Fourier Transform Symmetries

• by writing out the exponent from the Fourier transform, and recalling that it is made up of cosines and sines, can identify some interesting symmetries about the function.
  

  $$H(k) = \int_{-\infty}^{\infty} dx\, h(x) [\cos{2 \pi k x} -i\sin{ 2 \pi k x}]$$ (10.7)

  
  

• integral of product of even and odd functions of x over infinity is going to be 0, so if If $H(x)$ is real, know that $h_{real}(k)$ is an even function, and $h_{imaginary}(k)$ is an odd function to get a real Fourier transform product and zero the imaginary part of the Fourier transform.

  
  
  
  
  
  
  
  

• If assume that $\rho(x)$ is real, then should be able to sample only half of the data, and reconstruct the other half. Alternatively, use this information to understand other properties of the data.

• Useful to commit to memory.

  $h(x)$	$H(k)$
  	Real part	Imaginary part
  real	even	odd
  imaginary	odd	even