Correlation function bounds for aperiodic signals (Corresp.)

A basic signal design problem which arises in the construction of aperiodic signals with good correlation properties is how small can the peak value of the cross-correlation function be when the signals occupy approximately the same frequency spectra. The authors are not aware of any published results on this problem prior to Anderson´s work (in preparation). He derives a lower bound on the maximum in ν of the rms value of the convolution of f(x)e^iνx and g(x) where f and g are square integrable functions, the lower bound being expressed in terms of the energy bandwidth of f and g. In this correspondence, these results are used to obtain a lower bound on the maximum in τ and ν of the envelope of the crosscorrelation function between two real, bandpass, time-limited signals when one is frequency shifted by ν, assuming that the signals are in the same passband. The lower bound is expressed in terms of a notion of ϵ-approximete energy bandwidth. By using a similar approach, and defining a bandwidth measure given by Zakai (1960) a different lower bound is obtained. Furthermore, this approach yields an upper and lower bound on the rms value of the envelope of the cross-correlation function. For the special case when the signals have identical energy spectral densities, the rms value of the envelope of the cross-correlation function is related to the timebandwidth product of the signals and is independent of the phase characteristics of their Fourier transforms.