On the Maximal Rate of Non-Square STBCs from Complex Orthogonal Designs

A linear processing complex orthogonal design (LPCOD) is a ptimesn matrix epsiv, (pgesn) in k complex indeterminates x
₁
,x
₂
,...,x
_k
such that (i) the entries of epsiv are complex linear combinations of 0, plusmnx
_i
, i=1,...,k and their conjugates, (ii) epsiv
^H
epsiv=D, where epsiv
^H
is the Hermitian (conjugate transpose) of epsiv and D is a diagonal matrix with the (i,i)-th diagonal element of the form l
₁

⁽ⁱ⁾
|x
₁
|
²
+l
₂

⁽ⁱ⁾
|x
₂
|
²
+...+l
_k

⁽ⁱ⁾
|x
_k
|
²
where l
_j

⁽ⁱ⁾
,i=1,2,...,n, j=1,2,...,k are strictly positive real numbers and the condition l
₁

⁽ⁱ⁾
=l
₂

⁽ⁱ⁾
=...=l
_k

⁽ⁱ⁾
, called the equal- weights condition, holds for all values of i. For square designs it is known that whenever a LPCOD exists without the equal-weights condition satisfied then there exists another LPCOD with identical parameters with l
₁

⁽ⁱ⁾
=l
₂

⁽ⁱ⁾
=...=l
_k

⁽ⁱ⁾
=1. This implies that the maximum possible rate for square LPCODs without the equal-weights condition is the same as that of square LPCODs with equal-weights condition. In this paper, this result is extended to a subclass of non-square LPCODs. It is shown that, a set of sufficient conditions is identified such that whenever a non- square (p>n) LPCOD satisfies these sufficient conditions and do not satisfy the equal-weights condition, then there exists another LPCOD with the same parameters n, k and p in the same complex indeterminates with l
₁

⁽ⁱ⁾
=l
₂

⁽ⁱ⁾
=...=l
_k

⁽ⁱ⁾
=1.