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.

One-dimensional ESPRIT

At instance t, the M (complex -valued) output signals (measurements) y_m[t],, of the system are related to the K (complex -valued) input signals x_k[t], , aswhere n_m[t] 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 \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_{m,k} is denoted as. Collating the weights as and the M output signals at instance t as, where. Further, when the weight vectors are put into a Vandermonde matrix, and the K inputs at instance t into a vector , we can writeWith several measurements at instances and the notations , and , the model equation becomes

Dividing into virtual sub-arrays

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 (M-1) and \mathbf 0 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

The singular value decomposition (SVD) of \mathbf Y is given aswhere and are unitary matrices and is a diagonal matrix of size M \times T, that holds the singular values from the largest (top left) in descending order. The operator \dagger denotes the complex-conjugate transpose (Hermitian transpose). Let us assume that T \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 \mathbf Y 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 K columns of \mathbf U and \mathbf V, respectively and is a diagonal matrix comprising the K largest singular values. Thus, The SVD can be written aswhere, ⁣, and represent the contribution of the input signal x_k[t] to \mathbf Y. We term the signal subspace. In contrast,, , and represent the contribution of noise n_m[ t ] to \mathbf Y. 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 \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

Estimation of radial frequencies

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 \mathbf H are used to estimate the radial frequencies. Thus, after solving for \mathbf P in the relation, we would find the eigenvalues of \mathbf P, where , and the radial frequencies are estimated as the phases (argument) of the eigenvalues. Remark: In general, \mathbf S_1 is not invertible. One can use the least squares estimate. An alternative would be the total least squares estimate.

Algorithm summary

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

Choice of selection matrices

In the derivation above, the selection matrices and were used. However, any appropriate matrices \mathbf J_1 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 \mathbf A.

Generalized rotational invariance

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