13.1 Fourier transforms:

1. Chapter 13 Integral transforms 13.1 Fourier transforms:

2. 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:

3. Chapter 13 Integral transforms Properties of distribution:

4. Chapter 13 Integral transforms • The uncertainty principle: 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

5. Chapter 13 Integral transforms 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 =1 • is a Gaussian distribution centered on zero and with a root mean square deviation is a constant.

6. Chapter 13 Integral transforms Applications of Fourier transforms: (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.

7. 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 .

8. Chapter 13 Integral transforms • The Diracδ-function:

9. Chapter 13 Integral transforms Ex: Prove that

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

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

12. Chapter 13 Integral transforms Proof: • Relation of the δ-function to Fourier transforms

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

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

15. Chapter 13 Integral transforms

16. Chapter 13 Integral transforms

17. Chapter 13 Integral transforms Consider an amplitude-modulated radio wave initial, a message is represent by , then add a constant signal

18. 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.

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

20. Chapter 13 Integral transforms The Fourier transform of the convolution

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

22. 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

23. 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

24. 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.

25. Chapter 13 Integral transforms

26. 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:

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

28. 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:

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

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

31. Chapter 13 Integral transforms Standard Laplace transforms

32. Chapter 13 Integral transforms

33. Chapter 13 Integral transforms

34. Chapter 13 Integral transforms The inverse Laplace transform is unique and linear

35. Chapter 13 Integral transforms Laplace transforms of derivatives and integrals

36. Chapter 13 Integral transforms Other properties of Laplace transforms:

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