# Optimal random sampling for spectrum estimation in DASP applications

Andrzej Tarczynski; Dongdong Qu

International Journal of Applied Mathematics and Computer Science (2005)

- Volume: 15, Issue: 4, page 463-469
- ISSN: 1641-876X

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topTarczynski, Andrzej, and Qu, Dongdong. "Optimal random sampling for spectrum estimation in DASP applications." International Journal of Applied Mathematics and Computer Science 15.4 (2005): 463-469. <http://eudml.org/doc/207758>.

@article{Tarczynski2005,

abstract = {In this paper we analyse a class of DASP (Digital Alias-free Signal Processing) methods for spectrum estimation of sampled signals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimators of Fourier transforms of the analysed signals. The estimators are unbiased inside arbitrarily wide frequency ranges, regardless of how sparsely the signal samples are collected. In order to facilitate quality assessment of the estimators, we calculate their standard deviations. The optimal sampling scheme that minimises the variance of the resulting estimator is derived. The further analysis presented in this paper shows how sampling instant jitter deteriorates the quality of spectrum estimation. A couple of numerical examples illustrate the main thesis of the paper.},

author = {Tarczynski, Andrzej, Qu, Dongdong},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {random sampling; optimal sampling; spectral analysis; digital alias-free signal processing},

language = {eng},

number = {4},

pages = {463-469},

title = {Optimal random sampling for spectrum estimation in DASP applications},

url = {http://eudml.org/doc/207758},

volume = {15},

year = {2005},

}

TY - JOUR

AU - Tarczynski, Andrzej

AU - Qu, Dongdong

TI - Optimal random sampling for spectrum estimation in DASP applications

JO - International Journal of Applied Mathematics and Computer Science

PY - 2005

VL - 15

IS - 4

SP - 463

EP - 469

AB - In this paper we analyse a class of DASP (Digital Alias-free Signal Processing) methods for spectrum estimation of sampled signals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimators of Fourier transforms of the analysed signals. The estimators are unbiased inside arbitrarily wide frequency ranges, regardless of how sparsely the signal samples are collected. In order to facilitate quality assessment of the estimators, we calculate their standard deviations. The optimal sampling scheme that minimises the variance of the resulting estimator is derived. The further analysis presented in this paper shows how sampling instant jitter deteriorates the quality of spectrum estimation. A couple of numerical examples illustrate the main thesis of the paper.

LA - eng

KW - random sampling; optimal sampling; spectral analysis; digital alias-free signal processing

UR - http://eudml.org/doc/207758

ER -

## References

top- Beutler F. (1970): Alias-free randomly timed sampling of stochasticprocesses. - IEEE Trans. Inf. Theory, Vol. IT-16, No. 2, pp. 147-152. Zbl0193.15802
- Benedetto J. and Ferreira P.J.S.G. (2001): Modern Sampling Theory: Mathematics and Applications. - Boston, MA: Birkhaeuser. Zbl1017.94006
- Bilinskis I. and Mikelsons M. (1992): Randomized Signal Processing. - London: Prentice Hall. Zbl0925.94006
- Chen D. Shi and Allebach J.P. (1987): Analysis of error in reconstruction of two-dimensional signal from irregularly spaced samples. - Vol. ASSP-35, No.2, pp. 173-180.
- Feng P. and Bresler Y. (1996): Spectrum-blind minimum-rate sampling and reconstruction of multiband signals. - IEEE Int. Conf. Acoustics, Speech, and Signal Processing, ICASSP-96, Atlanta, GA, Vol. 3, pp. 1688-1691.
- Ferreira P.J.S.G. (1992): Incomplete sampling and the recovery of missing samples from oversampled band-limited signals. - IEEE Trans. Signal Process., Vol. 40, No.1, pp. 225-227.
- Herley C. and Wong P.W. (1999): Minimum rate sampling and reconstruction of signals with arbitrary frequency support. - IEEE Trans. Inf. Theory, Vol. 45, No. 5, pp. 1555-1564. Zbl0965.94005
- Kida T. and Mochizuki H. (1992): Generalized interpolatory approximation ofmulti-dimensional signals having the minimum measure of error. - IEICE Trans. Fundamentals, Vol. E75-A, No. 7, pp. 794-805.
- Landau H.J. (1967): Necessary density conditions for sampling and interpolation of certain entire functions. - Acta Math., Vol. 117, pp. 37-52. Zbl0154.15301
- Lomb N.R. (1976): Least-squares frequency analysis of unequally spaced data. - Astroph. Space Sc., Vol. 39, No. 2, pp. 447-462.
- Marvasti F. (1987): A Unified Approach to Zero-Crossing and Nonuniform Sampling of Single and Multidimensional Signals and Systems. - Oak Park IL: Nonuniform Publications.
- Marvasti F. (2001): Nonuniform Sampling, Theory and Practice. - New York: Kluwer. Zbl0987.00028
- Masry E. (1978): Alias-free sampling: An alternative conceptualization and its applications. - IEEE Trans. Inf. Theory, Vol. 24, No. 3, pp. 317-324. Zbl0384.62080
- Scargle J.D. (1982): Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data. - Astronom. J., Vol. 263, No. 1, pp. 835-853.
- Scargle J.D. (1989): Studies in astronomical time series analysis. III. Fourier transforms, autocorrelation functions, and gross-correlation functions of unevenly spaced data. - Astronom. J., Vol. 343, No. 1, pp. 874-887.
- Shapiro H.S. and Silverman R.A. (1960): Alias-free sampling of random noise. - SIAM J. Appl. Math., Vol. 8, No. 2, pp. 225-236. Zbl0121.14204
- Tarczyński A. (1997): Sensitivity of signal reconstruction. - IEEE Signal Process. Lett., Vol. 4, No. 7, pp. 192-194.
- Tarczyński A. and Allay N. (2004): Spectral analysis of randomly sampled signals: Suppression of aliasing and sampler jitter. - IEEE Trans. Signal Process.,Vol. 52, No. 12, pp. 3324-3334.
- Tarczyński A. and Cain G.D. (1997): Reliability of signal reconstruction from finite sets of samples. - 1997 Workshop Sampling Theory and Applications, SAMPTA'97, Aveiro, Portugal, Vol. 1, pp. 181-186.
- Tarczyński A. and Valimaki V. (1996): Modifying FIR and IIR filters for processing signals with lost samples. - Proc. IEEE Nordic Signal ProcessingSymposium, NORSIG'96, Helsinki, Finland, Vol. 1, pp. 359-362.
- Tarczyński A., Valimaki V. and Cain G.D. (1997): FIR filtering of nonuniformly sampled signals. - IEEE Int. Conf. Acoustics, Speech and Signal Processing, ICASSP'97, Munich, Germany, Vol. 3, pp. 2237-2240.
- Venkataramani R. and Bresler Y. (2001): Optimal sub-Nyquist nonuniform sampling and reconstruction for multiband signals. - IEEE Trans. Signal Process., Vol. 49, No. 10, pp. 2301-2313.
- Weisstein E.W. (2003): CRC Concise Encyclopedia of Mathematics. - London, UK, Chapman Hall CRC. Zbl1079.00009
- Wingham D.J. (1992): The reconstruction of a band-limited function and its Fourier transform from a finite number of samples at arbitrary locations by singular value decomposition. - IEEE Trans. Signal Process., Vol. 40,No. 3, pp. 559-570.
- Yen J.L. (1956): On nonuniform sampling of bandwidth-limited signals. - IRE Trans. Circ. Theory, Vol. 3, No. 4, pp. 251-257.

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