Lecture 1 — Continuous-Time Signals

#signals-and-systems #continuous-time #signals

1. Continuous-Time Signals

A continuous-time signal is a function of time

x (t) \in R or C, t \in R .

A signal typically represents or contains information.

Strictly speaking, $x$ denotes the whole signal while $x (t)$ denotes its value at time $t$ .

2. Standard Signals

Unit Step

The unit step or Heaviside step is

u (t) = {\begin{cases} 0, & t < 0 \\ 1, & t > 0 \end{cases}

Used to model something being switched on.

The exact value at $t = 0$ is usually unimportant:

u (0) = 0, 1, or \frac{1}{2}

may be chosen depending on convention.

Time reversal

u (- t)

represents a switch-off type signal.

Sign / Signum Function

sign (t) = {\begin{cases} - 1, & t < 0 \\ + 1, & t > 0 \end{cases}

Relationships with the unit step:

sign (t) = 2 u (t) - 1

sign (t) = u (t) - u (- t)

Rectangular Pulse

The standard rectangular pulse is

p (t) = {\begin{cases} 1, & 0 < t < 1 \\ 0, & t < 0 or t > 1 \end{cases}

Using step functions:

p (t) = u (t) - u (t - 1)

General idea

A finite pulse can often be constructed as

u (t - a) - u (t - b),

which turns the signal on at $t = a$ and off at $t = b$ .

Ramp Function

r (t) = {\begin{cases} t, & t > 0 \\ 0, & t < 0 \end{cases}

Equivalent representations:

r (t) = t u (t)

and

r (t) = \int_{- \infty}^{t} u (τ) d τ

So the ramp is the integral of the unit step.

Triangular Pulse

Λ (t) = {\begin{cases} t, & 0 \leq t \leq 1 \\ 2 - t, & 1 \leq t \leq 2 \\ 0, & t < 0 or t > 2 \end{cases}

It can be constructed using ramps:

Λ (t) = r (t) - 2 r (t - 1) + r (t - 2)

This is a useful general pattern:

Step functions create jumps, while ramp functions create changes in slope.

Sinc Function

The convention used in this course is

sinc (t) = {\begin{cases} \frac{\sin (π t)}{π t}, & t \neq 0 \\ 1, & t = 0 \end{cases}

because

lim_{t \to 0} \frac{\sin (π t)}{π t} = 1.

Zero crossings occur at

t = \pm 1, \pm 2, \pm 3, \dots

The sinc function becomes important for:

bandlimited signals
sampling
reconstruction from samples

3. Complex Numbers Refresher

A complex number can be written in rectangular form as

z = x + j y

or polar form as

z = r e^{j ϕ} .

Magnitude:

| z | = r = \sqrt{x^{2} + y^{2}}

Phase:

∠ z = ϕ .

Real and imaginary parts:

x = ℜ (z) = r \cos ϕ

y = ℑ (z) = r \sin ϕ .

Complex Conjugate

z^{*} = x - j y = r e^{- j ϕ} .

Important identity:

| z |^{2} = z z^{*}

because

(x + j y) (x - j y) = x^{2} + y^{2} .

Also,

ℜ (z) = \frac{z + z^{*}}{2}

and

ℑ (z) = \frac{z - z^{*}}{2 j}

4. Euler Representation of Sinusoids

Euler's formula gives

e^{j ϕ} = \cos ϕ + j \sin ϕ .

Therefore

\cos ϕ = \frac{e^{j ϕ} + e^{- j ϕ}}{2}

and

\sin ϕ = \frac{e^{j ϕ} - e^{- j ϕ}}{2 j}

These identities are fundamental throughout Signals and Systems.

Product-to-Sum Example

Using complex exponentials,

\cos α \cos β = \frac{1}{2} [\cos (α + β) + \cos (α - β)]

5. Exponential and Harmonic Signals

Real Exponential

x (t) = A e^{a t} .

A common one-sided exponential is

x (t) = A e^{a t} u (t)

so that the exponential exists only for $t > 0$ .

Complex Exponential

x (t) = A e^{j Ω t}

A complex exponential has constant magnitude and a rotating phase.

Harmonic Signals

Cosine:

\cos (Ω t) = ℜ (e^{j Ω t})

and

\cos (Ω t) = \frac{1}{2} (e^{j Ω t} + e^{- j Ω t})

Sine:

\sin (Ω t) = \frac{1}{2 j} (e^{j Ω t} - e^{- j Ω t})

A real sinusoid therefore contains both:

+ Ω and - Ω

frequency components.

Frequency Units

For continuous time, the course uses

Ω [rad/s] .

Relationship with ordinary frequency $F$ in Hz:

Ω = 2 π F

and therefore

F = \frac{Ω}{2 π} .

6. Phase and Complex Amplitude

Consider

\cos (Ω t + θ) .

Using exponentials,

\cos (Ω t + θ) = \frac{1}{2} (e^{j θ} e^{j Ω t} + e^{- j θ} e^{- j Ω t}) .

A = | A | e^{j θ},

then $A$ can be interpreted as a complex amplitude.

Thus

| A | \cos (Ω t + θ) = \frac{A}{2} e^{j Ω t} + \frac{A^{*}}{2} e^{- j Ω t}

This is important because many systems can be analyzed by determining what they do to

e^{j Ω t} .

7. Systems

A single-input single-output system transforms an input signal $x (t)$ into an output $y (t)$ :

x (t) ⟶ S ⟶ y (t) .

Mathematically,

y = S {x}

A system is therefore an operator or transformation.

This is analogous to linear algebra:

matrix : vector $\to$ vector
system : signal $\to$ signal

8. Elementary Systems

Scaling

y (t) = α x (t)

Time Delay

A delay of $T$ seconds:

y (t) = x (t - T)

Notation:

(D_{T} x) (t) = x (t - T) .

Important intuition

For

x (t - T),

the signal moves right by $T$ .

Differentiator

y (t) = \frac{d}{d t} x (t)

Integrator

y (t) = \int_{- \infty}^{t} x (τ) d τ

9. Building Systems from Blocks

Complex systems can be constructed using simple operations such as:

addition
scaling
integration
differentiation
feedback
delay

Integrator with Feedback

Suppose

e (t) = x (t) - α y (t)

and

y (t) = \int e (t) d t .

Then

y^{'} (t) = e (t) .

Therefore

y^{'} (t) = x (t) - α y (t)

and hence

y^{'} (t) + α y (t) = x (t)

The block diagram therefore corresponds directly to a differential equation.

10. First-Order RC Circuit

For the RC circuit from the lecture,

x (t) = R i (t) + y (t)

and

i (t) = C \frac{d y}{d t} .

Substituting:

x (t) = R C \frac{d y}{d t} + y (t) .

Therefore

R C y^{'} (t) + y (t) = x (t)

This has the same mathematical structure as the feedback-integrator system.

11. Numerical Simulation of a System

For

y^{'} (t) = e (t)

use the approximation

y^{'} (t) \approx \frac{y (t + Δ t) - y (t)}{Δ t} .

Therefore

y (t + Δ t) \approx y (t) + Δ t e (t)

Using discrete indices

y [i] = y (i Δ t),

a simulation can use

y[0] = 0

for i = 0,1,...
    e[i] = x[i] - αy[i]
    y[i+1] = y[i] + Δt e[i]
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