T t M ω g (5) f(τ)e jω(τ t) g(τ t) dτ (6) f(τ + t)e jωτ g(τ) dτ (7) e jωt V fi (ω, t). (8) V sym (ω, t) f, M ω/2 T t M ω/2 g (9) The overline g(t) denotes complex conjugation. The relations between the phases follow directly f(τ)e jωτ/2 e jω(τ t)/2 g(τ t) dτ (10) f(τ)e jω(τ t/2) g(τ t) dτ (11) f(τ + t/2)e jωτ g(τ t/2) dτ (12) e jωt/2 V fi (ω, t) (13) φ fi (ω, t) φ ti (ω, t) ωt φ sym (ω, t) ωt/2. (14) 2

3 1.1 Phase derivatives The phase derivatives along frequency are linked in the following way: t(ω, t) t φ ti φ fi t 2 φ sym, (15) where t(ω, t) denotes the absolute instantaneous time. Similarly, phase derivatives along time are linked in the following way: ω(ω, t) φ ti ω + φ fi ω 2 + φ sym, (16) where ω(ω, t) denotes the instantaneous frequency. Second derivatives do not depend on the phase convention anymore i.e. 2 φ ti 2 2 φ ti 2 Mixed phase derivatives are linked as follows: 2 φ fi 2 2 φ fi φ ti 2 φ fi φ sym 2 φ sym 2 (17) 2 φ sym 2. (18) 2 φ ti φ fi φ sym. (20) they do not depend on the order of the directions. In [1], the following quantities were defined: the chanelized instantaneous frequency as local group delay as (19) CIF(ω, t) φ ti (ω, t) (21) LGD(ω, t) φ ti (ω, t). (22) We define relative instantaneous frequency as RIF(ω, t) φ fi (ω, t) (23) Moreover, instantaneous slope was defined as a directional derivative in the direction of the unit vector α [α ω, α t ] which for α [0, 1] becomes for α [1, 0] becomes IS α (ω, t) IS [0,1] (ω, t) ω α t α 2 φ ti 2 2 φ fi 2 φ ti IS [1,0] (ω, t) (26) 2 φ fi 2 The authors of [2] describe the physical meaning of the mixed derivatives such that 0 (27) for constant sinusoidal components similarly for impulsive components. 3 (24) (25) 0 (28)

4 2 DGT DGT of signal f C L using window g phase convention pc will be denoted as where m n are frequency time indices respectivelly. Frequency-invariant phase is obtained as c pc (m, n) s(m, n)e iϕpc(m,n). (29) c fi (m, n) f, M mb T na g (30) f(l)e i2πmbl/l g(l an) (31) l0 for m 0,..., M 1 n 0,..., N 1 where M L/b is the number of channels, N L/a numer of time shifts, a is a hop factor in time b hop factor in frequency. T na is a circular translation operator (T na h)(l) h(l an) M mb is a modulation operator (M mb h)(l) h(l)e i2πmbl/l. l an is assumed to be evaluated modulo L. Time-invariant phase is obtained as c ti (m, n) f, T na M mb g (32) f(l)e i2πmb(l an)/l g(l an) (33) l0 f(l + an)e i2πmbl/l g(l) (34) l0 l + am is assumed to be evaluated modulo L. Symmetric phase is obtained as e i2πmbna/l c fi (m, n) (35) c sym (m, n) f, M mb/2 T na M mb/2 g (36) f(l)e i2πmb(l an/2)/l g(l an) (37) l0 e iπmbna/l c fi (m, n). (38) Note that the symmetric case introduces a phase discontinuity at the borders if b is odd. The relations between the phases follow directly for m, n as before. 2.1 Phase derivatives ϕ fi (m, n) ϕ ti (m, n) 2πmbna/L ϕ sym (m, n) πmbna/l (39) DGT amounts to STFT defined on L-periodic signals sampled uniformly with step 1. In the discrete case, we can only approximate the derivatives numerically. In this document, we abuse the notation slightly we denote the numerical approximation of a phase derivatives as ϕ(m,n) ϕ(m,n) respectivelly. We further assume that the derivatives are scaled such that the values are in samples. In the similar spirit as in sec.??, we list the relationships between the approximate phase derivatives. The phase derivatives along frequency are linked in the following way: n(m, n)a na ϕ ti ϕ fi an 2 ϕ sym, (40) where n(m, n) denotes the possibly noninteger instantaneous time index. 4

5 Similarly, phase derivatives along time are linked in the following way: m(m, n)b ϕ ti bm + ϕ fi where m(m, n) denotes the possibly noninteger instantaneous frequency index. Second derivatives do not depend on the phase convention anymore i.e. 2 ϕ ti 2 2 ϕ ti 2 Mixed phase derivatives are linked as follows: n(m, n) 2 ϕ fi 2 2 ϕ fi ϕ ti 2 ϕ fi ϕ sym bm 2 + ϕ sym, (41) 2 ϕ sym 2 (42) 2 ϕ sym 2. (43) m(m, n) 2 ϕ ti ϕ fi ϕ sym. (45) they do not depend on the order of the directions. 2.2 Matlab functions Function [tgrad,fgrad]gabphasegrad(...) returns tgrad fgrad which approximate the following: tgrad ϕ fi (44) RIF(m, n) (46) fgrad ϕ ti LGD(m, n) (47) We define this convention as relative as the time-frequency center of gravity functions contain only additions: n(m, n)a na + LGD(m, n) (48) m(m, n)b mb + RIF(m, n). (49) gabphasederiv computes approximates of the derivatives directly. (default), freqinv, symphase relative. 2.3 Algorithms It accepts flags freqinv The numerical differentiation is a well established part of numerical analysis. It however cannot be applied directly to phases as they live on a torus [ π, π[. For dense enough sampling, the phase unwrapping can fix that. The simplest algorithm for phase unwrapping adds or substracts 2π when an abs. value of a phase difference of two neighboring coefficients is greater than π. This however only works for dense enough sampling since it relies on the fact that the abs. difference of phase of two neighboring coefficients is less than π. In practise, a prior information is used to help the phase unwrapping task such as a constant factor phase increase is assumed in each channel m along time index n see chapter in [?] (note that ϕ ϕ denote frequency time invariant phases respectively). The phase wrapping problem arises in other fields such as like radar interferometry more advanced algorithms has been developed there. TBD: Auger Flrin algorithm TBD: Nelson algorithm 5

6 3 Reassignment Equations (40),(41) already give the formulas for the T-F reassignment [3, 4]. Recently, an adjustable version of the reassignment was introduced in [5]. The new time-frequency center of gravity depends on parameter µ R + is given by ( ) ( ) ω(ω, t) ω t(ω, t) t ( 1 t R(ω, t) + µi) R(ω, t) (50) where I is two-dimensional identity matrix, R(ω, t) is a negative of a relative displacement vector ( ) ( ) ω ω(ω, t) R(ω, t) (51) t t(ω, t) t R(ω, t) is a Jacobian matrix such that t R(ω, t) ( R(ω,t), ) R(ω,t) (52) Equation (??) can be rewritten such that ( t R(ω, t) + µi) ( ) ω(ω, t) t(ω, t) ( ) ( ω t ω(ω, t) t(ω, t) ) ( ) ω t (53) 4 Alternative conventions In the literature, two deviations from the conventions presented in (4),(8) (13) can be found. Namely the window g is sometimes time-reversed, then for fixed ω the formulas become convolutions the window is sometimes not conjugated. Both facts have no effect when using odd-symmetric real windows. References [1] D. J. Nelson, Instantaneous higher order phase derivatives, Digital Signal Processing, vol. 12, no. 2-3, pp , [2] K. R. Fitz S. A. Fulop, A unified theory of time-frequency reassignment, CoRR, vol. abs/ , [3] K. Kodera, R. Gendrin, C. Villedary, Analysis of time-varying signals with small bt values, Acoustics, Speech Signal Processing, IEEE Transactions on, vol. 26, pp , Feb [4] F. Auger P. Flrin, Improving the readability of time-frequency time-scale representations by the reassignment method, Signal processing, IEEE Transactions on, vol. 43, no. 5, pp , [5] F. Auger, E. Chasse-Mottin, P. Flrin, On phase-magnitude relationships in the shorttime fourier transform, Signal Processing Letters, IEEE, vol. 19, pp , May