Fourier Series and Transform

Name: Fourier Series and Transform
Rating: 4.6 (31 reviews)

Master Fourier Series and Transform, by gaining an amazing understanding behind the logic.

Created byAfterclap Team

Last updated 7/2023

English

What you'll learn

Learning the fundamentals of functions including "even-odd functions" ; "periodic functions" ; "trigonometric functions"
With the fundamentals of functions, exploring the building blocks of any possible function.
Being able to deal with complex numbers, complex number representations and calculations.
Being able to graph sinusoidal functions by knowing "amplitude" "argument" "period" "phase".

Course content

5 sections • 21 lectures • 6h 6m total length

Introduction to Course6:10

1.1 Fundamentals of Functions (Periodicity ; odd, even and trigonometric func.)16:31
1.2 Fundamentals of Complex Numbers (Euler's Formula)17:24
Explore complex numbers, their rectangular and polar forms, and derive Euler's formula e^{i x} = cos x + i sin x, including conjugates and magnitude–angle representations.
1.3 Orthogonal Functions24:21
this lecture shows that sine and cosine with period L are orthogonal: integral over 0 to L vanishes, and cosine–cosine integrals vanish when frequencies differ, while equal frequencies give L/2.
1.4 Orthogonal Functions - Part 213:21
1.5 The Single Period Idea13:47

1.6 Expressing the Idea in Mathematical Terms14:21
Express any periodic function as an infinite sum of sine and cosine terms with varying amplitudes and phases, and derive coefficients a_n and b_n in the Fourier series.
1.7 Finding out the coefficients a0, an and bn25:02
The lecture derives a0, an, and bn for Fourier series by projecting f(x) onto sine and cosine bases using period T integrals, expressing f as a sum of orthogonal functions.
1.8 Some Observations on the findings (symmetry)12:49
1.9 An Example19:42
Solve a Fourier series for f(x) = a x with period T, deriving a0, an, and bn; show a0 and an vanish and bn yields the sine series.
1.9.2 Example -214:30
Compute the Fourier series of a piecewise, period-2 function defined on 0–1 and 1–2; determine a0, an, bn via integration by parts, noting no symmetry about x axis.
1.10 Complex Fourier Series - 122:05
Explore complex Fourier series by converting cosine and sine to exponentials via Euler's relation and forming complex coefficients c_n from a_n and b_n to express F(x) as an exponential sum.
1.11 Complex Fourier Series - 211:23
1.12 Review of Fourier Series28:17
Review how sine and cosine form the building blocks of periodic functions, deriving amplitude, frequency, and phase from both trigonometric and complex Fourier series.
1.13 Example - 325:47
this lecture shows the complex Fourier series is equivalent to the trigonometric series for the given function, deriving c_n via integral using Euler's formula.

2.1 Fourier Transform - 124:05
Explore how Fourier transform extends Fourier series to non-periodic functions and the role of the complex coefficient C(k) in the integral representation of F(x).
2.2 Fourier Transform - 216:08
Explore the Fourier transform for non periodic functions, compare it with Fourier series, and learn to extract amplitude, frequency, and phase from C(K).
2.3 Fourier Transform - 3 (The idea of getting Fourier transform using trigo)3:57
Apply the Fourier transform to a non-periodic function using complex Fourier series, computing C(k) from an integral and assembling F(x) via e^{ikx}.
2.4 An Example and another approach to Fourier transform30:33
Apply complex and trig Fourier series to compute the Fourier transform of a nonperiodic function in this example, deriving F(x) from C(k) and from sine-cosine representations.

3.1 Idea of "discrete functions"8:48
Explore discrete functions and the discrete Fourier series, contrasting it with Fourier series for periodic functions and Fourier transform for non periodic signals, and see discrete data in practice.
3.2 Discrete Fourier Series (DFS)17:33
Explore discrete Fourier series and how to obtain dfs coefficients from sampled data using the continuous Fourier series, including synthesis and analysis formulas.

Requirements

Enthusiasm

Description

This Fourier Series course includes over 12 hours (estimation) of video-on-demand supported by fully detailed quizzes, formula sheets, and solutions.

While creating the course, we have spent a lot of time on its structure so that anyone with a basic math background could master Fourier Series and Transform with ease.

As Afterclap Academy always does, the topic is divided into smaller, easy to understand parts. So that, you can easily understand any topic no matter how complex it is.

In this course, what you are going to learn;

Graphical representation of trigonometric functions with variable amplitude, period, phase shift, and vertical displacement
Describing non-sinusoidal functions.
Graph of non-sinusoidal periodic functions
Integration of some functions that are built-in within Fourier Series and Transform
Find the coefficients of the Fourier series
Identify even and odd functions analytically and graphically

Also, you can ask your questions anytime you want. We will be replying your questions within 24 hours.

We expect our students to benefit from this course a 100% percent, to make that possible, we are giving-away 1 to 1 free lectures.

Without having any other lecture on the topic, you will be an expert. Thanks to neat explanation and smooth example solving.

In Afterclap, We Trust

Sincerely,

kavcar

Afterclap ACADEMY

Who this course is for:

Engineering Students (because the topic is included in Advanced Engineering Math)
Physics Students
Math Students
Basically, anyone looking for some fun Fourier learning process :)

Fourier Series and Transform

What you'll learn

Explore related topics

Course content

Introduction1 lecture • 6min

1- Preparation to Fourier Series and Transform (must knows)5 lectures • 1hr 25min

2- Fourier Series9 lectures • 2hr 54min

3- Fourier Transform4 lectures • 1hr 15min

Discrete Fourier (DFS, DFT)2 lectures • 26min

Requirements

Description

Who this course is for: