Fourier Transform: Effects of Sampling

• if Only truncated the data, note that image will not be exact representation, but still considering continuous integral over a function, might expect to do pretty well.

• What happens when start picking of singular points (sampling), move from integral to sum? Lots of data is now missing. Need rules to know how sparse sampling can be, and what is going to happen to the image.

• Model sampling as signal times a set of regularly spaced delta functions.

  
  
  
  
  
  
  
  

• sampling function
  

  $$u(k) = \Delta k \sum_{p=-\infty}^{\infty} \delta(k - p\Delta k)$$ (10.8)

  
  

• take the Fourier transform of the sampling function to get
  

  $$U(x) = {\cal F}(u(k)) = \Delta k \sum_{p=-\infty}^{\infty} \... ...} = \Delta k \sum_{p=-\infty}^{\infty}e^{i2\pi p \Delta k\, x}$$ (10.9)

  
  

• If $x \Delta k$ is not a nice integer number, sum is not going to add up to much, in fact adds upto zero. Proof is in a HW problem.

  
  
  
  
  
  
  
  

• Find out that
  

  $$U(x) = \sum_{q=-\infty}^{\infty} \delta (x - q/\Delta k)$$ (10.10)

  
  
  or Fourier transform of sampling function is another sampling function

• Let's draw the Fourier transform pair of sampling functions