Unit 1: Introduction to Signals

1. Introduction: Signals and Information Processing

In electrical, electronic, communications, and robotics engineering, a signal is formally defined as a mathematical function of one or more independent variables that carries information about the state or behavior of a physical system.

Common physical examples of signals include:

• The time-varying voltage $v(t)$ measured across a circuit capacitor or sensor load,
• The acoustic pressure variation $p(t)$ picked up by a robot microphone,
• The discrete pixel luminance matrix $I[x, y]$ captured by a machine vision camera sensor,
• The joint angular position $\theta[n]$ sampled by an optical shaft encoder at clock cycles $n$.

A system is a physical device or mathematical operator that takes an input signal $x(t)$ (excitation) and transforms it into an output signal $y(t)$ (response):
$y(t) = \mathcal{T}\{x(t)\} \quad \text{or} \quad y[n] = \mathcal{T}\{x[n]\}$

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+---------------------------------------------------------------------------------------------------+
|                           THE GENERAL SIGNAL & SYSTEM ARCHITECTURE                                |
+---------------------------------------------------------------------------------------------------+
|  PHYSICAL EXCITATION                SYSTEM OPERATOR T{.}                 PROCESSED RESPONSE       |
|  Input Signal x(t) or x[n]   ===>  [ Filter / Modulator / Controller ] ===> Output Signal y(t)     |
|                                                                                                   |
|  CONTINUOUS-TIME (CT):             DISCRETE-TIME (DT):                  SAMPLING INTERFACE:       |
|  t in R (Continuum)                 n in Z (Integers)                    x[n] = x(n * T_s)        |
|  Analog Voltages, Accelerations     Microcontroller Digital Registers    Analog-to-Digital (ADC)  |
+---------------------------------------------------------------------------------------------------+

2. Continuous-Time (CT) vs. Discrete-Time (DT) Signals

Signals are fundamentally categorized based on the nature of their independent variable:

2.1 Continuous-Time (CT) Signals

A signal $x(t)$ is continuous-time if its independent variable $t$ is continuous, taking values across a continuum of real numbers $t \in \mathbb{R}$. The signal value $x(t)$ is defined at every infinitesimal instant of time. Physical voltages, currents, biological EEG/ECG signals, and acoustic sound waves are inherently continuous-time.

2.2 Discrete-Time (DT) Signals

A signal $x[n]$ is discrete-time if its independent variable $n$ takes values only from a discrete, countable set of integers $n \in \mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$.

• By universal convention, discrete signals are written with square brackets $x[n]$.
• A discrete-time signal is often generated by uniformly sampling a continuous-time signal $x_a(t)$ at a sampling period $T_s$:
  $x[n] = x_a(n T_s), \quad n \in \mathbb{Z}$
• Notice that $x[n]$ is strictly undefined for non-integer values of $n$ (e.g. $x[1.5]$ is mathematically meaningless).

3. Transformations of the Independent Variable

Signal processing frequently modifies the time axis of signals via three elementary operations: time shifting, time reflection (reversal), and time scaling.

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+---------------------------------------------------------------------------------------------------+
|                        ELEMENTARY TIME-AXIS TRANSFORMATIONS OF x(t)                               |
+-------------------+-----------------------+-------------------------------------------------------+
| TRANSFORMATION    | MATHEMATICAL FORM     | PHYSICAL EFFECT / INTERPRETATION                      |
+-------------------+-----------------------+-------------------------------------------------------+
| Time Shifting     | x(t - t_0)            | Right-shift (Delay) if t_0 > 0; Left-shift if t_0 < 0 |
| Time Reflection   | x(-t)                 | Mirror flip about vertical axis t = 0 (Time Reversal) |
| Time Compression  | x(a * t), |a| > 1     | Signal plays faster; compressed by factor |a|         |
| Time Expansion    | x(a * t), 0 < |a| < 1 | Signal plays slower; stretched out by factor 1/|a|    |
+-------------------+-----------------------+-------------------------------------------------------+

3.1 Composite Transformations: $x(at + b)$

When applying combined scaling and shifting to evaluate $y(t) = x(at + b)$, the order of mathematical operations is critical:

• Method 1 (Shift then Scale):
  1. First shift $x(t)$ by $b$ to obtain intermediate signal $w(t) = x(t + b)$.
  2. Then scale the time variable $t$ by $a$: $w(at) = x(at + b)$.
• Method 2 (Scale then Shift - Factored Form):
  Factor out $a$: $y(t) = x\left( a \left( t + \frac{b}{a} \right) \right)$.
  1. First scale $x(t)$ by $a$ to form $v(t) = x(at)$.
  2. Then shift $v(t)$ by the effective amount $\frac{b}{a}$: $v\left(t + \frac{b}{a}\right) = x(at + b)$.

Example: To construct $y(t) = x(2t + 4)$, factor as $x(2(t + 2))$. The signal $x(t)$ is first compressed horizontally by a factor of 2, and then shifted to the left by 2 units.

4. Signal Classifications: Energy, Power, Periodicity, and Symmetry

4.1 Energy and Power Signals

The physical energy dissipated by a voltage signal $x(t)$ across a $1\,\Omega$ resistor over interval $[-T, T]$ leads to the definitions of total energy $E$ and average power $P$:

1. Continuous-Time Energy & Power:
   $\text{Total Energy: } E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt$
   $\text{Average Power: } P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 \, dt$

2. Discrete-Time Energy & Power:
   $\text{Total Energy: } E = \sum_{n=-\infty}^{\infty} |x[n]|^2$
   $\text{Average Power: } P = \lim_{N \to \infty} \frac{1}{2N + 1} \sum_{n=-N}^{N} |x[n]|^2$

3. Classification Categories:

   • Energy Signal: A signal with finite total energy ($0 < E < \infty$). Its average power is necessarily zero ($P = 0$). Examples: Single finite-duration pulses, exponentially decaying transients $e^{-at} u(t)$ ($a > 0$).
   • Power Signal: A signal with finite, non-zero average power ($0 < P < \infty$). Its total energy is infinite ($E = \infty$). Examples: Periodic sinusoids $\cos(\omega_0 t)$, unit step function $u(t)$, constant DC signals.
   • Neither Energy nor Power: Signals whose energy and power both diverge to infinity ($E = \infty$ and $P = \infty$). Examples: Growing ramps $t u(t)$, exponentially growing signals $e^{at} u(t)$ ($a > 0$).

4.2 Periodic vs. Aperiodic Signals

1. Continuous-Time Periodicity:
   A signal $x(t)$ is periodic if there exists a positive real constant $T > 0$ such that:
   $x(t + T) = x(t) \quad \forall t \in \mathbb{R}$
   The smallest positive value $T_0 > 0$ satisfying this condition is the fundamental period. The fundamental angular frequency is $\omega_0 = \frac{2\pi}{T_0}$ (or cyclic frequency $f_0 = \frac{1}{T_0}$).

   • The sum of two continuous periodic signals $x_1(t)$ and $x_2(t)$ with periods $T_1$ and $T_2$ is periodic if and only if the ratio of their periods is a rational number:
     $\frac{T_1}{T_2} = \frac{p}{q} \in \mathbb{Q} \quad (p, q \in \mathbb{Z}^+)$
     The fundamental period of the composite signal is the Least Common Multiple: $T_0 = q T_1 = p T_2$.

2. Discrete-Time Periodicity (Crucial Contrast):
   A discrete-time sequence $x[n]$ is periodic if there exists a positive integer $N \in \mathbb{Z}^+$ such that:
   $x[n + N] = x[n] \quad \forall n \in \mathbb{Z}$

   • Key Difference from Continuous Time: A continuous sinusoid $\cos(\omega_0 t)$ is always periodic for any real frequency $\omega_0$. However, a discrete sinusoid $\cos(\omega_0 n)$ is periodic if and only if its frequency $\omega_0$ is a rational multiple of $2\pi$:
     $\frac{\omega_0}{2\pi} = \frac{m}{N} \in \mathbb{Q} \quad (m, N \in \mathbb{Z}^+)$
     If this condition is met, the fundamental period is $N = m \left( \frac{2\pi}{\omega_0} \right)$ where $m$ is the smallest integer making $N$ an integer. If $\frac{\omega_0}{2\pi}$ is irrational (e.g. $\cos(3n)$), the sequence is strictly aperiodic!

