Abstract

A quadratic programming optimization procedure for designing asymmetric apodization windows tailored to the shape of time-domain sample waveforms recorded using a terahertz transient spectrometer is proposed. By artificially degrading the waveforms, the performance of the designed window in both the time and the frequency domains is compared with that of conventional rectangular, triangular (Mertz), and Hamming windows. Examples of window optimization assuming Gaussian functions as the building elements of the apodization window are provided. The formulation is sufficiently general to accommodate other basis functions.

© 2007 Optical Society of America

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References

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2005 (2)

2004 (1)

D. Marks, P. S. Carney, and S. A. Boppart, J. Biomed. Opt. 9, 1281 (2004).
[CrossRef] [PubMed]

2002 (2)

1998 (1)

1997 (1)

1995 (1)

H. C. Stankwitz, R. J. Dallaire and J. R. Fienup, IEEE Trans. Aerosp. Electron. Syst. 31, 267 (1995).
[CrossRef]

1981 (1)

S. M. Kay, Proc. IEEE 69, 1380 (1981).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

(a) Nondegraded time-domain sample waveforms, (b) degraded by a factor of 10, and (c) degraded by a factor of 30 with corresponding optimized apodization windows. (d) Nondegraded waveform with conventional apodization windows. (e)–(h) Corresponding frequency domain plots of windowing functions (low-frequency value normalized to 0 dB ).

Tables (1)

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Table 1 RMS difference (%) between the Apodized Degraded Waveforms and the Original Sample Waveform in the Time and Frequency ( 0.5 2.0 THz ) Domains a

Equations (6)

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w ( t ) = i = 1 M p i g i ( t ) ,
J ( p ) = t = 0 2 L 1 E { [ x a ( t ) x m ( t ) ] 2 } = t = 0 2 L 1 { { x m 2 ( t ) + σ 2 } w 2 ( t ) 2 x m 2 ( t ) w ( t ) + x m 2 ( t ) } ,
Q j , k ( t ) = exp [ 0.5 t 2 ( 1 s j 2 + 1 s k 2 ) ] ,
r j ( t ) = exp [ 0.5 t 2 s j 2 ] , j , k = 1 , , M .
[ 1 1 1 1 1 g 2 ( 2 L 1 ) g 3 ( 2 L 1 ) g M ( 2 L 1 ) ] p = [ 1 0 ] .
p i 0 , i = 2 , , M .

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