also known as the discrete-time Fourier transform (DTFT) of the sample sequence. As depicted, copies of are shifted by multiples of the sampling rate and combined by addition. For a band-limited function and sufficiently large it is possible for the copies to remain distinct from each other. But if the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous Any frequency component above is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies. In such cases, the customary interpolation techniques produce the alias, rather than the original component. When the sample-rate is pre-determined by other considerations (such as an industry standard), is usually filtered to reduce its high frequencies to acceptable levels before it is sampled. The type of filter required is a lowpass filter, and in this application it is called an anti-aliasing filter. Derivation as a special case of Poisson summation When there is no overlap of the copies (also known as "images") of , the term of can be recovered by the product: where: The sampling theorem is proved since uniquely determines All that remains is to derive the formula for reconstruction. need not be precisely defined in the region because is zero in that region. However, the worst case is when the Nyquist frequency. A function that is sufficient for that and all less severe cases is: where rect(•) is the rectangular function. Therefore: (from , above). The inverse transform of both sides produces the Whittaker–Shannon interpolation formula: which shows how the samples, can be combined to reconstruct Larger-than-necessary values of fs (smaller values of T), called oversampling, have no effect on the outcome of the reconstruction and have the benefit of leaving room for a transition band in which H(f) is free to take intermediate values. Undersampling, which causes aliasing, is not in general a reversible operation. Theoretically, the interpolation formula can be implemented as a low-pass filter, whose impulse response is sinc(t/T) and whose input is which is a Dirac comb function modulated by the signal samples. Practical digital-to-analog converters (DAC) implement an approximation like the zero-order hold. In that case, oversampling can reduce the approximation error. Shannon's original proof Poisson shows that the Fourier series in produces the periodic summation of , regardless of and . Shannon, however, only derives the series coefficients for the case . Virtually