4.3 Even and Odd Symmetry

1. Definitions:

   • Even Signal: Symmetric about the vertical axis:
     $x(-t) = x(t) \quad \text{or} \quad x[-n] = x[n]$
   • Odd Signal: Antisymmetric about the origin:
     $x(-t) = -x(t) \quad \text{or} \quad x[-n] = -x[n]$
     Property: For any continuous odd signal, $x(0) = -x(0) \implies x(0) = 0$.

2. Even-Odd Decomposition:
   Every arbitrary signal can be uniquely decomposed into the sum of an even component $x_e(t)$ and an odd component $x_o(t)$:
   $x(t) = x_e(t) + x_o(t)$
   where:
   $x_e(t) = \frac{x(t) + x(-t)}{2}, \quad x_o(t) = \frac{x(t) - x(-t)}{2}$

3. Definite Integral Properties:
   $\int_{-a}^{a} x_e(t) \, dt = 2 \int_{0}^{a} x_e(t) \, dt$
   $\int_{-a}^{a} x_o(t) \, dt = 0$

5. Elementary Standard Signals

5.1 The Continuous-Time Unit Impulse (Dirac Delta) $\delta(t)$

The Dirac delta $\delta(t)$ is not a conventional function in the classical sense, but a generalized function (distribution) defined by its operational effect under an integral:
$\delta(t) = 0 \quad \forall t \ne 0, \quad \text{and} \quad \int_{-\infty}^{\infty} \delta(t) \, dt = 1$

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|                                  THE SIFTING PROPERTY OF delta(t)                                 |
+---------------------------------------------------------------------------------------------------+
|  x(t) ^                                                                                           |
|       |                     x(t_0) * delta(t - t_0)                                               |
|       |                             ^                                                             |
|       |                             | Area = x(t_0)                                               |
|       |         .-------*-----------|----------.                                                  |
|       |       .-        |           |           -.      True Signal x(t)                          |
|       |     .-          |           |             -.                                              |
|       +----+------------+-----------+---------------+---------------------> Time t                |
|           -inf          0          t_0             +inf                                           |
|                                                                                                   |
|  Mathematical Formula:  int_{-inf}^{+inf} x(t) * delta(t - t_0) dt = x(t_0)                       |
+---------------------------------------------------------------------------------------------------+

• Fundamental Operational Properties:
  1. Sifting (Sampling) Property:
     $\int_{-\infty}^{\infty} x(t) \delta(t - t_0) \, dt = x(t_0)$
  2. Product Property:
     $x(t) \delta(t - t_0) = x(t_0) \delta(t - t_0)$
  3. Time-Scaling Property:
     $\delta(a t) = \frac{1}{|a|} \delta(t), \quad a \ne 0$
  4. Even Symmetry:
     $\delta(-t) = \delta(t)$
  5. Unit Doublet (Derivative of Impulse):
     $\int_{-\infty}^{\infty} x(t) \delta'(t - t_0) \, dt = -x'(t_0)$

5.2 The Discrete-Time Unit Impulse (Kronecker Delta) $\delta[n]$

Unlike the continuous Dirac delta, the discrete impulse is a true, simple function:
$\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \ne 0 \end{cases}$

• Discrete Sifting Property:
  $\sum_{n=-\infty}^{\infty} x[n] \delta[n - k] = x[k]$

5.3 The Unit Step Function $u(t)$ and $u[n]$

1. Continuous-Time Unit Step:
   $u(t) = \begin{cases} 1, & t > 0 \\ 0, & t < 0 \end{cases}$

   • Relationship with Impulse:
     $\delta(t) = \frac{d u(t)}{dt}, \quad u(t) = \int_{-\infty}^{t} \delta(\tau) \, d\tau$

2. Discrete-Time Unit Step:
   $u[n] = \begin{cases} 1, & n \ge 0 \\ 0, & n < 0 \end{cases}$

   • Relationship with Discrete Impulse:
     $\delta[n] = u[n] - u[n - 1], \quad u[n] = \sum_{k=-\infty}^{n} \delta[k] = \sum_{k=0}^{\infty} \delta[n - k]$

