Multiscale Signal Analysis and Modeling pp 105-137 | Cite as

# Coprime Sampling and Arrays in One and Multiple Dimensions

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## Abstract

This chapter gives an overview of the concept of coprime sampling. The basic idea is that a continuous-time (or spatial) signal is sampled simultaneously by two sets of samplers, with sampling rates 1∕*MT* and 1∕*NT* where *M* and *N* are coprime integers and *T*>0. One of the results is that it is possible to estimate the autocorrelation of the signal at a much higher rate (\(= 1/T\)) than the total sampling rate. Thus, any application which is based on autocorrelation will benefit from such sampling and reconstruction. One example is in array processing, in the context of estimation of direction of arrival (DOA) of sources. Traditionally, an array with *L* sensors would be able to identify *L* − 1 independent sources, but with a pair of coprime arrays, one can identify *O*(*L* ^{2}) sources. It is also shown how to use two DFT filter banks, one in conjunction with each sampling array, to produce a much denser tiling of the frequency domain than each filter bank would individually be able to do. This chapter also discusses the extension of coprime sampling to multiple dimensions by using sampling geometries that are defined based on lattices. In this context the generation of coprime pairs of integer matrices is a very interesting mathematical problem and is dealt with in detail. The use of coprime samplers in system identification is also elaborated upon. A brief review of fractionally spaced equalizers in digital communications, in the context of coprime sampling, is included.

## Keywords

Filter Bank Linear Time Invariant System Sparse Array Dense Tiling Integer Matrice## Notes

### Acknowledgements

This work was supported in parts by the Office of Naval Research grant N00014-08-1-0709 and the California Institute of Technology.

## References

- 1.Abramovich YI, Gray DA, Gorokhov AY, Spencer NK (1998) positive-definite toeplitz completion in DOA estimation for nonuniform linear antenna arrays. I. Fully augmentable arrays. IEEE Trans Signal Proc 46:2458–2471MathSciNetzbMATHCrossRefGoogle Scholar
- 2.Chen T, Vaidyanathan PP (1993) The role of integer matrices in multidimensional multirate systems. IEEE Trans Signal Proc SP-41:1035–1047CrossRefGoogle Scholar
- 3.Chen CY, Vaidyanathan, PP (2008) Minimum redundancy MIMO radars. IEEE Int Symp Circ Syst 45–48Google Scholar
- 4.Dudgeon, DE, Mersereau, RM (1984) Multidimensional digital signal processing, Prentice Hall, Inc., Englewoods CliffszbMATHGoogle Scholar
- 5.Horn RA, Johnson, CR (1985) Matrix analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
- 6.Kailath, T (1980) Linear Systems, Prentice Hall, Inc., Englewood CliffszbMATHGoogle Scholar
- 7.Hoctor RT, Kassam, SA (1990) The unifying role of the coarray in aperture synthesis for coherent and incoherent imaging. Proceedings of the IEEE. 78:735–752CrossRefGoogle Scholar
- 8.Moffet A (1968) Minimum-redundancy linear arrays. IEEE Trans Antenn Propag 16: 172–175CrossRefGoogle Scholar
- 9.Oppenheim AV, Schafer RW (1999) Discrete time signal processing, Prentice Hall, Inc., Englewood CliffsGoogle Scholar
- 10.Pal P, Vaidyanathan, PP (2011) Coprime sampling and the MUSIC algorithm. Proc of the 14th IEEE Digital Signal Processing Workshop, Sedona, AZGoogle Scholar
- 11.Pal P, Vaidyanathan PP (2010) Nested arrays: a novel approach to array processing with enhanced degrees of freedom. IEEE Trans Signal Process 58: 4167–4181MathSciNetCrossRefGoogle Scholar
- 12.Pal P, Vaidyanathan PP (2011) Coprimality of certain families of integer matrices, IEEE Trans Signal Process 59: 1481–1490MathSciNetCrossRefGoogle Scholar
- 13.Pillai SU, Bar-Ness, Y, Haber F (1985) A new approach to array geometry for improved spatial spectrum estimation. Proc IEEE 73: 1522–1524CrossRefGoogle Scholar
- 14.Proakis JG (1995) Digital communications. McGraw Hill, Inc., New YorkGoogle Scholar
- 15.Rajagopal R, Potter LC (2003) Multivariate MIMO FIR inverses. IEEE Trans Image Proc12: 458–465MathSciNetCrossRefGoogle Scholar
- 16.Skolnik MI (2001) Introduction to radar systems. McGraw-Hill, NYGoogle Scholar
- 17.Treichler JR, Fijalkow I, Johnson CR (1996) Fractionally spaced equalizers: how long should they be? IEEE Signal Process Mag 13:65–81CrossRefGoogle Scholar
- 18.Vaidyanathan PP, Pal P (2011) Sparse sensing with coprime samplers and arrays. IEEE Trans Signal Process 59:573–586MathSciNetCrossRefGoogle Scholar
- 19.Vaidyanathan PP, Pal P (2011) Theory of sparse coprime sensing in multiple dimensions. IEEE Trans Signal Process 59:3592–3608MathSciNetCrossRefGoogle Scholar
- 20.Vaidyanathan PP, Pal P (2011) A general approach to coprime pairs of matrices, based on minors. IEEE Trans Signal Process 59:3536–3548MathSciNetCrossRefGoogle Scholar
- 21.Vaidyanathan PP, Pal P (2010) System identification with sparse coprime sensing. IEEE Signal Process Lett 17:823–826CrossRefGoogle Scholar
- 22.Vaidyanathan PP, Pal P (2011) Coprime Sampling for System Stabilization with FIR Multirate Controllers. Proc IEEE Asilomar Conf on Signals, Systems, and CompuersGoogle Scholar
- 23.Vaidyanathan PP (1993) Multirate systems and filter bank. Prentice Hall, Inc., Englewood CliffsGoogle Scholar
- 24.Van Trees HL (2002) Optimum array processing: part IV of detection, estimation and modulation theory, Wiley Interscience, NYCrossRefGoogle Scholar
- 25.Vrcelj B, Vaidyanathan PP (2002) MIMO biorthogonal partners and applications, IEEE Trans Sig Process 50: 528–542CrossRefGoogle Scholar
- 26.Vrcelj B, Vaidyanathan PP (2003) Fractional biorthogonal partners in channel equalization and signal interpolation. IEEE Trans Signal Process 51:1928–1940MathSciNetCrossRefGoogle Scholar
- 27.Xia XG (1999) On estimation of multiple frequencies in undersampled complex valued waveforms. IEEE Trans Signal Process 47:3417–3419zbMATHCrossRefGoogle Scholar
- 28.Xia XG, Liu K (2005) A generalized Chinese remainder theorem for residue sets with errors and its application in frequency determination from multiple sensors with low sampling rates. IEEE Signal Process Lett 12:768–771CrossRefGoogle Scholar