13.1 Fourier transforms: Chapter 13 Integral transforms

The Fourier transform of f(t) Inverse Fourier transform of f(t) Ex: Find the Fourier transform of the exponential decay function and Sol:

Chapter 13 Integral transforms Properties of distribution:

The uncertainty principle: Chapter 13 Integral transforms Gaussian distribution: probability density function (1) is symmetric about the point the standard deviation describes the width of a curve (2) at falls to of the peak value, these points are points of inflection

Ex: Find the Fourier transform of the normalized Gaussian distribution. Sol: the Gaussian distribution is centered on t=0, and has a root mean square deviation Chapter 13 Integral transforms =1 is a Gaussian distribution centered on zero and with a root mean square deviation is a constant.

(1) Fraunhofer diffraction:When the cross-section of the object is small compared with the distance at which the light is observed the pattern is known as a Fraunhofer diffraction pattern. Applications of Fourier transforms: Chapter 13 Integral transforms

Ex: Evaluate for an aperture consisting of two long slits each of width 2b whose centers are separated by a distance 2a, a>b; the slits illuminated by light of wavelength. Chapter 13 Integral transforms

The Diracδ-function: Chapter 13 Integral transforms

Ex: Prove that Chapter 13 Integral transforms

consider an integral to obtain Chapter 13 Integral transforms Proof: Define the derivative of

Physical examples for δ-function: Chapter 13 Integral transforms (1)an impulse of magnitude applied at time (2)a point charge at a point (3) total charge in volume V unit step (Heviside) function H(t)

Chapter 13 Integral transforms Proof:  Relation of the δ-function to Fourier transforms

Chapter 13 Integral transforms for large becomes very large at t=0 and also very narrow about t=0 as

Properties of Fourier transforms: denote the Fourier transform of by or Chapter 13 Integral transforms

Consider an amplitude-modulated radio wave initial, a message is represent by, then add a constant signal

Chapter 13 Integral transforms Convolution and deconvolution Note: x, y, z are the same physical variable (length or angle), but each of them appears three different roles in the analysis.

Chapter 13 Integral transforms Ex: Find the convolution of the function with the function in the above figure. Sol:

Chapter 13 Integral transforms The Fourier transform of the convolution

The Fourier transform of the product is given by Chapter 13 Integral transforms

Ex: Find the Fourier transform of the function representing two wide slits by considering the Fourier transforms of (i) two δ-functions, at, (ii) a rectangular function of height 1 and width 2b centered on x=0

Chapter 13 Integral transforms Deconvolution is the inverse of convolution, allows us to find a true distribution f(x) given an observed distribution h(z) and a resolution unction g(y). Ex: An experimental quantity f(x) is measured using apparatus with a known resolution function g(y) to give an observed distribution h(z). How may f(x) be extracted from the measured distribution. the Fourier transform of the measured distribution extract the true distribution

Chapter 13 Integral transforms Correlation functions and energy spectra The cross-correlation of two functions and is defined by It provides a quantitative measurement of the similarity of two functions and as one is displaced through a distances relative to the other.

Chapter 13 Integral transforms

Parseval’s theorem: Ex: The displacement of a damped harmonic oscillator as a function of time is given by Find the Fourier transform of this function and so give a physical interpretation of Parseval’s theorem. Sol:

Chapter 13 Integral transforms Fourier transforms in higher dimensions: three dimensional δ-function:

Chapter 13 Integral transforms Ex: In three-dimensional space a function possesses spherical symmetry, so that. Find the Fourier transform of as a one-dimensional integral. Sol:

Chapter 13 Integral transforms 13.2 Laplace transforms: Laplace transform of a function f(t) is defined by define a linear transformation of

Ex: Find the Laplace transforms of the functions: Chapter 13 Integral transforms

Standard Laplace transforms

Chapter 13 Integral transforms

The inverse Laplace transform is unique and linear

Chapter 13 Integral transforms Laplace transforms of derivatives and integrals

Chapter 13 Integral transforms Other properties of Laplace transforms:

Chapter 13 Integral transforms Ex: Find the expression for the Laplace transform of Sol: