A Brief History of the Fourier Series and Fourier Transform in image processing

1. Sine function (sin x):

   $$\sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots, \quad -\infty < x < \infty$$

2. Cosine function (cos x):

   $$\cos x = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots, \quad -\infty < x < \infty$$

3. Tangent function (tan x):

   $$\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \frac{62x^9}{2835} + \cdots, \quad |x| < \frac{\pi}{2}$$

4. Secant function (sec x):

   $$\sec x = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots, \quad |x| < \frac{\pi}{2}$$

5. Cosecant function (csc x):

   $$\csc x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \frac{31x^5}{15120} + \cdots, \quad 0 < |x| < \pi$$

6. Cotangent function (cot x):

   $$\cot x = \frac{1}{x} – \frac{x}{3} – \frac{x^3}{45} – \frac{2x^5}{945} + \cdots, \quad 0 < |x| < \pi$$

The Trigonometric Series is a mathematical series that expresses periodic functions as sums of sines and cosines. It’s fundamental in Fourier analysis, which studies how functions can be represented as sums of simpler trigonometric functions. Below is an overview of the key concepts:

1. Periodic Functions

• A function $f(x)$ is called periodic if there exists a positive number $T$ such that $f(x + T) = f(x)$ for all $x$.
• The smallest such $T$ is called the fundamental period.

2. Trigonometric Functions

• The basic trigonometric functions are sine ($\sin$) and cosine ($\cos$).
• These functions have a period of $2\pi$, meaning $\sin(x + 2\pi) = \sin(x)$ and $\cos(x + 2\pi) = \cos(x)$.

3. Fourier Series

• Any periodic function $f(x)$ with period $2\pi$ can be expressed as a sum of sines and cosines in the form of a Fourier series:

• Here, $a_0$, $a_n$, and $b_n$ are the Fourier coefficients, which are determined by the function itself:

• These coefficients determine the amplitude and phase of the corresponding sine and cosine components.

4. Convergence of the Fourier Series

• If $f(x)$ is piecewise continuous and has a finite number of discontinuities, its Fourier series converges to $f(x)$ at points where $f(x)$ is continuous.
• At discontinuities, the Fourier series converges to the average of the left-hand and right-hand limits of the function.

5. Even and Odd Functions

• An even function satisfies $f(-x) = f(x)$ and has a Fourier series consisting only of cosine terms (since cosine is even).
• An odd function satisfies $f(-x) = -f(x)$ and has a Fourier series consisting only of sine terms (since sine is odd).

6. Applications of the Trigonometric Series

• Signal Processing: Decomposing signals into frequency components.
• Heat Equation: Solving partial differential equations in physics.
• Vibrations and Waves: Analyzing mechanical vibrations and waveforms.

Example: Fourier Series for a Simple Function

For a square wave function defined by:

The Fourier series would be:

This expansion is the trigonometric series representation of the square wave, illustrating how a periodic function can be expressed as a sum of simple trigonometric terms.

7. Complex Form of Fourier Series

• Fourier series can also be represented in a complex form using Euler’s formula $e^{ix} = \cos(x) + i\sin(x)$:

• Here, $c_n$ are complex Fourier coefficients.

The study of trigonometric series bridges various branches of mathematics and physics, providing tools for analyzing functions in terms of their frequency components.

The determination of Fourier coefficients is a critical step in expressing a periodic function as a Fourier series. These coefficients, which include $a_0$, $a_n$, and $b_n$, capture the amplitude of the respective sine and cosine components in the series. Here’s a detailed explanation of the advanced mathematical concepts involved in determining these coefficients.

1. Fourier Series Overview

A periodic function $f(x)$ with period $T$ can be represented as a Fourier series:

where:

• $a_0$ is the average (mean) value of the function over one period.
• $a_n$ and $b_n$ represent the amplitudes of the cosine and sine components, respectively.

2. Orthogonality of Sine and Cosine Functions

The key to determining the Fourier coefficients lies in the orthogonality properties of the sine and cosine functions:

These orthogonality conditions allow us to isolate each Fourier coefficient by multiplying the Fourier series by either $\cos\left(\frac{2\pi nx}{T}\right)$ or $\sin\left(\frac{2\pi nx}{T}\right)$ and integrating over one period.

