Convolution Theorem

• This is a very important concept. Can think of lots of things in MRI as ideal signal multiplied by something. (sampling, relaxation, data filter ...) What happens to multiplication when it goes through the Fourier transform
• multiplication of two functions in one domain leads to convolution of their Fourier transforms in the other domain.
  

  $${\cal F}\left(g(x)\cdot h(x)\right) = G(k) * H(k)$$ (10.4)

  
  

• where the convolution is defined by
  

  $$G(k) * H(k) \equiv \int_{-\infty}^{+\infty}dk'\,G(k')H(k-k')$$ (10.5)

  
  

• At any point $k$, you are integrating the overlap of one function at the origin with a flipped, shifted version of the other function.

• Consider the example from the book, followed by (IN-CLASS QUESTION)
  
  

• convolution is associative, so doesn't matter in which order that you do the convolutions
  $$\begin{eqnarray*} a(x) * (b(x) * c(x)) = (a(x) * b(x)) * c(x) \end{eqnarray*}$$