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Fall 2014
ECE 460- Optical Imaging
Homework 1
Due in class on Thursday September 9, 2014
1. Review of electromagnetic spectrum (20%)
a) Show a graph of the visible spectrum and other electromagnetic radiation (online resources are OK).
b) Pick your favorite three colors (e.g. red, green, and blue) and give their wavelength, frequency, angular
frequency, wave number, and photon energy.
c) Look up the spectral sensitivity of the eye. Plot it on the same graph as the spectrum of sun light.
Comment on how this affects our vision and perception of colors.
2.
a)
b)
c)
d)
e)
f)
Prove all the Fourier transform properties discussed in class:
Shift theorem
Parseval’s theorem
Similarity theorem
Convolution theorem
Correlation theorem
If F is the Fourier transform of f, what is the value F(0)? This is the “central ordinate” theorem.
(20%)
3. Calculate the Fourier transform of the following functions:
a) f ( x )  exp   x 2 / (2a 2 )  ; hint: you need to complete a square
b) ( x / a) , where  is the rectangular function, defined as:
1, if  a / 2  x  a / 2
 ( x / a)  
0, rest

c) f ( x)  sin(kx)
d) f ( x)  cos(kx)
e)
f ( x)  exp( x / a) and f ( x)  exp(  x / a ) , a>0; hint: does the integral converge?
(20%)
4. Redo problems in 2 for 2D functions (e.g., f ( x, y )  exp    x 2  y 2  / (2a 2 ) 
(20%)
5. Calculate the autocorrelation of the following functions
a) f ( x )  exp   x 2 / (2a 2 ) 
b) ( x / a)
c) f ( x)  sin(kx)
(20%)