[Paper Review] A Class of Maximal-Rate, Low-PAPR, Non-square Complex Orthogonal Designs

Smarajit Das, B. Sundar Rajan|ArXiv.org|Aug 10, 2008

Advanced Wireless Communication Techniques9 references3 citations

TL;DR

This paper proposes a novel construction of maximal-rate, non-square complex orthogonal designs (CODs) for MIMO systems with reduced peak-to-average power ratio (PAPR). By extending square CODs through a structured transformation and introducing coordinate-interleaved scaled CODs (CIS-CODs), the method achieves lower PAPR while maintaining full diversity and high spectral efficiency. Simulation results confirm superior performance under peak power constraints compared to existing maximal-rate codes.

ABSTRACT

Space-time block codes (STBCs) from non-square complex orthogonal designs are bandwidth efficient when compared with those from square real/complex orthogonal designs. Though there exists rate-1 ROD for any number of transmit antennas, rate-1 complex orthogonal designs (COD) does not exist for more than 2 transmit antennas. Liang (IEEE Trans. Inform. Theory, 2003) and Lu et al (IEEE Trans. Inform. Theory, 2005) have constructed a class of maximal rate non-square CODs where the rate is ${1/2}+\frac{1}{n}$ if number of transmit antennas $n$ is even and ${1/2}+\frac{1}{n+1}$ if $n$ is odd. In this paper, we present a simple construction for maximal rate non-square CODs obtained from square CODs which resembles the construction of rate-1 non-square RODs from square RODs. These designs are shown to be amenable for construction of a class of generalized CODs (called Coordinate-Interleaved Scaled CODs) with low peak-to-average power ratio (PAPR) having the same parameters as the maximal rate codes. Simulation results indicate that these codes perform better than the existing maximal rate codes under peak power constraint while performing the same under average power constraint.

Motivation & Objective

• To develop high-rate, bandwidth-efficient space-time block codes (STBCs) for MIMO systems with reduced peak-to-average power ratio (PAPR).
• To overcome the rate limitation of square complex orthogonal designs, which decay exponentially with transmit antennas.
• To generalize the construction of rate-1 non-square real orthogonal designs to complex orthogonal designs for maximal rate.
• To design a class of coordinate-interleaved scaled complex orthogonal designs (CIS-CODs) that minimize PAPR while preserving the same diversity and delay as existing maximal-rate codes.
• To provide a systematic method for constructing low-PAPR STBCs with fewer zero entries than prior constructions.

Proposed method

• A new construction method for maximal-rate non-square CODs is developed by extending square CODs using a transformation analogous to that used in rate-1 real orthogonal designs.
• The method generates CODs with rate $ \frac{1}{2} + \frac{1}{n} $ for even $ n $, and $ \frac{1}{2} + \frac{1}{n+1} $ for odd $ n $, matching the maximal rate of prior constructions.
• A subclass of linear complex orthogonal designs (LCODs) is introduced, where non-zero entries are either variables, their conjugates, or coordinate-interleaved combinations.
• Coordinate-interleaved scaled complex orthogonal designs (CIS-CODs) are constructed by replacing isolated variables with $ x_i $, and paired variables with $ \frac{x_i + x_j}{\sqrt{2}} $ and $ \frac{x_i - x_j}{\sqrt{2}} $, reducing PAPR.
• The PAPR of the resulting CIS-CODs depends on the Hamming weight $ w_l $ of the index $ l $, and the fraction of zero entries is minimized by selecting optimal $ l $.
• The construction is validated via simulation, comparing performance under both average and peak power constraints.

Experimental results

Research questions

• RQ1 Can a systematic method be developed to construct maximal-rate non-square complex orthogonal designs from square CODs, similar to the rate-1 real orthogonal design construction?
• RQ2 How can the PAPR of maximal-rate STBCs be reduced without sacrificing diversity gain or decoding delay?
• RQ3 What is the impact of coordinate interleaving on the PAPR and zero-entry distribution in non-square CODs?
• RQ4 For a given number of transmit antennas, which choice of index $ l $ minimizes the PAPR in the resulting CIS-COD?
• RQ5 How does the performance of the proposed CIS-CODs compare to existing maximal-rate codes under peak power and average power constraints?

Key findings

• The proposed construction achieves maximal rate $ \frac{1}{2} + \frac{1}{n} $ for even $ n $, and $ \frac{1}{2} + \frac{1}{n+1} $ for odd $ n $, matching the theoretical maximum of prior constructions.
• The CIS-CODs derived from the method have fewer zero entries than existing maximal-rate codes, with zero fraction decreasing significantly for $ n=4 $ (0%) and $ n=5 $ (8/75 ≈ 10.7%).
• Simulation results show that the CIS-CODs outperform existing maximal-rate codes under peak power constraints, particularly in high-SNR regimes.
• Under average power constraints, the performance of the CIS-CODs matches that of the original maximal-rate codes, confirming no rate or diversity loss.
• The PAPR of the CIS-CODs is shown to depend on the Hamming weight $ w_l $ of the index $ l $, with lower weights generally yielding lower PAPR.
• The fraction of zeros in the LCOD matrix $ M_n(l) $ is minimized when $ w_l $ is chosen appropriately, though the optimal $ w_l $ remains an open problem for general $ n $.