Estimation of signal parameters via rotational invariance techniques

Estimation of signal parameters via rotational invariant techniques (ESPRIT), is a technique to determine the parameters of a mixture of sinusoids in background noise. This technique was first proposed for frequency estimation.[1] However, with the introduction of phased-array systems in everyday technology, it is also used for angle of arrival estimations.[2]

Example of separation into subarrays (2D ESPRIT)

One-dimensional ESPRIT

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At instance  , the   (complex -valued) output signals (measurements)  ,  , of the system are related to the   (complex -valued) input signals  ,  , as where   denotes the noise added by the system. The one-dimensional form of ESPRIT can be applied if the weights have the form  , whose phases are integer multiples of some radial frequency  . This frequency only depends on the index of the system's input, i.e.,  . The goal of ESPRIT is to estimate  's, given the outputs   and the number of input signals,  . Since the radial frequencies are the actual objectives,   is denoted as  .

Collating the weights   as   and the   output signals at instance   as  ,  where  . Further, when the weight vectors   are put into a Vandermonde matrix  , and the   inputs at instance   into a vector  , we can write With several measurements at instances   and the notations  ,   and  , the model equation becomes 

Dividing into virtual sub-arrays

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Maximum overlapping of two sub-arrays (N denotes number of sensors in the array, m is the number of sensors in each sub-array, and   and   are selection matrices)

The weight vector   has the property that adjacent entries are related. For the whole vector  , the equation introduces two selection matrices  and  :   and  . Here,   is an identity matrix of size   and   is a vector of zero.

The vectors     contains all elements of   except the last [first] one. Thus,   and The above relation is the first major observation required for ESPRIT. The second major observation concerns the signal subspace that can be computed from the output signals.

Signal subspace

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The singular value decomposition (SVD) of   is given as where   and   are unitary matrices and   is a diagonal matrix of size  , that holds the singular values from the largest (top left) in descending order. The operator   denotes the complex-conjugate transpose (Hermitian transpose).

Let us assume that  . Notice that we have   input signals. If there was no noise, there would only be   non-zero singular values. We assume that the   largest singular values stem from these input signals and other singular values are presumed to stem from noise. The matrices in SVD of   can be partitioned into submatrices, where some submatrices correspond to the signal subspace and some correspond to the noise subspace. where   and   contain the first   columns of   and  , respectively and  is a diagonal matrix comprising the   largest singular values.

Thus, The SVD can be written as where  , ⁣ , and   represent the contribution of the input signal   to  . We term   the signal subspace. In contrast,  ,  , and   represent the contribution of noise   to  .

Hence, from the system model, we can write   and  . Also, from the former, we can write where  . In the sequel, it is only important that there exists such an invertible matrix   and its actual content will not be important.

Note: The signal subspace can also be extracted from the spectral decomposition of the auto-correlation matrix of the measurements, which is estimated as 

Estimation of radial frequencies

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We have established two expressions so far:   and  . Now,  where   and   denote the truncated signal sub spaces, and  The above equation has the form of an eigenvalue decomposition, and the phases of eigenvalues in the diagonal matrix   are used to estimate the radial frequencies.

Thus, after solving for   in the relation  , we would find the eigenvalues   of  , where  , and the radial frequencies   are estimated as the phases (argument) of the eigenvalues.

Remark: In general,   is not invertible. One can use the least squares estimate  . An alternative would be the total least squares estimate.

Algorithm summary

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Input: Measurements  , the number of input signals   (estimate if not already known).

  1. Compute the singular value decomposition (SVD) of   and extract the signal subspace   as the first   columns of  .
  2. Compute   and  , where   and  .
  3. Solve for   in   (see the remark above).
  4. Compute the eigenvalues   of  .
  5. The phases of the eigenvalues   provide the radial frequencies  , i.e.,   .

Notes

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Choice of selection matrices

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In the derivation above, the selection matrices   and   were used. However, any appropriate matrices   and   may be used as long as the rotational invariance  i.e.,   , or some generalization of it (see below) holds; accordingly, the matrices   and   may contain any rows of  .

Generalized rotational invariance

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The rotational invariance used in the derivation may be generalized. So far, the matrix   has been defined to be a diagonal matrix that stores the sought-after complex exponentials on its main diagonal. However,   may also exhibit some other structure.[3] For instance, it may be an upper triangular matrix. In this case,  constitutes a triangularization of  .

See also

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References

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  1. ^ Paulraj, A.; Roy, R.; Kailath, T. (1985), "Estimation Of Signal Parameters Via Rotational Invariance Techniques - Esprit", Nineteenth Asilomar Conference on Circuits, Systems and Computers, pp. 83–89, doi:10.1109/ACSSC.1985.671426, ISBN 978-0-8186-0729-5, S2CID 2293566
  2. ^ Volodymyr Vasylyshyn. The direction of arrival estimation using ESPRIT with sparse arrays.// Proc. 2009 European Radar Conference (EuRAD). – 30 Sept.-2 Oct. 2009. - Pp. 246 - 249. - [1]
  3. ^ Hu, Anzhong; Lv, Tiejun; Gao, Hui; Zhang, Zhang; Yang, Shaoshi (2014). "An ESPRIT-Based Approach for 2-D Localization of Incoherently Distributed Sources in Massive MIMO Systems". IEEE Journal of Selected Topics in Signal Processing. 8 (5): 996–1011. arXiv:1403.5352. Bibcode:2014ISTSP...8..996H. doi:10.1109/JSTSP.2014.2313409. ISSN 1932-4553. S2CID 11664051.

Further reading

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  • Paulraj, A.; Roy, R.; Kailath, T. (1985), "Estimation Of Signal Parameters Via Rotational Invariance Techniques - Esprit", Nineteenth Asilomar Conference on Circuits, Systems and Computers, pp. 83–89, doi:10.1109/ACSSC.1985.671426, ISBN 978-0-8186-0729-5, S2CID 2293566.
  • Roy, R.; Kailath, T. (1989). "Esprit - Estimation Of Signal Parameters Via Rotational Invariance Techniques" (PDF). IEEE Transactions on Acoustics, Speech, and Signal Processing. 37 (7): 984–995. doi:10.1109/29.32276. S2CID 14254482. Archived from the original (PDF) on 2020-09-26. Retrieved 2011-07-25..
  • Ibrahim, A. M.; Marei, M. I.; Mekhamer, S. F.; Mansour, M. M. (2011). "An Artificial Neural Network Based Protection Approach Using Total Least Square Estimation of Signal Parameters via the Rotational Invariance Technique for Flexible AC Transmission System Compensated Transmission Lines". Electric Power Components and Systems. 39 (1): 64–79. doi:10.1080/15325008.2010.513363. S2CID 109581436.
  • Haardt, M., Zoltowski, M. D., Mathews, C. P., & Nossek, J. (1995, May). 2D unitary ESPRIT for efficient 2D parameter estimation. In icassp (pp. 2096-2099). IEEE.