An efficient algorithm is proposed to extract components of a composite signal. The proposed approach has two stages of processing in which the time-frequency supports of the individual signal components are identified and then the individual components are estimated by performing a simple time-frequency domain incision on the identified support of the component. The use of a recently proposed time-frequency representation [1] significantly improves the performance of the proposed approach by providing very accurate description on the auto-Wigner terms of the composite signal. Then, simple fractional Fourier domain incision provides reliable estimates for each of the signal components in O(N log N) complexity for a composite signal of duration N

Arikan Orhan, Ozdemir, A.Kemal

Bilkent University Institutional Repository

A N  EFFICIENT ALGORITHM TO EXTRACT COMPONENTS OF A COMPOSITE SIGNAL Orhan Arzkan and A.  Kemal Ozdemir Department of Electrical and Electronics Engineering, Bilkent University, .4nkara, TR-06533 TURKEY. Phone & Fax: 90-312-2664307, e-mail: oarikanC!ee.bilkent.edu.tr and kozdemir@ee.bilkent.edu.tr ABSTRACT 4 n  efficient algorithm is proposed to extract components of a composite signal. The proposed approach has two stages of processing in which the time-frequency supports of the individual signal components are identified and then the in- dividual components are estimated by performing a simple time-frequency domain incision on the identified support of the component. The use of a recently proposed t i m e  frequency representation [l] significantly improves the per- formance of the proposed approach by providing very accu- rate description on the auto-Wigner terms of the composite signal. Then, simple fractional Fourier domain incision pro- vides reliable estimates for each of the signal components in O(N log N )  complexity for a composite signal of duration N .  1. INTRODUCTION Analysis of multi-component signals have been an active research area since the introduction of the time-frequency concepts. The search for the signal components which have compact time-frequency supports typically starts with the careful examination of the time-frequency distribution of the composite signal. The Wigner distribution is the most commonly used time-frequency analysis tool which provides the highest resolution time-frequency characterization of the signal. However, because of its bilinear nature, the sup- ports of the actual signal components may not be visible in the presence of cross-terms of the Wigner distribution. For instance, if the signal s ( t )  is composed of m signal com- ponents, z , ( t ) ,  1 5 i 5 m, then the corresponding Wigner distribution [2]  can be written as: 0-7803-6293-4/00/$10.00 02000 IEEE. where the high resolution auto-Wigner distributions cor- responding to  m individual signal components are accom- panied by m(m - 1)/2 cross-Wigner distributions [3]. As shown in Fig. 2(a), the cross-Wigner terms may partially or totally overlap with the auto-Wigner terms making it very difficult if not impossible to  detect and identify the time- frequency supports of the individual signal components. Since the cross-Wigner terms are oscillatory in nature [4],  2-D low pass filtering reduces the cross-term interfer- ence [ 5 ,  61. However, the resolution of the auto-Wigner terms may degrade considerably resulting in identification of significantly larger supports for the signal components. This not only causes extraction of more noisy signal com- ponent estimates but also signal components with closely spaced timefrequency supports to  be identified as only one signal component. Since the success of the component analysis is very much related to  the accurate identification smoothed Wigner distributions are not very suitable for the extraction of the signal components. The draMr-backs of smoothed-Wigner distributions in the analysis and ex- traction of individual signal components can be partially overcome with the use of signal dependent sliding window time-frequency representations [7]. However, the high com- plexity of the computation of these representations, and more importantly, the use of the same time domain filter- ing of all signal components occurring at the same time but different frequencies limits the success of these approaches. In this paper, the time-frequency supports of individ- ual components are identified by using a recently devel- oped time-frequency representation [l]. Since, in the new representation directional smoothing of arbitrarily chosen time-frequency regions is made