A parameter estimation approach to estimation of frequencies of sinusoids

It is shown that with uniform sampling, a sample of the sum of M sinusoids at time nT can be uniquely expressed as a linear combination of the 2M samples at times (n - 1)T,..., (n - 2M)T. The 2M coefficients of linear combination are also the coefficients of a 2M-order polynomial whose roots are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e^{ωᵢ}, ωᵢ</tex> being the frequencies of the sinusoids. Given the samples of sinusoids plus noise, a consistent estimate of the 2M coefficients are obtained by the instrumental variable method of parameter estimation. It is also possible to track time-varying frequencies by a recursive algorithm that exponentially weighs out the past data. Simulation examples of one and two sinusoids are given.