Equalisation for carrierless amplitude and phase modulation
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Carrierless amplitude and phase (CAP) modulation is generally regarded as a bandwidth efficient two-dimensional (2-D) passband line code. It is closely related to the pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) schemes. CAP has been proposed for various digital subscriber loop (DSL) systems over unshielded twisted pairs of copper wires. In this thesis, our main focus is on the minimum mean-square error (MMSE) performance of the ideal (i.e., infinite length) linear and non-linear (decision feedback) CAP receivers/equalisers in the presence of additive, coloured Gaussian noise, and/or data-like cross-talks. An in-depth analysis is given on the performance of both receiver structures. In the case of the linear receiver, one possible view of the overall CAP transceiver system which includes both data and cross-talk transmission paths is that it is a linear multiple-input multiple-output (MIMO) system. Accordingly, the existing MMSE results for a general MIMO system are applicable also to CAP systems. However, up to date, this approach was shown to be unsuccessful in the sense that the derived MMSE expressions are too complex and offer little insights. In our analysis, in order to find a more incisive MMSE expression, we reconsider the problem of minimisation of the MSEs at slicers. By exploiting the Hilbert transform pair relationship between the impulse responses of the inphase and quadrature transmit shaping filters, we are able to obtain an elegant and more meaningful MMSE expression, as well as the corresponding transfer functions of the optimum linear receive filters. In the case of the nonlinear, or decision feedback equaliser (DFE), receiver, we start our analysis with the receiver structure of a generic multidimensional (>/= 3) CAP-type system.This receiver consists of a bank of analog receive filters, the number of which equals the dimension of the CAP line code, and a matrix of cross-connected, infinite-length, baud-spaced feedback filters. It is shown that the optimum filters and the corresponding MMSE of the DFE receiver require the factorisation of a discrete-time channel spectral matrix. This mathematically intractable step can be avoided, however, when the DFE results are specialised to a standard 2-D CAP system where we are able to again exploit the Hilbert transform pair relationship to derive a further and more useful MMSE expression. Three sets of numerical studies are given on the MMSE performance of the CAP receivers. In the first set of studies. we model the sum of all crosstalks as an additive, Gaussian noise source and select three test transmission channels over which we compare the MMSE performance of the linear and DFE receiver structures. In the second set of studies, we compare the performance of the two receiver structures, but in a data-like cross-talk environment. The results demonstrate the importance of NEXT equalisation in the design of CAP receivers operating in a NEXT dominant environment. In the final set of studies which follows from the second set of studies, we investigate the relationship between the MMSE performance of the DFE receiver and system parameters which include excess bandwidth, data rate, CAP scheme. and relative phase between the received signal and the NEXT signal. The results show that data-like cross-talks can be effectively suppressed by using a large excess bandwidth (alpha > 1 in the case of a RC transmit shaping filter) alone.The relative phase also affect; the receiver performance. but to a lesser degree. In addition to the MMSE performance analysis. implementation issues of an adaptive linear CAP receiver are also considered. We propose a novel linear receiver by appending two fixed analog filters to the front-end of the existing adaptive linear receiver using fractionally-spaced equalisers (FSE). We show that if the analog filters are matched to the transmit shaping filters, then inphase and quadrature finite-length FSEs in the proposed receiver have the same NINISE solution. We further propose a modified least-mean-square (LMS) algorithm which takes advantage of this feature. The convergence analysis of the proposed LMS algorithm is also given. We show that the modified LMS algorithm converges approximately twice as fast as the standard LMS algorithm, given the same misadjustment, or alternatively, it halves the misadjustment, given the same initial convergence rate.
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