Upper bounds on aliasing error energy for multidimensional sampling of nonbandlimited signals

We present upper bounds on the 2-norm of the aliasing error in multidimensional Shannon sampling. Our bounds complement the previously known 1-norm upper bounds for the peak aliasing error. The proposed bounds provide a good estimate of the total error energy for any signal, rather than just for certain pathological extremals, as is the case with the 1-norm bounds which, as a result, tend to be too conservative for practical applications. The sampling representation is general, possibly multiband, and not restricted to bandlimited signals. Error bounds are phrased in terms of the energy of signal components that lie outside the assumed band-region. Therefore, they are easy to interpret and compute as is demonstrated for two practical signal classes, namely signals with exponential or polynomial out-of-band decay.