  An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on
generalized SAV approach for the Cahn-Hilliard equation by combining with
Fourier spectral method in space. It is proved that the modified energy
dissipation law is unconditionally preserved at discrete levels. Under a mild
ratio restriction, i.e., \Ass{1}: $0<r_k:=\tau_k/\tau_{k-1}< r_{\max}\approx
4.8645$, we establish a rigorous error estimate in $H^1$-norm and achieve
optimal second-order accuracy in time. The proof involves the tools of discrete
orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting
that the presented adaptive time-step scheme only requires solving one linear
system with constant coefficients at each time step. In our analysis, the
first-consistent BDF1 for the first step does not bring the order reduction in
$H^1$-norm. The $H^1$ bound of the numerical solution under periodic boundary
conditions can be derived without any restriction (such as zero mean of the
initial data). Finally, numerical examples are provided to verify our
theoretical analysis and the algorithm efficiency.
