  In this paper, we consider the generalized phase retrieval from affine
measurements. This problem aims to recover signals ${\mathbf x} \in {\mathbb
F}^d$ from the affine measurements $y_j=\norm{M_j^*\vx +{\mathbb b}_j}^2,\;
j=1,\ldots,m,$ where $M_j \in {\mathbb F}^{d\times r}, {\mathbf b}_j\in
{\mathbb F}^{r}, {\mathbb F}\in \{{\mathbb R},{\mathbb C}\}$ and we call it as
{\em generalized affine phase retrieval}. We develop a framework for
generalized affine phase retrieval with presenting necessary and sufficient
conditions for $\{(M_j,{\mathbf b}_j)\}_{j=1}^m$ having generalized affine
phase retrieval property. We also establish results on minimal measurement
number for generalized affine phase retrieval. Particularly, we show if
$\{(M_j,{\mathbf b}_j)\}_{j=1}^m \subset {\mathbb F}^{d\times r}\times {\mathbb
F}^{r}$ has generalized affine phase retrieval property, then $m\geq
d+\floor{d/r}$ for ${\mathbb F}={\mathbb R}$ ($m\geq 2d+\floor{d/r}$ for
${\mathbb F}={\mathbb C}$ ). We also show that the bound is tight provided
$r\mid d$. These results imply that one can reduce the measurement number by
raising $r$, i.e. the rank of $M_j$. This highlights a notable difference
between generalized affine phase retrieval and generalized phase retrieval.
Furthermore, using tools of algebraic geometry, we show that $m\geq 2d$ (resp.
$m\geq 4d-1$) generic measurements ${\mathcal A}=\{(M_j,b_j)\}_{j=1}^m$ have
the generalized phase retrieval property for ${\mathbb F}={\mathbb R}$ (resp.
${\mathbb F}={\mathbb C}$).
