Theory and practice of the pergressive Fourier analysis

Abstract

First it is demonstrated that through stepwise pergression of a constant analysing interval p throughout a sine the period T can be calculated even if p is not an integer multiple of T. The key is the fact that the two harmonics n=n₁=[r] and n= n₂=[r]+l produce parallel regression lines in the phase diagram with the standardized slope r=p:T. Through an iterated second analysis with an integer multiple of T as analysing interval the three sine parameters can be accurately calculated with the aid of the amplitude and phase diagrams. But it is also demonstrated that often the iteration of the analysis can be avoided. The parallelism also occurs when a curve is circasinusoidal, when‐e.g. the amplitude and the period vary at random around mean values or when single phase jumps at sufficient distance from each other occur, so that a certain parameter set which is contained in the circasinusoidal curve can be found. The parallelism occurs furthermore when an oscillation has an aperiodic amplitude function, as e.g. when it is monotonically damped. On the basis of these qualities two heuristic principles are defined through which oscillation estimates can be obtained. A signal ratio for the sine can be calculated and through the use of linear regression tests precise statistical criteria are obtained. All the properties of these methods are madiematically demonstrated and illustrated. A digital computer program and technique are given. The pergressive Fourier analysis often requires longer calculations than other methods, but it produces also more information which is important when a time dependence of the parameters or hidden transient interferences are contained in the data.