Unit 3 - Notes

ECE220

Unit 3: Fourier series representation of periodic signals

1. Introduction

1.1 Overview

In the analysis of Linear Time-Invariant (LTI) systems, signals are often complex. To simplify analysis, it is beneficial to decompose signals into basic building blocks. For LTI systems, complex exponentials ( $e^{st}$ ) are eigenfunctions, meaning the output of the system is a scaled version of the input.

Fourier Series provides the mathematical framework to represent any periodic signal as an infinite sum of harmonically related sinusoidal components (or complex exponentials).

1.2 Fundamental Definitions

Periodic Signal: A continuous-time 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 $T$ for which the equation holds.
Fundamental Frequency:
- $\omega_0 = \frac{2\pi}{T_0}$ (radians per second)
- $f_0 = \frac{1}{T_0}$ (Hertz)

2. Fourier Series Representation of Continuous Time Periodic Signals

There are three primary forms of the Fourier Series representation.

2.1 Trigonometric Fourier Series

Any periodic signal $x(t)$ satisfying the Dirichlet conditions can be expanded as:

x(t) = a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t) \right]

Evaluation of Coefficients:

DC Component ( $a_0$ ): Represents the average value of the signal over one period.
$a_0 = \frac{1}{T_0} \int_{T_0} x(t) \, dt$
Cosine Coefficients ( $a_n$ ):
$a_n = \frac{2}{T_0} \int_{T_0} x(t) \cos(n\omega_0 t) \, dt$
Sine Coefficients ( $b_n$ ):
$b_n = \frac{2}{T_0} \int_{T_0} x(t) \sin(n\omega_0 t) \, dt$

Note: $\int_{T_0}$ denotes integration over any interval of length $T_0$ (e.g., $-T_0/2$ to $T_0/2$ or $0$ to $T_0$ ).

2.2 Compact Trigonometric (Polar) Form

By combining sine and cosine terms, the series can be expressed using only cosines with phase angles:

x(t) = A_0 + \sum_{n=1}^{\infty} A_n \cos(n\omega_0 t + \theta_n)

Relationships:

$A_0 = a_0$
Amplitude: $A_n = \sqrt{a_n^2 + b_n^2}$
Phase: $\theta_n = -\tan^{-1}\left(\frac{b_n}{a_n}\right)$

2.3 Exponential Fourier Series (Most Common)

Using Euler’s identity ( $e^{j\theta} = \cos\theta + j\sin\theta$ ), the series becomes more compact mathematically. This is the standard form used in Signal Processing.

Synthesis Equation (Reconstruction):

x(t) = \sum_{k=-\infty}^{\infty} C_k e^{j k \omega_0 t}

Analysis Equation (Finding Coefficients):

C_k = \frac{1}{T_0} \int_{T_0} x(t) e^{-j k \omega_0 t} \, dt

Relationship to Trigonometric Coefficients:

$C_0 = a_0$
$C_k = \frac{1}{2}(a_k - j b_k)$
$C_{-k} = \frac{1}{2}(a_k + j b_k)$
$|C_k| = \frac{1}{2}\sqrt{a_k^2 + b_k^2}$ (Magnitude Spectrum)

3. Convergence of the Fourier Series

The Fourier Series approximation does not converge to the original signal $x(t)$ for every signal. The signal must behave "reasonably." The sufficient conditions for convergence are known as the Dirichlet Conditions.

3.1 The Dirichlet Conditions

A periodic signal $x(t)$ has a convergent Fourier series representation if:

Absolutely Integrable: Over any period $T_0$ , the signal must be absolutely integrable.
$\int_{T_0} |x(t)| \, dt < \infty$
Example of failure: $x(t) = 1/t$ over $0 < t \leq 1$ .
Finite Variations: In any finite time interval, $x(t)$ must have a finite number of maxima and minima.
Example of failure: $x(t) = \sin(1/t)$ near $t=0$ .
Finite Discontinuities: In any finite time interval, $x(t)$ must have a finite number of discontinuities, and each discontinuity must be finite in amplitude.

3.2 Convergence at Discontinuities

If $x(t)$ has a discontinuity at $t = t_0$ , the Fourier Series converges to the average of the limits from the left and the right:

x_{FS}(t_0) = \frac{x(t_0^+) + x(t_0^-)}{2}

3.3 Gibbs Phenomenon

When approximating a signal with discontinuities (like a Square Wave) using a finite number of terms ( $N$ ), oscillations occur near the discontinuity. As $N \to \infty$ :

The ripples become narrower.
The amplitude of the overshoot does not decrease; it remains approximately 9% larger than the jump size.
This is known as the Gibbs Phenomenon.

4. Properties of Continuous Time Fourier Series

Let $x(t)$ and $y(t)$ be periodic signals with period $T_0$ and fundamental frequency $\omega_0$ . Let their Fourier Series coefficients be $C_k$ and $D_k$ respectively.

