Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2008, Article ID 672941, 13 pages doi:10.1155/2008/672941
Research Article A Window Width Optimized S-Transform
Ervin Sejdic,´ 1 Igor Djurovic,´ 2 and Jin Jiang1
1 Department of Electrical and Computer Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9 2 Electrical Engineering Department, University of Montenegro, 81000 Podgorica, Serbia and Montenegro
Correspondence should be addressed to Jin Jiang, [email protected]
Received 14 May 2007; Accepted 15 November 2007
Recommended by Sven Nordholm
Energy concentration of the S-transform in the time-frequency domain has been addressed in this paper by optimizing the width of the window function used. A new scheme is developed and referred to as a window width optimized S-transform. Two opti- mization schemes have been proposed, one for a constant window width, the other for time-varying window width. The former is intended for signals with constant or slowly varying frequencies, while the latter can deal with signals with fast changing frequency components. The proposed scheme has been evaluated using a set of test signals. The results have indicated that the new scheme can provide much improved energy concentration in the time-frequency domain in comparison with the standard S-transform. It is also shown using the test signals that the proposed scheme can lead to higher energy concentration in comparison with other standard linear techniques, such as short-time Fourier transform and its adaptive forms. Finally, the method has been demon- strated on engine knock signal analysis to show its effectiveness.
Copyright © 2008 Ervin Sejdic´ et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. INTRODUCTION vided by the S-transform is referenced to the time origin, and therefore provides supplementary information about spectra In the analysis of the nonstationary signals, one often needs which is not available from locally referenced phase infor- to examine their time-varying spectral characteristics. Since mation obtained by the continuous wavelet transform [5]. time-frequency representations (TFR) indicate variations of For these reasons, the S-transform has already been consid- the spectral characteristics of the signal as a function of time, eredinmanyfieldssuchasgeophysics[6–8], cardiovascular they are ideally suited for nonstationary signals [1, 2]. The time-series analysis [9–11], signal processing for mechanical ideal time-frequency transform only provides information systems [12, 13], power system engineering [14], and pattern about the frequency occurring at a given time instant. In recognition [15]. other words, it attempts to combine the local information Even though the S-transform is becoming a valuable tool of an instantaneous-frequency spectrum with the global in- for the analysis of signals in many applications, in some formation of the temporal behavior of the signal [3]. The cases, it suffers from poor energy concentration in the time- main objectives of the various types of time-frequency anal- frequency domain. Recently, attempts to improve the time- ysis methods are to obtain time-varying spectrum functions frequency representation of the S-transform have been re- with high resolution and to overcome potential interferences ported in the literature. A generalized S-transform, proposed [4]. in [12], provides greater control of the window function, and The S-transform can conceptually be viewed as a hybrid the proposed algorithm also allows nonsymmetric windows of short-time Fourier analysis and wavelet analysis. It em- to be used. Several window functions are considered, includ- ploys variable window length. By using the Fourier kernel, it ing two types of exponential functions: amplitude modu- can preserve the phase information in the decomposition [5]. lation and phase modulation by cosine functions. Another The frequency-dependent window function produces higher form of the generalized S-transform is developed in [7], frequency resolution at lower frequencies, while at higher where the window scale and shape are a function of fre- frequencies, sharper time localization can be achieved. In quency. The same authors introduced a bi-Gaussian window contrast to wavelet transform, the phase information pro- in [8], by joining two nonsymmetric half-Gaussian windows. 2 EURASIP Journal on Advances in Signal Processing
Since the bi-Gaussian window is asymmetrical, it also pro- TFRs for these signals are given, respectively, as shown in [16] duces an asymmetry in the time-frequency representation, with higher-time resolution in the forward direction. As a re- sult, the proposed form of the S-transform has better perfor- 1 dφ(t) ITFR(t, f ) = 2πA(t)δ f − ,(1) mance in detection of the onset of sudden events. However, 2π dt in the current literature, none has considered optimizing the energy concentration in the time-frequency domain directly, dχ( f ) ITFR(t, f ) = 2πG( f )δ t + ,(2) that is, to minimize the spread of the energy beyond the ac- df tual signal components. The main approach used in this paper is to optimize the width of the window used in the S-transform. The optimiza- where ITFR stands for an ideal time-frequency represen- tion is performed through the introduction of a new parame- tation. These two representations are ideally concentrated ter in the transform. Therefore, the new technique is referred along the instantaneous frequency, (dφ(t)/dt)/2π,andon to as a window width optimized S-transform (WWOST). group delay −dχ( f )/df . Simplest examples of these signals The newly introduced parameter controls the window width, are the following: a sinusoid with A = const. and dφ(t)/dt = andtheoptimalvaluecanbedeterminedintwoways.The const. depicted in Figure 1(a); and a Dirac pulse x2(t) = first approach calculates one global, constant parameter and δ(t −t0) shown in Figure 1(b). The ideal time-frequency rep- this is recommended for signals with constant or very slowly resentations are depicted in Figures 1(c) and 1(d). These two varying