  For a given finitely generated shift invariant (FSI) subspace $\cW\subset
L^2(\R^k)$ we obtain a simple criterion for the existence of shift generated
(SG) Bessel sequences $E(\cF)$ induced by finite sequences of vectors $\cF\in
\cW^n$ that have a prescribed fine structure i.e., such that the norms of the
vectors in $\cF$ and the spectra of $S_{E(\cF)}$ is prescribed in each fiber of
$\text{Spec}(\cW)\subset \T^k$. We complement this result by developing an
analogue of the so-called sequences of eigensteps from finite frame theory in
the context of SG Bessel sequences, that allows for a detailed description of
all sequences with prescribed fine structure. Then, given $0<\alpha_1\leq
\ldots\leq \alpha_n$ we characterize the finite sequences $\cF\in\cW^n$ such
that $\|f_i\|^2=\alpha_i$, for $1\leq i\leq n$, and such that the fine spectral
structure of the shift generated Bessel sequences $E(\cF)$ have minimal spread
(i.e. we show the existence of optimal SG Bessel sequences with prescribed
norms); in this context the spread of the spectra is measured in terms of the
convex potential $P^\cW_\varphi$ induced by $\cW$ and an arbitrary convex
function $\varphi:\R_+\rightarrow \R_+$.
