The TT-transform stands for time-time transform and has been derived as
an inverse Fourier transform of the time-frequency S-transform. So far, it has been
proposed that the diagonal of the TT-transform can be used for signal characterisation.
We show here an alternative and simplified derivation of the TT-transform
which enables a better understanding of this transform.Peer ReviewedPostprint (published version

Dañobeitia, Juan José

Manuel Lázaro, Antonio

Simon, Carine

English

The TT-transform stands for time-time transform and has been derived as an inverse Fourier transform of

the time-frequency S-transform. So far, it has been proposed that the diagonal of the TT-transform can be used for

signal characterisation. We show here an alternative and simplified derivation of the TT-transform which enables a

better understanding of this transform.Peer Reviewe

UPCommons

The diagonal of the TT transform: computation and meaning

UPCommons. Portal del coneixement obert de la UPC

Instrumentation Viewpoint 8 60m7Abstract - The TT-transform stands for time-time transform and has been derived as an inverse Fourier transform of the time-frequency S-transform. So far, it has been proposed that the diagonal of the TT-transform can be used for signal characteri-sation. We show here an alternative and simplified derivation of the TT-transform which enables a better understanding of this transform.Keywords - TT-transform, S-transform, time-frequency localisation, time-time anal-ysisI. INTRODUCTIONIn disciplines such as music or geophysics, signals are non stationary. The need for processing such signals has led to the appearance of several types of time varying frequency filters, such as the short time Fourier transform [1], wavelets [2], and more recently the S-transform (ST) [3]. These transforms introduce re-dundancy passing from a 1D time signal to a 2D time-frequency (or time-scale) signal. In 2003, [4] introduced a new transform based on the ST and called it the TT-transform (TT). It includes redundancy in time passing from a 1D time signal to a 2D time-time signal. Until now, this transform has seen little application [5] and in general, interest has mainly been focused on the diagonal part. The aimof this correspondence is to show an simpler way of computing the diagonal of the TT and to give a clear interpretation of it.In the next section, the S- and TT-transforms will be reviewed. Section 3 will demonstrate a simplified way to compute the latter. Section 4 will show exam-ples and the last section will conclude this paper.II. THE S- AND TT-TRANSFORMSThe ST of u(t) isw(t,f ) being a 1-mean window, generally a Gaussian with a variance of 1/f .The TT, [4], is the inverse Fourier transform(FT) w.r.t. f of S(tf ) :Both transforms are easily invertible.III. COMPUTING THE DIAGONAL OF THE TT-TRANSFORMIn the applications using the TT, only its diagonal part has been used. We will hence concentrate on this part. For details on the rest of the transform, see [5].We have shown, [5], that where F-1 is the inverse FT, U(f) the FT of u(t) and G(f) = -kp2|f|, k being a constant. This is an important result as, by using this formula, not only can we completely forget about the use of the ST and thus simplify the computation of the TT but we can also much more easily understand its behaviour. Indeed, we see that the diagonal terms of the TT are just a frequency filtered version of theoriginal signal, putting more emphasis on high frequencies.IV. EXAMPLE OF APPLICATIONThe example, Fig. 1, top, is a sum of sines. The bottom plot show the diagonal terms of its TT.On its TT, Fig. 2, it is seen how most of the information is centred on the diagonal part.On Fig. 3, the FFT of the diagonal of the TT as well as the FFT of the original signal is shown. The first one has been normalised to facilitate the comparison. Here, we can clearly see how the TT emphasises high frequencies at the cost of low frequencies.V. CONCLUSIONIn this presentation, we show that computing the diagonal elements of the TT-transform of a signal is equivalent to frequency filtering it; the equivalent filter gives more emphasis to high frequencies with respect to low ones. As well as al-lowing a clear interpretation of the meaning of the diagonal of the TTtransform,this work thus gives a much simpler and more direct way to compute it.REFERENCES[1] D. Gabor, “Theory of communication”, Journal of Institution of Electrical Engineering, vol. 93, pp. 429–457, 1946.[2] S. Mallat, A wavelet tour of signal processing, U. London, Ed. Academic press, 1998.[3] R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: The S transform”, IEEE Trans. SignalProcessing, vol. 44, no. 4, pp. 998–1001, 1996.[4] C. Pinnegar and L. Mansinha, “A method of time-time analysis: the TT-transform”, Digital Signal Processing, vol. 13, pp. 588–603, 2003.[5] C. Simon, M. Schimmel and JJ. Dañobeitia, “On the TT-Transform and Its Diagonal Elements”, IEEE Trans. Signal Processing,vol. 56, no. 11, pp. 5709-5713, 2008. Fig. 1. Original signal and the diagonal of its TTS(t.f) = ʃ  u(t)w(t - t, f)e-2ipftdtTT(t,t) = ʃ    S(t,f )e2ipftdtT(t,t) = F-1 {U(f)G(f)}Fig. 1. Original signal and the diagonal of its TTFig.2. TT of the signalFig.3. FFT of the signal and of the diagonal of its TTTHE DIAGONAL OF THE TT TRANSFORm: COmPUTATION AND mEANINGC. Simon1, JJ. Dañobeitia1 and A. mànuel2 (1)  Unitat de Tecnologia marina (UTm-CSIC)barcelona, csimon@cmima.csic.es(2)  Centre Tecnológic de Vilanova i la Geltrú (CTVG-UPC) 

The TT-transform stands for time-time transform and has been derived as

an inverse Fourier transform of the time-frequency S-transform. So far, it has been

proposed that the diagonal of the TT-transform can be used for signal characterisation.

