On bandlimited signals with minimal product of effective spatial and spectral widths

It is known that signals (which could be functions of space or time) belonging to 𝕃2-space cannot be localized simultaneously in space/time and frequency domains. Alternatively, signals have a positive lower bound on the product of their effective spatial andeffective spectral widths, for simplicity, hereafter called the effective space-bandwidthproduct (ESBP). This is the classical uncertainty inequality (UI), attributed to many, but, from a signal processing perspective, to Gabor who, in his seminal paper, established the uncertainty relation and proposed a joint time-frequency representation in which the basis functions have minimal ESBP. It is found that the Gaussian function is the only signal that has the lowest ESBP. Since the Gaussian function is not bandlimited, no bandlimited signal can have the lowest ESBP. We deal with the problem of determining finite-energy, bandlimited signals which have the lowest ESBP. The main result is as follows. By choosing the convolution product of a Gaussian signal (with σ as the variance parameter) and a bandlimited filter with a continuous spectrum, we demonstrate that there exists a finite-energy, bandlimited signal whose ESBP can be made to be arbitrarily close (dependent on the choice of σ) to the optimal value specified by the UI