Understanding Fourier Transforms: Finite and Infinite Cosine and Sine Transform Formulas

The Fourier transforms, both finite and infinite, are pivotal tools in signal processing, engineering, and physics. This article delves into the intricacies of the Fourier Cosine and Sine Transforms, offering a comprehensive overview of their formulas and applications.

Introduction

Fourier transforms allow the decomposition of a function into its constituent frequencies. They are crucial for analyzing periodic and non-periodic signals. The Fourier Cosine and Sine Transforms are specific variations of these transforms, each with its unique applications.

Infinite Fourier Transforms

There are two main categories of infinite Fourier transforms: the Fourier Cosine Transform (FCT) and the Fourier Sine Transform (FST).

Fourier Cosine Transform (FCT)

Formula: The Fourier Cosine Transform of a function f(x) is given by:

Fcω (frac{1}{2}) U(int_0^infty {f(x)cos(ωx)dx}

Description: The FCT is particularly useful for functions that are even and defined over the entire real line. Application: Commonly used in temperature distribution analysis and other physical phenomena.

Fourier Sine Transform (FST)

Formula: The Fourier Sine Transform of a function f(x) is given by:

Fsω (frac{1}{2}) U(int_0^infty {f(x)sin(ωx)dx}

Description: The FST is ideal for functions that are odd and defined over the entire real line. Application: Widely used in solving partial differential equations and in the analysis of odd periodic functions.

Finite Fourier Transforms

When functions are defined over a finite interval, the finite Fourier transforms come into play, specifically the Finite Fourier Cosine Transform (FCT) and the Finite Fourier Sine Transform (FST).

Finite Fourier Cosine Transform (FCT)

Formula: The FCT for a function f(x) defined on the interval [0, L] is given by:

Fc(ωk) (frac{2}{L}) (int_0^L {f(x)cosleft(frac{nπx}{L}right)dx}

Description: This transform is especially useful in analyzing functions defined over a fixed interval. Application: Particularly useful in signal processing and in solving problems involving periodic signals within a fixed period.

Finite Fourier Sine Transform (FST)

Formula: The FST for a function f(x) defined on the interval [0, L] is given by:

Fs(ωk) (frac{2}{L}) (int_0^L {f(x)sinleft(frac{nπx}{L}right)dx}

Description: This transform is applicable to functions that are odd and defined within the interval [0, L]. Application: Useful in the analysis of odd functions and in solving differential equations within a finite interval.

Inverse Transforms

To fully understand the transforms, we must also consider their inverse processes:

Inverse Fourier Cosine Transform

Formula: The inverse Fourier Cosine Transform is given by:

f(x) (frac{1}{2} int_0^infty {F_c(ω)cos(ωx)} dω}

Inverse Fourier Sine Transform

Formula: The inverse Fourier Sine Transform is given by:

f(x) (frac{1}{pi} int_0^infty {F_s(ω)sin(ωx)} dω}

Key Formulas for Fourier Transforms

The key formulas for infinite and finite Fourier transforms are summarized below:

Infinite Fourier transform of f(x)

Let f(x) be a function defined on (?∞, ∞) and be piecewise continuous in each finite partial interval and absolutely integrable over (?∞, ∞). Then, the Fourier transform is defined as:

(hat{f}(k) frac{1}{sqrt{2π}} int_{?∞}^{∞} e^{ikx} f(x) dx)

Some authors also define the Fourier transform as: (hat{f}(k) frac{1}{2π} int_{?∞}^{∞} f(x) e^{?ikx} dx) (hat{f}(k) frac{1}{sqrt{2π}} int_{?∞}^{∞} f(x) e^{?ikx} dx)

Finite Fourier sine Transforms

The finite Fourier sine transformation of f(x) when 0 is defined as:

(hat{f_s}(k) int_0^L f(x) sin(frac{πkx}{L}) dx)

where k is an integer.

Finite Fourier cosine Transforms

The finite Fourier cosine transform of f(x) where 0 is defined as:

(hat{f_c}(k) int_0^L f(x) cos(frac{πkx}{L}) dx)

where k is an integer.

Integral Transform Representation

The integral transform of a function ft is defined by the equation:

(hat{f}(k) int_a^b f(t) K(k,t) dt)

In this representation, K(k,t) is called the kernel of the integral transform, and f(t) is referred to as the inverse transform of f(k).

Finite vs. Infinite Fourier Transforms

Depending on the interval of integration: (K(k,t) e^{ikt}, a -L, b L)

This corresponds to the Finite Fourier transform of f(x).

(K(k,t) e^{ikt}, a -∞, b ∞)

This corresponds to the Infinite Fourier transform of f(x).

Conclusion

In conclusion, the Fourier transforms, both finite and infinite, are fundamental tools in understanding and analyzing various functions and signals. Whether dealing with even or odd functions, or specific intervals, the Fourier transforms provide a powerful methodology to break down complex functions into their constituent frequencies, making them invaluable in fields ranging from engineering to physics.