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Drexel University Medical Imaging Systems I Fall 2002 Homework 2 Due October 16, 2002 (note later date because of Columbus day). Problem 1. g = f*rect(t+1/2) + f*rect(t1/2). Show that g = f*h, and find h. Plot h. Problem 2. Find the Fourier transforms of g(t) = tri(t1) + tri(t+1), h(t) = tri(t1) – tri(t+1). Plot the two functions and their Fourier transforms. Problem 3. We define square(t) as the periodic function, with period 1, which is equal to rect(2t) for the period –1/2 < t < 1/2. Plot square(t) over an interval [3, 3]. Find the Fourier series expansion coefficients for square(t). Problem 4. The function sinc^{2}(t) is sampled with the folowing periods. (a) 1, (b) 0.75, (c) 0.5. Do any of the sampling periods satisfy the sampling theorem? Find the Fourier transforms of all sampled functions & plot. Problem 5. f(x, y) = exp(x)*(1+y^{2})^{2}. Find the Fourier transform. Problem 6. f, g, h are two dimensional functions, and g = f*h. f_{1}(x) = f(Ax), with h_{1}(x) = h(Ax), Find a formula for f_{1}*h_{1}. Problem 7. Find the 2D FT of f(x, y) = rect(x+y)rect(xy). Plot it in some suitable way. Compare with the plot of the FT of rect(x)rect(y) Problem 8. The figure on the right is a ‘top view’ of a funcion f(x, y), with gray denoting value 0 and black the value 1. The function is zero outside the domain shown. Each of the black squares stands for a unit rect function with appropriate shifts. For example, the square on the upper left is rect(x+1.5)rect(y–1.5). (a) Write a formula for f(x, y) as a sum of rect functions. (b) What is the value of F(0, 0)? (c) Find the FT of f(x, y) and plot. References: Available through Drexel library (http://www.engnetbase.com/). Ziemer, R. E. “Convolution Integral” The Engineering Handbook. Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 Jenkins, W.K. “Fourier Series, Fourier Tran forms, and the DFT”, Digital Signal Processing Handbook, Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999. This article is about 30 pages long, but only the first 10 are relevant. Book (not on the web): Ronald Bracewell, The Fourier Transform and its Applications, McGrawHill, 1999 and 1986 editions are in stock at Barnes and Noble. This book is very good, but costs a lot of money. 