Estimation of signal parameters via rotational invariance techniques explained

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. However, with the introduction of phased-array systems in everyday technology, it is also used for angle of arrival estimations.^[1]

One-dimensional ESPRIT

\omega _{k} . This frequency only depends on the index of the system's input, i.e., $k$. The goal of ESPRIT is to estimate \omega _{k} 's, given the outputs $y\_m[t]$ and the number of input signals, $K$. Since the radial frequencies are the actual objectives, $a\_$ is denoted as $a\_m(\backslash omega\_k)$.

Collating the weights $a\_m(\backslash omega\_k)$ as

a(\omega _{k)} = [ 1 e ^{- j\omega _{k}} e ^{- j 2 \omega _{k}} ... e ^{- j(M -1 )\omega _{k}} ] ^{\top} and the $M$ output signals at instance $t$ as y[t] = [y _{1[t] y} _{2[t]} ... y ^{\top} _{M[t]]} , $$\backslash mathbf\; y[t]\; =\; \backslash sum\_^K\; \backslash mathbf\; a(\backslash omega\_k)\; x\_k[t]\; +\; \backslash mathbf[t],$$where n[t] = [n _{1[t] n} _{2[t]} ... n ^{\top} _{M[t]]} . Further, when the weight vectors a(\omega _{1),} a(\omega _{2), ...,} a(\omega _{K)} are put into a Vandermonde matrix A = [ a(\omega _{1)} a(\omega _{2)} ... a(\omega _{K)]} , and the K inputs at instance $t$ into a vector x[t] = [x _{1[t]} ... x ^{\top} _{k[t]]} , we can write$$\backslash mathbf\; y[t]\; =\; \backslash mathbf\; A\backslash ,\; \backslash mathbf\; x[t]+\backslash mathbf\; n[t].$$With several measurements at instances $t\; =\; 1,\; 2,\; \backslash dots,\; T$ and the notations $\backslash mathbf\; =\; [\backslash ,\backslash mathbf\{y\}[1]\; \backslash ,\backslash \; \backslash mathbf[2]\; \backslash ,\backslash \; \backslash dots\; \backslash ,\backslash \; \backslash mathbf[T\; ]\backslash ,]$, $\backslash mathbf\; =\; [\backslash ,\backslash mathbf\{x\}[1]\; \backslash ,\backslash \; \backslash mathbf[2]\; \backslash ,\backslash \; \backslash dots\; \backslash ,\backslash \; \backslash mathbf[T\; ]\backslash ,]$ and $\backslash mathbf\; =\; [\backslash ,\; \backslash mathbf\{n\}[1]\; \backslash ,\backslash \; \backslash mathbf[2]\; \backslash ,\backslash \; \backslash dots\; \backslash ,\backslash \; \backslash mathbf[T\; ]\backslash ,]$, the model equation becomes$$\backslash mathbf\; =\; \backslash mathbf\; \backslash mathbf\; +\; \backslash mathbf.$$

Dividing into virtual sub-arrays

The weight vector $\backslash mathbf\; a(\backslash omega\_k)$ has the property that adjacent entries are related.$$[\backslash mathbf\{a\}(\backslash omega\_k)]\_\; =\; e^\; [\backslash mathbf\{a\}(\backslash omega\_k)]\_$$For the whole vector

a(\omega _{k)} , the equation introduces two selection matrices J _{1} and J _{2} : J _{1} = [ I _{M-1} and J _{2} = [ 0 I _{M-1} ] . Here, I _{M-1} is an identity matrix of size $(M-1)$ and 0 is a vector of zero.

The vectors

J _{1} a(\omega _{k)} [ J _{2} a(\omega _{k)]} contains all elements of a(\omega _{k)} except the last [first] one. Thus, J _{2} a(\omega _{k)} = e ^{- j\omega _{k}} J _{1} a(\omega _{k)} 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

The singular value decomposition (SVD) of $\backslash mathbf\; Y$ is given as$$\backslash mathbf\; =\; \backslash mathbf\; U\; \backslash mathbf\; \backslash Sigma\; \backslash mathbf\; V^\backslash dagger$$where $\backslash mathbf\; U\; \backslash in\backslash mathbb\; C^$ and $\backslash mathbf\; V\; \backslash in\; \backslash mathbb\; C^$ are unitary matrices and $\backslash mathbf\; \backslash Sigma$ is a diagonal matrix of size $M\; \backslash times\; T$, that holds the singular values from the largest (top left) in descending order. The operator $\backslash dagger$ denotes the complex-conjugate transpose (Hermitian transpose).

