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.

(complex-valued) output signals (measurements)

(complex-valued) input signals

denotes the noise added by the system.

, whose phases are integer multiples of some radial frequency

This frequency only depends on the index of the system's input, i.e.

and the number of input signals,

Since the radial frequencies are the actual objectives,

output signals at instance

has the property that adjacent entries are related.

, the equation introduces two selection matrices

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.

The singular value decomposition (SVD) of

, that holds the singular values from the largest (top left) in descending order.

largest singular values stem from these input signals and other singular values are presumed to stem from noise.

can be partitioned into submatrices, where some submatrices correspond to the signal subspace and some correspond to the noise subspace.

represent the contribution of the input signal

In the sequel, it is only important that there exists such an invertible matrix

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

denote the truncated signal sub spaces, and

The above equation has the form of an eigenvalue decomposition, and the phases of the eigenvalues in the diagonal matrix

are estimated as the phases (argument) of the eigenvalues.

An alternative would be the total least squares estimate.

, the number of input signals

The rotational invariance used in the derivation may be generalized.

has been defined to be a diagonal matrix that stores the sought-after complex exponentials on its main diagonal.

[3] For instance, it may be an upper triangular matrix.