Fourier Analysis and Transforms

Fourier Analysis

• A powerful tool for analyzing periodic functions.
• Based on the idea that any periodic function can be represented as a sum of sines and cosines.

Fourier Series

• Represents a periodic function f(t) with period T as an infinite sum of sine and cosine functions.
• Defined as: $f(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n\omega t) + \sum_{n=1}^{\infty} b_n \sin(n\omega t)$.
• $\omega = \frac{2\pi}{T}$ represents the fundamental angular frequency.
• $a_0 = \frac{1}{T} \int_{0}^{T} f(t) dt$ is the average value of the function.
• $a_n = \frac{2}{T} \int_{0}^{T} f(t) \cos(n\omega t) dt$ represents the cosine coefficients.
• $b_n = \frac{2}{T} \int_{0}^{T} f(t) \sin(n\omega t) dt$ is the sine coefficients.

Complex Form of Fourier Series

• Expressed as: $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega t}$.
• $c_n = \frac{1}{T} \int_{0}^{T} f(t) e^{-jn\omega t} dt$ is the complex coefficients.

Fourier Transform

• An extension of Fourier analysis to non-periodic functions.
• Transforms a function from the time domain to the frequency domain.

Definition of Fourier Transform

• Defined as: $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$.
• $F(\omega)$ is the function's representation in the frequency domain.
• $\omega$ is the angular frequency.

Inverse Fourier Transform

• Used to retrieve the original function from the frequency domain to the time domain.
• Expressed as: $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega$.

Properties of Fourier Transform

• Linearity: $F{af(t) + bg(t)} = aF(\omega) + bG(\omega)$
• Scaling: $F{f(at)} = \frac{1}{|a|} F(\frac{\omega}{a})$
• Time Shifting: $F{f(t - t_0)} = e^{-j\omega t_0} F(\omega)$
• Frequency Shifting: $F{e^{j\omega_0 t} f(t)} = F(\omega - \omega_0)$
• Convolution: $F{(f * g)(t)} = F(\omega)G(\omega)$
• Differentiation in Time: $F{\frac{df(t)}{dt}} = j\omega F(\omega)$
• Integration in Time: $F{\int_{-\infty}^{t} f(\tau) d\tau} = \frac{F(\omega)}{j\omega} + \pi F(0) \delta(\omega)$

Applications of Fourier Analysis

• Signal Processing: filter analysis/design and audio/video compression.
• Image Analysis: edge detection, pattern recognition.
• Communications: Signal modulation and demodulation.
• Physics: Includes vibration analysis and spectroscopy.
• Medicine: biomedical signal analysis and medical imaging.
• Engineering; systems analysis and control.

Example: Fourier Series of a Square Wave

• Periodic square wave $f(t)$ with period $T$ is defined as 1 for $01$ for $T/2
• The Fourier series coefficients are computed as follows:
• $a_0 = 0$ as the function is odd
• $a_n = 0$ because the function is odd
• $b_n = \frac{4}{n\pi}$ when n is odd
• $b_n = 0$ when n is even
• Therefore, the Fourier series for the square wave is $f(t) = \sum_{n=1,3,5...}^{\infty} \frac{4}{n\pi} \sin(n\omega t)$
• In which $\omega = \frac{2\pi}{T}$.

Matrices

• Rectangular arrays of numbers, symbols, or expressions.
• Arranged in rows (horizontal) and columns (vertical).

Matrix General Form

• Expressed as: $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$
• $a_{ij}$ denotes the element in the i-th row and j-th column.
• m represents the number of rows.
• n represents the number of columns.

Types of Matrices

• Square Matrix: Rows equals columns ($m = n$).
• Row Matrix: Only one row ($m = 1$).
• Column Matrix: Only one column ($n = 1$).
• Zero Matrix: All elements are zero.
• Diagonal Matrix: Square matrix with all non-diagonal elements as zero.
• Identity Matrix: Diagonal matrix with all diagonal elements as 1, denoted by $I$.
• $I = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix}$
• Transpose of a Matrix: Rows and columns are interchanged, denoted by AT or A’.
• If $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$, then $A^T = \begin{bmatrix} a_{11} & a_{21} \ a_{12} & a_{22} \end{bmatrix}$
• Symmetric Matrix: Equal to its transpose ($A = A^T$).
• Skew-Symmetric Matrix: Equal to the negative of its transpose ($A = -A^T$).
• Triangular Matrix:
• Upper Triangular Matrix: Square matrix with elements below the main diagonal as zero.
• Lower Triangular Matrix: Square matrix with elements above the main diagonal as zero.

Matrix Operations

• Addition/Subtraction: Possible if the dimensions are the same.
• If $A = [a_{ij}]$ and $B = [b_{ij}]$, then $A + B = [a_{ij} + b_{ij}]$ and $A - B = [a_{ij} - b_{ij}]$.
• Scalar Multiplication: Multiplying each matrix element by a scalar.
• If $A = [a_{ij}]$ and c is a scalar, $cA = [ca_{ij}]$.
• Matrix Multiplication: Defined if the number of columns in A equals the number of rows in B.
• ($AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$ represents the element calculation in the i-th row and j-th column of AB.

Determinant of a Matrix

• Scalar value calculated from a square matrix elements.
• For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is $|A| = ad - bc$.
• Used for solving linear equations and finding eigenvalues

Inverse of a Matrix

• Matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$.
• A matrix is invertible if the determinant is non-zero.
• The inverse of a $2 \times 2$ is $A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$, provided that $ad - bc \neq 0$.

Rank of a Matrix

• The maximum number of linearly independent rows (or columns). Provides information about linear equations in a system.

Eigenvalues and Eigenvectors

• Key properties that satisfy the equation $Av = \lambda v$, where $\lambda$ is the eigenvalue.
• Used in stability analysis, vibration analysis, and principal component analysis.

Explore Fourier analysis, a tool for analyzing periodic functions by representing them as sums of sines and cosines. Learn about Fourier series, its complex form, and the Fourier transform for non-periodic functions.