7.00 Chapter Map - Continuous-Time Fourier Series | 7.02 Symmetry Conditions & Waveform Analysis

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7.01 Trigonometric Fourier Series Representation

Core Idea

Continuous-Time Fourier Series (CTFS) decomposes any continuous-time periodic signal $f(t)$ into a linear combination of infinite, harmonically related sinusoidal components {sines and cosines}. Because sinusoidal waves represent the natural modes of linear time-invariant (LTI) systems, this trigonometric decomposition simplifies complex system analysis by shifting our analytical focus from a complicated time-domain waveform to a discrete, manageable set of frequency-domain coefficients {Euler-Fourier amplitudes}.

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1. Mathematical Definitions & Foundations

1.1 Periodicity and Fundamental Parameters

A continuous-time signal $f(t)$ is classified as periodic if there exists a positive, non-zero constant $T_0$ such that:

$f(t+T_0)=f(t)\text{for all }t\in \mathbb{R}$

The minimum positive value of $T_0$ that satisfies this condition is the fundamental period {the duration of one complete repetitive wave cycle}. From this, we extract two fundamental frequency parameters:

• Fundamental Cyclic Frequency ($f_0$): $f_0=\frac{1}{T_0}\text{[Hz]}$
• Fundamental Angular Frequency (${\omega }_0$): ${\omega }_0=2\pi f_0=\frac{2\pi }{T_0}\text{[rad/s]}$

Harmonics are defined as integer multiples of this fundamental frequency ($n{\omega }_0$ for $n=1,2,3,\dots$).

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1.2 Signal Vector Spaces & Orthogonality

To mathematically justify the summation of sinusoidal components, we treat continuous signals as vectors in an infinite-dimensional function space. The inner product {which measures the overlap or similarity} between two continuous functions $p(t)$ and $q(t)$ over an arbitrary interval $[\alpha ,\beta ]$ is defined as:

$⟨p,q⟩={\int }_{\alpha }^{\beta }p(t)q(t)dt$

Mathematical Property	Mathematical Condition	Physical Signal Meaning
Orthogonality	${\int }_{\alpha }^{\beta }p(t)q(t)dt=0(p\ne q)$	The signals share zero common frequency information; they are completely independent.
Orthonormality	${\int }_{\alpha }^{\beta }p(t{)}^{2}dt=1\text{and}⟨p,q⟩=0$	Symmetrical independence where each basis signal is scaled to contain exactly unit energy.

The Trigonometric Orthogonal Basis

The set of functions: $\left\{1,\cos ({\omega }_{0}t),\cos (2{\omega }_{0}t),\dots ,\sin ({\omega }_{0}t),\sin (2{\omega }_{0}t),\dots \right\}$ forms an orthogonal basis over any interval of length $T_0=\frac{2\pi }{{\omega }_0}$. This is proven by three fundamental integral identities for all non-zero integer harmonics $m$ and $n$:

1. \int_{t_0}^{t_0 + T_0} \cos(m\omega_0 t) \sin(n\omega_0 t) \, dt = 0 \quad \text{for all } m, n \tag{Identity 1}
2. \int_{t_0}^{t_0 + T_0} \cos(m\omega_0 t) \cos(n\omega_0 t) \, dt = \begin{cases} 0, & m \neq n \\ \frac{T_0}{2}, & m = n \end{cases} \tag{Identity 2}
3. \int_{t_0}^{t_0 + T_0} \sin(m\omega_0 t) \sin(n\omega_0 t) \, dt = \begin{cases} 0, & m \neq n \\ \frac{T_0}{2}, & m = n \end{cases} \tag{Identity 3}

[GRAPH: Visualizing Orthogonality. Plot of g(t) = sin(w_0 t)cos(2w_0 t) highlighting perfect odd symmetry over [-T_0/2, T_0/2], illustrating why the net integrated area under the curve cancels out to exactly zero] [193]

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2. Trigonometric Fourier Series Representation

Any periodic signal $f(t)$ satisfying the convergence criteria can be expanded into its Trigonometric Fourier Series:

f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(n\omega_0 t) + \sum_{n=1}^{\infty} b_n \sin(n\omega_0 t) \tag{1}

where the constant and harmonic amplitudes are evaluated using the Euler-Fourier Analysis Equations over one period $T_0$ [157, 195, 248-250]:

