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