possible, the interference of cross-Wigner terms can be greatly reduced with negli- gible distortion on the auto-Wigner terms. Therefore the reliable detection and high resolution identification can be performed very easily on the new tirne-frequency represen- t a t  ion. of the timefrequency supports of the signal components, 2. AN EFFICIENT ALGORITHM FOR THE IDENTIFICATION AND EXTRACTION OF SIGNAL COMPONENTS Time-frequency based extraction of the individual signal components of a given multi-component signal can be con- 697 ducted in two stages. In the first stage detection and iden- tification of the individual signal components is performed on the time-frequency plane. Then, the signal components are estimated based on the obtained time-frequency infor- mation on them. As it is explained in the previous sec- tion, high resolution and accurate description of the t i m e  frequency content of the individual signal components is essential in the over-all performance of the component ex- traction. Since, the currently used timefrequency repre- sentations do not provide such a description, the second stage of processing becomes significantly involved to  pro- vide reasonable results [8, 91. In the following, we propose to  use a recently introduced time-frequency representation in the first stage of the analysis. Since, this new represen- tations provides the required time-frequency information very precisely, the signal components can be extracted very efficiently. In order to  demonstrate the efficiency of the new t i m e  frequency representation, the fivecomponent signal whose Wigner distribution is shown in Fig. 2(a) is analyzed as de- tailed in [l] .  4 s  shown in Fig. 2(c), signal components can be easily detected and their supports can be accurately de- scribed. The supports of the individual signal components can be identified either manually or automatically by using adaptive thresholding methods. In the second stage of processing, the obtained informa- tion on the supports of the individual signal components is used to  design proper time-frequency incision techniques to  extract the components directly from the signal. To demon- strate the required processing for the signal component ex- traction, consider the supports of auto-terms of the Wigner distribution of a composite signal as shown in Fig. 3. In or- der to  extract the signal component which is localized at the center of the time-frequency plane, a time-frequency inci- sion around this component should be performed. Among many alternatives, the simplest incision can be performed by first applying a frequency domain mask Hl(f) to S ( f )  whose support is the same as the frequency axis projection of the signal component. Then, to the result a t imedomain mask, whose support is the projection of the signal compo- nent on the time-axis, can be applied to approximate the signal component. This way, the estimated signal compo- nent will have its timefrequency support approximately limited into the dashed-box around the desired signal com- ponent. Formally, the component estimate is obtained by: &(t )  = h2(t)[hl( t )  * s(t)]  =: x , ( t )  In a more general case, if the supports of the auto- components in the timefrequency plane are as shown in Fig. 4 ,  then it is not possible to  extract z , ( t )  from s ( t ) ,  by successive masking in frequency and time domains. Because in this case there does not exist a rectangular region in the time-frequency plane, which contains only the the support of the ith auto-component but not the others. However, a viable solution in this case is first to  translate the origin of the time-frequency plane to  approximate center ( t t ,  f t )  of the ith auto-component as shown in Fig. 4 .  The required translation can be performed as: S ( t )  = s ( t  + t")e-J2"". (3) Note that the ith component of the signal S ( t )  is ?,(t) = zs ( t  + t ,)e-32"t'*. Then the fractional Fourier transform (FrFT) [lo] of this signal is &, ( t )  E { . P d } ( t )  Ka,  ( t ,  t')S(t') dt' , (4) where a,  = 24,/7r is the order of the FrFT and K,,(t,t ') is the kernel of the transformation given in [lo]. Since the WD of the a:h order FrFT of a signal is the same as the WD of the original signal rotated by angle of a,7r/2 in the clock-wise direction [lo], the W D  of S a ,  ( t )  is aligned with one of the axis as shown in Fig. 4. Thus after the elementary operations of translation and rotation in the time-frequency plane, the WD of 5r,a,(t) fits into a compact rectangular region as shown in Fig. 4(c). Therefore, as it was the case for the WD in Fig. 3 ,  the ith component of s ( t )  can be extracted in the transform domain by successive masking J as: ( 5 )  where h z ( t )  is the dual of time-domain mask and h l ( t )  is the inverse Fourier transform of the dual of frequency do- main mask H I ( ! ) .  