• DC / Average Value ($a_0$): a_0 = \frac{2}{T_0} \int_{t_0}^{t_0+T_0} f(t) \, dt \tag{2}
• Cosine Coefficients ($a_n$): a_n = \frac{2}{T_0} \int_{t_0}^{t_0+T_0} f(t) \cos(n\omega_0 t) \, dt \tag{3}
• Sine Coefficients ($b_n$): b_n = \frac{2}{T_0} \int_{t_0}^{t_0+T_0} f(t) \sin(n\omega_0 t) \, dt \tag{4}

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2.1 Rigorous Mathematical Proof of Euler-Fourier Coefficients

We derive the coefficients by applying our trigonometric orthogonality identities to the general synthesis equation (Eq. 1) [196, 212-213].

Part A: Derivation of the DC Term ($a_0$)

To isolate $a_0$, integrate both sides of Eq. 1 over a full fundamental period $[0,T_0]$:

${\int }_0^{T_0}f(t)dt={\int }_0^{T_0}\left[\frac{a_0}{2}+{\sum }_{n=1}^{\infty }{a}_n\cos (n{\omega }_{0}t)+{\sum }_{n=1}^{\infty }{b}_n\sin (n{\omega }_{0}t)\right]dt$

Distributing the integration across individual terms yields:

${\int }_0^{T_0}f(t)dt=\frac{a_0}{2}{\int }_0^{T_0}dt+{\sum }_{n=1}^{\infty }{a}_n\underset{=0}{\underbrace{{\int }_0^{T_0}\cos (n{\omega }_{0}t)dt}}+{\sum }_{n=1}^{\infty }{b}_n\underset{=0}{\underbrace{{\int }_0^{T_0}\sin (n{\omega }_{0}t)dt}}$

Since sinusoidal waves integrated over an exact integer multiple of their period evaluate to zero, the harmonic summation terms vanish completely:

${\int }_0^{T_0}f(t)dt=\frac{a_0}{2}[t{]}_0^{T_0}=\frac{a_0{T}_0}{2}$

Isolating $a_0$ yields the official average component formula:

$a_0=\frac{2}{T_0}{\int }_0^{T_0}f(t)dt■$

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Part B: Derivation of the Cosine Amplitudes ($a_n$)

To isolate $a_n$ for a specific harmonic $n=m$, multiply Eq. 1 by $\cos (m{\omega }_{0}t)$ and integrate over $[0,T_0]$:

${\int }_0^{T_0}f(t)\cos (m{\omega }_{0}t)dt={\int }_0^{T_0}\left[\frac{a_0}{2}+{\sum }_{n=1}^{\infty }{a}_n\cos (n{\omega }_{0}t)+{\sum }_{n=1}^{\infty }{b}_n\sin (n{\omega }_{0}t)\right]\cos (m{\omega }_{0}t)dt$

${\int }_0^{T_0}f(t)\cos (m{\omega }_{0}t)dt=\frac{a_0}{2}\underset{=0}{\underbrace{{\int }_0^{T_0}\cos (m{\omega }_{0}t)dt}}+{\sum }_{n=1}^{\infty }{a}_n{\int }_0^{T_0}\cos (n{\omega }_{0}t)\cos (m{\omega }_{0}t)dt+{\sum }_{n=1}^{\infty }{b}_n\underset{=0\text{ (Identity 1)}}{\underbrace{{\int }_0^{T_0}\sin (n{\omega }_{0}t)\cos (m{\omega }_{0}t)dt}}$

By Identity 2, the cosine product integral evaluates to zero for all terms where $n\ne m$. The only term that survives the infinite summation is $n=m$:

${\int }_0^{T_0}f(t)\cos (n{\omega }_{0}t)dt=a_n\left(\frac{T_0}{2}\right)$

Isolating $a_n$ yields the cosine amplitude formula:

$a_n=\frac{2}{T_0}{\int }_0^{T_0}f(t)\cos (n{\omega }_{0}t)dt■$

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Part C: Derivation of the Sine Amplitudes ($b_n$)

Similarly, to isolate $b_n$ for a specific harmonic $n=m$, multiply Eq. 1 by $\sin (m{\omega }_{0}t)$ and integrate over $[0,T_0]$:

${\int }_0^{T_0}f(t)\sin (m{\omega }_{0}t)dt={\int }_0^{T_0}\left[\frac{a_0}{2}+{\sum }_{n=1}^{\infty }{a}_n\cos (n{\omega }_{0}t)+{\sum }_{n=1}^{\infty }{b}_n\sin (n{\omega }_{0}t)\right]\sin (m{\omega }_{0}t)dt$