frequency components. The second approach calcu- graphs are compared with the TFRs obtained by the standard lates the time-varying parameter, and it is more suitable for S-transform in Figures 1(e) and 1(f). signals with fast varying frequency components. For the sinusoidal case, the frequencies surrounding The proposed scheme has been tested using a set of syn- (dφ(t)/dt)/2π also have a strong contribution, and from (1), thetic signals and its performance is compared with the stan- it is clear that they should not have any contributions. Sim- dard S-transform. The results have shown that the WWOST ilarly, for the Dirac function, it is expected that all the fre- enhances the energy concentration. It is also shown that the quencies have the contribution but only for a single time in- WWOST produces the time-frequency representation with a stant. Nevertheless, it is clear that the frequencies are not only higher concentration than other standard linear techniques, contributing during a single time instant as expected from such as the short-time Fourier transform and its adaptive (2), but the surrounding time instants also have strong en- forms. The proposed technique is useful in many applica- ergy contribution. tions where enhanced energy concentration is desirable. As The examples presented here are for illustrations only, an illustrative example, the proposed algorithm is used to since a priori knowledge about the signals is assumed. In analyze knock pressure signals recorded from a Volkswagen most practical situations, the knowledge about a signal is Passat engine in order to determine the presence of several limited and the analytical expressions similar to (1)and(2) signal components. are often not available. However, the examples illustrate a This paper is organized as follows. In Section 2, the con- point that some modifications to the existing S-transform cept of ideal time-frequency transform is introduced, which algorithm, which do not assume a priori knowledge about can be compared with other time-frequency representations the signal, may be useful to achieve improved performance including transforms proposed here. The development of in time-frequency energy concentration. Such improvements the WWOST is covered in Section 3. Section 4 evaluates the only become possible after modifications to the width of the performance of the proposed scheme using test signals and window function are made. also the knock pressure signals. Conclusions are drawn in Section 5. 3. THE PROPOSED SCHEME
2. ENERGY CONCENTRATION IN 3.1. Standard S-transform TIME-FREQUENCY DOMAIN
The standard S-transform of a function x(t)isgivenbyan The ideal TFR should only be distributed along frequencies integral as in [5, 7, 12] for the duration of signal components. Thus, the neighbor- ing frequencies would not contain any energy; and the energy contribution of each component would not exceed its dura- +∞ tion [3]. Sx(t, f ) = x(τ)w t − τ, σ( f ) exp (− j2πfτ)dτ (3) For example, let us consider two simple signals: an −∞ FM signal, x1(t) = A(t)exp(jφ(t)), where |dA(t)/dt| | | dφ(t)/dt and the instantaneous frequency is defined as with a constraint f (t) = (dφ(t)/dt)/2π; and a signal with the Fourier = transform given as X( f ) G( f )exp(j2πχ( f )), where the spectrum is slowly varying in comparison to phase +∞ | | | | w t − τ, σ( f ) dτ = 1. (4) dG( f )/df dχ( f )/df . Further, A(t)andG(t) The ideal −∞ Ervin Sejdicetal.´ 3
1 1
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Figure 1: Comparison of the ideal time-frequency representation and S-transform for the two simple signal forms: (a) 30 Hz sinusoid; (b) sample Dirac function; (c) ideal TFR of a 30 Hz sinusoid; (d) ideal TFR of a Dirac function; (e) TFR by standard S-transform for a 30 Hz sinusoid; and (f) TFR by standard S-transform of the Dirac delta function.
A window function used in S-transform is a scalable Gaus- The disadvantage of the current algorithm is the fact that sian function defined as the window width is always defined as a reciprocal of the ff 2 frequency. Some signals would benefit from di erent win- = 1√ − t dow widths. For example, for a signal containing a single si- w t, σ( f ) exp 2 . (5) σ( f ) 2π 2σ ( f ) nusoid, the time-frequency localization can be considerably The advantage of the S-transform over the short-time improved if the window is very narrow in the frequency do- Fourier transform (STFT) is that the standard deviation σ( f ) main. Similarly, for signals containing only a Dirac impulse, is actually a function of frequency, f ,definedas it would be beneficial for good time-frequency localization to have very wide window in the frequency domain. 1 σ( f ) = . (6) | f | 3.2. Window width optimized S-transform Consequently, the window function is also a function of time and frequency. As the width of the window is dictated by the A simple improvement to the existing algorithm for the S- frequency, it can easily be seen that the window is wider in transform can be made by modifying the standard deviation the time domain at lower frequencies, and narrower at higher of the window to frequencies. In other words, the window provides good lo- calization in the frequency domain for low frequencies while providing good localization in time domain for higher fre- = 1 σ( f ) p . (7) quencies. | f | 4 EURASIP Journal on Advances in Signal Processing
