Date of Award
Master of Engineering (ME)
Electrical and Computer Engineering
K. Max Wong
Timothy N. Davidson
In this thesis the linear precoder which minimizes the bit error rate (BER) is derived for block transmission systems in which zero forcing (ZF) equalization and threshold detection are applied. Because the bit error rate for block transmission is a highly non-linear function of the precoder parameters, its minimization has been regarded as being difficult to implement. Therefore designers have attempted to find low BER precoders indirectly by optimizing alternative objectives, such as minimizing the Mean Square Error (MMSE), or maximizing the received signal-to-noise ratio (SNR). However, these precoders do not minimize the BER directly, and it is this problem which is the subject of the thesis.
The block transmission systems considered in this thesis employ block by block processing at the receiver, and therefore elimination of inter-block interference (IEI) is desirable. We will design Minimum BER (MBER) precoders for two schemes which eliminate IEI, namely zero padding (ZP) and cyclic prefix (CP). Based on the bit error rate formula derived in the thesis, an analytic solution for the MBER precoder at moderate-to-high SNRs is derived via a two-stage optimization process using Jensen's inequality. At moderate-to-high SNRs, the bit error rate is a convex function of the autocorrelation matrix which is, itself, a function of the precoder matrix because of the use of a zero-forcing equalizer. Simulations and analyses are given to the two sets of precoders based on ZP and CP respectively to verify the optimal precoders derived. The BER improvement of the ZP-MBER/CP-MBER precoders over other ZP/CP precoders is substantial, and the ZP-MBER precoder is superior to the CPMBER precoder in performance. The latter also outperforms the scheme of discrete multitone (DMT) with water filling power loading, and cyclic prefix orthogonal frequency division multiplexing (CP-OFDM). The CP-MBER precoder is shown to be a two-stage modification IV of the transmitting scheme for tandard DMT. Firstly the water filling algorithm is replaced by the MMSE power loading algorithm suggested in the thesis, and secondly, the power loading is augmented by multiplication with a DFT matrix. It is shown that the CP-MBER precoder does not require more power to transmit a cyclic prefix than the CP-OFDM or water-filling DMT precoder.
A simple test which determines whether the SNR is high enough for our MBER precoder design to be optimal is provided. Furthermore, methods to guarantee sufficient SNR are suggested. One can either increase the transmitting power, or drop sub-channels and hence avoid transmission on the sub-channels which correspond to the small eigen-values of the channel. The MBER precoder design after dropping sub-channels is also discussed.
For the precoders which are characterized by an arbitrary unitary matrix in their solutions, the optimal unitary matrix which minimizes the BER is designed. Therefore, minimization of BER is achieved within the optimal solution set of the precoders. Applications of the design are discussed. Simulations and analytic evaluations are presented to show the BER improvement provided by the optimal unitary matrix
Ding, Yanwu, "Minimum BER Block Precoders" (2001). Open Access Dissertations and Theses. Paper 3682.