After obtaining an estimate for Z,+, ( t ) ,  an estimate of x, ( t )  can be easily computed by reversing the operations of translation and rotation in the time-frequency plane: In practice the required fractional Fourier transform can be directly carried on the given Nyquist rate samples of the composite signal s ( t )  by using the algorithm given in [ l l ] .  As shown in [ll], the complexity of the fractional Fourier transform is the same as FFT. Therefore, the overall com- plexity of the proposed signal component extraction algo- rithm is O(N1ogN) for a component whose time domain support is of approximately N samples in duration. The required incision in the more general case shown in Fig. 4 can also be performed by using fractional Fourier domain filtering techniques given in [12, 13, 141. However, the proposed techniques in [12, 13, 141 are for noise suppres- sion. Therefore, there is a need for improvement in these techniques to  suppress both the noise and the other signal components. We are currently working on these improve- ments and planning to  report on the obtained results and their comparisons with the simple incision technique used in this work. 3. SIMULATIONS In this section we investigate the performance of the pro- posed algorithm by conducting computer simulations on a complicated composite test signal which is composed of 5 chirp signals with Gaussian envelopes. 4s  shown in Fig. 1, it is not possible to  identify individual signal components of the composite signal. The corresponding Wigner distri- bution shown in Fig. 2(a) is very much cluttered with the cross-terms. Because of the significant overlaps between the 698 cross-terms and the auto-terms, the auto-terms shown in Fig. 2(b) cannot be identified. 4 s  shown in Fig. 2 ( c ) ,  by us- ing the first stage of the processing a significantly improved time-frequency representation of the composite signal can be obtained. 4s seen from this figure, as a result of the utilized directional filtering technique [l], the cross terms of the Wigner distribution are highly attenuated with little distortion on the auto-Wigner terms. As shown in Fig. 2(d), the error in the estimated auto-Wigner terms is negligible. Therefore, as a result of the first stage of processing, very accurate detection and support-identification of the signal components can be achieved. To illustrate the performance of the second stage of pro- cessing, we present results on the extraction of two chirp components of the composite signal shown in Fig. 1. The es- timated signal component corresponding to  the chirp com- ponent of the original signal near the origin of the time- frequency plane is shown in Fig. 5 (a). This result is ob- tained by performing time-frequency domain incision on a rotated time-frequency plane obtained by using fractional Fourier transformation of order 0.5 corresponding to  n / 4  radians of rotation. The error in the estimated signal com- ponent is shown in Fig. 5(b). As seen from this figure, the extracted signal component is a very close approximation of the original signal component with a normalized error of E, = 8.2 x lo-* which is defined as: where z and 2, are the actual and estimated signal com- ponents in vector notation. The result of the estimated signal component corre- sponding to  the shorter chirp component with a time cen- ter right below the origin is shown in Fig. 6(a) .  This re- sult is obtained by first translating the origin of the time- frequency plane to  the center of the chirp component. Then the time-frequency domain incision over the estimated sup- port of the signal component is performed on a rotated time-frequency plane obtained by using fractional Fourier transformation of order 0.5 corresponding to  7r/4 radians of rotation. The difference plot of the estimated and ac- tual signal component is shown in Fig. 6 ( b )  to illustrate the accuracy of the algorithm. As seen from this figure, the extracted signal Component is a very close approximation of the original signal component with a normalized error of E, = 4.8 4. CONCLUSIONS ,4 turo-stage processing algorithm is proposed for the ex- traction of components of a composite signal. Based on a set of simulations, it is shown that the proposed two stage processing algorithm provides highly accurate estimates for the individual signal components. The use of a recently pro- posed time-frequency representation to  detect and identify the time-frequency donlain supports of the signal compo- nents play the key role in the success of the proposed ap- proach. In the second stage, the use of fractional Fourier domain incision greatly increases the efficiency of the algo- rithm. 