## Channel modelling and fractionally-spaced MMSE equalisers for broadband channels.

##### Author

##### Date

2018##### Permanent Link

http://hdl.handle.net/10092/15014##### Thesis Discipline

Electrical and Electronic Engineering##### Degree Grantor

University of Canterbury##### Degree Level

Doctoral##### Degree Name

Doctor of PhilosophyIn response to the increasing demand for high data rate communications, broadband (BB) wireless systems utilising several gigahertz (GHz) of bandwidth will be used in future generations of wireless networks. The characteristics of BB wireless channels differ from those of narrowband (NB) channels. Three of the key differences are as follows:

1. The response of communication system components (e.g., connectors and anten- nas in each direction) and propagation mechanisms (e.g., diffraction, scattering, reflection from and transmission through obstacles) over a wide frequency band are frequency selective. As a result, per-path pulse distortion in BB channels is more than NB channels. In NB channel modelling the frequency dependency of physical channel effects are usually ignored and the frequency dependency of antennas and connectors are minimised using matching circuits and appropriate antenna design. The residual frequency dependency does not cause considerable approximation error because variations of the electric parameters of materials over a narrow frequency band are not significant. 2. The delay spread of BB channels in terms of symbol interval are much longer than those of NB channels. This is due to the fact that the duration of a BB pulse is shorter than that of a NB pulse. 3. A BB channel consists of many clusters of propagation paths. The clustering effect tends not to be observable in indoor NB channel measurements because the pulse length1 of electromagnetic pulses used in NB systems are usually large compared to the geometrical size of the scatterers and the differences between the lengths of paths travelled by individual pulses. Therefore, a large number of received pulses overlap at the receiver to make a single fading multipath compo- nent. But, when the pulse bandwidth is large, its duration (and consequently, its pulse length) is short and the number of overlapping pulses at the receiver are fewer.

Per-path pulse distortion occurs in channels with large fractional bandwidths. Longer delay spread and the clustering phenomena are more observable in channels with large absolute bandwidths. In channel modelling there are no particular values for absolute or fractional bandwidths in order to be used for classifying channels with respect to the significance of per-pulse distortion effects, channels’ delay spreads or their clustering effects. This is so because the significance of these effects not only depend on the channel’s fractional and absolute bandwidth but also on the communication environment. Therefore, in this thesis, we use the term BB to refer to those channels that in their mathematical models none of the above mentioned effects is ignored. The design of an appropriate receiver and its theoretical and simulated perfor- mance analysis depends on the adopted model for the channel. In this thesis, certain issues regarding channel modelling and receiver designing for BB systems are consid- ered.

Considering the above mentioned differences of NB and BB channels, we derive a mathematical formula for a lowpass received signal from a BB channel which in- cludes pulse distorting effects of the physical channel due to frequency selectivity and wideband Doppler effect. For a given pulse shaping filter, we characterise the small- est signal space that includes the set of all received signals from a BB channel. In particular, it is shown that an appropriate model for a BB channel is a fractionally- spaced tapped delay-line (TDL) model in which the tap delays are shorter than the symbol interval. Then, using a realizable front-end receiver filter and the properties of orthonormal bases, a set of variables is extracted that constitute a sufficient statistic for any optimum receiver. The geometry of Hilbert spaces and the theory of shift- invariant subspaces of finite-energy signals are used to prove the sufficiency of the extracted statistic. Our approach does not suffer from the ideality of the Shannon sampling theorem and the corresponding sampling models for communication chan- nels. We show that using realisable filters the performance of the ideal lowpass filter can be achieved.

As a case study of the per-path pulse-distorting effects of a physical channel, we analyse the effects of lossy dielectric walls on BB pulses by using the basic principles of electromagnetics and frequency domain methods. The frequency-dependent param- eters of commonly used building materials are used to analyse the effects of multiple reflections and transmissions, material distortion, and interpulse interference (IPI) on BB pulse waveforms. The possibility of polarisation-dependent distortion (PDD) is discussed. Various thicknesses of walls and angles of incidence are considered. The distortion due to each effect is quantified in terms of maximum correlation coefficients (MCCs). The overall effect of the wall is modelled as a TDL filter based on the MCC. Using our model derivation approach, the sources of the multicluster and the soft-onset phenomena of BB channel models are explained. This part of the thesis proposes a theoretical approach, by using laws of classical electromagnetics, to derive models for indoor channels where reflection from and transmission through walls or partitions are major propagation mechanisms. The theoretical approach can be used to complement and validate the experimental channel modelling approaches.

It is well known that in a linear transmission system with unknown channel infor- mation, where a basic pulse shape with non-zero excess bandwidth is used for transfer- ring information symbols, a sufficient statistic for any optimum detection method can only be obtained by sampling the received signal at rates higher than or equal to the Nyquist rate for the received signal2. In this case the channel observed by a receiver is a fractionally-spaced (FS) TDL channel. Specially, in BB channels where there exists significant pulse distortion due to the frequency dependency of the BB channel effects, the necessity of adopting a FS channel model and employing a FS equaliser becomes more prominent.

Another feature of BB channels is their longer delay spread compared to NB chan- nels when measured in terms of their corresponding symbol intervals. The longer delay spread of BB channels increases the sensitivity of adaptive receivers to perturbations when used for these channels.

The final part of the thesis is devoted to spectral analysis of FS minimum mean- square error (MMSE) equalisers. While many aspects of the FS MMSE equalisers have been studied since their first appearance in the literature in 1970s, some aspects of them are not completely understood. For example, behaviour of a FS MMSE equaliser is usually speculated based on the eigenvalues of the correlation matrix of the input FS samples of the received signal. A characterisation of the equaliser in terms of its transfer function (TF) that includes the effects of the front end prefilter and sampler is not available in the literature. In Chapter 4, we derive the TF of a FS MMSE equaliser under the general system model described in Chapter 2. Then, the TF is used to analyse the behaviour of the adaptive FS MMSE equaliser.

An adaptive FS MMSE equaliser, implemented using the least-mean-square (LMS) algorithm, suffers from instability. Stabilisation of adaptive FS MMSE equalisers has a long history. A major problem in dealing with FS sampling receivers is the non- stationarity of the received sampled process, even if the channel itself is time invariant. In communication systems that use a linear modulation scheme for transmission, the FS samples of the received signal from a wide-sense stationary (WSS) channel constitute a wide-sense cyclostationary (WSCS) time series. Hence, standard Fourier transform techniques cannot be used directly to study the spectral characteristics of the received FS samples or to derive the TF of the corresponding MMSE receiver. In this thesis, an expression for the TF of the FS MMSE equaliser is derived. Due to the WSCS nature of the input sampled signal, the FS equaliser’s TF is periodically time varying. Using the TF, the sources of instability of the FS LMS algorithm are characterized. The obtained results improve the existing knowledge in the literature regarding the sources of instability of FS MMSE equalisers. Based on the analysis performed, sufficient conditions are provided to increase the stability and guarantee convergence of the FS LMS algorithm.

For channels with longer delay spread, such as BB channels, the corresponding TDL equaliser needs to be sufficiently long in order to perform satisfactorily. The LMS algorithm is more sensitive to perturbation when the equaliser delay line is longer. The effects of equaliser length and sampling phase on stability of the LMS algorithm are explained. The waveform level simulations of communication systems validates the theoretical results.