PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Analysis → Harmonic Analysis ↓


Fourier Transforms


Introduction

The Fourier transform is a powerful mathematical tool used to analyze the frequencies present in a signal or function. Its importance spans across various fields such as engineering, physics, and mathematics. Using the Fourier transform, we can transform a function from its original domain (often time) to the frequency domain, revealing the various frequencies that make up the original signal.

Historical context

The concept of the Fourier transform originated from the work of Jean-Baptiste Joseph Fourier, a French mathematician and physicist. In the early 19th century, Fourier introduced the idea that a function could be represented or approximated by a sum of simpler trigonometric functions. His work laid the foundation for what we now call Fourier series in the study of periodic functions, and by extension, Fourier transforms for more general cases.

Mathematical formulation

The Fourier transform of a continuous function f(t), where t denotes time, is given by the integral:

f(ω) = ∫[−∞, ∞] f(t) e^(−iωt) dt

Here, ω (omega) is the angular frequency and i is the imaginary unit. The exponential term e^(−iωt) is a compact representation of the oscillations, which combines both sine and cosine functions due to Euler's formula e^(ix) = cos(x) + i sin(x).

Inverse Fourier transform

To return from the frequency domain to the time domain, we use the inverse Fourier transform:

f(t) = (1/2π) ∫[−∞, ∞] F(ω) e^(iωt) dω

This complementary pair of transformations allows conversion between time and frequency representations, and provides insight into the nature of the original function.

Examples and visualizations

Example 1: A simple sine wave

Consider a sine wave given by f(t) = sin(2πft), where f is the frequency of the sine wave. The Fourier transform of this sine wave results in a delta function at frequencies f and -f, indicating the presence of these specific frequency components in the function.

frequency domain -F F

This view shows two spikes at f and -f, which represent the oscillatory components of the sine wave.

Example 2: Mixed signals

Real-world signals are usually more complex, involving multiple frequencies. Consider a signal that adds two sine waves: f(t) = sin(2πf₁t) + 0.5sin(2πf₂t).

frequency domain -f₂ -F₁ F₁ F₂

In this case, the Fourier transform shows four spikes, indicating the presence of two positive and two negative frequency components. The heights of the spikes are proportional to the amplitudes of the corresponding sine waves.

Properties of Fourier transform

The Fourier transform has several important properties that make it a versatile tool:

  • Linearity: The transform of a sum of functions is the sum of their transforms.
  • Time shifting: If a function is shifted in time, its Fourier transform is multiplied by a phase factor.
  • Frequency shifting: Frequency shifting produces modulation in the time domain.
  • Scaling: Compressing a function in the time domain expands it in the frequency domain and vice versa.

Applications of Fourier transform

The Fourier transform is widely used in a variety of applications, including:

  • Signal processing: Analyzing the frequencies of audio, speech, and image signals.
  • Quantum mechanics: The study of wave functions in the context of position and momentum space.
  • Control systems: Designing systems that respond optimally to various inputs.
  • Communication systems: Modulation and demodulating of signals in transmission systems.

Conclusion

The Fourier transform provides a comprehensive view of the frequency components that make up a signal. It helps analyze and interpret functions in a different domain, which are not easily observable in their original domain. Mastery of the Fourier transform and its properties is essential for mathematicians, engineers, and scientists working with complex systems and signals. Through rigorous study and practice, the Fourier transform becomes a powerful lens for detecting the intricate dance of oscillations hidden in seemingly simple functions.

Delving deeper into Fourier transforms gives a chance to explore complex concepts like Fourier series, Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT), each of which expands the horizons of analysis and computation. A journey into the world of Fourier analysis, which has its roots in harmonic observations, reveals the delicate balance between time and frequency, providing solutions and answers to both classical and modern challenges in mathematics and science.


PHD → 2.5.2


U
username
0%
completed in PHD

← Prev (2.5.1)
Fourier Series
Next (2.5.3) →
Distribution Theory

Comments 



Fourier Transforms

https://www.buddymath.com/phd/2.5.2?bmal=en