Property	Time Domain Signal	Fourier Series Coefficients (Frequency Domain)
Linearity	$A x(t) + B y(t)$	$A C_k + B D_k$
Time Shifting	$x(t - t_0)$	$e^{-j k \omega_0 t_0} C_k$
Frequency Shifting	$e^{j M \omega_0 t} x(t)$	$C_{k-M}$
Time Reversal	$x(-t)$	$C_{-k}$
Time Scaling	$x(\alpha t)$	Coefficients remain $C_k$ , but fundamental freq becomes $\alpha \omega_0$ .
Multiplication	$x(t)y(t)$	$\sum_{l=-\infty}^{\infty} C_l D_{k-l}$ (Discrete Convolution)
Conjugation	$x^*(t)$	$C_{-k}^*$
Differentiation	$\frac{dx(t)}{dt}$	$(jk\omega_0) C_k$
Integration	$\int_{-\infty}^{t} x(\tau) d\tau$	$\frac{1}{jk\omega_0} C_k$ (if $C_0=0$ )

4.1 Parseval’s Theorem (Power Relation)

The average power of a periodic signal is the sum of the average powers of its harmonic components.

\frac{1}{T_0} \int_{T_0} |x(t)|^2 \, dt = \sum_{k=-\infty}^{\infty} |C_k|^2

4.2 Symmetry Properties

Even Symmetry: If $x(t) = x(-t)$ , the coefficients $C_k$ are purely real and even ( $C_k = C_{-k}$ ). In Trig form, $b_n = 0$ .
Odd Symmetry: If $x(t) = -x(-t)$ , the coefficients $C_k$ are purely imaginary and odd ( $C_k = -C_{-k}$ ). In Trig form, $a_n = 0$ .
Half-Wave Symmetry: If $x(t) = -x(t \pm T_0/2)$ , the series contains only odd harmonics ( $C_k = 0$ for even $k$ ).

5. Software Simulation of Frequency Spectrum of Periodic Signals

The frequency spectrum visualizes the content of the signal. It consists of:

Magnitude Spectrum: Plot of $|C_k|$ vs. $k\omega_0$ .
Phase Spectrum: Plot of $\angle C_k$ vs. $k\omega_0$ .

5.1 Python Simulation Example (Square Wave)

Below is a Python script using numpy and matplotlib to generate the spectrum of a Square Wave.

Theoretical Expectation:
For a Square wave of amplitude 1 defined from $-\pi$ to $\pi$ :

$C_k = \frac{\sin(k\pi/2)}{k\pi/2}$ for odd $k$ .
$C_k = 0$ for even $k$ (except potentially DC depending on offset).

PYTHON

import numpy as np
import matplotlib.pyplot as plt

def square_wave_spectrum(n_harmonics):
    """
    Simulates the Line Spectrum of a Square Wave
    x(t) = 1 for 0 < t < T/2, -1 for T/2 < t < T
    """
    
    # Harmonics indices (k)
    # We include negative and positive frequencies for exponential series
    k = np.arange(-n_harmonics, n_harmonics + 1)
    
    # Initialize coefficients array
    Ck = np.zeros(len(k), dtype=complex)
    
    # Calculate Coefficients for Square Wave
    # Formula: C_k = -2j / (k*pi) for odd k, 0 for even k, 0 for k=0 (if zero mean)
    for i, val in enumerate(k):
        if val == 0:
            Ck[i] = 0 # DC component is 0 for symmetric square wave
        elif val % 2 != 0: # Odd harmonics only
            Ck[i] = -2j / (np.pi * val)
        else:
            Ck[i] = 0
            
    # Magnitude and Phase
    magnitude = np.abs(Ck)
    phase = np.angle(Ck)
    
    # Plotting
    plt.figure(figsize=(12, 6))
    
    # Magnitude Spectrum
    plt.subplot(1, 2, 1)
    plt.stem(k, magnitude, basefmt=" ")
    plt.title('Magnitude Spectrum |C_k|')
    plt.xlabel('Harmonic index k')
    plt.ylabel('Amplitude')
    plt.grid(True)
    
    # Phase Spectrum
    plt.subplot(1, 2, 2)
    plt.stem(k, phase, basefmt=" ")
    plt.title('Phase Spectrum < C_k')
    plt.xlabel('Harmonic index k')
    plt.ylabel('Phase (radians)')
    plt.grid(True)
    
    plt.tight_layout()
    plt.show()

# Run simulation for 10 harmonics
square_wave_spectrum(10)

5.2 Observations from Simulation

Discrete Nature: The spectrum of a periodic signal is a Line Spectrum (discrete spikes), not a continuous curve.
Spacing: The lines are spaced by $\omega_0$ . As the period $T$ increases, $\omega_0$ decreases, and the spectral lines get closer together.
Decay: The magnitude of coefficients decreases as $k$ increases ( $|C_k| \propto 1/k$ for square wave). This confirms that high-frequency components contribute less to the signal's shape but are crucial for sharp transitions (edges).