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Estimation of signal parameters via rotational invariance techniques

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Example of separation into subarrays (2D ESPRIT)

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]

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 , , aswhere 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 writeWith 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, andThe 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 aswhere 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 aswhere , ⁣, 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 writewhere . 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.