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In our study of discrete-time signals and signal processing, there are five very important signals that we will use to both illustrate signal processing concepts, and also to probe or test signal processing systems: the delta function , the unit step function , the unit pulse function , the real exponential function , sinusoidal functions , and complex exponential functions . This module will consider the first four; sinusoids and complex exponentials are particularly important, so a separate model will cover them. Each of these signals will be introduced as infinite-length signals, but they all have straightforward finite-length equivalents.
The discrete-time delta function
The delta function is probably the simplest nontrivial signal. It is represented mathematically with (no surprise) the Greek letter delta: $\delta[n]$. It takes the value 0 for all time points, except at the time point 0 where it peaks up to the value 1:$\delta[n]=\begin{cases}1&n=0 \\ 0&\textrm{otherwise}\end{cases}$
The unit step function
The unit step function can be thought of like turning on a switch. Usually identified as $u[n]$, it is $0$ for all $n \lt 0$, and then at $n=0$ it "switches on" and is $1$ for all $n \geq 0$: $u[n]=\begin{cases}1&n \lt 0\\ 1&n\geq 0\end{cases}$:
The unit pulse function
The unit pulse $p[n]$ is very similar to the unit step function in how it "switches on" from 0 to 1, but then it also "switches off" at a later time. We will say it "switches on" at time $N_1$, and "off" at time $N_2$: $p[n] = \begin{cases}0&n\lt N_1 \\ 1&N_1 \leq n \leq N_2 \\ 0&n\gt N_2\\ \end{cases}$
The real exponential function
Finally, we have the real exponential function, which takes a real number $a$ (that we are going to assume is positive) and raises it to the power of $n,$ where $n$ is the time index: $r[n] = a^n$, $a\in R$, $a\geq 0$. So at $n=0$, $r[n]=a^0$, at $n=1$ it equals $a$, is $a^2$ at $n=2$, and so on. As the name suggests, the signal will exponentially increase or decrease, depending on the value of $a$.Read also:
OpenStax, Discrete-time signals and systems. OpenStax CNX. Oct 07, 2015 Download for free at https://legacy.cnx.org/content/col11868/1.2
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