Homework2_Solutions

# Homework2_Solutions - I HOMEWORK#2 SOLUTIONS 1 M T F =...

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I. HOMEWORK #2 SOLUTIONS 1) MTF = | H ( u ) | H (0) a) H 1 ( u ) = F{ h 1 ( x ) } Note: F{ e - ax 2 } = p π a e - π 2 u 2 a H 1 ( u ) = 5 πe - 5 π 2 u 2 MTF = 5 πe - 5 π 2 u 2 5 πe - 5 π 2 0 2 = e - 5 π 2 u 2 b) MTF total = MTF 1 · MTF 2 MTF 2 = e - 10 π 2 u 2 by the same methods used in part a. MTF total = e - 5 π 2 u 2 · e - 10 π 2 u 2 = e - 15 π 2 u 2 c) The system blurring increases (image quality decreases) exponentially. 2) To solve for the FWHM, we need to find the point where the exponential term equals 1/2 and double it. a) h 1 ( x ) = e - x 2 2 = 0 . 5 - x 2 2 = ln (0 . 5) x 2 = 2 ln (2) x = p 2 ln (2) FWHM = 2 x = 2 p 2 ln (2) h 2 ( x ) = e - x 2 200 = 0 . 5 - x 2 200 = ln (0 . 5) x 2 = 200 ln (2) x = p 200 ln (2) FWHM = 2 x = 2 p 200 ln (2) b) FWHM total = p FWHM 2 1 + FWHM 2 2 FWHM total = q (2 p 2 ln (2)) 2 + (2 p 200 ln (2)) 2 1

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FWHM total = p 8 ln (2) + 800 ln (2) FWHM total = p 808 ln (2) c) System two ( h 2 ( x )) has the largest affect on the overall FWHM. d) System two ( h 2 ( x )) also dominates the resolution of the system, as evidenced by having the larger FWHM. The resolution of a system is always limited by its lowest- resolution component.
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