3. Derivation of Fourier Coefficients

a) Coefficient $a_0$

• $a_0$ represents the constant (DC) component of the function, or the average value of the function over one period.
• To derive $a_0$, integrate $f(x)$ over one period:

This formula is derived by recognizing that integrating the entire Fourier series over one period, only the constant term contributes since the integrals of the sine and cosine terms over one period are zero due to their orthogonality.

b) Coefficients $a_n$ and $b_n$

• $a_n$ is the coefficient of the cosine term and $b_n$ is the coefficient of the sine term.
• To derive $a_n$, multiply both sides of the Fourier series by $\cos\left(\frac{2\pi nx}{T}\right)$ and integrate over one period:

• Similarly, to derive $b_n$, multiply both sides of the Fourier series by $\sin\left(\frac{2\pi nx}{T}\right)$ and integrate over one period:

These integrals project the function onto the respective cosine and sine functions, effectively isolating the corresponding Fourier coefficients.

4. Complex Fourier Series and Coefficients

• The Fourier series can also be expressed in its complex form:

where $c_n$ are complex Fourier coefficients given by:

• This complex form simplifies the analysis and derivation process by unifying the sine and cosine terms into a single exponential function using Euler’s formula:

• The relationship between the complex coefficients and the real coefficients is:

This form is especially useful in signal processing and systems analysis, where complex numbers simplify the mathematics of handling oscillatory behavior.

5. Parseval’s Theorem

• Parseval’s theorem provides a relationship between the Fourier coefficients and the total energy of the function. For a periodic function $f(x)$ with period $T$:

• In the complex form:

• Parseval’s theorem confirms that the sum of the squares of the Fourier coefficients (which represent the power in each frequency component) equals the total power of the function, providing a bridge between the time domain and the frequency domain.

6. Symmetry Properties and Simplification

• Even and Odd Functions:
  • If $f(x)$ is an even function ($f(-x) = f(x)$), all $b_n$ coefficients will be zero because the sine function is odd, leaving only the cosine terms in the Fourier series.
  • If $f(x)$ is an odd function ($f(-x) = -f(x)$), all $a_n$ coefficients will be zero, leaving only the sine terms in the Fourier series.
• Half-Range Expansions:
  • For functions defined on $[0, T/2]$ rather than the full period $T$, half-range Fourier series expansions can be used:
    • Cosine Series: If $f(x)$ is extended as an even function.
    • Sine Series: If $f(x)$ is extended as an odd function.

7. Practical Considerations and Convergence

• The convergence of the Fourier series depends on the smoothness and continuity of the function. For smooth functions, the Fourier series converges rapidly, while for functions with discontinuities, the series may exhibit Gibbs phenomenon, where overshoots occur near the discontinuities.
• The rate of convergence of the Fourier coefficients $a_n$ and $b_n$ also provides insight into the function’s smoothness. For instance, if the function is piecewise smooth, the Fourier coefficients decrease in magnitude as $n$ increases, typically at a rate proportional to $1/n$ or faster.

The Fourier series can be elegantly represented using the exponential series through the use of complex numbers and Euler’s formula. This representation provides a unified framework for analyzing periodic functions and is particularly powerful in applications involving complex analysis, signal processing, and differential equations. Below is a comprehensive exploration of these advanced concepts.

1. Review of Euler’s Formula

• Euler’s formula is a fundamental relationship in complex analysis that connects exponential functions to trigonometric functions:

• Similarly, the inverse relationship is:

2. Complex Exponential Fourier Series

• Instead of representing a periodic function $f(x)$ using sines and cosines, we can use complex exponentials. For a function with period $T$, the Fourier series can be written as:

• Here, $c_n$ are complex Fourier coefficients, and $e^{i \frac{2\pi nx}{T}}$ are complex exponentials that generalize sine and cosine functions.
• The complex Fourier coefficients $c_n$ are calculated as:

• This representation allows us to encapsulate both the amplitude and phase information of the original function in a single complex number.