Let us assume that $T\; \backslash geq\; M$. Notice that we have $K$ input signals. If there was no noise, there would only be $K$ non-zero singular values. We assume that the $K$ largest singular values stem from these input signals and other singular values are presumed to stem from noise. The matrices in SVD of $\backslash mathbf\; Y$ can be partitioned into submatrices, where some submatrices correspond to the signal subspace and some correspond to the noise subspace.where $\backslash mathbf\_\backslash mathrm\; \backslash in\; \backslash mathbb^$ and $\backslash mathbf\_\backslash mathrm\; \backslash in\; \backslash mathbb^$ contain the first $K$ columns of $\backslash mathbf\; U$ and $\backslash mathbf\; V$, respectively and $\backslash mathbf\_\backslash mathrm\; \backslash in\; \backslash mathbb^$is a diagonal matrix comprising the $K$ largest singular values.

Thus, The SVD can be written as$$\backslash mathbf\; =\; \backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf\; \backslash Sigma\_\backslash mathrm\; \backslash mathbf\; V\_\backslash mathrm^\backslash dagger\; +\; \backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf\; \backslash Sigma\_\backslash mathrm\; \backslash mathbf\; V\_\backslash mathrm^\backslash dagger,$$where $\backslash mathbf\_\backslash mathrm$, ⁣$\backslash mathbf\_\backslash mathrm$, and $\backslash mathbf\_\backslash mathrm$ represent the contribution of the input signal $x\_k[t]$ to $\backslash mathbf\; Y$. We term $\backslash mathbf\_\backslash mathrm$ the signal subspace. In contrast, $\backslash mathbf\_\backslash mathrm$, $\backslash mathbf\_\backslash mathrm$, and $\backslash mathbf\_\backslash mathrm$ represent the contribution of noise $n\_m[t\; ]$ to $\backslash mathbf\; Y$.

Hence, from the system model, we can write $\backslash mathbf\; \backslash mathbf\; =\; \backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf\; \backslash Sigma\_\backslash mathrm\; \backslash mathbf\; V\_\backslash mathrm^\backslash dagger$ and $\backslash mathbf\; =\; \backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf\backslash Sigma\_\backslash mathrm\; \backslash mathbf\; V\_\backslash mathrm^\backslash dagger$. Also, from the former, we can write$$\backslash mathbf\; U\_\backslash mathrm=\; \backslash mathbf\; \backslash mathbf,$$where

{ F } = X V _{S} \Sigma ^{-1} _{S} . In the sequel, it is only important that there exist such an invertible matrix $\backslash mathbf\; F$ 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$$\backslash mathbf\_\backslash mathrm\; =\; \backslash frac\backslash sum\_^T\; \backslash mathbf[t]\; \backslash mathbf[t]\; ^\backslash dagger=\backslash frac\backslash mathbf\; \backslash mathbf^\backslash dagger=\; \backslash frac\backslash mathbf\; U\; \backslash mathbf\; U^\backslash dagger=\; \backslash frac\backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf\; \backslash Sigma\_\backslash mathrm^2\; \backslash mathbf\; U\_\backslash mathrm^\backslash dagger+\backslash frac\backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf\; \backslash Sigma\_\backslash mathrm^2\; \backslash mathbf\; U\_\backslash mathrm^\backslash dagger.$$

Estimation of radial frequencies

We have established two expressions so far: $\backslash mathbf\; J\_2\; \backslash mathbf\; A\; =\; \backslash mathbf\; J\_1\; \backslash mathbf\; A\; \backslash mathbf\; H$ and $\backslash mathbf\; U\_\backslash mathrm\; =\; \backslash mathbf\; \backslash mathbf$. Now, $$\backslash begin\backslash mathbf\; J\_2\; \backslash mathbf\; A\; =\; \backslash mathbf\; J\_1\; \backslash mathbf\; A\; \backslash mathbf\; H\backslash implies\backslash mathbf\; J\_2\; (\backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf^)=\; \backslash mathbf\; J\_1\; (\backslash mathbf\; U\_\backslash mathrm\; \backslash mathbf^)\; \backslash mathbf\; H\backslash implies\backslash mathbf\; S\_2\; =\; \backslash mathbf\; S\_1\; \backslash mathbf,\backslash end$$where

S _{1} = J _{1} U _{S} and S _{2} = J _{2} U _{S} denote the truncated signal sub spaces, and $$\backslash mathbf\; =\; \backslash mathbf^\; \backslash mathbf\; H\; \backslash mathbf.$$The above equation has the form of an eigenvalue decomposition, and the phases of eigenvalues in the diagonal matrix H are used to estimate the radial frequencies.

