falls off rapidly at diffraction limits. The sampling theorem also applies to post-processing digital images, such as to up or down sampling. Effects of aliasing, blurring, and sharpening may be adjusted with digital filtering implemented in software, which necessarily follows the theoretical principles. Critical frequency To illustrate the necessity of , consider the family of sinusoids generated by different values of in this formula: With or equivalently , the samples are given by: . That sort of ambiguity is the reason for the strict inequality of the sampling theorem's condition. Sampling of non-baseband signals As discussed by Shannon: A similar result is true if the band does not start at zero frequency but at some higher value, and can be proved by a linear translation (corresponding physically to single-sideband modulation) of the zero-frequency case. In this case the elementary pulse is obtained from sin(x)/x by single-side-band modulation. That is, a sufficient no-loss condition for sampling signals that do not have baseband components exists that involves the width of the non-zero frequency interval as opposed to its highest frequency component. See Sampling (signal processing) for more details and examples. For example, in order to sample the FM radio signals in the frequency range of 100–102 MHz, it is not necessary to sample at 204 MHz (twice the upper frequency), but rather it is sufficient to sample at 4 MHz (twice the width of the frequency interval). A bandpass condition is that X(f) = 0, for all nonnegative f outside the open band of frequencies: for some nonnegative integer N. This formulation includes the normal baseband condition as the case N=0. The corresponding interpolation function is the impulse response of an ideal brick-wall bandpass filter (as opposed to the ideal brick-wall lowpass filter used above) with cutoffs at the upper and lower edges of the specified band, which is the difference between a pair of lowpass impulse responses: Other generalizations, for example to signals occupying multiple non-contiguous bands, are possible as well. Even the most generalized form of the sampling theorem does not have a provably true converse. That is, one cannot conclude that information is necessarily lost just because the conditions of the sampling theorem are not satisfied; from an engineering perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost. Nonuniform sampling The sampling theory of Shannon