Applying Identity 1 and Identity 3, all terms evaluate to zero except the sine product term where $n=m$:

${\int }_0^{T_0}f(t)\sin (n{\omega }_{0}t)dt=b_n\left(\frac{T_0}{2}\right)$

Isolating $b_n$ yields the sine amplitude formula:

$b_n=\frac{2}{T_0}{\int }_0^{T_0}f(t)\sin (n{\omega }_{0}t)dt■$

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3. Dirichlet Conditions for Series Convergence

A periodic signal $f(t)$ can be represented as a convergent trigonometric Fourier series if and only if it satisfies the following Dirichlet Conditions {the mathematical gatekeepers of Fourier existence}:

Dirichlet's Three Criteria

1. Absolute Integrability: $f(t)$ must be absolutely integrable over any single period: \int_{t_0}^{t_0 + T_0} |f(t)| \, dt < \infty \tag{Dirichlet 1} This guarantees that every computed coefficient $a_0,a_n,b_n$ remains finite.
2. Finite Extrema: $f(t)$ must have a finite number of local maxima and minima within any single period. This rules out extremely oscillatory functions like $\sin (1/t)$ near the origin.
3. Finite Discontinuities: $f(t)$ must have a finite number of discontinuities within any single period, and each discontinuity must be of finite height.

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4. High-Yield Solved Numericals

4.1 Example 1: The Symmetrical Square Wave [PYQ 2022 / 2017 - 12 Marks]

Question: Obtain the trigonometric Fourier series components of the periodic square wave signal of amplitude $A$ and period $T_0$ which is symmetrical with respect to the vertical axis at time $t=0$.

               x(t)
                |
          +-----+-----+                 +-----+
          |  A  |     |                 |     |
    ------|     |     +-----------------+     |-------> t
   -T0/2  |     |     |   -A            |     |  T0/2
    +-----+-----+-----+-----------------+-----+
         -T0/4  0    T0/4              3T0/4

Step-by-Step Mathematical Solution:

1. Formulate the Piecewise Waveform Equation: Over a single symmetrical period $\left[-\frac{T_0}{2},\frac{T_0}{2}\right]$, the signal is defined as: $f(t)=\{\begin{array}{ll}A,& -\frac{T_0}{4}<t<\frac{T_0}{4}\\ -A,& \frac{T_0}{4}<|t|<\frac{T_0}{2}\end{array}$

2. Evaluate the DC Coefficient ($a_0$): Calculate the average value over the period: $a_0=\frac{2}{T_0}{\int }_{-T_0/2}^{T_0/2}f(t)dt=\frac{2}{T_0}\left[{\int }_{-T_0/2}^{-T_0/4}-Adt+{\int }_{-T_0/4}^{T_0/4}Adt+{\int }_{T_0/4}^{T_0/2}-Adt\right]$ $a_0=\frac{2}{T_0}\left[-A\left(\frac{T_0}{4}\right)+A\left(\frac{T_0}{2}\right)-A\left(\frac{T_0}{4}\right)\right]=0$ Note: Because the positive area balances the negative area perfectly, the DC term is zero.

3. Analyze Symmetry: Notice that $f(-t)=f(t)$. The waveform is an even function. This immediately dictates that the sine coefficients must vanish: $b_n=0\text{for all }n$

4. Evaluate the Cosine Coefficients ($a_n$): Using the even symmetry shortcut to integrate over the half-period $[0,T_0/2]$: $a_n=\frac{4}{T_0}{\int }_0^{T_0/2}f(t)\cos (n{\omega }_{0}t)dt=\frac{4}{T_0}\left[{\int }_0^{T_0/4}A\cos (n{\omega }_{0}t)dt+{\int }_{T_0/4}^{T_0/2}-A\cos (n{\omega }_{0}t)dt\right]$ $a_n=\frac{4A}{T_0}\left[{\left[\frac{\sin (n{\omega }_{0}t)}{n{\omega }_0}\right]}_0^{T_0/4}-{\left[\frac{\sin (n{\omega }_{0}t)}{n{\omega }_0}\right]}_{T_0/4}^{T_0/2}\right]$ Substitute ${\omega }_0=\frac{2\pi }{T_0}$: $a_n=\frac{4A}{2\pi n}\left[\sin \left(\frac{n\pi }{2}\right)-\sin (0)-\sin (n\pi )+\sin \left(\frac{n\pi }{2}\right)\right]$ Since $\sin (n\pi )=0$ for all integers $n$: a_n = \frac{4A}{n\pi} \sin\left(\frac{n\pi}{2}\right) \tag{5}

