A discrete-time signal is represented as a sequence of numbers:
Here n is an integer, and x[n] is the nth sample in the sequence. Discrete-time signals are often obtained by sampling continuous-time signals. In this case the nth sample of the sequence is equal to the value of the analogue signal xa(t) at time t = nT:
The sampling period is then equal to T, and the sampling frequency is fs = 1=T .
For this reason, although x[n] is strictly the nth number in the sequence, we often refer to it as the nth sample. We also often refer to \the sequence x[n]" when we mean the entire sequence. Discrete-time signals are often depicted graphically as follows:
(This can be plotted using the MATLAB function stem.) The value x[n] is unde_ned for noninteger values of n. Sequences can be manipulated in several ways. The sum and product of two sequences x[n] and y[n] are de_ned as the sample-by-sample sum and product respectively. Multiplication of x[n] by a is de_ned as the multiplication of each sample value by a. A sequence y[n] is a delayed or shifted version of x[n] if
with n0 an integer.
This sequence is often referred to as a discrete-time impulse, or just impulse. It plays the same role for discrete-time signals as the Dirac delta function does for continuous-time signals. However, there are no mathematical complications in its defnition.
An important aspect of the impulse sequence is that an arbitrary sequence can be represented as a sum of scaled, delayed impulses.