On the Design of Channel Shortening Demodulators for Iterative Receivers in Linear Vector Channels

Abstract

We consider the problem of designing demodulators for linear vector channels with memory that use reduced-size trellis descriptions for the received signal. We assume an overall iterative receiver, and use interference cancellation (IC) based on the soft information provided by the outer decoder, to mitigate the parts of the signal that are not covered by the reduced-size trellis description. In order to reach a trellis description, a linear filter is applied as front-end to compress the signal structure into a small trellis. This process requires three parameters to be designed: (i) the front-end filter, (ii) the feedback filter through which the IC is done, and (iii) a target response which specifies the trellis.Demodulators of this form have been studied before under then name channel shortening (CS), but the interplay between CS, IC and the trellis-search process has not been adequately addressed in the literature. In this paper, we analyze two types of CS demodulators that are based on the Forney and Ungerboeck detection models, respectively. The parameters are jointly optimized based on a generalized mutual information (GMI) function. We also introduce a third type of CS demodulator that is in general suboptimal, but has closed-form solutions. Moreover, signal to noise ratio (SNR) asymptotic properties are analyzed and we show that the third CS demodulator asymptotically converges to the optimal CS demodulator in the sense of GMI-maximization. Index TermsChannel shortening, intersymbol interference, prefilter, front-end filter, feedback filter, target response, generalized mutual information, achievable information rate, Forney model, Ungerboeck model. The authors are with the Department of Electrical and Information Technology, Lund University, Lund, Sweden (email: firstname.lastname@eit.lth.se). This paper has been presented in part [53] at the IEEE 26th annual international symposium on personal, indoor and mobile radio communications (PIMRC), Hongkong, Sep. 2015.

Approximate Noise-Whitening in MIMO Detection via Banded-Inverse Extension

“…MMSE-QRD, is proposed and shows superior performance in MIMO detection. In what follows, we extend the MMSE-QRD to cope with the coloured noise in (5), and elaborate the pre-processes needed.…”

“…Surprisingly, the maximum-entropy distribution is still Gaussian, and the unique solution R, whose inverse is banded, is called as the 'banded-inverse extension' (BIE) of C. Although BIE has been used for approximating non-stationary random auto-regressive (AR) processes [1], [4], applying it for approximate matrix inverse in a multi-input and multi-output (MIMO) system, to our best knowledge, has not been considered before. One similar concept is channelshortening (CS) [5], which approximates a general MIMO channel by a banded one to reduce the detection-complexity. However, CS operates on the MIMO channel but not the noise covariance matrix, and cannot be applied for noise-whitening.…”

“…The analytical tool assessing the final performance is based on GMI, which establishes a lower bound of the achievable rate that can be attained with a mismatched MIMO detection [5], due to the approximate noise-whitening. Further, as R −1 is ν-banded with the BIE, it takes the conventional design in [3] with a diagonal R as a special case by setting ν = 0.…”

Approximate Noise-Whitening in MIMO Detection via Banded-Inverse Extension

“…One similar concept is channel-shortening (CS) [5], which approximates a general MIMO channel by a banded one to reduce the detection-complexity. However, CS operates on the MIMO channel but not the noise covariance matrix, and cannot be applied for noise-whitening.…”

Closed-Form Expressions for Channel Shortening Receivers Using $A\ Priori$ Information

“…Channel shortening has been extended to the optimization of the transmit filters [3] and a channel shortening receiver with an additional return filter using a priori information has been introduced in [4], [5] although no closed-form expressions have been provided for the return filter and the target response so far. In [6], the authors observe that further constraints must be put on the return channel shortening filter in order for the receiver to perform interference cancellation in a turbo equalization scheme. These additional constraints do not allow a closed-form derivation of the channel shortening solution with a priori information.…”