5. REFERENCES [l] 4 .  K. Ozdemir and 0. Arikan, “A high resolution time frequency representation with significantly re- duced cross-terms,” submitted t o  IEEE Int. Conf. Acoust. Speech Signal Process., Jun. 2000. 121 T .  A. C. hl. Claasen and W. F.  G. hlecklenbrauker, “The Wigner distribution - A tool for time-time fre- quency signal analysis, Part I: Continuous-time sig- nals,” Philips J .  Res., vol. 35, no. 3, pp. 217-250, 1980. [3] L. Cohen, “Time-frequency distributions - A review,” Proc. IEEE, vol. 77, pp. 941-981, July 1989. [4] P. Flandrin, “Some features of timefrequency rep- resentations of multicomponent signals,” Proc. IEEE Int. Conf. Acoust. Speech Signal Process., vol. 3, [5] H. I. Choi and W. J.  Williams, “Improved time- frequency representation of multicomponent signals us- ing exponential kernels,” IEEE Trans. Acoust., Speech, and Signal Process., vol. ASSP-37, pp. 862-871, June 1989. [6] Y. Zhao, L. E. Atlas, and R. J. hlarks, “The use of coneshaped kernels for generalized timefreuqeny rep- resentations of nonstationary signals,” IEEE Trans. Acoust.; Speech, and Signal Process., vol. ASSP-38, 1990. [7] R. G. Baraniuk and D. L. Jones, “An adaptive optimal-kernel time-frequency representation,” IEEE Trans. Signal Process., vol. 43, pp. 2361-2371, Oct. 1995. [8] J. Jeong and W. J.  Williams, “Timevarying filtering and signal sythesis using the extended discretetime Wigner distribution,” in Proc. ISSPA 90; Signal Pro- cessing Theories, Implementations and Applications, (Gold Coast, Australia), pp. 895-898, Aug. 1990. [9] F.  Hlawatsch, A. H. Costa, and W. Krattenhaler, “Time-frequency signal synthesis with time-frequency extrapolation and don’t-care regions,” IEEE Trans. Signal Process., vol. 42, pp. 2513-2520, Sept. 1994. [lo] L. B. Almedia, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process., vol. 42, pp. 3084-3091, Nov. 1994. [ l l]  H. hl. Ozaktas, 0. Arikan, hl. ,4. Kutay, and G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process., vol. 44, pp. 2141-2150, Sept. 1996. [12] hl. A. Kutay, H. hl. Ozaktas, 0. Arikan, andL.  Onural, “Optimal filtering in fractional fourier domains,” IEEE Trans. Signal Process., vol. 45, pp. 1129-1143, hlay 1997. [13] hl. F. Erden and H. hl. Ozaktas, “Synthesis of gen- eral linear systems with repeated filtering in consecu- tive fractional fourier domains,” J .  Opt. Soc. Am. A ,  vol. 15, no. 1647-1657, 1998. [14] hl. A. Kutay, hl. F. Erden, H. hl. Ozaktas, 0. Arikan, 0 .  Giileryiiz, and C.  Candan, “Space-bandwidth- efficient realizations of linear systems,” Optics letters, pp. 41B.4.1-41B.4.4, 1984. vol. 23, pp. 1069-1071, 1998. 699 I Y I -5 0 5 lime Figure 1: The time domain representation of a multi- component signal s ( t ) ,  which is composed of 5 linear- frequency modulated chirp signals. 1 5  1 0 5  - 5 0 01 0 005 0 -0 005 -0 01 time time Figure 2: (a) The Wigner distribution of the signal s ( t )  shown in Fig. 1, (b) the auto-components of the Wigner dis- tribution, (c) the slices of the Wigner distribution smoothed by using the data-adaptive directional filtering algorithm in [l], (d) the difference of the smoothed slices from the auto- components of the Wigner distribution. f I I I I I I I I  I Figure 3: The extraction of the component centered at the origin of the timefrequency plane by using frequency and time domain masks. Figure 4: The supports of the Wigner distribution (only auto-terms) of s ( t )  and its various transforms: (a) the WD of s ( t ) ,  (b) the WD of S ( t )  = s ( t+t , )e -32" f* t ,  (c) the WD of ia, = Fal[i](t), (d) the WD of 2 , ( t )  = Faf [h2(hl * & , ) ] ( t ) .  Figure 5 :  (a) The estimate of the long chirp component in Fig. 2 (b) which is near the origin of the time-frequency plane, (b) the difference of the estimate from the actual signal component. Figure 6: (a) The estimate of the short chirp component in Fig. 2(b) with the time center ,just below the origin, (b) the difference of the estimate from the actual signal component. 700 

Efficient algorithm to extract components of a composite signal

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Efficient algorithm to extract components of a composite signal

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