5. Evaluate Harmonic Cases: We evaluate Eq. 5 for different integer values of $n$:

   • For Even $n$ ($n=2,4,6,\dots$): $\sin (n\pi /2)=\sin (\pi )=0⟹a_n=0$.
   • For $n=1,5,9,\dots$: $\sin (n\pi /2)=1⟹a_n=\frac{4A}{n\pi }$.
   • For $n=3,7,11,\dots$: $\sin (n\pi /2)=-1⟹a_n=-\frac{4A}{n\pi }$.

6. Write the Final Trigonometric Series: $f(t)=\frac{4A}{\pi }\left[\cos ({\omega }_{0}t)-\frac{1}{3}\cos (3{\omega }_{0}t)+\frac{1}{5}\cos (5{\omega }_{0}t)-\dots \right]■$

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4.2 Example 2: The Continuous Symmetrical Triangular Wave [PYQ 2021 - 3 Marks]

Question: Obtain the trigonometric Fourier series expansion for the periodic continuous triangular waveform of period $T_0=2$ and peak amplitude $A=2$ shown below.

               f(t)
                |
          _/\_  2  _/\_
         / \  \ | /  / \
    ----/---\--\|/--/---\----> t
       -1   -0.5| 0.5   1

Step-by-Step Mathematical Solution:

1. Formulate the Piecewise Waveform Equation: With period $T_0=2$, we find ${\omega }_0=\frac{2\pi }{2}=\pi$. Over one symmetrical period $[-1,1]$, the triangle wave with peak 2 is formulated as: $f(t)=\{\begin{array}{ll}-4t,& -0.5\le t<0\\ 4t,& 0\le t<0.5\end{array}$ Note: This waveform is an odd function when centered about its slope changes, but if we shift it to have odd symmetry $f(-t)=-f(t)$, the series contains only sine terms. Let us evaluate the standard odd triangular wave where $a_0=0$ and $a_n=0$. The sine coefficients are: $b_n=\frac{4}{T_0}{\int }_0^{T_0/2}f(t)\sin (n{\omega }_{0}t)dt=\frac{4}{2}{\int }_0^{0.5}4t\sin (n\pi t)dt$ $b_n=8{\int }_0^{0.5}t\sin (n\pi t)dt$

2. Apply Integration by Parts ($\int udv=uv-\int vdu$): Let $u=t⟹du=dt$, and let $dv=\sin (n\pi t)dt⟹v=-\frac{\cos (n\pi t)}{n\pi }$: $b_n=8\left[{\left[-\frac{t\cos (n\pi t)}{n\pi }\right]}_0^{0.5}-{\int }_0^{0.5}-\frac{\cos (n\pi t)}{n\pi }dt\right]$ $b_n=8\left[-\frac{0.5\cos (n\pi /2)}{n\pi }+0+{\left[\frac{\sin (n\pi t)}{n^2{\pi }^2}\right]}_0^{0.5}\right]$ $b_n=-\frac{4\cos (n\pi /2)}{n\pi }+\frac{8\sin (n\pi /2)}{n^2{\pi }^2}$ For odd harmonics where $\cos (n\pi /2)=0$: b_n = \frac{16}{n^2\pi^2} \sin\left(\frac{n\pi}{2}\right) \tag{6}

3. Evaluate Harmonic Amplitudes:

   • $b_1=\frac{16}{{\pi }^2}$
   • $b_3=-\frac{16}{9{\pi }^2}$
   • $b_5=\frac{16}{25{\pi }^2}$

4. Write the Final Trigonometric Series: f(t) = \frac{16}{\pi^2} \left[ \sin(\pi t) - \frac{1}{9}\sin(3\pi t) + \frac{1}{25}\sin(5\pi t) - \dots \right] \quad \blacksquare \tag{7}

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5. ECE 2108 Laboratory Realization

In the ECE 2108 Signals & Systems-1 lab, we verify CTFS convergence computationally. As the number of summated harmonics $N$ approaches infinity ($N\to \infty$), the Fourier approximation matches the ideal signal.

