Covariance Based Linear Precoding for Full Rank Channels

We make two contributions to the field of FDD transmit processing for multi-user MISO systems: first the concept of covariance based linear precoding is extended to spatio-temporal channels with full rank covariance matrices. Within, the variance true approximation of all signal components allows the formulation of the receive signal exclusively basing upon covariance knowledge and thus the derivation of optimal linear precoding solutions. Second, the derived chip level signal model demonstrates, that the spatial transmit processing on the pilot channel has pivotal influence on the performance of precoding approaches for data channels in general. Through the consideration of the receive filter in the mobile devices, the effects of transmitter side processing on pilot channels can be included into the optimization of precoders for data channels yielding drastic performance enhancements with respect to state of the art techniques. I. I NTRODUCTION For broadcast channels, see e.g. [1], receive processing has proven clearly suboptimum due to the lack of cooperation among the receive terminals. In the case of cooperative transmitters, precoding can be employed to jointly transform the different users’ data signals prior to transmission. This way, the channels act as equalizers of the precoding filters, rendering complex receiver structures dispensable. In complete analogy to spatio-temporal receive processing, linear precoding methods for multi-user MISO systems can be classified into three categories: Approaches that aim at the maximization of the desired signal power at the receiver, in the sequel called transmit matched filters(TxMF) as introduced in e.g. [2] or [3], interference suppression schemes called transmit zero forcing techniques (TxZF) [4], [5], [6] and the transmit Wiener filter (TxWF) [7], [8], [9], that trades the objectives of TxMF and TxZF in anmean square error (MSE) optimal sense. Note, that the following investigations will spare nonlinear precoding techniques such as Tomlinson Harashima precoding , lattice basis reductionor vector precoding . A general precondition of the above mentioned techniques is the full channel state information(CSI) at the transmitter. This assumption is usually non-critical in TDD systems, where downlink CSI can be obtained from uplink measurements, i.e. via the pilot information transmitted by the mobile station (MS). If upand downlink are separated in the frequency domain though (FDD), the corresponding channels no longer share the favorable property of full reciprocity with respect to the complex fading coefficients. Thus full transmitter side CSI only can be obtained through the feedback of the complete channel information from the mobile units which in the regarded standards (e.g. HSDPA [10]) is not considered. The design of linear precoding filters therefore must be based on partial CSI. In opposite to the complex channel coefficients, a set of longterm parameters including the angle of departure of all paths, the path delays, and the average path gains are almost independent of the carrier frequency [11]. Thus these parameters are reciprocal even in FDD systems and can be obtained from uplink measurements. While the path delays can be adopted directly, the path covariance matrices that are characterizing the spatial channels and the mean path powers can be obtained through a conversion of uplink measurements [12]-[16]. This partial reciprocity of FDD channels can serve as a basis for the design of linear precoding techniques. Thus we investigate a system with partial CSI at the transmitter andmaximum ratio combining(MRC) rake receivers at the mobile units, which adapt to the channel via measurements on a pilot channel. The linear precoding thus has to adapt to the resulting combination of channel and receive filter. For such a system [17] introduced a TxZF and [18] investigated the design of GSM filters in the frequency domain under comparable circumstances. The following sections extend the theory introduced in [19] or [20] to channels with full rank covariance matrices and gives solutions for the TxMF, the TxZF and the TxWF. To this end Section II derives an analytical expression for the chip level representation of the rake filter output signal of the system depicted in Fig. 1. Introducing variance true approximations for all signal components Section III prepares the derivation of the mentioned linear filters for the system with partial CSI at the transmitter in Section IV where the different linear filters are derived through the closed analytical optimization of the corresponding objective functions. As the rake receiving mobile station will adapt its filter coefficients to the performed channel estimation, the precoding solution through the consideration of the receive filter will inherently depend upon the spatial processing of the pilot channel. Therefore, Section V investigates three prominent examples of pilot transmission schemes in multi-user MISO systems with special focus on thecommon pilot channels(CPICH) introduced in [21]. Finally evaluations demonstrate the significant performance enhancement obtained by full rank processing and provides comparisons of different spatial pilot channels. II. SIGNAL MODEL With χ, L, Q, F denoting the spreading factor, the orders of the precoding filters, the channels, and the rake receivers respectively, this section establishes a signal model for ŝk[n] in the K user CDMA system withNa transmit antennas as s1[n] p1[n] x1[n] h1 [n]