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
Signal subspace
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
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
Input: Measurements :=[\,\mathbf {y} [1]\,\ \mathbf {y} [2]\,\ \dots \,\ \mathbf {y} [T]\,]}
, the number of input signals (estimate if not already known).
- Compute the singular value decomposition (SVD) of and extract the signal subspace as the first columns of .
- Compute and , where and .
- Solve for in (see the remark above).
- Compute the eigenvalues of .
- The phases of the eigenvalues provide the radial frequencies , i.e., .