1 There are two ways to determine the optimal value of p. 0.9 One is to determine a global, constant value of p for the en- tire signal. The other is to determine a time-varying p(t), 0.8 which depends on each time instant considered. The first ap- 0.7 proach is more suitable for signals with the constant or slowly 0.6 varying frequency components. In this case, one value of p will suffice to give the best resolution for all components. 0.5 The time-varying parameter is more appropriate for signals 0.4 with fast varying frequency components. In these situations, depending on the time duration of the signal components, 0.3 Normalized amplitude it would be beneficial to use lower value of p (somewhere 0.2 in the middle of the particular component’s interval), and 0.1 to use higher values of p for the beginning and the end of the component’s interval, so the component is not smeared 0 −2 −1.5 −1 −0.500.511.52 in the time-frequency plane. It is important to mention that Time (s) both proposed schemes for determining the parameter p are the special cases of the algorithm which would evaluate the p = 0.5 parameter on any arbitrary subinterval, rather than over the p = 1 p = 2 entire duration of the signal. ff Figure 2: Normalized Gaussian window for di erent values of p. 3.2.1. Algorithm for determining the time-invariant p The algorithm for determining the optimized time-invariant Based on the above equation, the new S-transform can be value of p is defined through the following steps. represented as (1) For p selected from a set 0 3.2.2. Algorithm for determining p(t) In this section, the performance of the proposed scheme is examined using a set of synthetic test signals first. Further- The time-varying parameter p(t)isrequiredforsignalswith more, the analysis of signals from an engine is also given. components having greater or abrupt changes. The algo- The first part includes two cases: (1) a simple case involving rithm for choosing the optimal p(t) can be summarized three slowly varying frequencies and (2) more complicated through the following steps. cases involving multiple time-varying components. The goal (1) For p selected from a set 0 p popt(t) The optimization for synthetic signals is performed on the S (t, f ) = S (t, f ). (17) x x setofvaluesdefinedby The main difference between the two techniques lies in ={ ∈ N ≤ ≤ } step (3). For the time invariant case, a single value of p is σgw n/128 : n ,1 n 128 (20) chosen, whereas in the time-varying case, an optimal value of p(t)isafunctionoftime.AsitisdemonstratedinSection 4, and both the time-invariant and time-varying values of σgw the time-dependent parameter is beneficial for signals with are calculated. the fast varying components. 4.1. Synthetic test signals 3.2.3. Inverse of the WWOST Example 1. The first test signal is shown in Figure 3(a).Ithas Similarly to the standard S-transform, the WWOST can be the following analytical expression: used as both an analysis and a synthesis tool. The inversion 2 2 x1(t) = cos 132πt +14πt +cos 10πt − 2πt procedure for the WWOST resembles that of the standard (21) S-transform, but with one additional constraint. The spec- +cos 30πt +6πt2 , p trum of the signal obtained by averaging Sx (t, f )overtime must be normalized by W(0, f ), where W(α, f ) represents where the signal exists only on the interval 0 ≤ t<1. The sig- the Fourier transform (from t to α) of the window function, nal consists of three slowly varying frequency components. w(t, σ( f )). Hence, the inverse WWOST for a signal, x(t), is It is analyzed using the STFT (Figure 3(b)), ASTFT with defined as time-invariant optimum value of σgw (Figure 3(c)), stan- +∞ +∞ dard S-transform (Figure 3(d)), and the proposed algorithm 1 p x(t) = Sx (τ, f )exp(j2πft)dτ df. (18) (Figure 3(f)). A Gaussian window is also used in the analy- −∞ −∞ W(0, f ) sis by the STFT, with standard deviations equal to 0.05. The 6 EURASIP Journal on Advances in Signal Processing 5 100 0 50 Amplitude Frequency (Hz) −5 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (a) (b) 100 100 50 50 Frequency (Hz) Frequency (Hz) 0 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (c) (d) 1 100 0.5 50 Frequency (Hz) Amplitude of CM 0 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (e) (f) p Figure 3: Test signal x1(t): (a) time-domain representation; (b) STFT of x1(t); (c) ASTFT of x1(t) with σopt;(d)Sx (t, f )ofx1(t) with p = 1 p (standard S-transform); (e) concentration measure CM(p); (f) Sx (t, f )ofx1(t) with the optimal value of p = .57. optimum value of standard deviation for the ASTFT is calcu- signals with slowly varying frequencies, it is expected that lated to be σopt = 0.094. The colormap used for plotting the smaller values of p will produce the best energy concentra- time-frequency representations in Figure 3 and all the subse- tion. In this example, the optimal value, found according to quent figures is a linear grayscale with values from 0 to 1. (12), is determined numerically to be 0.57. The standard S-transform, shown in Figure 3(d), depicts Based on the visual inspection of the time-frequency rep- all three components clearly. However, only the first two resentations shown in Figure 3, it can be concluded that the components have relatively good concentration, while the proposed algorithm achieves higher concentration among third component is completely smeared in frequency. As the considered representations. To confirm this fact, a per- shown in Figure 3(b), the STFT provides better energy con- formance measure given by centration than the standard S-transform. The ASTFT, de- − +∞ +∞ 1 picted in Figure 3(c), shows a noticeable improvement for all ΞTF = TF(t, f ) dt df (22) three components. The results with the proposed scheme is −∞ −∞ shown in Figure 3(f) for p = 0.57. The value of p is found ac- cording to (12). For the determined value of p, the first two is used for measuring the concentration of the representa- components have higher concentration even than the ASTFT, tion, where |TF(t, f )| is a normalized time-frequency repre- while the third component has approximately the same con- sentation. The performance measure is actually the concen- centration. tration measure proposed in (9). A more concentrated rep- In Figure 3(e), the normalized concentration measure is resentation will produce a higher value of ΞTF. Table 1 sum- depicted. The obtained results verify the theoretical predic- marizes the performance measure for the STFT, the ASTFT, tions from Section 3.2. For this class of signals, that is, the the standard S-transform, and the WWOST. Ervin Sejdicetal.