Discrete-Time Signals and Systems

We assume that we derived a discrete-time signal from a continuous time signal via sampling. Given $f(t)$ to be a continuous time signal, $f(nT)$ is the value of $f(t)$ at $t=nT$. The discrete-time signal $f[n]$ is defined only for $n$ an integer. So if we derive $f[n]$ from $f(t)$ by sampling every $T$ seconds, where $T$ is the sample period, we get:

$$f(nT) = f(t)\vert _{t=nT}$$

$$f[n] = f(nT) = f(t) \vert _{t=nT}$$

We will not necessarily assume that $f[n]$ is a discrete amplitude signal. A signal that is both discrete time and discrete amplitude is known as a digital signal. You will see these in later communications courses a well-known example of a digital signal is music on a compact disk.

Note that a discrete-time signal need not be generated by explicitly sampling a continuous-time signal. Some signals are inherently discrete time, such as computer bit sequences, and some signals are implicitly sampled, such as the daily DJIA or yearly temperature averages.

Discrete-Time Signals and Systems

Remember the square brackets!

Read (skim) the example of Euler integration in the book but we will cover difference equations in Chapter 10 and when we discuss the Z-transform. Euler integration approximates the area under a curve $x(t)$ by the sum of rectangular areas.

Discrete-Time Unit Step Function

$$u[n] = \left\{ \begin{array}{cc} 1, & n\ge 0 \\ 0, & n < 0 \end{array} \right.$$

Notice that here, the unit step is defined at $n=0$, unlike for continuous time.

The time-shifted unit step function $u[n - n_0]$ is:

$$u[n - n_0] = \left\{ \begin{array}{cc} 1, & n \ge n_0 \\ 0, & n < n_0 \end{array} \right.$$

Discrete-Time Unit Impulse Function

$$\delta [n] = \left\{ \begin{array}{cc} 1, & n = 0 \\ 0, & n \ne 0 \end{array} \right.$$

Here, there is no difficulty in defining the impulse as we had in continuous time.

Shifted unit impulse:

$$\delta [n - n_0] = \left\{ \begin{array}{cc} 1, & n = n_0 \\ 0, & n \ne n_0 \end{array} \right.$$

The summation is the discrete-time analog of the running integral in continuous time, and the first difference is the analog of the derivative. With these analogies, the unit impulse has essentially the same behavior in discrete and continuous time, including the sifting property.

$$\begin{eqnarray*} \mbox{Continuous time} & \qquad & \mbox{Discrete time} \\ u... ... x(t_0) & & \sum_{n=-\infty}^\infty x[n]\delta[n-n_0] = x[n_0]\ \end{eqnarray*}$$

Recall that continuous-time signals could be represented by an equation (which might be defined in regions) or a graph. Discrete-time signals can be represented in these ways, but also using a table. For example:

$$x[n] = u[n]-u[n-4]$$

$n$	$\le -1$	0	1	2	3	$\ge 4$
$x[n]$	0	1	1	1	1	0