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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
RR\mathbb{R} \rightarrow \mathbb{R}

Definition 3

A discrete signal
x[n]x[n]
maps
ZR\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[nN]=f[N]δ[nN]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)=limΔ0δΔ(t)δΔ(t)={1Δ,if 0t<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(tτ)=f(τ)δ(tτ)f(t)\delta(t-\tau) = f(\tau)\delta(t-\tau)
  • δ(at)=1aδ(t)\delta(at) = \frac{1}{|a|}\delta(t)

Definition 6

The unit step is defined as
u[n]={1if n00else.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(tτ)x(t - \tau)
    : Shift a signal right by
    τ\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(τt)=x((tτ))y(t) = x(\tau - t) = x(-(t-\tau))
flips
xx
and shifts it right by
τ\tau
timesteps. This is equivalent to shifting
xx
left by
τ\tau
timesteps and then flipping it.

Convolution

Definition 7

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

Definition 8

The convolution of two signals in continuous time
(xh)(t)=x(τ)h(tτ)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.
  • (xδ)[n]=x[n](x*\delta)[n] = x[n]
  • x[n]δ[nN]=x[nN]x[n]*\delta[n-N]=x[n-N]
  • (xh)[n]=(hx)[n](x*h)[n] = (h*x)[n]
  • x(h1+h2)=xh1+xh2x * (h_1 + h_2) = x*h_1 + x*h_2
  • x(h1h2)=(xh1)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(tτ)x(t-\tau)
is transformed to
y(tτ)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,zC)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