´ 7 Table 1: Performance measure for the three time-frequency trans- Table 2: Performance measures for the time-frequency representa- forms. tions considered in Example 2. TFR ΞTF TFR ΞTF STFT 0.0119 STFT 0.0108 ASTFT 0.0131 ASTFT with σopt 0.0115 Standard S-transform 0.0080 ASTFT with σopt(t) 0.0119 WWOST 0.0136 WWOST with p 0.0116 WWOST with p(t) 0.0124 The value of the performance measure for the standard S- transform is the lowest, followed by the STFT. The WWOST values of standard deviation provide good concentration of produces the highest value of ΞTF, and thus achieves a TFR the linear FM component and the stationary parts of the sec- with the highest energy concentration amongst the trans- ond component, the transition region of the second compo- forms considered. nent is smeared in time. Figure 4(f) represents the signal optimized S-transform Example 2. The signal in the second example contains mul- obtained by using p(t). A significant improvement in the en- tiple components with faster time-varying spectral contents. ergy concentration is easily noticeable in comparison to the The following signal is used: standard S-transform. All components show improved en- ergy concentration in comparison to the S-transform. Fur- x2(t) = cos 40π(t − 0.5) arctan (21t − 10.5) ther, a comparison of the representations obtained by the − 20π ln (21t − 10.5)2 +1 /21 + 120πt (23) proposed implementation of the S-transform and the STFT +cos 40πt − 8πt2 , shows that both components have higher energy concentra- tion in the representation obtained by the WWOST with where x2(t) exists only on the interval 0 ≤ t<1. This p(t). signal consists of two components. The first has a transi- As mentioned previously, for this type of signals it is tion region from lower to higher frequencies, and the sec- more appropriate to use the time-varying p(t) rather than ond is a linear chirp. In the analysis, the time-frequency a single constant p valueinordertoachievebetterconcen- transformations that employ a constant window exhibit a tration of the nonstationary data. By comparing Figures 4(d) conflicting issue between good concentration of the tran- and 4(f), the component with the fast changing frequency sition region for the first component versus good con- has better concentration with p(t) than a fixed p,whichis centration for the rest of the signal. In order to numer- calculated according to (12), while the linear chirp has simi- ically demonstrate this problem, the signal is again ana- lar concentration in both cases. lyzed using the STFT (Figure 4(a)), ASTFT with the opti- It would be beneficial to quantify the results by eval- mal time-invariant value of σgw (Figure 4(c)), ASTFT with uating the performance measure again. The performance the optimal time-varying value of σgw (Figure 4(e)), stan- measure is given by (22) and the results are summarized dard S-transform (Figure 4(b)), the proposed algorithm in Table 2. A higher value of the performance measure for with both time-invariant (Figure 4(d)), and time-varying p WWOST with p(t) reconfirms that the time-varying algo- (Figure 4(f)). A Gaussian window is used for the STFT, with rithm should be used for the signals with fast changing com- σ = 0.03. The optimum time-invariant value of the standard ponents. Also, it is worthwhile to examine the value of (22) deviation for the ASTFT is determined to be σopt = 0.055. for the STFT and the ASTFT. The time-frequency represen- The standard deviation of the Gaussian window used tations of the signal obtained by the STFT and ASTFT al- should be small in order for the STFT to provide relatively gorithms achieve smaller values of the performance measure good concentration in the transition region. However, as the than WWOST. This supports the earlier conclusion that the value of the standard deviation decreases, so is the concentra- WWOST produces more concentrated energy representation tion of the rest of the signal. To a certain extent, the standard than the STFT and ASTFT. The WWOST with the time- S-transform is capable of producing a good concentration invariant value of p produces higher concentration than the around the instantaneous frequencies at the lower frequen- ASTFT with the optimum time-invariant value of σgw,and cies and also in the transition region for the first component. the WWOST with p(t) produces higher concentration than However, at the high frequencies, the standard S-transform the ASTFT with the optimum time-varying value of the exhibits poor concentration for the first component. The σgw. WWOST with a time-invariant p enhances the concentra- tion of the linear chirp, as shown in Figure 4(d). However the Example 3. Another important class of signals are those with concentration of the transition region of the first component crossing components that have fast frequency variations. A has deteriorated in comparison to the standard S-transform. representative signal as shown in Figure 5(a) is given by The concentration obtained with the WWOST with the time- 2 x3(t) = cos 20π ln (10t +1) +cos 48πt +8πt (24) invariant p for this transition region is equivalent to the poor concentration exhibited by the STFT. Even though the with x3(t) = 0 outside the interval 0 ≤ t<1. For this ASTFT with both time-invariant and time-varying optimum class of signals, similar conflicting issues occur as in the 8 EURASIP Journal on Advances in Signal Processing 100 100 50 50 Frequency (Hz) Frequency (Hz) 0 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (a) (b) 100 100 50 50 Frequency (Hz) Frequency (Hz) 0 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (c) (d) 100 100 50 50 Frequency (Hz) Frequency (Hz) 0 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (e) (f) p Figure 4: Comparison of different algorithms: (a) STFT of x2(t); (b) Sx (t, f )ofx2(t) with p = 1 (standard S-transform); (c) ASTFT of x2(t) p p with σopt = 0.055; (d) Sx (t, f )ofx2(t) with p = 0.73; (e) ASTFT of x2(t) with σopt(t); (f) Sx (t, f )ofx2(t) with the optimal p(t). previous example; however, here exists an additional con- high frequencies, while the linear chirp has significantly straint, that is, the crossing