Introduction to Signals and Systems

# Types of Signals

### Definition 1

A signal is a function of one or more variables

### Definition 2

A continuous signal
$x(t)$
maps
$\mathbb{R} \rightarrow \mathbb{R}$

### Definition 3

A discrete signal
$x[n]$
maps
$\mathbb{Z} \rightarrow \mathbb{R}$

## Properties of the Unit Impulse

### Definition 4

The unit impulse in discrete time is defined as
$\delta[n] = \begin{cases} 1 & \text{if } n = 0\\ 0 & \text{else } \end{cases}$
• $f[n]\delta[n] = f[0]\delta[n]$
• $f[t]\delta[n-N] = f[N]\delta[n-N]$

### Definition 5

The unit impulse in continuous time is the dirac delta function
$\delta(t)=lim_{\Delta\rightarrow 0}\delta_{\Delta}(t) \qquad \delta_{\Delta}(t)=\begin{cases} \frac{1}{\Delta}, & \text{if } 0 \leq t < \Delta \\ 0 & \text{else.} \end{cases}$
• $f(t)\delta(t) = f(0)\delta(t)$
• $f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)$
• $\delta(at) = \frac{1}{|a|}\delta(t)$

### Definition 6

The unit step is defined as
$u[n] = \begin{cases} 1 & \text{if } n \geq 0\\ 0 & \text{else.} \end{cases}$

# Signal transformations

Signals can be transformed by modifying the variable.
• $x(t - \tau)$
: Shift a signal right by
$\tau$
steps.
• $x(-t)$
: Rotate a signal about the
$t=0$
• $x(kt)$
: Stretch a signal by a factor of
$k$
These operations can be combined to give more complex transformations. For example,
$y(t) = x(\tau - t) = x(-(t-\tau))$
flips
$x$
and shifts it right by
$\tau$
timesteps. This is equivalent to shifting
$x$
left by
$\tau$
timesteps and then flipping it.

# Convolution

### Definition 7

The convolution of two signals in discrete time
$(x*h)[n] = \sum_{k=-\infty}^{\infty}{x[k]h[n-k]}$

### Definition 8

The convolution of two signals in continuous time
$(x*h)(t) = \int_{-\infty}^{\infty}{x(\tau)h(t-\tau)d\tau}$
While written in discrete time, these properties apply in continuous time as well.
• $(x*\delta)[n] = x[n]$
• $x[n]*\delta[n-N]=x[n-N]$
• $(x*h)[n] = (h*x)[n]$
• $x * (h_1 + h_2) = x*h_1 + x*h_2$
• $x * (h_1 * h_2) = (x * h_1) * h_2$

# Systems and their properties

### Definition 9

A system is a process by which input signals are transformed to output signals

### Definition 10

A memoryless system has output which is only determined by the input's present value

### Definition 11

A causal system has output which only depends on input at present or past times

### Definition 12

A stable system produces bounded output when given a bounded input. By extension, this means an unstable system is when
$\exists$
a bounded input that makes the output unbounded.

### Definition 13

A system is time-invariant if the original input
$x(t)$
is transformed to
$y(t)$
, then
$x(t-\tau)$
is transformed to
$y(t-\tau)$

### Definition 14

A system
$f(x)$
is linear if and only if for the signals
$y_1(t) = f(x_1(t)), y_2(t) = f(x_2(t))$
, then scaling (
$f(a x_1(t)) = a y(t)$
and superposition (
$f(x_1(t) + x_2(t)) = y_1(t) + y_2(t)$
) hold.
Notice: The above conditions on linearity require that
$x(0) = 0$
because if
$a = 0$
, then we need
$y(0) = 0$
for scaling to be satisfied

### Definition 15

The impulse response of a system
$f[x]$
is
$h[n] = f[\delta[n]]$
, which is how it response to an impulse input.

### Definition 16

A system has a Finite Impulse Response (FIR) if
$h[n]$
decays to zero in a finite amount of time

### Definition 17

A system has an Infinite Impulse Response (IIR) if
$h[n]$
does not decay to zero in a finite amount of time

# Exponential Signals

Exponential signals are important because they can succinctly represent complicated signals using complex numbers. This makes analyzing them much easier.
$x(t) = e^{st}, x[n] = z^n (s, z \in \mathbb{C})$

### Definition 18

The frequency response of a system is how a system responds to a purely oscillatory signal