  Craigen introduced and studied {\it signed group Hadamard matrices}
extensively in \cite{Craigenthesis, Craigen}. Livinskyi \cite{Ivan}, following
Craigen's lead, studied and provided a better estimate for the asymptotic
existence of signed group Hadamard matrices and consequently improved the
asymptotic existence of Hadamard matrices. In this paper, we introduce and
study signed group orthogonal designs. The main results include a method for
finding signed group orthogonal designs for any $k$-tuple of positive integer
and then an application to obtain orthogonal designs from signed group
orthogonal designs, namely, for any $k$-tuple $\big(u_1, u_2, ..., u_{k}\big)$
of positive integers, we show that there is an integer $N=N(u_1, u_2, ...,
u_k)$ such that for each $n\ge N$, a full orthogonal design (no zero entries)
of type $\big(2^nu_1,2^nu_2,...,2^nu_{k}\big)$ exists . This is an alternative
approach to the results obtained in \cite{EK}.
