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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:$\delta[n]=\begin{cases}1&n=0 \\ 0&\textrm{otherwise}\end{cases}$
Image
The discrete-time delta function.
In a variety of important settings, we will often see the delta function shifted by a particular time value. The delta function $\delta[n-m]$ is 0, except for a peak of 1 at time $m$:
Image
A time-shifted discrete-time delta function $\delta[n-m]$, where $m=9$.
One of the reasons the shifted delta function is so useful is that we can use it to select, or sample, a value of another signal at some defined time value. Suppose we have some signal $x[n]$, and we would like to isolate that signal's value at time $m$. What we can do is multiply that signal by a shifted delta signal. We can say $y[n]=x[n]\delta[n-m]$, but since that $y[n]$ will be zero for all $n$ except at $n=m$, it is equivalent to express it as $y[n]=x[m]\delta[n-m]$, where now $x[m]$ is no longer a function, but a constant. The following figure shows how this operation isolates a particular time sample of $x[n]$:
Image Image Image
Using a time-shifted delta function to isolate a sample of the signal $x[n]$.

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 $n \geq 0$: $u[n]=\begin{cases}1&n \lt 0\\ 1&n\geq 0\end{cases}$:
Image
The unit step function.
As with the delta function, it will also be useful for us to shift the step function:
Image
A shifted step function $[n-m]$ with $m=5$.
And, as you might have guessed, we can use a shifted step function in a similar way to the delta function by multiplying it with another signal. Whereas the delta function selected a single value of a certain signal (zeroing out the rest), the step function isolates a portion of a signal after a given time. Below, a step function is used to zero out all the values of $x[n]$ for $n\lt 5$, keeping the rest:
Image Image Image
A shifted step function can be used to zero out all values of a signal before a certain time index.
Supposing a signal $x[n]$ were not causal, setting $m$ to zero and performing the operation $x[n]u[n]$ would zero out all values of $x[n]$ before $n=0$, thereby making the result causal.

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 $N_2$: $p[n] = \begin{cases}0&n\lt N_1 \\ 1&N_1 \leq n \leq N_2 \\ 0&n\gt N_2\\ \end{cases}$
Image
The unit pulse function $p[n]$, here with $N_1 = −5$ and $N_2= 3$.
Of course, rather than use the above piece-wise notation, it is also possible to express the pulse as the difference of two step functions: $p[n] = u[n-N_1]- u[n-(N_2+1)]$.

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 $a$.
Image
For $a\gt 1$, the real exponential function increases with time. Here $a=1.1$.
Image
For $0 \lt a\lt 1$, the real exponential function decreases with time (or we could say it increases exponentially as the time index decreases). Here $a=.9$.

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Source:  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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