Thus, after solving for

P in the relation S _{2} = S _{1} P , we would find the eigenvalues $\backslash lambda\_1,\backslash \; \backslash lambda\_2,\backslash \; \backslash ldots,\backslash \; \backslash lambda\_K$ of P , where λ _{k} = \alpha _{k} e ^{j \omega _{k}} , and the radial frequencies \omega _{1, \omega} _{2, \ldots, \omega} _{K} are estimated as the phases (argument) of the eigenvalues.

Remark: In general,

S _{1} is not invertible. One can use the least squares estimate { P} = ( S ^{\dagger} _{1} { S ^{-1} _{1 })} S ^{\dagger} _{1} { S _{2}} . An alternative would be the total least squares estimate.

Algorithm summary

Input: Measurements $\backslash mathbf\; :=\; [\backslash ,\backslash mathbf\{y\}[1]\; \backslash ,\backslash \; \backslash mathbf[2]\; \backslash ,\backslash \; \backslash dots\; \backslash ,\backslash \; \backslash mathbf[T\; ]\backslash ,]$, the number of input signals $K$ (estimate if not already known).

1. Compute the singular value decomposition (SVD) of $\backslash mathbf\; =\; \backslash mathbf\; U\; \backslash mathbf\; \backslash Sigma\; \backslash mathbf\; V^\backslash dagger$ and extract the signal subspace $\backslash mathbf\_\backslash mathrm\; \backslash in\; \backslash mathbb^$

as the first $K$ columns of $\backslash mathbf\; U$.

1. Compute $\backslash mathbf\_1\; =\; \backslash mathbf\; J\_1\; \backslash mathbf\; U\_\backslash mathrm$ and $\backslash mathbf\_2\; =\; \backslash mathbf\; J\_2\; \backslash mathbf\; U\_\backslash mathrm$

, where

J _{1} = [ I _{M-1} 0] and J _{2} = [ 0 I _{M-1} ] .

1. Solve for $\backslash mathbf$

in $\backslash mathbf\; S\_2\; =\; \backslash mathbf\; S\_1\; \backslash mathbf$ (see the remark above).

1. Compute the eigenvalues $\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots,\; \backslash lambda\_K$

of $\backslash mathbf$.

1. The phases of the eigenvalues $\backslash lambda\_k\; =\; \backslash alpha\_k\; \backslash mathrm^$

provide the radial frequencies $\backslash omega\_k$, i.e., $\backslash omega\_k\; =\; \backslash arg\; \backslash lambda\_k$

Notes

Choice of selection matrices

In the derivation above, the selection matrices

J _{1} = [ I _{M-1} 0] and J _{2} = [ 0 I _{M-1} ] were used. However, any appropriate matrices $\backslash mathbf\; J\_1$ and J _{2} may be used as long as the rotational invariance $($i.e., J _{2} a(\omega _{k)} = e ^{- j\omega _{k}} J _{1} a(\omega _{k)} $)$, or some generalization of it (see below) holds; accordingly, the matrices $\backslash mathbf\; J\_1\; \backslash mathbf\; A$ and $\backslash mathbf\; J\_2\; \backslash mathbf\; A$ may contain any rows of $\backslash mathbf\; A$.

Generalized rotational invariance

The rotational invariance used in the derivation may be generalized. So far, the matrix

H has been defined to be a diagonal matrix that stores the sought-after complex exponentials on its main diagonal. However, H may also exhibit some other structure.^[2] For instance, it may be an upper triangular matrix. In this case, $\backslash mathbf\; =\; \backslash mathbf^\; \backslash mathbf\; H\; \backslash mathbf$constitutes a triangularization of P .

See also

• Multiple signal classification
• Generalized pencil-of-function method
• Independent component analysis

Further reading

• .
• Roy . R. . Kailath . T. . Esprit - Estimation Of Signal Parameters Via Rotational Invariance Techniques . 1989 . IEEE Transactions on Acoustics, Speech, and Signal Processing . 37 . 7 . 984–995 . 10.1109/29.32276 . 14254482 . 2011-07-25 . 2020-09-26 . https://web.archive.org/web/20200926212409/https://www.vtvt.ece.vt.edu/research/references/uwb/ranging_mobile_location/esprit.pdf . dead . .
• A. M.. Ibrahim. M. I.. Marei. S. F.. Mekhamer. M. M.. Mansour . Electric Power Components and Systems . 39. 1. 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. 10.1080/15325008.2010.513363 . 64–79. 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.

Notes and References

1. 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. - https://ieeexplore.ieee.org/abstract/document/5307000
2. 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 . 10.1109/JSTSP.2014.2313409 . 1403.5352 . 2014ISTSP...8..996H . 11664051 . 1932-4553.