  Let $\mu$ denot the infinite convolution generated by
$\{(N_k,B_k)\}_{k=1}^\infty$ given by $$ \mu
=\delta_{{N_1}^{-1}B_1}\ast\delta_{(N_1N_2)^{-1}B_2}\ast\dots\ast\delta_{(N_1N_2\cdots
N_k)^{-1}B_k} *\cdots. $$ where $B_k$ is a complete residue system for each
integer $k>0$. We write $$ \nu_{>k}=\delta_{N_{k+1}^{-1} B_{k+1}} *
\delta_{(N_{k+1} N_{k+2})^{-1} B_{k+2}} * \cdots. $$ Since the elements in
$B_k$ may have very large absolute values, the infinite convolution may not be
compactly supported. In this paper, we study the necessary and sufficient
conditions for such infinite convolutions being a spectral measure.
  Generally, for such infinite convolutions, the necessary conditions for
spectrality mainly depend on the properties of the polynomials generated by the
complete residue systems. The main result shows that if every $B_k$ satisfies
uniform discrete zero condition, and $\{\nu_{>k}\}_{k=1}^\infty$ is {\it
tight}, then $\# B_k | N_k$ for all integers $k\geq 2$. For some special
complete residue systems $\{B_k\}_{k=1}^\infty$, we provide the necessary and
sufficient conditions for $\mu$ being a spectral measure.
