Medical Imaging Systems I




НазваниеMedical Imaging Systems I
Дата конвертации16.02.2013
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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(t-1/2). Show that g = f*h, and find h. Plot h.

Problem 2. Find the Fourier transforms of g(t) = tri(t-1) + tri(t+1), h(t) = tri(t-1) – 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 sinc2(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+y2)-2. Find the Fourier transform.

Problem 6. f, g, h are two dimensional functions, and g = f*h. f1(x) = f(Ax), with h1(x) = h(Ax), Find a formula for f1*h1.

Problem 7. Find the 2D FT of f(x, y) = rect(x+y)rect(x-y). 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, McGraw-Hill, 1999 and 1986 editions are in stock at Barnes and Noble. This book is very good, but costs a lot of money.

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