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)eiνxand 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.