We show here an alternative and simplified derivation of the TT-transform

which enables a better understanding of this transform.Peer Reviewe

The TT-transform stands for time-time transform and has been derived as an inverse Fourier transform of
the time-frequency S-transform. So far, it has been proposed that the diagonal of the TT-transform can be used for
signal characterisation. We show here an alternative and simplified derivation of the TT-transform which enables a
better understanding of this transform.Peer ReviewedPostprint (published version

The diagonal of the TT transform: computation and meaningC. Simon (1), JJ. Dañobeitia (1) and A. Mànuel (2)(1) Unitat de Tecnologia Marina (UTM-CSIC) Barcelona, csimon@cmima.csic.es(2) Centre Tecnológic de Vilanova i la Geltrú (CTVG-UPC)Abstract: The TT-transform stands for time-time transform and has been derived as an inverse Fourier transform of  the time-frequency S-transform. So far, it has been proposed that the diagonal of the TT-transform can be used for  signal characterisation. We show here an alternative and simpliﬁed derivation of the TT-transform which enables a better understanding of this transform. Keywords: TT-transform, S-transform, time-frequency localisation, time-time analysis1. INTRODUCTIONIn disciplines such as music or geophysics,  signals are non stationary.  The need for processing such signals has led to the appearance of several types of time varying frequency filters, such as the short time Fourier  transform  [1],  wavelets  [2],  and  more  recently  the  S-transform  (ST)  [3].  These  transforms introduce redundancy passing from a 1D time signal to a 2D time-frequency (or time-scale) signal. In 2003, [4] introduced a new transform based on the ST and called it the TT-transform (TT). It includes redundancy in time passing from a 1D time signal to a 2D time-time signal. Until now, this transform has seen little application [5] and in general, interest has mainly been focused on the diagonal part. The aim of this correspondence is to show an simpler way of computing the diagonal of the TT and to give a clear interpretation of it.In the next section, the S- and TT-transforms will be reviewed. Section 3 will demonstrate a simpliﬁed way to compute the latter. Section 4 will show examples and the last section will conclude this paper.2. THE S- AND TT-TRANSFORMSThe ST of u(t) is , w(t,f) being a 1-mean window, generally a Gaussian with a variance of 1/f .The TT, [4], is the inverse Fourier transform(FT) w.r.t. f of S(¿,f) : .Both transforms are easily invertible.3. COMPUTING THE DIAGONAL OF THE TT-TRANSFORMIn the applications using the TT, only its diagonal part has been used. We will hence concentrate on this part. For details on the rest of the transform, see [5].We have shown, [5], that  where  is the inverse FT, U(f) the FT of u(t) and , k being a constant.This is an important result as, by using this formula, not only can we completely forget about the use of the ST and thus simplify the computation of the TT but we can also much more easily understand its behaviour. Indeed, we see that the diagonal terms of the TT are just a frequency filtered version of the original signal, putting more emphasis on high frequencies.4. EXAMPLE OF APPLICATIONThe example, Fig. 1, top, is a sum of sines.  The bottom plot show the diagonal terms of its TT.Fig. 1. Original signal and the diagonal of its TTOn its TT, Fig. 2,  it is seen how most of the information is centred on the diagonal part.Fig.2. TT of the signalOn Fig. 3, the FFT of the diagonal of the TT as well as the FFT of the original signal is shown. The first one has been normalised to facilitate the comparison. Here, we can clearly see how the TT  emphasises high frequencies at the cost of low frequencies.Fig.3. FFT of the signal and of the diagonal of its TT5. CONCLUSIONIn this presentation, we show that computing the diagonal elements of the TT-transform of a signal is equivalent to frequency filtering it; the equivalent filter gives more emphasis to high frequencies with respect to low ones. As well as allowing a clear interpretation of the meaning of the diagonal of the TT-transform, this work thus gives a much simpler and more direct way to compute it.REFERENCES[1] D. Gabor, “Theory of communication”, Journal of Institution of Electrical Engineering, vol. 93, pp. 429–457, 1946.[2] S. Mallat, A wavelet tour of signal processing, U. London, Ed. Academic press, 1998.[3] R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: The S transform”, IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 998–1001, 1996.[4] C. Pinnegar and L. Mansinha, “A method of time-time analysis: the TT-transform”, Digital Signal Processing, vol. 13, pp. 588–603, 2003.[5] C. Simon, M. Schimmel and JJ. Dañobeitia, “On the TT-Transform and Its Diagonal Elements”, IEEE Trans. Signal Processing, vol. 56, no. 11, pp. 5709-5713, 2008.

Martech 2009 Third Marine Technology Workshop, 19-20 november 2009, Vilanova i la Geltrú, Barcelona.-- 1 page, 3 figuresThe TT-transform stands for time-time transform and has been derived as an inverse Fourier transform of the time frequency S-transform. So far, it has been proposed that the diagonal of the TT transform can be used for signal characterisation. We show here an alternative and simplified derivation of the TT-transform which enables a better understanding of this transformPeer reviewe

Mànuel-Làzaro, Antoni

Digital.CSIC

The diagonal of the TT transform: Computation and meaning

https://upcommons.upc.edu/bitstream/2117/8772/1/n8p60.pdf

The diagonal of the TT transform: computation and meaning

Abstract

Similar works

Full text

Available Versions

UPCommons

UPCommons. Portal del coneixement obert de la UPC

UPCommons

UPCommons. Portal del coneixement obert de la UPC

Digital.CSIC