G.-S. HU ET AL.

Copyright © 2011 SciRes. CS

111

The voltage signal divided into 5 time segments. In the

first time interval, the voltage constrains only a fre-

quency: 50 Hz. At time 0.3, the voltage constrains an-

other harmonic: 150 Hz. In the time interval 0.6 1.4t

,

the voltage constrains a linear time changing harmonic.

Then at 1.4 s, the harmonic, 350 Hz, added to the signal.

Finally, the voltage retain normally at 1.7 s.

The sampling frequency is 1000 Hz. Figure 2 (a) is

the improved chirplet transform plot of (16). Figure 2(b)

is the contour plot of the signal (a) using (13) and (14).

Form Figure 2(b), we find the ICT contour illustrates the

work frequency 50 Hz, two harmonic frequency 150 Hz

and 350 Hz, and the linear time changing frequency 50 +

0.2 t Hz.

Moreover, the contour in Figure 2(b) clearly shows

the harmonic occurrence times and durations.

In order to investigate the influence of initial phase,

we modulating the above simulating voltage signal with

pi/3 phase. From Figure 3, we find that the initial phase

does not influence the harmonic detection.

5. Conclusions

The chirplet transform is the generalization form of fast

Fourier transform, short-time Fourier transform, and

wavelet transform. It has the most flexible time fre-

quency window and successfully used in practices.

However, the chirplet transform has not inherent recon-

structing formulae. So we proposed the improved chir-

plet transform (ICT) and constructed the inverse ICT.

Finally, the power of the improved chirplet transform is

apparent from the above examples.

6. References

[1] R. G. Stockwell, L. Mansinha and R. P. Lowe, “Location

of the Complex Spectrum: The S-Transform,” IEEE

Transactions on Signal Processing, Vol. 44, No. 4, 1996,

pp. 998-1001. doi:10.1109/78.492555

[2] D. Gabor, “Theory of Communication,” Journal of Insti-

tution of Electrical Engineers, Vol. 93, No. 3, 1946, pp.

429-457.

[3] R. N. Bracewell, “The Fourier Transform and Its Appli-

cations,” McGraw-Hill, New York, 1978.

[4] S. Mallat, “A Wavelet Tour of Signal Processing,” 2nd

Edition, Academic Press, Waltham, 2001.

[5] L. Cohen, “Time-Frequency Distributions—A Review,”

Proceedings of the IEEE, Vol. 77, No. 7, 1989, pp.

941-981. doi:10.1109/5.30749

[6] F. Hlawatsch and G. F. Boudreuax-Bartels, “Linear and

Quadratic Time-Frequency Signal Representations,” Pro-

ceedings of Signal Processing Magazine, Vol. 9, No. 2,

1992, pp. 21-67.

[7] M. V. Chilukur and P. K. Dash, “Multiresolution S-Trans-

form-Based Fuzzy Recognition System for Power Quality

Events,” IEEE Transactions on Power Delivery, Vol. 19,

No. 1, 2004, pp. 323-330.

doi:10.1109/TPWRD.2003.820180

[8] S. Mann and S. Haykin, “The Chirplet Transform: Physi-

cal Considerations,” IEEE Transactions on Signal Proc-

essing, 1995, Vol. 43, No. 11, pp. 2745-2761.

doi:10.1109/78.482123

[9] G.-S. Hu, F.-F. Zhu and Y.-J. Tu, “Power Quality Dis-

turbance Detection and Classification Using Chirplet

Transform,” Lecture Notes in Computer Science, Vol.

4247, 2006, pp. 34-41. doi:10.1007/11903697_5

[10] Z. Ren, G. S. Hu, W. Y. Huang and F. F. Zhu, “Motor

Fault Signals Denosing Based on Chirplet Transform,”

Transactions of China Electrotechnical Society, Vol. 17,

No. 3, 2002, pp. 59-62.

[11] G. S. Hu and F. F. Zhu, “Location of slight Fault in Elec-

tric Machine Using Trigonometric Spline Chirplet Trans-

forms,” Power System Technology, Vol. 27, No. 2, 2003,

pp. 28-31.