This module looks at the basic circular convolution relationship between two sets of Fourier coefficients.
Signals can be composed by a superposition of an infinite number of sine and cosine functions. The coefficients of the superposition depend on the signal being represented and are equivalent to knowing the function itself.
Details the Continuous-Time Fourier Series.
This module discusses the covergence of the Fourier Series to show that it can be a very good approximation for all signals.
Using Euler's relations to define the complex Fourier series.
The module goes through the steps of deriving the Fourier coefficient equation from the general Fourier series equation.
This module shows how to derive the scintillating and useful Fourier transform.
Study of ordinary differential equations, including modeling of physical problems and interpretation of their solutions. Standard solution methods for single first-order equations, including graphical and numerical methods. Higher-order forced linear equations with constant coefficients. Complex numbers and exponentials. Matrix methods for first-order linear systems with constant coefficients. Non-linear autonomous systems; phase plane analysis. Fourier series; Laplace transforms.
Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.
The Dirichlet conditions are the sufficient conditions to guarantee existence and convergence of the Fourier series or the Fourier transform.
This module defines eigenvalues and eigenvectors and explains a method of finding them given a matrix. These ideas are presented, along with many examples, in hopes of leading up to an understanding of the Fourier Series.
This module defines eigenvalues and eigenvectors and explains a method of finding them given a matrix. These ideas are presented, along with many examples, in hopes of leading up to an understanding of the Fourier Series.
This module discusses signal equality. It looks at Gibb's phenomenon, and defines equality in the mean square and pointwise.
Lists the four Fourier transforms and when to use them.
Continues 18.100. Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals.
This module derives the Discrete-Time Fourier Series (DTFS), which is a fourier series type expansion for discrete-time, periodic functions. The module also takes some time to review complex sinusoids which will be used as our basis.
Learn how to make waves of all different shapes by adding up sines or cosines. Make waves in space and time and measure their wavelengths and periods. See how changing the amplitudes of different harmonics changes the waves. Compare different mathematical expressions for your waves.
A brief introduction to Fourier Series starting from the normal modes of an oscillating string. The concept is then extended to Fourier's integral theorem.
Shows how to use Fourier series to approximate a square wave, as opposed to the sinusoidal waves seen previously.
This module will introduce the Fourier Series and its Fourier coefficients using the concepts of eigenfunctions and basis. We will show several examples of how to decompose a signal and find the Fourier coefficients.