5.4 The Unit Ramp Function $r(t)$ and $r[n]$

$r(t) = t u(t) = \begin{cases} t, & t \ge 0 \\ 0, & t < 0 \end{cases} \implies r(t) = \int_{-\infty}^{t} u(\tau) \, d\tau, \quad u(t) = \frac{d r(t)}{dt}$

6. Exponential and Sinusoidal Signals

6.1 Real Exponential Signals: $x(t) = C e^{a t}$

• Decaying Exponential ($a < 0$): Energy signal ($E < \infty, P = 0$). Models passive RC circuit discharge or robot joint damping decay.
• Growing Exponential ($a > 0$): Neither energy nor power signal. Models unstable open-loop systems.
• Constant DC Signal ($a = 0$): Power signal with $P = C^2$.

6.2 Complex Exponential Signals: $x(t) = e^{s t}$

Let $s = \sigma + j\omega_0 \in \mathbb{C}$:
$x(t) = e^{(\sigma + j\omega_0)t} = e^{\sigma t} e^{j\omega_0 t} = e^{\sigma t} \left( \cos(\omega_0 t) + j\sin(\omega_0 t) \right)$

• If $\sigma = 0$: $x(t) = e^{j\omega_0 t} = \cos(\omega_0 t) + j\sin(\omega_0 t)$, representing pure, undamped harmonic oscillation with fundamental period $T_0 = \frac{2\pi}{|\omega_0|}$ and average power $P = 1$.
• If $\sigma < 0$: Exponentially damped sinusoid (stable underdamped physical response).
• If $\sigma > 0$: Exponentially growing sinusoid (unstable flutter/resonance).

6.3 Discrete-Time Complex Exponentials: $x[n] = e^{j\omega_0 n}$

• Periodicity in Frequency:
  $e^{j(\omega_0 + 2\pi) n} = e^{j\omega_0 n} e^{j 2\pi n} = e^{j\omega_0 n} \cdot 1 = e^{j\omega_0 n}$
  Unlike continuous-time signals where different $\omega_0$ always yield distinct signals, discrete complex exponentials are identical when their frequencies differ by any integer multiple of $2\pi$.
  • All unique discrete frequencies lie within a fundamental interval of length $2\pi$, typically $[-\pi, \pi)$ or $[0, 2\pi)$.
  • Low frequencies occur near $\omega_0 = 0, \pm 2\pi, \pm 4\pi, \dots$.
  • Highest frequency of oscillation occurs at $\omega_0 = \pm \pi$.

7. Summary Comparison of Signal Classifications

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+---------------------------+-----------------------------------+-----------------------------------+
| CHARACTERISTIC            | CONTINUOUS-TIME (CT)              | DISCRETE-TIME (DT)                |
+---------------------------+-----------------------------------+-----------------------------------+
| Independent Variable      | t in R (Continuum)                | n in Z (Integers)                 |
| Unit Impulse              | Dirac delta(t) (Infinite height)  | Kronecker delta[n] = 1 at n = 0   |
| Step-Impulse Relation     | delta(t) = d/dt [u(t)]            | delta[n] = u[n] - u[n-1]          |
| Total Energy E            | int_{-inf}^{+inf} |x(t)|^2 dt     | sum_{n=-inf}^{+inf} |x[n]|^2      |
| Average Power P           | lim 1/(2T) int_{-T}^{+T} |x(t)|^2 dt| lim 1/(2N+1) sum_{-N}^{+N} |x[n]|^2|
| Sinusoid Periodicity      | ALWAYS periodic for any omega_0   | Periodic ONLY if omega_0/(2pi) in Q|
| Distinct Frequencies      | Distinct for all real omega_0     | Identical for omega_0 + 2*pi*k    |
| Highest Rate of Oscil.    | Approaches infinity as omega -> inf| Occurs at omega_0 = +/- pi        |
| Even / Odd Decomposition  | x(t) = x_e(t) + x_o(t)            | x[n] = x_e[n] + x_o[n]            |
+---------------------------+-----------------------------------+-----------------------------------+