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Unit Sample

2 June, 2016 - 15:17

The second-most important discrete-time signal is the unit sample, which is defined to be

\delta (n)=\left\{\begin{matrix} 1\:if\:n=0 & & \\ 0\:otherwise & & \end{matrix}\right.


Figure 5.8 The unit sample.

Examination of a discrete-time signal's plot, like that of the cosine signal shown in Figure 5.7(Cosine), reveals that all signals consist of a sequence of delayed and scaled unit samples. Because the value of a sequence at each integer m is denoted by s(m) and the unit sample delayed to occur at m is written δ(n - m), we can decompose any signal as a sum of unit samples delayed to the appropriate location and scaled by the signal value.

s(n)=\sum_{m=-\infty }^{\infty } \left ( s(m)\delta (n-m) \right )

This kind of decomposition is unique to discrete-time signals, and will prove useful subsequently.