Novel selective transform for non-stationary signal processing

Transcription

1 Novel selective transform for non-stationary signal processing Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Czech Technical University in Prague Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

2 Contents Introduction 2 Basis functions for a new multispectral transformation 3 Fourier and Zolotarev Transforms 4 Applications Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

3 Non-stationary signals Non-stationary signals occur almost everywhere, in broadcasting, in seismic physics, and in numerous other interdisciplinary fields such as multimedia, telecommunications, radar, sonar, vibration analysis, speech processing, and medical diagnosis (as EEG, or ECG signals). click to play Scientists have been studying the time-frequency phenomena of such signals for a considerable period. piano tone g Figure: A non-stationary signal Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

4 Non-stationary signals There are several approaches available for addressing non-stationary signals, including the short time Fourier Transform (STFT), the fractional Fourier Transform (FRFT), the Wavelet Transform (WT), and the Hilbert-Huang Transform (HHT). A large group of non-linear multiresolution transforms is also available but very often they suffer from the difficult interpretation we have introduced a novel tool - Discrete Zolotarev Transform (DZT) - 29 Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

5 Discrete Zolotarev Transform Basis functions for Discrete Zolotarev Transform are derived from the family of Zolotarev polynomials, which are iso-extremal within two disjoint intervals Symmetrical 3 2 Antisymmetrical w w Figure: Symmetrical polynomial Z, (w,.52) and antisymmetrical iso-extremal function based on polynomial S, (w,.995) Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

6 Discrete Zolotarev Transform R. Spetik, Discrete Zolotarev Transform, Doctoral thesis, Czech Technical University in Prague, 26 p., 29. (DZT) is signal dependent transform, representing clearly the non-stationary segments (DZT) partially avoids Heisenberg principle of uncertainty, reduces spectral leakages (DZT)provides a non-stationarity measure STFT Wavelets Hilbert-Huang DZT inversion yes yes, but... no yes time resolution limited yes yes yes frequency resolution yes bad or n/a floating components yes Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

7 Discrete Zolotarev Transform In this lecture few words on mathematical problems concerning the iso-extremal polynomials construction of basis functions and definition of the Discrete Zolotarev Transform numerical experiments and applications to several signals Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

8 Symmetrical Zolotarev polynomials - approximation ( ) 2 ( w 2 )(w 2 κ 2 dzm,m (w) ) = 4 m 2 w 2 ( Zm,m 2 dw (w)) Z m,m (w,k) w Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

9 The second order differential equation ( ) 2 dy(w) ( w 2 )(w 2 κ 2 ) = 4 m 2 w 2 ( y 2 (w)) dw [ w(w 2 κ 2 ) ( w 2 ) d 2 y dw 2 w dy dw ( 2w 2 κ 2 ) y(w) T m κ 2 = ] +κ 2 ( w 2 ) dy dw + 4 m2 w 3 y = m b(2l)w 2l l=.. is a polynomial of variable x Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

10 An alternative representation ( 2w 2 κ 2 ) y(w) T m κ 2 = m b(2l)w 2l = l= m a(2l)t 2l (w) l=.. is developed in terms of Chebyshev polynomials Inserting y(w) = m a(2l)t 2l (w) in differential equation l= [ w(w 2 κ 2 ) ( w 2 ) d 2 y dw 2 w dy dw we obtain... ] +κ 2 ( w 2 ) dy dw + 4 m2 w 3 y = Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

11 Recursive algorithm for the coefficients a(2l) given m,κ initialisation a(2m) = ( k 2 ) m a(2m + 2) = a(2m + 4) = recursive body (for l = m to [ m 2 l 2] a(2l) = + [ 3(m 2 (l+) 2 )+(2l+2)(2l+)κ 2] a(2l+2) + [ 3(m 2 (l+2) 2 )+(2l+4)(2l+5)κ 2] a(2l+4) + [ m 2 (l+3) 2] a(2l+6) end ) Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

12 Basis function Zcos(φ) A symmetrical Zolotarev polynomial of the st kind ( ) m Z m,m (w,κ) = = 2m µ= a(µ)t µ (w) = m a(2l)t 2l (w). l= m b(2l)w 2k l= provides the basis Zolotarev cosine function m Zcos(φ) = a(2l) cos(2lφ). l= Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

13 Basis function Zcos(φ) Zcos(φ) φ a(µ) µ Figure: Zolotarev cosine function Zcos(φ) and its coefficients Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

14 Linear differential equation For a general Zolotarev polynomial of the st kind Z p,q (w) the second order linear differential equation is valid [ ] d f(w) d f(w) g(w) dw g(w) dw Z p,q(w) + n 2 Z p,q (w) = () where f(w) = ( w 2 )(w w p )(w w s ), and g(w) = (w w m ) Z,7 (w,.65) ws wm wp w Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

