  Bigraded Toda hierarchy $L_1^M(n)=L_2^N(n)$ is generalized to
$L_1^M(n)=L_2^{N}(n)+\sum_{j\in \mathbb
Z}\sum_{i=1}^{m}q^{(i)}_n\Lambda^jr^{(i)}_{n+1}$, which is the analogue of the
famous constrained KP hierarchy $L^{k}=
(L^{k})_{\geq0}+\sum_{i=1}^{m}q_{i}\partial^{-1}r_i$. It is known that
different bosonizations of fermionic KP hierarchy will give rise to different
kinds of integrable hierarchies. Starting from the fermionic form of
constrained KP hierarchy, bilinear equation of this generalized bigraded Toda
hierarchy (GBTH) are derived by using 2--component boson--fermion
correspondence. Next based upon this, the Lax structure of GBTH is obtained.
Conversely, we also derive bilinear equation of GBTH from the corresponding Lax
structure.
