Additional Filters

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Know that there are other things such as relaxation modifying the data, and we could apply additional modification to the data.
Generally, take all other signal modifiers in -space as extra multipliers of the signal, and refer to them as filters

$\displaystyle \hat{s}_{m, filter}(k)$ $\textstyle =$ $\displaystyle s_m(k) \cdot H_{filter}(k) = s(k) \cdot H_{ws}(k) \cdot H_{filter}(k)$

$\textstyle \equiv$ $\displaystyle s(k) \cdot H_{ws,filter}(k)$ (13.5)

where refers to windowing, sampling, and filtering
Of course, all of these filters get convolved with the spin density to give us the final reconstructed image

$\displaystyle {\tilde{\rho}}_{filter}(x)$ $\textstyle =$ $\displaystyle \hat{\rho}(x) * h_{filter}(x) = \rho(x) * h_{ws}(x) * h_{filter}(x)$

$\textstyle \equiv$ $\displaystyle \rho(x) * h_{ws,filter}(x)$ (13.6)

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Michael Thompson 2003-11-21

$\displaystyle \hat{s}_{m, filter}(k)$	$\textstyle =$	$\displaystyle s_m(k) \cdot H_{filter}(k) = s(k) \cdot H_{ws}(k) \cdot H_{filter}(k)$
	$\textstyle \equiv$	$\displaystyle s(k) \cdot H_{ws,filter}(k)$	(13.5)

$\displaystyle {\tilde{\rho}}_{filter}(x)$	$\textstyle =$	$\displaystyle \hat{\rho}(x) * h_{filter}(x) = \rho(x) * h_{ws}(x) * h_{filter}(x)$
	$\textstyle \equiv$	$\displaystyle \rho(x) * h_{ws,filter}(x)$	(13.6)