Part 1 - Signals

Introduction

In this series we’ll cover what we mean with transforms, signals and systems. How they relate and are used in the real world.

Signals

In this part we’ll try to understand signals, classify these. Perform different signal operations and lastly understand and use signal models. Let’s first define what a signal is

Definition

A signal is a set of information or data. Any physical quantity that varies over time, space or any other variable or variables.

We will usually define signals with mathematical functions.

Signal classifications

There are different types of signals and representations of signals. Let’s list these:

• Continuous VS Discrete (Time)
• Continuous VS Discrete (Amplitude)
• Periodic VS Aperiodic
• Deterministic VS Stochastic

We’ll properly define each of these, let’s start with the time representation:

Continuous VS Discrete (Time)

As we can see, the discrete representation, is points that are spread with a time interval $T$.

Continuous VS Discrete (Amplitude)

As we can see, this is quantization, analog $\to$ digital.

Even, Odd & Periodic

Let’s also define what an even, odd and periodic functions are.

An even function is symmetrical about the vertical axis. Mathematically this means: $$ f(t) = f(-t) \newline f[k] = f[-k] $$ An odd function is anti-symmetrical about the vertical axis. Mathematically this means: $$ f(t) = -f(-t) \newline f[k] = -f[-k] $$

A periodic function has a fundamental period (minimum), $T_0$. Which also means it has a fundamental frequency, $f_0 = \dfrac{1}{T_0}$. $$ f(t) = f(t + nT_0) \newline f[k] = f[k + nK_0] $$

We sometimes define the fundamental frequency in angular velocity instead of Hz, which means, $\omega_0 = 2\pi f_0 = \dfrac{2\pi}{T_0}$.

Deterministic VS Stochastic

These are quite easy to define.

Deterministic signal: Its physical description is known completely (mathematical or graphical).

Stochastic signal: Values are only known in probabilistic terms.

Energy VS Power

We’ll see these later on, but let’s define them.

An energy signal is a signal whose energy is finite and power is zero.

A power signal is a signal whose power is finite and energy is infinite.

Signal energy

Real signal: $$ E_f = \int_{-\infty}^{\infty} f^2(t)\ dt $$

Complex signal: $$ E_f = \int_{-\infty}^{\infty} |f^2(t)|\ dt $$

$$ 0 < E_f < \infty $$

Signal power

Real signal: $$ P_f = \lim_{T \to \infty} \dfrac{1}{T} \int_{- \tfrac{T}{2}}^{\tfrac{T}{2}} f^2(t)\ dt $$

Complex signal: $$ P_f = \lim_{T \to \infty} \dfrac{1}{T} \int_{- \tfrac{T}{2}}^{\tfrac{T}{2}} |f^2(t)|\ dt $$

$$ 0 < P_f < \infty $$

Signal operations

Now that we have defined what signals are, what operations can we perform? Since they are mathematical functions, we can perform a whole row of operations.

Let’s start in the time-continuous world. We’ll list all the operations we can perform.

• Amplitude scaling (Gain)

• DC (Offset)

• Time scaling

• Reflection (Time inversion)

• Time shift

Let’s go through them all and define them.

Amplitude scaling

$$ f(t) \newline \Phi(t) = A \cdot f(t) $$

$$ A > 1 \ | \ \text{Amplification} \newline 0 < A < 1 \ | \ \text{Attenuation} \newline A < 0 \ | \ \text{Amplitude reversal} $$

DC (Offset)

$$ f(t) \newline \Phi(t) = f(t) + B $$

Time scaling

Given our input signal $$ f(t) $$

We can perform different operations.

Compression & expansion

$$ \Phi(t) = f(2t) $$

$$ \Phi(t) = f(\dfrac{t}{2}) $$

Reflection

$$ \Phi(t) = f(-t) $$

Time shift

$$ \Phi(t) = f(t \pm T) $$

Summary of operations

Signal models

We’ll now cover how we can (usually) model these signals, we’ll look at three functions which model signals.

These are:

• Unit step function

• Unit impulse function (also called the Dirac delta function)

• Exponential function

Unit step function

The unit step function is defined as:

$$ u(t) = \begin{cases} 1 & t \geq 0 \newline 0 & t < 0 \end{cases} $$

In the discrete case: $$ u[k] = \begin{cases} 1 & k \geq 0 \newline 0 & k < 0 \end{cases} $$

This means we can represent rectangular signals as linear combination of the unit step function. For example:

$$ f(t) = u(t - 2) - u(t - 4) $$

Unit impulse function (Dirac delta function)

We define dirac delta function as the following: $$ \delta(t) = 0 \ | \ t \neq 0 $$

$$ \int_{-\infty}^{\infty} \delta(t)\ dt = 1 $$

In the discrete case: $$ \delta[k] = \begin{cases} 1 & k = 0 \newline 0 & k \neq 0 \end{cases} $$

We’ll see that we can define discrete time-signals with this function! But the main power with the dirac delta function is property to sample/sift:

Suppose we have a function, $\phi(t)$, which is continuous at $t = 0$. We can perform: $$ \phi(t)\delta(t) = \phi(0)\delta(t) $$

$$ \int_{-\infty}^{\infty} \phi(t)\delta(t)\ dt = \phi(0) \int_{-\infty}^{\infty} \delta(t)\ dt= \phi(0) $$

This also is true if, $\phi(t)$, is continuous at $t = T$ $$ \phi(t)\delta(t - T) = \phi(T)\delta(t - T) $$

$$ \int_{-\infty}^{\infty} \phi(t)\delta(t - T)\ dt = \phi(T) $$

The area under the product of a function with an impulse, $\delta(t)$, is equal to the value of that function at the instant where the unit impulse is located.

Exponential function

We define the exponential function with complex numbers:

$$ e^{st} \ | \ s = \sigma + j\omega $$

This means: $$ e^{st} = e^{t(\sigma + j\omega)} = e^{t\sigma + j\omega t} = e^{t\sigma} \cdot e^{j\omega t} = e^{t\sigma}(cos \omega t + j sin \omega t) $$

We have some special cases where we get:

1. A constant $k = ke^{0t} \ | \ (s = 0)$
2. A monotonic exponential $e^{\sigma t} \ | \ (\omega = 0, s = \omega)$
3. A sinusoid $cos \omega t \ | \ (\sigma = 0, s = \pm j\omega)$
4. A exp. varying $e^{\sigma t} cos \omega t \ | \ (s = \omega \pm j\omega)$

Discrete time

Operations for discrete time behave the same.