15 Basis function Zsin(φ) If we look for the second solution of eq. () having zeros at w = ±, by substitution w 2 S p,q (w,κ) we arrive to the differential equation f(w) w 2 g(w) d dw [ ] f(w) d w 2 S p,q (w,κ) + g(w) dw +n 2 S p,q (w,κ) =. We have developed the backward recursion for coefficients a + (l) of Zolotarev polynomial of the fourth kind S p,q (w,κ) S p,q (w,κ) = ( ) p n a + (l)u l (w). l= (2) Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

16 Recursive algorithm for the coefficients a + (l) given p, q, κ initialisation n = p + q, u = 2p + 2n + 2 K(κ) w p = 2 cd 2 (u κ), w s = 2 cn 2 (u κ) w q = w p + w s, w m = w s + 2 sn(u κ)cn(u κ) Z(u κ) 2 dn(u κ) α(n) = α(n+i) = for i =, 2,..., 5 Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

17 ...coefficients a + (l)... cont. (for m = n+2 to 3) 8c() = n(n + 2) (m + 3)(m + 5) 4c(2) = 3w m [n(n+2) (m + 2)(m + 4)]+(m+3)(2m + 7)(w m w q ) 8c(3) = 3[n(n + 2) (m+)(m + 3)]+2w m [(n+) 2 w m (m + 2) 2 w q ] 4(m+2)(m + 3)(w p w s w m w q ) 2c(4) = 3[(n+) 2 w m (m + ) 2 w q ] (m + ) 2 (w m w q ) +2w m [(n + ) 2 w 2 m (m+) 2 w p w s ] 8c(5) = 3[n(n + 2) (m )(m + )]+2w m [(n+) 2 w m m 2 w q ] 4m(m )(w p w s w m w q ) 4c(6) = 3w m [n(n+2) (m 2)m]+(m )(2m 3)(w m w q ) 8c(7) = n(n + 2) (m 3)(m ) α(m 3) = 6 ( ) l c(l)α(m + 4 l) c(7) (end loop on m) l= Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

18 coefficients a + (l)... cont. normalisation s(n) = n α(k)(k + ) k= (for l = to n) a + (l) = ( ) p (n+) α(l) s(n) (end loop on l) Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

19 Basis function Zsin(φ) For p = 2m and q = the function w 2 S 2m, (w,κ) is an odd function and has following form (3) w 2 S 2m, (w,κ) = 2m+ w 2 µ= a + (µ)u µ (w) = = m+ w 2 a + (2l )U 2l (w). It provides with w = cosφ definition of Zolotarev sine function l= m+ Zsin(φ) = a + (2l ) sin(2lφ). l= Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

20 Basis function Zsin(φ) Zsin(φ) a + (µ) φ µ Figure: Zolotarev sine functions Zsin(φ) and its coefficients Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

21 Application of Zolotarev polynomials in Signal Processing in Digital filter design Vlcek M., Zahradnik P.: Almost Equiripple Low-Pass FIR Filters, Circuits, Systems, and Signal Processing, Birkhauser, April 23, vol. 32, no. 2, pp Zahradnik P., Vlcek M.: Perfect Decomposition Narrow-Band FIR Filter Banks, IEEE Transactions on Circuits and Systems II: Express Briefs, October 22, vol. 59, no., pp Zahradnik P., Vlcek M.: Equiripple Approximation of Half-Band FIR Filters, IEEE Trans. on Circuits and Systems, vol. 56, no. 2, December 29, pp recent research in Multispectral Analysis of Non-stationary Signals - following slides Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

22 Fourier and Zolotarev basis functions For exp(i2πlt) = cos(2πlt) + i sin(2πlt) a band-limited signal s(t) can be expressed as s(t) = N l= N S(l) exp(i2πlt), where the coefficients S(l) represent the frequency spectrum of the signal S(l) = exp( i2πlt), s(t) =, l > N. (3) Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

23 Fourier and Zolotarev basis functions For the basis functions Zexp(i2πlt) = Zcos(2πlt) + i Zsin(2πlt) a band-limited signal s(t) can be expressed as s(t) = N l= N S Z (l)cexp(i2πlt). Functions Cexp(i2πlt) form a biorthogonal basis to Zexp(i2πlt) ( ) {Zexp(i2πlt)} N l= N = {Cexp(i2πlt)} N l= N Coefficients S Z (l) define the Zolotarev spectrum S Z (l) = Zexp( i2πlt), s(t) =, l > N. Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

