ELECTRONICS LETTERS 7th December 2000 Vol. 36 No. 25 Constrained surplus energy adaptive blind CDMA detection Teng Joon Lim, Yu Gong and B. Farhang-Boroujeny The blind minimum output energy (MOE) adaptive detector for code division multiple access (CDMA) signals requires exact knowledge of the received spreading code of the desired user. This requirement can be relaxed by constraining the so-called surplus energy of the adaptive tap- weight vector, but the ideal constraint value is not easily obtained in practice. An algorithm is proposed to adaptively track this value and hence to approach the best possible performance for this class of CDMA detector. Introduction: An adaptive linear detector for CDMA that does not require training sequences, and is in that sense ‘blind’, was introduced in [1] and has come to be known as the MOE detector. It is based on the assumption that the receiver is interested in demodulating only one user, as is the case for a mobile terminal in a cellular system, and knows the spreading code and propagation channel of that desired user. Its key fea- ture is that the filter tap-weight vector is split into fixed and variable components, i.e. c 1 = s 1 + x 1 , where the subscript 1 indicates that the first user is considered the desired one (without loss of generality), s 1 is the channel-distorted or received spreading code of user 1, and x 1 is orthog- onal to s 1 . The component s 1 is fixed, and x 1 is adapted (at the symbol rate), in order to minimise the output energy E|c 1 H y| 2 , y being the filter input vector. When s 1 is perfectly known, the MOE and MMSE tap-weight vectors differ only by a scaling factor. However, in the presence of channel esti- mation or synchronisation errors, we must use 1 (≠ s 1 ) in its place, and constrain x 1 to be orthogonal to 1 . In this case, the MOE solution c 1,0 will generally be orthogonal to s 1 , resulting in signal cancellation. Under the assumption that 1 is closer to s 1 than to the interference sub-space, it was shown in [1] that the constraint ||x 1 || 2 = χ limits signal suppression while still enabling interference suppression. However, the value of χ which gives the best performance cannot be computed easily. We address this problem by iteratively finding the MMSE value of ν 1 , which is a Lagrange multiplier related to χ (see [1]), and using that in the constrained surplus-energy MOE adaptive algorithm. The new algo- rithm introduces little extra complexity, and is shown through simula- tions to have better performance. Constrained surplus energy MOE detector: Known technique: We assume a K-user CDMA system, where the symbol-rate baseband received signal vector is given by where A is the channel matrix, whose columns represent the received spreading sequences of all users, d(i) is the column vector of symbols contained in y(i), and n(i) is a circularly symmetric, zero-mean complex Gaussian noise vector with correlation function E[n(i)n H (i – j)] = σ 2 δ(j)I, δ(j) being the Kronecker delta function. Referring to eqn. (32) of [1], the MOE tap-weight vector is given by where R = A H A and γ = ν 1 + σ 2 . The expression for γ suggests that the surplus energy constraint effectively injects white noise of variance ν 1 to the received signal, and hence changes the cost function from c 1 H (R + σ 2 I)c 1 to c 1 H (R + γI)c 1 . With this observation, an adaptive constrained surplus-energy MOE algorithm is easily found to be where μ is the LMS step size, z 1 (i) = c 1 H (i)y(i) is the detector output and P = I – s 1 s 1 H is the projection matrix for the subspace orthogonal to s 1 . Eqn. 3 is slightly different to eqn. (45) of [1], because it explicitly ensures orthogonality between x 1 and 1 whereas Honig et al. suggest a reorthogonalisation ‘on occasion’. Proposed method: The excess energy χ is a function of the Lagrange multiplier ν 1 , as shown in eqn. (35) of [1]. Furthermore, χ appears in the MOE minimisation problem through the constraint equation ||c 1 || 2 = χ + 1. Therefore, optimising ν 1 is akin to finding an ‘optimal’ constraint, a problem that is not commonly encountered. We circumvent this problem by limiting ourselves to using eqn. 3 as the weight-adaptation equation, but allowing ν 1 to be adapted in order to minimise the MSE Note that ν 1 only affects ξ 1 implicitly through the adaptation of x 1 . The steepest-descent algorithm will adapt ν 1 according to [2] where λ is the step size for the adaptation of ν 1 (i), and is different in general from μ. In eqn. 3, the previously fixed ν 1 will be replaced by ν 1 (i). Using the chain rule of differentiation, we obtain where the second line is obtained by expanding eqn. 4 and also using eqn. 3, and the third comes from observing that x 1 H (i – 1)s 1 = 0. The approximation in the second line is due to the use of instantaneous val- ues in place of ensemble averages. In addition, the averaging factor 1/N is ignored since it does not affect the algorithm. sˆ sˆ sˆ Fig. 1 SINR learning curves for different ν 1 —n— ν 1 = 0 —×— ν 1 = 0.1 —s— ν 1 = 10 —✩— new algorithm sˆ Fig. 2 Average learning curves of ν 1 —s— ν 1 start adapting at 0 —+— ν 1 start adapting at 10