components. The time-frequency higher concentration in comparison to the other represen- analysis is performed using the STFT (Figure 5(b)), the tations. It is also interesting to note how p(t)variesbetween ASTFT with the time-varying σgw (Figure 5(c)), the standard 0.6 and 1.0 as a function of time shown in Figure 5(e).Inpar- S-transform (Figure 5(d)), and the proposed algorithm for ticular, p(t) is close to 1 at the beginning of the signal in order the S-transform (Figure 5(f)). In the STFT, a Gaussian win- to achieve good concentration of the high-frequency compo- dow with a standard deviation of 0.02 is used. Due to the nent. As time progresses, the value of p(t)decreasesinorder time-varying nature of the frequency components present in to provide a good concentration at the lower frequencies. To- the signal, the time-varying algorithm is used in the calcula- wards the end of the signal, p(t) increases again to achieve a tion of the WWOST in order to determine the optimal value good time localization of the signal. of p. In Section 3, it has been stated that for the components The representation obtained by the STFT depicts good with faster variations, it is recommended that the time- concentration of the higher frequencies, while having rela- varying algorithm with the WWOST be used. In order to tively poor concentration at the lower frequencies. An im- substantiate that statement, the performance measure imple- provement in the concentration of the lower frequencies mented in the previous examples is used again and the results is obtained with the ASTFT algorithm. The standard S- are shown in Table 3. The optimized time-invariant value of transform is capable of providing better concentration for the parameter popt for this signal, found according to (12), is the high frequencies, but for the linear chirp, the concentra- determined numerically to be 0.71. These performance mea- tion is equivalent to that of the STFT. sures verify that the time-varying algorithm should be used From the time-frequency representation obtained by the for the faster varying components. For comparison purposes, WWOST, it is clear that the concentration is preserved at the performance measures for the representations given by Ervin Sejdicetal.´ 9 2 100 0 50 Amplitude Frequency (Hz) −2 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (a) (b) 100 100 50 50 Frequency (Hz) Frequency (Hz) 0 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (c) (d) 1 100 0.5 50 Amplitude Frequency (Hz) 0 0 00.20.40.60.81 00.20.40.60.81 Time (s) Time (s) (e) (f) Figure 5: Time-frequency analysis of signal with fast variations in frequency: (a) time-domain representation; (b) STFT of x3(t); (c) ASTFT p p of x3(t) with σopt(t); (d) Sx (t, f )ofx3(t) with p = 1 (standard S-transform); (e) p(t); (f) Sx (t, f )ofx3(t) with the optimal p(t). Table 3: Performance measures for the time-frequency representa- signals were considered. It would be beneficial to compare tions considered in Example 3. the performance of the considered algorithms in the pres- ence of additive white Gaussian noise in order to understand TFR Ξ (noise-free) Ξ (SNR = 25 dB) TF TF whether the proposed algorithm is capable of providing the STFT 0.0106 0.0100 enhanced performance in noisy environment. Hence, the sig- ASTFT with σ 0.0121 0.0114 opt nal x3(t) is contaminated with the additive white Gaussian ASTFT with σopt(t) 0.0122 0.0113 noise and it is assumed that SNR = 25 dB. The results of such WWOST with p 0.0122 0.0110 an analysis are summarized in Table 3. Even though, the per- WWOST with p(t) 0.0126 0.0116 formance has degraded in comparison to the noiseless case, the WWOST with p(t) still outperforms the other considered representations. the STFT and its time-invariant (σopt = 0.048) and time- varying adaptive algorithms are calculated as well. By com- 4.2. Demonstration example paring the values of the performance measure for different time-frequency transforms, these values confirm the earlier In order to illustrate the effectiveness of the proposed statement which assures that each algorithm for the WWOST scheme, the method has been applied to the analysis of en- produces more concentrated time-frequency representation gine knocks. A knock is an undesired spontaneous autoigni- in its respective class than the ASTFT. tion of the unburned air-gas mixture causing a rapid in- In the analysis performed so far, it was assumed that the crease in pressure and temperature. This can lead to seri- signal-to-noise ratio (SNR) is infinity, that is, the noise-free ous problems in spark-ignition car engines, for example, 10 EURASIP Journal on Advances in Signal Processing 1 3 2 0 1 Amplitude (kPa) −1 Frequency (kHz) 0 0246 0246 Time (ms) Time (ms) (a) (b) 3 3 2 2 1 1 Frequency (kHz) Frequency (kHz) 0 0 0246 0246 Time (ms) Time (ms) (c) (d) 3 3 2 2 1 1 Frequency (kHz) Frequency (kHz) 0 0 0246 0246 Time (ms) Time (ms) (e) (f) Figure 6: Time-frequency analysis of engine knock pressure signal (17th trial): (a) time-domain representation; (b) STFT; (c) ASTFT with p p p σopt(t); (d) Sx (t, f ) with p = 1 (standard S-transform); (e) Sx (t, f ) with p = 0.86; (f) Sx (t, f ) with the optimal p(t). environment pollution, mechanical damages, and reduced formed on the set of values defined by σgw ={0.01n : n ∈ energy efficiency [21, 22]. In this paper, a focus will be on N and 1 ≤ n ≤ 744} milliseconds. the analysis of knock pressure signals. A comparison of these representations show that the It has been previously shown that high-pass filtered pres- WWOST performs significantly better than the standard S- sure signals in the presence of knocks can be modeled as transform. The presence of several signal components can be multicomponent FM signals [22]. Therefore, the goal of this easily identified with the WWOST, but rather difficult with analysis is to illustrate how effectively the proposed WWOST the standard S-transform. In addition, both proposed algo- can decouple these components in time-frequency represen- rithms produce higher