Unit 1 - Notes

ECE220

Unit 1: Introduction to Signals

1. Continuous-Time and Discrete-Time Signals

Signals are mathematical functions that convey information about the behavior or attributes of some phenomenon. They are broadly classified based on the nature of the independent variable (usually time).

Continuous-Time (CT) Signals

A signal $x(t)$ is a continuous-time signal if the independent variable $t$ is continuous.

Definition: The signal is defined for every instant of time within a range.
Representation: $x(t)$ , where $(\cdot)$ denotes continuous variables.
Examples: Voltage across a resistor, audio recordings on a vinyl record, temperature in a room over a day.

Discrete-Time (DT) Signals

A signal $x[n]$ is a discrete-time signal if the independent variable $n$ is discrete.

Definition: The signal is defined only at integer values of the independent variable.
Representation: $x[n]$ , where $[\cdot]$ denotes discrete integers.
Origin: Often derived from sampling a CT signal: $x[n] = x(nT_s)$ , where $T_s$ is the sampling period.
Examples: Stock market indices at closing, pixels in a digital image, digital audio samples.

2. Energy and Power Signals

Signals are classified based on the "size" or strength of the waveform.

Energy Signals

A signal is an Energy signal if it has finite total energy ( $0 < E < \infty$ ) and zero average power ( $P = 0$ ). These are typically aperiodic signals that decay to zero (transients).

CT Energy Formula:
$E = \int_{-\infty}^{\infty} |x(t)|^2 dt$
DT Energy Formula:
$E = \sum_{n=-\infty}^{\infty} |x[n]|^2$

Power Signals

A signal is a Power signal if it has finite non-zero average power ( $0 < P < \infty$ ) and infinite total energy ( $E = \infty$ ). These are typically periodic or random signals that do not decay.

CT Power Formula:
$P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt$
DT Power Formula:
$P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$

Summary Table

Signal Type	Energy ( $E$ )	Power ( $P$ )	Typical Characteristic
Energy	Finite	0	Non-periodic / Deterministic
Power	Infinite	Finite	Periodic / Random
Neither	Infinite	Infinite	Unbounded (e.g., $x(t) = t$ )

3. Transformations of the Independent Variable

Manipulating the time variable ( $t$ or $n$ ) transforms the signal geometry.

Time Shifting

Displaces the signal along the time axis.

Equation: $y(t) = x(t - t_0)$
Delay: If $t_0 > 0$ , the signal shifts to the right.
Advance: If $t_0 < 0$ , the signal shifts to the left.

Time Scaling

Expands or compresses the signal in time.

Equation: $y(t) = x(at)$
Compression: If $|a| > 1$ , the signal plays faster (compresses).
Expansion: If $0 < |a| < 1$ , the signal plays slower (expands).
Note for DT: In DT, $x[kn]$ is known as decimation (down-sampling), which may result in loss of information if not band-limited.

Time Reversal (Reflection)

Reflects the signal across the vertical axis (y-axis).

Equation: $y(t) = x(-t)$
This is a special case of scaling where $a = -1$ .

Precedence Rule

When multiple transformations occur (e.g., $y(t) = x(at - b)$ ), standard order of operations matters:

Shift first: $x(t) \to x(t-b)$
Scale second: Replace $t$ with $at$ $\to x(at - b)$

4. Periodic Signals

A periodic signal repeats its pattern indefinitely.

Continuous-Time Periodicity

A signal $x(t)$ is periodic if there exists a positive value $T$ such that:

x(t) = x(t + T) \quad \text{for all } t

Fundamental Period ( $T_0$ ): The smallest positive value of $T$ .
Fundamental Frequency: $f_0 = 1/T_0$ or $\omega_0 = 2\pi/T_0$ .

Discrete-Time Periodicity

A signal $x[n]$ is periodic if there exists an integer $N$ such that:

x[n] = x[n + N] \quad \text{for all } n

Crucial Difference: A CT sinusoid is always periodic. A DT sinusoid $x[n] = \cos(\omega_0 n)$ is periodic only if $\frac{\omega_0}{2\pi}$ is a rational number (ratio of integers).
Fundamental Period ( $N$ ): Given $\frac{\omega_0}{2\pi} = \frac{m}{N}$ , the period is $N$ (after reducing the fraction).

5. Even and Odd Signals

This classification refers to the symmetry of the signal.

Even Signal (Symmetric)

Ideally symmetric about the vertical axis ( $t=0$ ).

x(t) = x(-t)

Examples: $\cos(t)$ , $t^2$ , constant $C$ .

Odd Signal (Anti-symmetric)

Symmetric about the origin.

x(t) = -x(-t)

Examples: $\sin(t)$ , $t$ , $t^3$ .
Note: An odd signal must be zero at the origin ( $x(0) = 0$ ).

