In many contexts, a discrete time (DT) system is really part of a larger continuous time (CT) system. For example, a digital recording system takes an analog sound, digitizes it, possibly processes the digital signals, and plays back an analog sound for people to listen to. In practical systems, DT signals obtained are usually uniformly sampled versions of CT signals. If is a CT signal, then the sampling circuit used before an analog-to-digital converter will transform it to a DT signal: where T is the sampling period. Before sampling, the input signal is normally run through a so-called Nyquist filter which removes frequencies above the "folding frequency" 1/(2T); this guarantees that no information in the filtered signal will be lost. Without filtering, any frequency component above the folding frequency (or Nyquist frequency) is aliased to a different frequency (thus distorting the original signal), since a DT signal can only support frequency components lower than the folding frequency. Impulse response and convolution Let represent the sequence And let the shorter notation represent A discrete system transforms an input sequence, into an output sequence, In general, every element of the output can depend on every element of the input. Representing the transformation operator by , we can write: Note that unless the transform itself changes with n, the output sequence is just constant, and the system is uninteresting. (Thus the subscript, n.) In a typical system, y[n] depends most heavily on the elements of x whose indices are near n. For the special case of the Kronecker delta function, the output sequence is the impulse response: For a linear system, must satisfy: And the time-invariance requirement is: In such a system, the impulse response, , characterizes the system completely. That is, for any input sequence, the output sequence can be calculated in terms of the input and the impulse response. To see how that is done, consider the identity: which expresses in terms of a sum of weighted delta functions. Therefore: where we have invoked for the case and . And because of , we may write: Therefore: {| | | |- | | (commutativity) |} which is the familiar discrete convolution formula. The operator can therefore be interpreted as proportional to a weighted average of the function x[k]. The weighting function is h[−k], simply shifted by amount n. As n changes, the weighting function emphasizes different parts of the input function.