ESTIMATING THE PARAMETERS OF EXPONENTIALLY DAMPED OR UNDAMPED SINUSOIDAL SIGNALS IN NOISE

We address the problem of estimating the parameters of a signal embedded in noise. The signal is composed of samples of a sum of M exponentially damped or undamped sinusoidal signals. That is, s(n) the signal samples, are given by the formula (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where a(,k) are the amplitudes of the exponentials. The problem is to estimate the value of M, {a(,k)} and {s(,k)} from a short record of the data samples which are corrupted by noise. We suggest improvements to the original approach proposed by Prony. He observed that the samples s(n) obey an M th order difference equation and that from the roots of the characteristic polynomial of the difference equation, the parameters {s(,k)} can be determined. We present three methods to improve this approach when the signal is noise corrupted. The two key ideas used in the improvements are listed below in the order of importance. (1) By using a higher order (greater than M) difference equation formed with the data samples, we try to explain some of the noise in the data by a few additional exponentials, which are subsequently eliminated. (2) The signal samples s(n) when embedded in a (Toeplitz or Hankel) matrix result in a rank defficient matrix. This is a special property of the exponential signals and is independent of the signal parameters. We make use of this knowledge regarding the signal to improve the signal to noise ratio (SNR) in the data. This is accomplished in a natural way using singular value decomposition (SVD) of a matrix. The improvement in performance of our methods over traditional methods is demonstrated with computer simulation