concentration than the STFT and the tation. A knock pressure signal recorded from a 1.81 Volk- corresponding class of the ASTFT. This is accurately depicted swagen Passat engine at 1200 rpm is considered. Note that the through the results presented in Table 4. The best concen- signal is high-pass filtered with a cutoff frequency of 3000 Hz. tration is achieved with the time-varying algorithm, while The sampling rate is fs = 100 kHz and the signal contains 744 the time invariant value p produces slightly higher concen- samples. tration than the ASTFT with the time-invariant value of The performance of the proposed scheme in this case is σgw (σopt = 0.2 milliseconds for the signal in Figure 6 and evaluated by comparing it with that of the STFT, the ASTFT, σopt = 0.19 milliseconds for the signal in Figure 7). and the standard S-transform. The results are shown in Fig- The direct implication of the results is that the WWOST ures 6 and 7. These results represent two sample cases from could potentially be used for the knock pressure signal anal- fifty trials. For the STFT, a Gaussian window, with a standard ysis. A major advantage of such an approach in compari- deviation of 0.3 milliseconds, is used for both cases. The op- son to some existing methods is that the signals could be timization of the standard deviation for the ASTFT is per- modeled based on a single observation, instead of multiple Ervin Sejdicetal.´ 11 1 3 2 0 1 Amplitude (kPa) Frequency (kHz) −1 0 0246 0246 Time (ms) Time (ms) (a) (b) 3 3 2 2 1 1 Frequency (kHz) Frequency (kHz) 0 0 0246 0246 Time (ms) Time (ms) (c) (d) 3 3 2 2 1 1 Frequency (kHz) Frequency (kHz) 0 0 0246 0246 Time (ms) Time (ms) (e) (f) Figure 7: Time-frequency analysis of engine knock pressure signal (48th trial): (a) time-domain representation; (b) STFT; (c) ASTFT with p p p σopt(t); (d) Sx (t, f ) with p = 1 (standard S-transform); (e) Sx (t, f ) with p = 0.87; (f) Sx (t, f ) with the optimal p(t). Table 4: Concentration measures for the two sample trials. The STFT and the ASTFT are valuable signal decom- position-based representations, which can achieve good en- Trial Ξ Ξ Ξ Ξ = Ξ Ξ STFT ASTFTσopt ASTFTσopt(t) p 1 popt popt(t) ergy concentration for a wide variety of signals. However, 17th trial 0.0057 0.0059 0.0068 0.0054 0.0065 0.0074 throughout this paper, it is shown that the proposed opti- 48th trial 0.0052 0.0054 0.0060 0.0053 0.0058 0.0069 mization of the window width used in the S-transform is beneficial, and in presented cases, it outperforms other stan- dard linear techniques, such as the STFT and the ASTFT. realizations required by some other time-frequency methods It is also crucial to mention that the WWOST is designed such as Wigner-Ville distribution [23], since the WWOST to achieve better concentration in the class of the time- does not suffer from the cross-terms present in bilinear trans- frequency representations based on the signal decomposi- forms. tion. In comparison to the standard S-transform or the STFT, 4.3. Remarks the WWOST does have a higher computational complexity. The algorithm for the WWOST is based on an optimiza- It should be noted that, in some cases, when implementing tion procedure and requires a parameter tuning. However, the proposed algorithm, it may be beneficial to window the when compared to the transforms of similar group algo- p signal before evaluating Sx (t, f ) in step (1). This additional rithms (e.g., ASTFT), the WWOST has almost the same de- step diminishes the effects of a discrete implementation. As gree of complexity. shown in [24], wideband signals might lead to some irregular The sampled data version of the standard S-transform results unless they are properly windowed. and their MATLAB implementations have been discussed in 12 EURASIP Journal on Advances in Signal Processing several publications [5, 7, 25–27]. The WWOST is a straight- puters in Cardiology, pp. 383–386, Lund, Sweden, September forward extension of the standard S-transform. Therefore, 1997. the sampled data version of the WWOST also follows the [10] G. Livanos, N. Ranganathan, and J. Jiang, “Heart sound anal- steps presented in earlier publications. ysis using the S-transform,” in Computers in Cardiology,pp. 587–590, Cambridge, Mass, USA, September 2000. [11] E. Sejdic´ and J. Jiang, “Comparative study of three time- 5. CONCLUSION frequency representations with applications to a novel correla- tion method,” in Proceedings of the IEEE International Confer- In this paper, a scheme for improvement of the energy ence on Acoustics, Speech and Signal Processing (ICASSP ’04), concentration of the S-transform has been developed. The vol. 2, pp. 633–636, Montreal, Quebec, Canada, May 2004. scheme is based on the optimization of the width of the win- [12]P.D.McFadden,J.G.Cook,andL.M.Forster,“Decomposi- dow used in the transform. The optimization is carried out tion of gear vibration signals by the generalized S-transform,” by means of a newly introduced parameter. Therefore, the Mechanical Systems and Signal Processing, vol. 13, no. 5, pp. developed technique is referred to as a window width opti- 691–707, 1999. mized S-transform (WWOST). Two algorithms for parame- [13]A.G.Rehorn,E.Sejdic,´ and J. Jiang, “Fault diagnosis in ma- ter optimization have been developed: one for finding an op- chine tools using selective regional correlation,” Mechanical timal constant value of the parameter p for the entire signal; Systems and Signal Processing, vol. 20, no. 5, pp. 1221–1238, while the other is to find a time-varying parameter. The pro- 2006. posed scheme is evaluated and compared with the standard [14] P.K. Dash, B. K. Panigrahi, and G. Panda, “Power quality anal- ysis using S-transform,” IEEE Transactions on Power Delivery, S-transform by using a set of synthetic test signals. The re- vol. 18, no. 2, pp. 406–411, 2003. sults have shown that the WWOST can achieve better energy [15] E. Sejdic´ and J. 