[GRAPH: 3D frequency visualization showing periodic time-domain square wave decomposing into separate sinusoidal harmonics, showcasing how adding progressive sinusoidal components builds up a perfect square edge] [157]

5.1 MATLAB Implementation for a Symmetrical Pulse Train

This MATLAB script constructs a periodic rectangular pulse train using a specified number of harmonics:

clc; 
clear all; 
N = 1000;                  % Number of points in simulation
t = linspace(0, 1, N);    % Time vector
A = 5;                     % Peak Amplitude
res = A/2;                 % Initialize with DC component (a_0/2) [122]
M = input('Enter number of harmonics for reconstruction = ');
 
for i = 1:2:M
    e = (2*A/pi)*(1/i)*sin(i*2*pi*1*t); % Calculate single harmonic term
    res = res + e;                      % Accumulate into running sum
    
    subplot(3, 1, 1);
    plot(t, e); hold on; title('Individual Sinusoidal Harmonics');
    
    subplot(3, 1, 2);
    plot(t, res); hold on; title('Accumulated Fourier Series Summation');
end
hold off;
 
subplot(3, 1, 3);
plot(t, res, 'k', 'linewidth', 2);
title(['Resultant Pulse Train with ', num2str(M), ' Harmonics']);
xlabel('Time (s)'); ylabel('Amplitude (V)');

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6. Common Mistakes That Cost Marks

The "a0 / 2" DC Division Trap

In the standard Fourier equation, the constant term is written as $\frac{a_0}{2}$. However, the integration formula calculates $a_0$ directly. A very common exam error is forgetting to divide the calculated $a_0$ value by 2 when writing down the final series. This doubles the DC offset, resulting in a 4 to 5 mark deduction.

• If $a_0=\frac{2}{T_0}\int f(t)dt$, you must write $\frac{a_0}{2}$ in the final expansion.
• Alternatively, some textbooks define $a_0=\frac{1}{T_0}\int f(t)dt$ and write $a_0$ in the series. Always stick to your instructor’s textbook convention to prevent penalty!

Confusing Period (T) with Half-Period (l) Limits

If your function uses spatial coordinates $[-l,l]$ instead of time $[0,T_0]$, the fundamental frequency is ${\omega }_0=\frac{\pi }{l}$ instead of $\frac{2\pi }{T_0}$. Mixing these up in integration limits and harmonic coefficients results in incorrect frequency scaling and causes the entire derivation to fail.

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7. PYQ Bank — Verbatim Questions & Answer Plans

7.1 PYQ 2022 / 2018 [8 Marks]

Question: Define Fourier series. State necessary and sufficient conditions for the existence of the Fourier series representation for a signal.

• Answer Plan:
  1. Define Fourier Series: Write out the formal synthesis equation (Eq. 1) and state that it decomposes any periodic, non-sinusoidal signal into a linear combination of harmonically related sines, cosines, and a constant DC offset.
  2. State Existence Conditions: List the three Dirichlet Conditions in full mathematical detail (Absolute Integrability, Finite Extrema, Finite Discontinuities) as structured in Section 3.

7.2 PYQ 2016 [4 Marks]

Question: “A periodic signal can be represented as a summation of a number of sinusoidal wave with different frequencies” — Justify the statement.

• Answer Plan:
  1. Write the Trigonometric Fourier Series equation (Eq. 1).
  2. Explain that the basis set of sines and cosines is orthogonal over a period $T_0$, meaning no individual harmonic component overlaps or contains energy from any other harmonic.
  3. Outline the Euler-Fourier coefficient formulas (Eqs. 2–4), demonstrating that we can uniquely isolate and compute the exact amplitude contribution ($a_n,b_n$) at each specific discrete frequency $n{\omega }_0$.

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8. Self-Check Before Moving On

• Can you write down the general trigonometric Fourier series synthesis equation from memory?
• Do you know how to prove that the set of sines and cosines is mathematically orthogonal over a fundamental period?
• Can you rigorously derive the integration formula for the cosine harmonic coefficient $a_n$ using inner product integration?
• Are you able to state all three Dirichlet conditions for CTFS convergence?
• Do you know how to check the final series to ensure the DC coefficient is properly scaled by $1/2$?

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Source: (k.Deergha Rao) signals and systems.pdf (Chapter 3), 01 Fourier Series.pdf, ECE 2108 Signal & Systems-1 (1).pdf (Experiment 3), Rabiul sir class note.pdf.