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)x(t)
maps
R→R\mathbb{R} \rightarrow \mathbb{R}
​

Definition 3

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

Properties of the Unit Impulse

Definition 4

The unit impulse in discrete time is defined as
​
δ[n]={1if n=00else \delta[n] = \begin{cases} 1 & \text{if } n = 0\\ 0 & \text{else } \end{cases}
​
  • ​
    f[n]δ[n]=f[0]δ[n]f[n]\delta[n] = f[0]\delta[n]
    ​
  • ​
    f[t]δ[n−N]=f[N]δ[n−N]f[t]\delta[n-N] = f[N]\delta[n-N]
    ​

Definition 5

The unit impulse in continuous time is the dirac delta function
​
δ(t)=limΔ→0δΔ(t)δΔ(t)={1Δ,if 0≤t<Δ0else.\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)δ(t)=f(0)δ(t)f(t)\delta(t) = f(0)\delta(t)
    ​
  • ​
    f(t)δ(t−τ)=f(τ)δ(t−τ)f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)
    ​
  • ​
    δ(at)=1∣a∣δ(t)\delta(at) = \frac{1}{|a|}\delta(t)
    ​

Definition 6

The unit step is defined as
​
u[n]={1if n≥00else.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−τ)x(t - \tau)
    : Shift a signal right by
    Ï„\tau
    steps.
  • ​
    x(−t)x(-t)
    : Rotate a signal about the
    t=0t=0
    ​
  • ​
    x(kt)x(kt)
    : Stretch a signal by a factor of
    kk
    ​
These operations can be combined to give more complex transformations. For example,
y(t)=x(τ−t)=x(−(t−τ))y(t) = x(\tau - t) = x(-(t-\tau))
flips
xx
and shifts it right by
Ï„\tau
timesteps. This is equivalent to shifting
xx
left by
Ï„\tau
timesteps and then flipping it.

Convolution

Definition 7

The convolution of two signals in discrete time
​
(x∗h)[n]=∑k=−∞∞x[k]h[n−k](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)=∫−∞∞x(τ)h(t−τ)dτ(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∗δ)[n]=x[n](x*\delta)[n] = x[n]
    ​
  • ​
    x[n]∗δ[n−N]=x[n−N]x[n]*\delta[n-N]=x[n-N]
    ​
  • ​
    (x∗h)[n]=(h∗x)[n](x*h)[n] = (h*x)[n]
    ​
  • ​
    x∗(h1+h2)=x∗h1+x∗h2x * (h_1 + h_2) = x*h_1 + x*h_2
    ​
  • ​
    x∗(h1∗h2)=(x∗h1)∗h2x * (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)x(t)
is transformed to
y(t)y(t)
, then
x(t−τ)x(t-\tau)
is transformed to
y(t−τ)y(t-\tau)
​

Definition 14

A system
f(x)f(x)
is linear if and only if for the signals
y1(t)=f(x1(t)),y2(t)=f(x2(t))y_1(t) = f(x_1(t)), y_2(t) = f(x_2(t))
, then scaling (
f(ax1(t))=ay(t)f(a x_1(t)) = a y(t)
and superposition (
f(x1(t)+x2(t))=y1(t)+y2(t)f(x_1(t) + x_2(t)) = y_1(t) + y_2(t)
) hold.
Notice: The above conditions on linearity require that
x(0)=0x(0) = 0
because if
a=0a = 0
, then we need
y(0)=0y(0) = 0
for scaling to be satisfied

Definition 15

The impulse response of a system
f[x]f[x]
is
h[n]=f[δ[n]]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]h[n]
decays to zero in a finite amount of time

Definition 17

A system has an Infinite Impulse Response (IIR) if
h[n]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)=est,x[n]=zn(s,z∈C)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