3: Fourier Series and Laplace Transform

In this unit we will learn two new ways to represent certain types of functions, and these will help us solve linear time invariant (LTI) DE’s with these functions as inputs.

We start with Fourier series, which are a way to write periodic functions as sums of sinusoids. In Unit Two we learned how to solve a constant coefficient linear ODE with sinusoidal input. Now using Fourier series and the superposition principle we will be able to solve these equations with any periodic input.

Next we will study the Laplace transform. This operation transforms a given function to a new function in a different independent variable. For example, the Laplace transform of $f(t) = \cos(3t)$ is $F(s) = s/ (s^2 + 9)$. If we think of $f(t)$$ as an input signal, then the key fact is that its Laplace transform $F(s)$ represents the same signal viewed in a different way. The Laplace transform converts a DE for the function $x(t)$ into an algebraic equation for its Laplace transform $X(s)$. Then, once we solve for $X(s)$ we can recover $x(t)$.

In the course of this unit, two important ideas will be introduced. The first is the convolution product of two functions. At first meeting this operation may seem a bit strange. Nonetheless, as we will see, it arises naturally, and the Laplace transform will allow us to work easily with it.

The second important idea is the delta function. Up to now all inputs to our systems have caused small changes in a small amount of time. An impulse is an input that causes a sudden jump in the system. For example, a sharp blow to a mass will cause its momentum to jump. The delta function is a mathematical idealization of an impulse and one which allows us to handle DE’s with these types of inputs.

Last Modification: 2020-09-16 Wed 22:56

Captured On: [2019-11-15 Fri 13:45]
Source: Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare

Fourier Series: Basics

Captured On [2019-11-15 Fri 13:49] Source Fourier Series: Basics | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 Which is the minimum period of a constant function? 1.1 Front Which is the minimum period of a constant function? 1.2 Back There is no minimal period, we don’t allow $P = 0$ to be periodic. Also, for any $P$ value is a period, but no minimal...

Operations on Fourier Series

Captured On [2019-11-23 Sat 18:11] Source Operations on Fourier Series | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is an even function? 1.1 Front What is an even function? Let $f(t)$ be a function, and write examples 1.2 Back $f(t)$ is even if $f(-t) = f(t)$ for all $t$ $t^2, t^4, t^6, \dots$, any even power of $t$ $\cos(at)$ power series for $\cos(at)$ has only even powers of $t$ A constant function 2 What is the integral of even function on a ‘balanced’ interval?...

ODE's with Periodic Input, Resonance

Captured On [2019-11-29 Fri 13:34] Source [[https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/odes-with-periodic-input-resonance/index.htm][ODE’s with Periodic Input, Resonance | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare]] 1 How can we solve this ODE? 1.1 Front How can we solve this ODE? $\ddot{x} + 9.1 x = f(t)$, where $f(t)$ is a odd square wave of period $2\pi$ with $f(t) = 1$ for $0 \lt t \lt \pi$ 1.2 Back Use the Fourier Series of $f(t)$ ${\displaystyle f(t) = \frac{4}{\pi} \sum_{n \text{ odd}}^{\infty} \frac{\sin(nt)}{n}}$ So the DE: \({\displaystyle \ddot{x} + 9....

Step and Delta Functions: Integrals and Generalized Derivatives

Captured On [2019-12-06 Fri 14:05] Source Step and Delta Functions: Integrals and Generalized Derivatives | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare This session looks closely at discontinuous functions and introduces the notion of an impulse or delta function. The goal is to use these functions as the input to differential equations. Step functions and delta functions are not differentiable in the usual sense, but they do have what we will call generalized derivatives, which are suitable for use in DE’s....

Unit Step and Unit Impulse Response

Captured On [2019-12-17 Tue 13:01] Source Unit Step and Unit Impulse Response | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare In this session we study differential equations with step or delta functions as input. For physical systems, this means that we are looking at discontinuous or impulsive inputs to the system. 1 When happens the pre-initial conditions in a DE? 1.1 Front When happens the pre-initial conditions in a DE?...

Convolution

Captured On [2019-12-27 Fri 13:46] Source Convolution | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the definition of the convolution of 2 functions? 1.1 Front What is the definition of the convolution of 2 functions? One-sided convolution 1.2 Back It’s a function defined by this integral ${\displaystyle (f * g)(t) = \int_{0^-}^{t^+} f(\tau) g(t - \tau) \dd{\tau}}$ for $t \gt 0$...

Laplace Transform: Basics

Captured On [2019-12-30 Mon 17:43] Source Laplace Transform: Basics | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the definition of Laplace Transform? 1.1 Front What is the definition of Laplace Transform? 1.2 Back The Laplace transform of a function $f(t)$ of a real variable $t$ is another function depending on a new variable $s$, which is in general complex....

Partial Fractions and Inverse Laplace Transform

Captured On [2020-01-02 Thu 18:40] Source Partial Fractions and Inverse Laplace Transform | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is a rational function? 1.1 Front What is a rational function? 1.2 Back Is a function that is the ratio of two polynomials ${\displaystyle \frac{s + }{s^2 + 7s + 9}}$ 2 When can we say that a rational function is proper?...

Laplace Transform: Solving Initial Value Problems

Captured On [2020-01-06 Mon 19:03] Source Laplace Transform: Solving Initial Value Problems | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 Compute with the definition the Laplace transform of $f’$ 1.1 Front Compute with the definition the Laplace transform of $f’$ ${\displaystyle \mathcal{L}(f’)(s)}$ Don’t do by heart 1.2 Back ${\displaystyle \mathcal{L}(f’) = \int_{0^-}^{\infty} f’(t) e^{-st} \dd{t}}$ Integrating by parts $u = e^{-st}$, $\dd{u} = -s e^{-st} \dd{t}$ $\dd{v} = f’(t) \dd{t}$, $v = f(t)$ ${\displaystyle f(t) e^{-st} \bigg|_{0^-}^{\infty} + s \int_{0^-}^{\infty} f(t) e^{-st} \dd{t} = -f(0^-) + s F(s)}$...

Transfer (System) and Weight Functions, Green's Formula

Captured On [2020-01-10 Fri 13:16] Source Transfer (System) and Weight Functions, Green’s Formula | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the transfer function for any LTI? 1.1 Front What is the transfer function for any LTI? In terms of unit impulse response 1.2 Back ${\displaystyle W(s) = \mathcal{L}(w(t))}$ where $w(t)$ is the unit impulse response 2 What is the system function?...

Poles, Amplitude Response, Connection to ERF

Captured On [2020-01-11 Sat 17:01] Source Poles, Amplitude Response, Connection to ERF | Unit III: Fourier Series and Laplace Transform | Differential Equations | Mathematics | MIT OpenCourseWare 1 When can we say that a rational function is in reduced form? 1.1 Front When can we say that a rational function is in reduced form? 1.2 Back If the numerator $q(s)$ and the denominator $p(s)$ have no roots in common, then $q(s)/p(s)$ is in reduced form...