24 Discrete Zolotarev Transform The Zolotarev basis can be transferred from continuous time domain into the discrete time domain ( Zexp i 2πln ) ( ) ( ) 2πln 2πln = Zcos + i Zsin N N N Then the band-limited signal s(n) is expressed as s(n) = N l= N ( S Z (l)cexp i 2πln ). N Zolotarev spectrum is defined as ( S Z (l) = Zexp i 2πln ), s(n) =, l > N. N Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

25 Discrete Zolotarev Transform Using a matrix notion, the band-limited signal s(n) can be written as s = W C S Z, where W C contains the Zolotarev synthesis functions and S Z consists of Zolotarev spectrum coefficients. Thus Zolotarev spectra are given as S Z = W Z s, where W Z contains the Zolotarev analysis functions and it is biorthogonal W Z W C = E. Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

26 Analysis functions of DZT - part 4 Zolotarev spectrum Z [ ] Real part of analysis ADZT function l = spectrum index l [ ] Imaginary part of analysis ADZT function l = l = l = l = l = l = n [ ] l = l = l = l = l = n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

27 Synthesis functions of DZT - part 4 Zolotarev spectrum Z [ ] Real part of analysis ADZT function l = spectrum index l [ ] Imaginary part of analysis ADZT function l = l = l = l = l = l = n [ ] l = l = l = l = l = n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

28 Analysis functions of DZT - part 2 4 Zolotarev spectrum Z [ ] Real part of synthesis ADZT function l = spectrum index l [ ] Imaginary part of synthesis ADZT function l = l = l = l = l = l = n [ ] l = l = l = l = l = n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

29 Synthesis functions of DZT - part 2 4 Zolotarev spectrum Z [ ] Real part of synthesis ADZT function l = spectrum index l [ ] Imaginary part of synthesis ADZT function l = l = l = l = l = l = n [ ] l = l = l = l = l = n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

30 Suppression of spectral leakage by Approximated Discrete Zolotarev Transform (ADZT) 5 Discrete Fourier Transform (DFT) S(i) [ ] spectral index i [ ] Approximated Discrete Zolotarev Transform (ADZT) 5 Z(i) [ ] spectral index i [ ] ( s(n) = sin 2πn ), N = 32, n =,...N N Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

31 Suppression of spectral leakage by Approximated Discrete Zolotarev Transform (ADZT) 5 Discrete Fourier Transform (DFT) S(i) [ ] spectral index i [ ] Approximated Discrete Zolotarev Transform (ADZT) 5 Z(i) [ ] spectral index i [ ] ( s(n) = sin.2 2πn ), N = 32, n =,...N N Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

32 Suppression of spectral leakage by Approximated Discrete Zolotarev Transform (ADZT) 5 Discrete Fourier Transform (DFT) S(i) [ ] spectral index i [ ] Approximated Discrete Zolotarev Transform (ADZT) 5 Z(i) [ ] spectral index i [ ] ( s(n) = sin.3 2πn ), N = 32, n =,...N N Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

33 Suppression of spectral leakage by Approximated Discrete Zolotarev Transform (ADZT) 5 Discrete Fourier Transform (DFT) S(i) [ ] spectral index i [ ] Approximated Discrete Zolotarev Transform (ADZT) 5 Z(i) [ ] spectral index i [ ] ( s(n) = sin.8 2πn ), N = 32, n =,...N N Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

34 Suppression of spectral leakage by Approximated Discrete Zolotarev Transform (ADZT) Discrete Fourier Transform (blue) and Approximated Discrete Zolotarev Transform (red) S(i) and Z(i) [ ] S(i) and Z(i) [ ] S(i) and Z(i) [ ] S(i) and Z(i) [ ] spectral index i [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

35 Narrowband and broadband STFT spectrograms of a test signal I 2 Sum of Gaussian impulse, Dirac impulse, Unit step and sum of two sines s(n) [ ] frequency [Hz] n [ ] Narrowband STFT frequency [Hz] n [ ] Broadband STFT n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

36 Adaptive STFT spectrogram and STADZT zologram 2 Sum of Gaussian impulse, Dirac impulse, Unit step and sum of two sines y(n) [ ] frequency [Hz] n [ ] STFT with optimal window length (MESP) frequency [Hz] n [ ] STADZT n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

37 Narrowband and broadband STFT spectrograms of a test signal II Sum of two sines and two dirac impulses s(n) [ ] 5 frequency [Hz] n [ ] Narrowband STFT frequency [Hz] n [ ] Broadband STFT n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