3. Relationship to the Real Fourier Series

• The real Fourier series is typically expressed as:

• The relationship between the complex coefficients $c_n$ and the real coefficients $a_n$ and $b_n$ is given by:

• This shows that the real Fourier series is just a decomposition of the complex exponential Fourier series into its real and imaginary parts.

4. Advantages of the Exponential Fourier Series

• Simplification of Mathematical Operations: Using exponential functions simplifies differentiation, integration, and multiplication of Fourier series. This is because the derivative and integral of $e^{ix}$ are straightforward and follow standard rules for exponential functions.
• Phase and Amplitude Representation: The complex Fourier coefficients $c_n$ inherently contain information about both the amplitude and phase of the corresponding frequency component, making them very useful in signal processing.
• Symmetry and Fourier Transform: The exponential Fourier series provides a natural transition to the Fourier transform, which is used for non-periodic functions and is crucial in the analysis of signals in the frequency domain.

5. Parseval’s Theorem in Complex Form

• Parseval’s theorem, which relates the total energy of the function to the sum of the squares of its Fourier coefficients, also applies to the complex Fourier series. In its complex form, Parseval’s theorem is:

• This theorem is used in many applications, including signal processing, where it quantifies the power in each frequency component of the signal.

6. Example: Complex Exponential Fourier Series

• Consider the function $f(x) = x^2$ defined on the interval $[-\pi, \pi]$. To compute the complex Fourier series, we first determine the Fourier coefficients $c_n$ by evaluating the integral:

• This integral can be solved using integration by parts or by recognizing the relationship between Fourier coefficients and the original function.

7. Generalization to Multivariable Functions

• The exponential Fourier series concept can be extended to functions of multiple variables. For example, in two dimensions, a periodic function $f(x,y)$ can be expressed as:

• This is particularly useful in the study of wave phenomena, quantum mechanics, and other areas where multidimensional periodic functions arise.

8. Fourier Transform: The Continuous Extension

• The Fourier transform is the continuous extension of the Fourier series, applicable to non-periodic functions. It is defined as:

• The inverse Fourier transform is:

• The Fourier transform is a cornerstone in modern signal analysis, allowing functions to be analyzed in the frequency domain.

When dealing with periodic impulses, the Fourier series becomes particularly interesting. Impulses are signals that are essentially “spikes” at specific points, usually represented by the Dirac delta function $\delta(t)$. These impulses are idealized as having an infinite amplitude and zero width, yet an area of one under the curve.

The Dirac Delta Function and the Fourier Series

The Dirac delta function, $\delta(t)$, is not a function in the traditional sense but a “distribution” that captures the idea of a signal that is infinitely narrow and infinitely high, yet integrates to one. In a Fourier series, impulses can be represented using these delta functions.

For a periodic impulse train (a series of equally spaced impulses), the Fourier series can represent it as a sum of complex exponentials:

Here, $\delta_T(t)$ represents an impulse train with period $T$, and $f_0 = \frac{1}{T}$ is the fundamental frequency.

Sifting Property of the Delta Function

The sifting property of the delta function is a crucial concept when working with Fourier series and impulses. The sifting property states that for any function $f(t)$:

This means that when a delta function is integrated against another function, it “sifts out” the value of that function at the location of the impulse. This property is extensively used in signal processing to extract specific values from a signal.

Fourier Series Representation of an Impulse Train

Consider an impulse train composed of impulses spaced by $T$:

The Fourier series representation of this impulse train can be written as:

Here, $f_0 = \frac{1}{T}$ is the fundamental frequency. This series shows that the impulse train in the time domain corresponds to an infinite series of harmonics in the frequency domain, each spaced by the fundamental frequency $f_0$.

Applying the Sifting Property

The sifting property can be applied to Fourier series by multiplying the series by a function $f(t)$ and integrating:

This property is used in signal analysis, filtering, and modulation techniques where specific frequency components are targeted.

Fourier series and the concept of impulses are interconnected through the use of the Dirac delta function. The sifting property is a powerful tool that allows us to extract information from signals. In the context of Fourier analysis, it enables us to isolate specific frequency components, making Fourier series an essential technique in both theoretical and applied mathematics, especially in fields like signal processing, communication systems, and control theory.