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Response to the Letter to Editor by Dr. M. Feldman Entitled “A Signal Decomposition or Lowpass Filtering with Hilbert Transform”
Mechanical Systems and Signal Processing 25 (2011) 3204
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Response to the Letter to Editor by Dr. M. Feldman Entitled ‘‘A Signal Decomposition or Lowpass Filtering with Hilbert Transform’’ With gratitude and appreciation, we respond to the above letter by Dr. M. Feldman. The comments made in the letter are encouraging for our future work along this line of research. We are also excited to see such a letter in immediate response to our original paper [1]. The alternative proof provided in the letter is indeed concise but rich in physical meaning. The formal introduction of a lowpass filter based on the decomposition theorem [1] is greatly appreciated. Following are our specific responses for clarification: 1. The application of the decomposition theorem is not limited to a summation of single harmonics; the theorem is also applicable to the decomposition of non-overlapping narrowband signals as Fig. 1 in [1] indicated. 2. Eq. (1) seems exact only when x(t) is an amplitude-modulated, sinusoidal function with a constant frequency [2]. It is unclear whether Eq. (1) is true in a general sense, which is a key to the generalization of the decomposition theorem from stationary to non-stationary signals. Based on the recently completed, independent study [3], Eq. (1) can be extended to the case where x(t) is a frequency-modulated harmonic signal with spectrum overlapping among signal components over the time duration. This extension can be mathematically proven when Hilbert transform is performed in phase domain instead of time domain. 3. It appears that Eq. (2) needs to be revised into Re[HxY] ReY~ þ Im[HxY] ImY~ ¼ s(t). In addition, Eq. (2) may be true only for limited Y functions such as the complex exponential function in [1]. 4. The decomposition theorem indeed functions like a ‘‘perfect’’ lowpass filter for the two-part decomposition of a signal. In fact, the decomposition is equivalent to a suite of adaptive bandpass filters in general cases. The term ‘‘adaptive filter’’ is attributed to the fact that bisecting frequencies vary with time, depending upon the specific characteristics of signals [3].
References [1] G. Chen, Z. Wang, A signal decomposition theorem with Hilbert transform and its application to narrow band time series with closely spaced frequency components, Mechanical Systems and Signal Processing doi:10.1016/j.ymssp.2011.02.002. [2] S. Hahn, Hilbert Transforms in Signal Processing, Artech House, Inc., MA, 1996, pp. 91–92. [3] Z. Wang, G. Chen, Analytical mode decomposition of time series with overlapping and rapidly modulated frequencies with Hilbert transform, Mechanical Systems and Signal Processing (2011).
G.D. Chen n, Z.C. Wang Department of Civil, Architectural, and Environmental Engineering, Missouri University of Science and Technology, 328 Butler-Carlton Hall, 1401 N Pine Street, Rolla, MO 65409-0030, USA E-mail address: [email protected] (G.D. Chen)
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DOI of original article: 10.1016/j.ymssp.2011.04.016 Corresponding author. Tel.: þ 1 573 341 4462; fax: þ 1 573 341 6215.
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.05.010
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Response to the Letter to Editor by Dr. M. Feldman Entitled ‘‘A Signal Decomposition or Lowpass Filtering with Hilbert Transform’’ With gratitude and appreciation, we respond to the above letter by Dr. M. Feldman. The comments made in the letter are encouraging for our future work along this line of research. We are also excited to see such a letter in immediate response to our original paper [1]. The alternative proof provided in the letter is indeed concise but rich in physical meaning. The formal introduction of a lowpass filter based on the decomposition theorem [1] is greatly appreciated. Following are our specific responses for clarification: 1. The application of the decomposition theorem is not limited to a summation of single harmonics; the theorem is also applicable to the decomposition of non-overlapping narrowband signals as Fig. 1 in [1] indicated. 2. Eq. (1) seems exact only when x(t) is an amplitude-modulated, sinusoidal function with a constant frequency [2]. It is unclear whether Eq. (1) is true in a general sense, which is a key to the generalization of the decomposition theorem from stationary to non-stationary signals. Based on the recently completed, independent study [3], Eq. (1) can be extended to the case where x(t) is a frequency-modulated harmonic signal with spectrum overlapping among signal components over the time duration. This extension can be mathematically proven when Hilbert transform is performed in phase domain instead of time domain. 3. It appears that Eq. (2) needs to be revised into Re[HxY] ReY~ þ Im[HxY] ImY~ ¼ s(t). In addition, Eq. (2) may be true only for limited Y functions such as the complex exponential function in [1]. 4. The decomposition theorem indeed functions like a ‘‘perfect’’ lowpass filter for the two-part decomposition of a signal. In fact, the decomposition is equivalent to a suite of adaptive bandpass filters in general cases. The term ‘‘adaptive filter’’ is attributed to the fact that bisecting frequencies vary with time, depending upon the specific characteristics of signals [3].
References [1] G. Chen, Z. Wang, A signal decomposition theorem with Hilbert transform and its application to narrow band time series with closely spaced frequency components, Mechanical Systems and Signal Processing doi:10.1016/j.ymssp.2011.02.002. [2] S. Hahn, Hilbert Transforms in Signal Processing, Artech House, Inc., MA, 1996, pp. 91–92. [3] Z. Wang, G. Chen, Analytical mode decomposition of time series with overlapping and rapidly modulated frequencies with Hilbert transform, Mechanical Systems and Signal Processing (2011).
G.D. Chen n, Z.C. Wang Department of Civil, Architectural, and Environmental Engineering, Missouri University of Science and Technology, 328 Butler-Carlton Hall, 1401 N Pine Street, Rolla, MO 65409-0030, USA E-mail address: [email protected] (G.D. Chen)
n
DOI of original article: 10.1016/j.ymssp.2011.04.016 Corresponding author. Tel.: þ 1 573 341 4462; fax: þ 1 573 341 6215.
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.05.010