38 Adaptive STFT spectrogram and STADZT zologram Sum of two sines and two dirac impulses s(n) [ ] 5 frequency [Hz] n [ ] STFT with optimal window length (MESP) frequency [Hz] n [ ] STADZT n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

39 Narrowband and broadband STFT spectrograms of a test signal III 2 Sum of two frequency modulated signals s(n) [ ] frequency [Hz] n [ ] Narrowband STFT frequency [Hz] n [ ] Broadband STFT n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

40 Adaptive STFT spectrogram and STADZT zologram 2 Sum of two frequency modulated signals s(n) [ ] frequency [Hz] n [ ] STFT with optimal window length (MESP) frequency [Hz] n [ ] STADZT n [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

41 Narrowband and broadband STFT spectrograms of an ECG signal R Electrocardiograph signal [ ].5 P T Q S Narrowband STFT 6 spectral index [ ] Broadband STFT QRS spectral index [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

42 Adaptive STFT spectrogram and STADZT zologram R Electrocardiograph signal [ ].5 P T Q S Adaptive STFT with optimal window length (MESP) 6 QRS 5 spectral index [ ] spectral index [ ] STADZT QRS P Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

43 STADZT zologram with added leading and trailing zeros R Electrocardiograph signal [ ].5 P T Q S STADZT without added by leading and trailing zeros 6 spectral index [ ] 5 QRS spectral index [ ] 2 QRS 8 P STADZT added by leading and trailing zeros T Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

44 ECG signal - STFT spectrogram signal [ ] ECG signal wlen = 28, lpz = tpz = spectral index [ ] Sum of S [ ] Sum of S [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

45 ECG signal - STADZT zologram with added leading and trailing zeros signal [ ] ECG signal wlen = 28, lpz = tpz = spectral index [ ] Sum of Z [ ] 4 Sum of Z [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

46 ECG signal - STADZT k-gram with added leading and trailing zeros signal [ ] ECG signal wlen = 28, lpz = tpz = spectral index [ ] Sum of Zk [ ] Sum of Zk [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

47 Broadband and narrowband STFT spectrograms and adaptive STFT spectrogram signal [ ].2 Speech czech word "osum" "O" "S" "UM" Narrowband STFT spectral index [ ] spectral index [ ] spectral index [ ] Broadband STFT Adaptive STFT Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

48 STFT spectrogram and STADZT zolograms with and without added leading and trailing zeros signal [ ].2 "O" Speech czech word "osum" "S" "UM" STFT wlen = 28 spectral index [ ] spectral index [ ] spectral index [ ] STADZT without added leading and trailing zeros wlen = 28, lpz = tpz = STADZT added leading and trailing zeros wlen = 28, lpz = tpz = Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

49 Speech - STFT spectrogram signal [ ].2 Speech czech word "osum" wlen = spectral index [ ] Sum of S [ ] 2 Sum of S [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

50 Speech - STADZT zologram signal [ ].2 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Z [ ] Sum of Z [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

51 Speech - STADZT k-gram signal [ ].2 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Zk [ ] 4 Sum of Zk [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

52 Speech - STADZT zologram with added leading and trailing zeros signal [ ].2 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Z [ ] 5 Sum of Z [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

53 Speech - STADZT k-gram with added leading and trailing zeros signal [ ].2 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Zk [ ] 4 Sum of Zk [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

54 Detail of vowel o - STFT spectrogram signal [ ].2 Speech czech word "osum" wlen = spectral index [ ] Sum of S [ ] Sum of S [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

55 Detail of vowel o - ADZT zologram with added leading and trailing zeros signal [ ].2 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Z [ ] 5 Sum of Z [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

56 Detail of vowel o - ADZT k-gram with added leading and trailing zeros signal [ ].2 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Zk [ ] Sum of Zk [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

57 Detail of consonant m - STFT spectrogram signal [ ].5 Speech czech word "osum" wlen = spectral index [ ] Sum of S [ ] 5 Sum of S [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

58 Detail of consonant m - ADZT zologram with added leading and trailing zeros signal [ ].5 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Z [ ] 2 Sum of Z [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

59 Detail of consonant m - ADZT k-gram with added leading and trailing zeros signal [ ].5 Speech czech word "osum" wlen = 28, lpz = tpz = spectral index [ ] Sum of Zk [ ] 4 Sum of Zk [ ] Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23

60 Final remarks We perform basic research in the domain Novel selective transforms for non-stationary signal processing... recent results are available at We invite partners who are willing to cooperate with us in particular research projects. Miroslav Vlcek, Pavel Sovka, Vaclav Turon, Radim Spetik Novel selective transform, RWTH Aachen, November 28, 23