Integral Transform Method Chapter 15 Integral Transform Method

Outline 15.1 Error Function 15.2 Applications of the Laplace Transform 15.3 Fourier Integral 15.4 Fourier Transforms 15.5 Fast Fourier Transform

Error Function Error function Complementary error function The functions are related by the identity

Error Function (cont’d.) Useful Laplace transforms

Applications of the Laplace Transform Assume that the operational properties of Laplace transforms of one-variable functions discussed in Chapter 4 apply to functions of two variables For example ℒ It follows that ℒ ℒ

Applications of the Laplace Transform (cont’d.) Example: Transform of a PDE Find the Laplace transform of the wave equation From the transform properties of partial derivatives, ℒ ℒ Which becomes ℒ ℒ or

Fourier Integral The Fourier integral of a function f defined on is given by where and

Fourier Integral (cont’d.) The Fourier integral of an even function on is the cosine integral where The Fourier integral of an odd function on is the sine integral

Fourier Integral (cont’d.) The Fourier integral also possesses an equivalent complex form, or exponential form

Fourier Transforms An integral transform and its inverse is a transform pair If ℒ , then the inverse Laplace transform is the contour integral ℒ If f(x) is transformed into F() by an integral transform then the function can be recovered by the inverse transform

Fourier Transforms (cont’d.) The Fourier integral is the source of three new integral transforms

Fourier Transforms (cont’d.) Examining the transforms of derivatives will aid in applying the concepts to BVPs Fourier transform Fourier sine transform Fourier cosine transform The choice of which transform to use on a given BVP depends on the type of BC specified at zero

Fast Fourier Transform Consider a function f that is defined and continuous on the interval are equally spaced points on the interval Corresponding function values represent a discrete sampling of f

Fast Fourier Transform (cont’d.) Sampling (cont’d.) T is the sampling rate, or length of sampling interval is the fundamental angular frequency 2p is the fundamental period If f is continuous at T, the sample of f at T is the product of the function f and the Dirac delta function, and the discrete version of f, or discrete signal is

Fast Fourier Transform (cont’d.) Sampling (cont’d.) Applying the Fourier transform and the sifting property of the Dirac delta function, we arrive at the discrete Fourier transform (DFT) Consider the function values f(x) at N equally spaced points, The finite (discrete) Fourier series gives

Fast Fourier Transform (cont’d.) Sampling (cont’d.) The finite (discrete) Fourier series and letting gives

Fast Fourier Transform (cont’d.) Sampling (cont’d.) This system in matrix notation is

Fast Fourier Transform (cont’d.) Sampling (cont’d.) Let the N  N matrix be If is the matrix of complex conjugates of and I denotes the identity matrix, we have It follows that

Fast Fourier Transform (cont’d.) If a signal is band-limited (the range of frequencies of the signal like in a band ), the signal can be reconstructed by sampling two times for every cycle of the highest frequency present

Fast Fourier Transform (cont’d.) The DFT can be written as Or, for simplicity of notation, Recall the matrix notation

Fast Fourier Transform (cont’d.) The key to the Fast Fourier Transform (FFT) is properties of and matrix factorization If we can write Fn in the following way and is the identity matrix and P is the permutation matrix that rearranges c