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Pinnegar, “Time-frequency and time-time filtering with the S-transform and TT-transform,” DigitalSignalProcessing, vol. 15, no. 6, pp. 604–620, 2005. EURASIP Journal on Wireless Communications and Networking Special Issue on Fairness in Radio Resource Management for Wireless Networks Call for Papers Radio resource management (RRM) techniques (such as ad- • RRM framework and QoS architecture mission control, scheduling, sub-carrier allocation, channel • Complexity and scalability issues assignment, power allocation, and rate control) are essential • Experimental and implementation results and issues for maximizing the resource utilization and providing qual- • Fairness in multiple-antenna transmission/reception ity of service (QoS) in wireless networks. systems In many cases, the performance metrics (e.g., overall • Fairness in end-to-end QoS provisioning throughput) can be optimized if opportunistic algorithms are employed. However, opportunistic RRM techniques al- Authors should follow the EURASIP Journal on Wire- ways favor advantaged users who have good channel condi- less Communications and Networking manuscript for- tions and/or low interference levels. The problem becomes mat described at the journal site http://www.hindawi.com/ even worse when the wireless terminals have low mobility journals/wcn/. Prospective authors should submit an elec- since the channel conditions become slowly varying (or even tronic copy of their complete manuscript through the jour- static), which might lead to long-term unfairness. The prob- nal Manuscript Tracking System at http://mts.hindawi.com/, lem of fair resource allocation is more challenging in multi- according to the following timetable: hop wireless networks (e.g., mesh and multihop cellular net- works). 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Topics include (but are Applied Science, Memorial University of Newfoundland, not limited to): St. John’s, NF, Canada A1B 3V6; [email protected] • Fairness of scheduling schemes in wireless networks Alagan Anpalagan, Department of Electrical and • Tradeoff between maximizing the overall throughput Computer Engineering, Ryerson University, Toronto, ON, and achieving throughput fairness Canada M5G 2C5; [email protected] • RRM fairness: problem definition and solution tech- Kwang-Cheng Chen, niques Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan; • Fairness performance in emerging wireless systems [email protected] (WiMAX, ad hoc networks, mesh networks, etc.) • Cross-layer RRM design with fairness Zhu Han, Department of Electrical and Computer • Short-term and long-term fairness requirements Engineering, College of Engineering, Boise State University • Adaptive RRM to support fairness Boise, Idaho, USA; [email protected] • Fairness in cooperative wireless communications Ekram Hossain, Department of Electrical and Computer • Issues and approaches for achieving fairness in multi- Engineering, University of Manitoba, Winnipeg, MB, hop wireless networks Canada R3T 2N2; [email protected] Hindawi Publishing Corporation http://www.hindawi.com EURASIP Journal on Advances in Signal Processing Special Issue on Signal Processing Advances in Robots and Autonomy Call for Papers The capabilities of robots and autonomous systems have in- Authors should follow the EURASIP Journal on Ad- creased dramatically over the past years. This success story vances in Signal Processing manuscript format described partly depends on advances in signal processing which pro- at http://www.hindawi.com/journals/asp/. Prospective au- vide appropriate and efficient analysis of sensor data and thors should submit an electronic copy of their complete enable autonomy. A key element of the transition of signal manuscript through the journal Manuscript Tracking Sys- processing output to its exploitation inside robots and au- tem at http://mts.hindawi.com/, according to the following tonomous systems is the way uncertainty is managed: uncer- timetable: tainty originating from insufficient sensor data, uncertainty about effects of future autonomous actions and, in the case of Manuscript Due August 1, 2008 distributed sensors and actuators (like for a team of robots), uncertainty about communication lines. The aim of this spe- First Round of Reviews November 1, 2008 cial issue is to focus on recent developments that allow pass- Publication Date February 1, 2009 ing this transition path successfully, showing either where signal processing is used in robotics and autonomy or where robotics and autonomy had special demands that had not Guest Editors been fulfilled by signal processing before. Frank Ehlers, NATO Undersea Research Centre (NURC), Topics of interest include, but are not limited to: Viale S. Bartolomeo 400, 19138 La Spezia, Italy; • Autonomous navigation: [email protected] ◦ Outdoor navigation using geo-information and Fredrik Gustafsson, Department of Electrical Engineering, dedicated indoor navigation solutions Linköping University, 58183 Linköping, Sweden; ◦ Collision avoidance/sense and avoid ◦ Dynamic feature maps, and simultaneous local- [email protected] ization and mapping (SLAM) Matthijs Spaan, Instituto de Sistemas e Robótica, Instituto • Path planning: Superior Técnico, Avenue Rovisco Pais 1, 1049-001 Lisboa, ◦ Proactive, based on open-loop optimization Portugal; [email protected] ◦ Reactive, based on adaptive control or model predictive control (MPC) ◦ Probabilistic approaches for maximizing the ex- pected future information • Exploration: ◦ Networked teams of robots ◦ Sensor networks which mix static sensors with autonomous moving ones ◦ Distributed algorithms and communication aspects The special issue will focus on the one hand on the devel- opment and comparison of algorithmic approaches and on the other hand on their currently ever-widening range of ap- plications in any platform: underwater, surface, ground, and airborne. Special interest lies in probabilistic approaches and setups of distributed sensors and actuators. Hindawi Publishing Corporation http://www.hindawi.com EURASIP Journal on Image and Video Processing Special Issue on Patches in Vision Call for Papers The smallest primitive employed for describing an image is • Approachesthatdrawuponpreviousworkinstruc- the pixel. However, analyzing an image as an ensemble of tural pattern recognition in order to improve current patches (i.e., spatially adjacent pixels/descriptors which are patch-based algorithms treated collectively as a single primitive), rather than indi- • Novel applications that extend the concept of patch- vidual pixels/descriptors, has some inherent advantages (i.e., based analysis to other, hitherto, nonconventional ar- computation, generalization, context, etc.) for numerous im- eas of image and video processing, computer vision, age and video content extraction applications (e.g., match- and pattern recognition ing, correspondence, tracking, rendering, etc.). Common de- • Novel techniques for estimating dependencies between scriptors in literature, other than pixels, have been contours, patches in the same image (e.g., 3D rotations) to im- shape, flow, and so forth. prove matching/correspondence algorithmic perfor- Recently, many inroads have been made into novel tasks mance in image and video content extraction through the employ- ment of patch-based representations with machine learning Authors should follow the EURASIP Journal on Image and and pattern recognition techniques. Some of these novel ar- Video Processing manuscript format described at the jour- eas include (but are not limited to): nal site http://www.hindawi.com/journals/ivp/.Prospective • authors should submit an electronic copy of their complete Object recognition/detection/tracking manuscripts through the journal Manuscript Tracking Sys- • Event recognition/detection tem at http://mts.hindawi.com/, according to the following • Structure from motion/multiview timetable: In this special issue, we are soliciting papers from the im- age/video processing, computer vision, and pattern recogni- Manuscript Due May 1, 2008 tion communities that expand and explore the boundaries of First Round of Reviews August 1, 2008 patch representations in image and video content extraction. Relevant topics to the issue include (but are not limited Publication Date November 1, 2008 to): • Novel methods for identifying (e.g., SIFT, DoGs, Har- Guest Editors ris detector) and employing salient patches • Simon Lucey, Robotics Institute, Carnegie Mellon Techniques that explore criteria for deciding the size University, Pittsburgh, PA, USA; [email protected] and shape of a patch based on image content and the application Tsuhan Chen, ECE Department, Carnegie Mellon • Approaches that explore the employment of multiple University, Pittsburgh PA, USA; [email protected] and/or heterogeneous patch sizes and shapes during the analysis of an image • Applications that explore how important relative patch position is, and whether there are advantages in allow- ing those patches to move freely or in a constrained fashion • Novel methods that explore and extend the concept of patches to video (e.g. space-time patches/volumes) Hindawi Publishing Corporation http://www.hindawi.com EURASIP Journal on Bioinformatics and Systems Biology Special Issue on Applications of Signal Processing Techniques to Bioinformatics, Genomics, and Proteomics Call for Papers The recent development of high-throughput molecular ge- Guest Editors netics technologies has brought a major impact to bioin- Erchin Serpedin, Department of Electrical and Computer formatics, genomics, and proteomics. Classical signal pro- Engineering, Texas A&M University, College Station, TX cessing techniques have found powerful applications in ex- 77843-3128, USA; [email protected] tracting and modeling the information provided by genomic and proteomic data. This special issue calls for contributions Javier Garcia-Frias, Department of Electrical and to modeling and processing of data arising in bioinformat- Computer Engineering, University of Delaware, Newark, DE ics, genomics, and proteomics using signal processing tech- 19716-7501, USA; [email protected] niques. Submissions are expected to address theoretical de- Yufei Huang, Department of Electrical and Computer velopments, computational aspects, or specific applications. Engineering, University of Texas at San-Antonio, TX However, all successful submissions are required to be tech- 78249-2057, USA; [email protected] nically solid and provide a good integration of theory with practical data. Ulisses Braga Neto, Department of Electrical and Suitable topics for this special issue include but are not Computer Engineering, Texas A&M University, College limited to: Station, TX 77843-3128, USA; [email protected] • Time-frequency representations • Spectral analysis • Estimation and detection • Stochastic modeling of gene regulatory networks • Signal processing for microarray analysis • Denoising of genomic data • Data compression • Pattern recognition • Signal processing methods in sequence analysis • Signal processing for proteomics Authors should follow the EURASIP Journal on Bioinfor- matics and Systems Biology manuscript format described at the journal site http://www.hindawi.com/journals/bsb/. 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Basic multimedia applications ing System at http://mts.hindawi.com/ according to the fol- are already available for current generation devices, and more lowing timetable: complex broadcasting services are under development or ex- pected to be launched soon, among which mobile and inter- Manuscript Due March 1, 2008 active television (ITV). Among the many challenging issues opened by these developments is the problem of personaliza- First Round of Reviews June 1, 2008 tion of such services: adaptation of the content to the techni- Publication Date September 1, 2008 cal environment of the users (device and network type) and to their individual preferences, providing personalized assis- tance for selecting and locating interesting programes among Guest Editors an overwhelming number of proposed services. 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