Decomposition Property

Any arbitrary signal $x(t)$ can be decomposed into a sum of an even part and an odd part:

x(t) = x_e(t) + x_o(t)

Even Part: $x_e(t) = \frac{1}{2} [x(t) + x(-t)]$
Odd Part: $x_o(t) = \frac{1}{2} [x(t) - x(-t)]$

6. Exponential and Sinusoidal Signals

These are the fundamental building blocks (basis functions) for signal analysis (Fourier Series/Transform).

Real Exponential

x(t) = Ce^{at}

If $a > 0$ : Growing exponential.
If $a < 0$ : Decaying exponential.

Complex Exponential

x(t) = e^{j\omega_0 t}

Using Euler’s Identity:

e^{j\omega_0 t} = \cos(\omega_0 t) + j\sin(\omega_0 t)

This represents a rotating phasor in the complex plane.
It is a Power signal.

Sinusoidal Signals

x(t) = A \cos(\omega_0 t + \phi)

$A$ : Amplitude
$\omega_0$ : Angular frequency (rad/sec)
$\phi$ : Phase angle

DT Complex Exponential

x[n] = e^{j\omega_0 n}

Properties differ from CT; high frequencies in DT are near $\pi$ (oscillating), while low frequencies are near $0$ and $2\pi$ .

7. The Unit Impulse and Unit Step Functions

These are "Singularity Functions" used to model system responses.

Unit Step Function ( $u(t)$ )

Used to represent a signal that switches on at $t=0$ .

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

Unit Impulse Function ( $\delta(t)$ ) - Dirac Delta

Ideally a pulse with infinite height and zero width, but area equals 1.

Definition: $\delta(t) = 0$ for $t \neq 0$ , and $\int_{-\infty}^{\infty} \delta(t) dt = 1$ .
Sifting Property: Used to extract values from a function.
$\int_{-\infty}^{\infty} x(t)\delta(t - t_0) dt = x(t_0)$

Discrete-Time Impulse ( $\delta[n]$ )

Much simpler than the CT version.

\delta[n] = \begin{cases} 1, & n = 0 \\ 0, & n \neq 0 \end{cases}

Relationship

CT: $\delta(t) = \frac{d}{dt}u(t)$ and $u(t) = \int_{-\infty}^{t} \delta(\tau) d\tau$
DT: $\delta[n] = u[n] - u[n-1]$ and $u[n] = \sum_{k=-\infty}^{n} \delta[k]$

8. Operations on Signals

Beyond time transformations, we perform arithmetic operations on signal amplitudes.

Amplitude Scaling: $y(t) = A x(t)$ . Amplifies or attenuates the signal.
Addition: $y(t) = x_1(t) + x_2(t)$ . Used in mixing signals (e.g., audio mixing).
Multiplication: $y(t) = x_1(t) \cdot x_2(t)$ . Used in modulation (AM radio) and windowing.
Differentiation: $y(t) = \frac{dx(t)}{dt}$ . Represents rate of change.
Integration: $y(t) = \int_{-\infty}^{t} x(\tau) d\tau$ . Accumulation of the signal over time.

9. Software Simulation of Basic Operations

Below is a generic Python implementation (using NumPy and Matplotlib) to simulate elementary signals and operations.

Generating Elementary Signals

PYTHON

import numpy as np
import matplotlib.pyplot as plt

# Time vector
t = np.linspace(-5, 5, 1000)
n = np.arange(-5, 6)

# 1. Unit Step u(t)
u_t = np.where(t >= 0, 1, 0)

# 2. Unit Impulse delta[n] (Discrete)
delta_n = np.where(n == 0, 1, 0)

# 3. Sinusoid x(t) = A cos(wt + phi)
A, w, phi = 2, 2*np.pi, 0
sine_wave = A * np.cos(w * t + phi)

# 4. Exponential (Decaying)
exp_signal = np.exp(-0.5 * t) * u_t  # Multiplied by u(t) to start at 0

# Plotting
plt.figure(figsize=(10, 8))

plt.subplot(2, 2, 1)
plt.plot(t, u_t)
plt.title('Continuous Unit Step u(t)')
plt.grid(True)

plt.subplot(2, 2, 2)
plt.stem(n, delta_n)
plt.title('Discrete Unit Impulse delta[n]')
plt.grid(True)

plt.subplot(2, 2, 3)
plt.plot(t, sine_wave)
plt.title('Sinusoidal Signal')
plt.grid(True)

plt.subplot(2, 2, 4)
plt.plot(t, exp_signal)
plt.title('Real Exponential Signal')
plt.grid(True)

plt.tight_layout()
plt.show()

Simulating Signal Operations (Addition & Scaling)