We determine all signals giving equality for the discrete Hirschman uncertainty principle. We single out the case where the entropies of the time signal and its Fourier transform are equal. These signals (up to scalar multiples) form an orthonormal basis giving an orthogonal transform that optimally packs a finite-duration discrete-time signal. The transform may be computed via a fast algorithm due to its relationship to the discrete Fourier transform.

We have previously developed an uncertainty measure that is suitable for finitely-supported (N samples) discrete-time signals. A specific instance of our measure has been termed the "discrete Hirschman (1957) uncertainty principal" in the literature, and we have adopted this terminology for our more general measure. We compare the optimal signals of this discrete version to the already determined optimal signals of the (continuous-time) Hirschman uncertainty principal. From our comparison, we conclude that a basic premise in signal processing, that if we sample densely enough, the discrete-time case directly corresponds to the continuous-time case, is not correct in this instance. The arithmetic of N, which seems to have no analog in continuous time, is crucial to the construction of the Hirschman optimal discrete representation. We suggest that more work in this important area be performed to determine what impact this has, and to find out how widespread this problem may be.

We use a new uncertainty measure, H/sub p/, that predicts the compactness of digital signal representations to determine a good (non-orthogonal) set of basis vectors. The measure uses the entropy of the signal and its Fourier transform in a manner that is similar to the use of the signal and its Fourier transform in the Heisenberg uncertainty principle. The measure explains why the level of discretization of continuous basis signals can be very important to the compactness of representation. Our use of the measure indicates that a mixture of (non-orthogonal) sinusoidal and impulsive or "blocky" basis functions may be best for compactly representing signals.