Signal interpolation using numerically robust differential operators

In our work on frequency estimation based on local signal behavior [15] for testing purposes we needed a signal φ(t) which over some disjoint intervals of (continuous) time In is equal to a corresponding linear combination fn(t) of up to N sine waves, possibly damped and phase shifted, of (normalized) frequencies smaller than π. The signal should also satisfy the following constraints: φ(t) should contain a minimal amount of out-of-band energy, i.e., the energy of its Fourier transform φ̂(ω) outside interval [−π, π] should be as small as possible; φ(t) should fit within an as narrow envelope as possible; φ(t) should also have a finite support in the time domain, which is as short as possible. Clearly, these are mutually conflicting requirements and we want to look for a compromise solution which is nevertheless good in all of these respects. A computationally efficient method for producing such a signal can be useful for designing novel digital modulation schemas which satisfy stringent conditions on out of band leakage and envelope properties of the generated signal. The method we propose in this paper employs some special, numerically robust linear differential operators, called the chromatic derivatives, which were introduced relatively recently, and which we believe hold yet unexplored promise in signal and image processing. Key–Words: numerical differentiation, chromatic derivatives, local signal representation, signal interpolation, digital modulation