Optimum Detection of Walsh-Hadamard Multiplexed Antipodal Signals over Rayleigh Fading Channels Athanassios C. Iossifides Department of Electronics Alexander Technological Educational Institute (TEI) of Thessaloniki Thessaloniki, Greece aiosifidis@el.teithe.gr Abstract—This paper proposes and analyzes a maximum aposteriori probability (MAP) coherent detector for Walsh- Hadamard multiplexed binary antipodal signals, based on vector detection at the code-length size of the Walsh-Hadamard codes. Tight bounds on the bit error probability that are analytically derived or numerically computed, together with simulation results, show significant performance gain over symbol-by- symbol MAP detection for either one-dimensional or two- dimensional system configurations that lead to pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) rectangular constellations, respectively. In addition, the proposed system with MAP vector detection, presents great performance enhancement compared to standard binary phase shift keying (BPSK) modulation under identical information bit rate and mean power constraints, at a cost of higher instant power and moderate detection complexity. Keywords-Maximum Aposteriori Probability (MAP) Detection; Walsh-Hadamard codes; Pulse Amplitude Modulation (PAM); Quadrature Amplitude Modulation (QAM) I. INTRODUCTION Walsh-Hadamard codes have lately been under thorough investigation for possible use in communication systems. Among numerous applications and proposals, they are commonly used in contemporary mobile communication systems (UMTS 3G or 3.5G) in terms of orthogonal variable spreading factor (OVSF) codes for multiplexing different data streams e.g. [1-2], or as an M−ary modulation technique in 3G cdma2000 systems [3]. Recently, a coded modulation or multiplexing scheme based on Walsh-Hadamard codes was described and analyzed [4]. Antipodal information symbols are spread by different Walsh codes which are then summed up over the block (length of the code) level prior to transmission. A MAP symbol-by-symbol detector was proposed there and analysis including co-ordinate interleaving techniques was presented in order to provide efficient symbol constellations. Based on the approach of [4], and restricting our attention to simple rectangular produced constellations, a MAP vector detector is analyzed in this paper. The bit error probability (BEP) of the system is computed either by simulation or by upper and lower bounds analytically or numerically derived. Although, in general, more than one data streams may be in parallel transmitted with this scheme (introducing information spreading as in UMTS code multiplexing), a single data stream that uses the full Walsh-Hadamard codeset is considered here, so that direct comparison with standard BPSK is possible. The rest of the paper is organized as follows: Section II and III describe the system configuration, section IV presents performance evaluation, section V presents the results, and, finally, section VI concludes the paper. II. ONE-DIMENSIONAL SYSTEM DESCRIPTION A. Transmitter The transmitter of the system is identical to the one used in [4] and [5]. However, different notation is used in order to analyze the system in a more convenient matrix/vector form. The information data sequence {di }, di ∈{0,1}, is mapped to a bipolar sequence {bi} (referred to as information symbols for the rest of the paper), where 2 1 i i b d = - , leading to a standard BPSK modulation of unity power. The antipodal information symbols are grouped in groups of M = 2 n (n positive integer) and are multiplexed using a Walsh-Hadamard matrix of order M, i.e. after serial-to-parallel conversion of the M information symbols, each one is spread with a code of length M, i.e. with a line of the normalized Walsh-Hadamard matrix, given by 1 1 1 1 1 M M M M M M - - - - ⎛ ⎞ = ⎜ ⎟ - ⎝ ⎠ W W W W W , with 2 1 1 1 1 1 2 ⎛ ⎞ = ⎜ ⎟ - ⎝ ⎠ W , (1) and, subsequently, column wise addition of the matrix elements takes place. The channel symbols sk produced by the afore- mentioned procedure for each block of M information bits, are , 1 , 1 M k i ik i s bw k M = = ≤ ≤ ∑ , (2) where wi,k are the elements of the normalized Walsh-Hadamard matrix. The channel symbols sk arising by this procedure can take the values −M, −M + 2, …, 0, …, M − 2, M, divided by the normalization factor M , thus leading to (M +1)−PAM scheme where the number of occurrences of each channel symbol value has been evaluated in [4] as 2 2 2 2 2 2 , , 1, , ( ) 0, otherwise M M M M M k M i i Ps i M - ⎧⎛ ⎞ =- - + ⎪ ⎜ ⎟ = = + ⎨ ⎝ ⎠ ⎪ ⎩ … . (3) 978-1-4577-0024-8/11/$26.00 ©2011 IEEE 2011 18th International Conference on Telecommunications 978-1-4577-0023-1/11/$26.00 ©2011 IEEE 316