On the Minimum Decoding Delay of Balanced Complex Orthogonal Designs

Abstract

Complex orthogonal design (COD) with parameter [p, n, k] is a combinatorial design used in space-time block codes (STBCs). For STBC, n is the number of antennas, k/p is the rate, and p is the decoding delay. A class of rate 1/2 COD called balanced complex orthogonal design (BCOD) has been proposed by Adams et al., and they constructed BCODs with rate k/p = 1/2 and decoding delay p = 2 m for n = 2m. Furthermore, they prove that the constructions have optimal decoding delay when m is congruent to 1, 2, or 3 module 4. They conjecture that for the case m ≡ 0 (mod 4), 2 m is also a lower bound of p. In this paper, we prove this conjecture.

On the criteria for designing complex orthogonal space-time block codes

“…They presented a new class of rate 1/2 GCODs based on BCODs that satisfies design criteria C4-C7 [3]. In [14], Liu et al define the standard form for BCODs, and prove the conjecture that the minimum coding delay for BCODs with 2m columns is 2 m .…”

“…Fortunately, there does exist a special class of CODs that performs perfectly with respect to all the seven design criteria. To begin the construction, we first introduce the BCOD, which was presented for the first time by Adams et al [3], and simplified by Liu et al [14]. Definition 6.…”

“…Definition 7. [14] We say a BCOD G[2k, 2m, k] is in standard form iff it is already in B j form for some 1 j k. Note that the columns of a BCOD have correspondingly complementary zero patterns. This is the key property that can help us to obtain an elegant design via our new method.…”

“…Liu et al [14] determined the minimum decoding delay for BCODs, and hence, for this class of APGCODs. Theorem 11.…”

“…Theorem 11. [14] The minimum decoding delay for BCODs with n columns is 2 n 2 . Thus, the decoding delay for the new APGCOD with 2m columns is 2 m .…”