Fourier Series and Transforms: Analyzing Signals in Time and Frequency

Table of Contents

1. Introduction
2. Motivation: Why Fourier Analysis?
3. Fourier Series: Periodic Function Decomposition
4. Orthogonality of Trigonometric Functions
5. Fourier Coefficients
6. Convergence of Fourier Series
7. Even and Odd Functions
8. Complex Exponential Form of Fourier Series
9. The Fourier Transform
10. Properties of the Fourier Transform
11. Inverse Fourier Transform
12. The Dirac Delta and Fourier Theory
13. Discrete Fourier Transform (DFT) and FFT
14. Applications in Physics and Engineering
15. Conclusion

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1. Introduction

Fourier series and transforms allow us to represent functions in terms of sinusoids and complex exponentials. This is not only mathematically elegant, but also practical for solving differential equations, analyzing signals, and understanding physical systems in terms of their frequency content.

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2. Motivation: Why Fourier Analysis?

Many physical systems exhibit oscillatory behavior — vibrations, waves, circuits, or signals.

Decomposing a function into a sum (or integral) of simpler, periodic functions helps us:

• Analyze behavior in the frequency domain
• Solve partial differential equations
• Understand resonance, filtering, and spectral content

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3. Fourier Series: Periodic Function Decomposition

Let \( f(x) \) be periodic with period \( 2\pi \). Then:

\[
f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)
\]

This expresses \( f(x) \) as a weighted sum of harmonics.

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4. Orthogonality of Trigonometric Functions

The key to Fourier decomposition is orthogonality:

\[
\int_{-\pi}^{\pi} \cos(nx)\cos(mx) dx =
\begin{cases}
0, & n \neq m \
\pi, & n = m \neq 0
\end{cases}
\]

\[
\int_{-\pi}^{\pi} \sin(nx)\cos(mx) dx = 0
\]

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5. Fourier Coefficients

The coefficients are computed as:

\[
a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx) dx, \quad
b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx) dx
\]

\[
a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx
\]

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6. Convergence of Fourier Series

Fourier series converges:

• Pointwise for piecewise continuous functions
• Uniformly if the function is continuous and has a continuous derivative

The Gibbs phenomenon refers to overshoots near discontinuities.

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7. Even and Odd Functions

• For even functions, \( b_n = 0 \)
• For odd functions, \( a_n = 0 \)

This symmetry simplifies the computation of Fourier series.

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8. Complex Exponential Form of Fourier Series

Using Euler’s formula:

\[
f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}
\]

Where:

\[
c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} dx
\]

This form is convenient in physics and electrical engineering.

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9. The Fourier Transform

For a non-periodic function \( f(t) \), the Fourier transform is:

\[
\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt
\]

It generalizes Fourier series to arbitrary functions.

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10. Properties of the Fourier Transform

• Linearity: \( \mathcal{F}\{af + bg\} = aF + bG \)
• Time shift: \( f(t – t_0) \leftrightarrow e^{-i\omega t_0} F(\omega) \)
• Frequency shift: \( e^{i\omega_0 t}f(t) \leftrightarrow F(\omega – \omega_0) \)
• Differentiation: \( \frac{d^n f}{dt^n} \leftrightarrow (i\omega)^n F(\omega) \)

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11. Inverse Fourier Transform

Recovers \( f(t) \) from \( F(\omega) \):

\[
f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega
\]

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12. The Dirac Delta and Fourier Theory

The Dirac delta function \( \delta(t) \) is defined as:

\[
\delta(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega t} d\omega
\]

Used to express impulse responses and fundamental identities in transform theory.

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13. Discrete Fourier Transform (DFT) and FFT

In computation, we use:

\[
X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i kn/N}
\]

• DFT converts discrete data from time to frequency
• FFT is an efficient algorithm to compute DFT in \( O(N \log N) \) time

Used in signal processing, communications, and image analysis.

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14. Applications in Physics and Engineering

• Signal processing: filters, compression, analysis
• Quantum mechanics: wavefunctions in momentum space
• Acoustics: harmonic content of sound
• Optics: diffraction and interference patterns
• Heat and wave equations: solution via separation and transform

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15. Conclusion

Fourier series and transforms bridge time and frequency domains. They enable a deep understanding of oscillatory phenomena and play a vital role in solving differential equations, analyzing systems, and processing data.

In physics, they reveal symmetry, periodicity, and conserved quantities across a wide spectrum of domains.

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