  In 1949, Denis Gabor introduced the ``complex signal'' (nowadays called
``analytic signal'') by combining a real function $f$ with its Hilbert
transform $Hf$ to a complex function $f+ iHf$. His aim was to extract phase
information, an idea that has inspired techniques as the monogenic signal and
the complex dual tree wavelet transform. In this manuscript, we consider two
questions: When do two real-valued bases or frames $\{f_{n} : n\in\mathbb{N}\}$
and $\{g_{n} : n\in\mathbb{N}\}$ form a complex basis or frame of the form
$\{f_{n} + i g_{n}: n\in\mathbb{N}\}$? And for which bounded linear operators
$A$ forms $\{f_{n} + i A f_{n} : n\in\mathbb{N}\}$ a complex-valued orthonormal
basis, Riesz basis or frame, when $\{f_{n} : n\in\mathbb{N}\}$ is a real-valued
orthonormal basis, Riesz basis or frame? We call this approach
\emph{rebricking}. It is well-known that the analytic signals don't span the
complex vector space $L^{2}(\mathbb{R}; \mathbb{C})$, hence $H$ is not a
rebricking operator. We give a full characterization of rebricking operators
for bases, in particular orthonormal and Riesz bases, Parseval frames, and
frames in general. We also examine the special case of finite dimensional
vector spaces and show that we can use any real, invertible matrix for
rebricking if we allow for permutations in the imaginary part.
