The discrete versions of the unit impulse (or sample) \(\delta[n]\), unit step \(u[n]\), and unit ramp \(r[n]\) are defined as
For integer index \(n\),
\(\boxed{\delta[n] = \begin{cases}0\,, & n \neq 0 \\ 1\,, & n = 0\end{cases}}\)
\(\boxed{u[n] = \begin{cases}0\,, & n < 0 \\ 1\,, & n \geq 0\end{cases}}\)
\(\boxed{r[n] = n\,u[n] = \begin{cases}0\,, & n < 0 \\ n\,, & n \geq 0\end{cases}}\)
The discrete impulse is just a unit sample at \(n = 0\). It is completely free of the kind of ambiguities associated with analog impulse \(\delta(t)\) at \(t = 0\). The discrete unit step \(u[n]\) also has a well defined, unique value of \(u[0] = 1\) (unlike its analog counterpart). The signal \(x[n] = An\,u[n] = A\,r[n]\) describes a discrete ramp whose slope \(A\) is given by \(x[k] - x[k - 1]\), the difference between adjacent sample values.
Properties of the Discrete Impulse
The product and sifting properties of discrete impulse are analogous to their analog counterparts.
Product Property
The product of a signal \(x[n]\) with the impulse \(\delta[n - k]\) results in
\(\boxed{x[n]\,\delta[n - k] = x[k]\,\delta[n - k]}\)
This is just an impulse with strength \(x[k]\).
Sifting Property
The impulse \(\delta[n]\) extracts the value \(x[k]\) from signal \(x[n]\) at the impulse location \(n = k\).
\(\boxed{\sum_{n = -\infty}^{\infty} x[n]\,\delta[n - k] = x[k]}\)
Signal Representation by Impulses
Any discrete signal \(x[n]\) may be expressed as a sum of shifted impulses \(\delta[n - k]\) whose strengths \(x[k]\) correspond to the signal values at \(n = k\).
\(\boxed{x[n] = \sum_{k = -\infty}^{\infty} x[k]\,\delta[n - k]}\)
The discrete unit step \(u[n]\) may be expressed as a train of shifted impulses \(\delta[n - k]\) or as the cumulative sum of discrete impulse \(\delta[n]\).
\(\boxed{u[n] = \sum_{k = 0}^{\infty} \delta[n - k] = \sum_{k = -\infty}^{n} \delta[k]}\)
The discrete ramp \(r[n]\) may be expressed as a train of shifted impulses \(\delta[n - k]\) or as the cumulative sum of discrete unit step \(u[n]\).
\(\boxed{r[n] = \sum_{k = 0}^{\infty} k\,\delta[n - k] = \sum_{k = -\infty}^{n} u[k]}\)