Fourier Transform Shift Theorem

• What if the $k$-space data is shifted? (timing error, gradient area error?)

• Introduce $\hat{\rho}$ which is the image that you get back, artifacts included, and $s_m(k)$ which is the signal that you measure, complete with errors and physical effects added in. $\hat{\rho}(x) = {\cal F}^{-1} (s_{m}(k)$.

• $\hat{\rho}(x)$ is found by taking the inverse Fourier transform of $s_m(k-k_{0})$
  

  $$\hat{\rho}(x) = {\cal F}^{-1} \left(s_m(k - k_{0})\right)$$

  $$= \int_{-\infty}^{\infty} dk\, s_m(k-k_{0}) e^{i 2 \pi k x}$$

  $$= e^{ i 2 \pi k_{0} x} \int_{-\infty}^{\infty} dk'\, s_m(k') e^{i 2 \pi k' x}$$

  $$= e^{ i 2 \pi k_{0} x} \hat{\rho}_{expected}(x)$$ (10.2)

  
  

• A shift in one domain leads to a linear phase change in the other domain. (IN-CLASS QUESTION)