  Approximate Message Passing (AMP) is an efficient iterative
parameter-estimation technique for certain high-dimensional linear systems with
non-Gaussian distributions, such as sparse systems. In AMP, a so-called Onsager
term is added to keep estimation errors approximately Gaussian. Orthogonal AMP
(OAMP) does not require this Onsager term, relying instead on an
orthogonalization procedure to keep the current errors uncorrelated with (i.e.,
orthogonal to) past errors. \LL{In this paper, we show the generality and
significance of the orthogonality in ensuring that errors are "asymptotically
independently and identically distributed Gaussian" (AIIDG).} This AIIDG
property, which is essential for the attractive performance of OAMP, holds for
separable functions. \LL{We present a simple and versatile procedure to
establish the orthogonality through Gram-Schmidt (GS) orthogonalization, which
is applicable to any prototype. We show that different AMP-type algorithms,
such as expectation propagation (EP), turbo, AMP and OAMP, can be unified under
the orthogonal principle.} The simplicity and generality of OAMP provide
efficient solutions for estimation problems beyond the classical linear models.
\LL{As an example, we study the optimization of OAMP via the GS model and GS
orthogonalization.} More related applications will be discussed in a companion
paper where new algorithms are developed for problems with multiple constraints
and multiple measurement variables.
