Patrick Robertson

A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain

For estimating the states or outputs of a Markov process, the symbol-by-symbol MAP algorithm is optimal. However, this algorithm, even in its recursive form, poses technical difficulties because of numerical representation problems, the necessity of nonlinear functions and a high number of additions and multiplications. MAP like algorithms operating in the logarithmic domain presented in the past solve the numerical problem and reduce the computational complexity, but are suboptimal especially at low SNR (a common example is the max-log-MAP because of its use of the max function). A further simplification yields the soft-output Viterbi algorithm (SOVA). We present a log-MAP algorithm that avoids the approximations in the max-log-MAP algorithm and hence is equivalent to the true MAP, but without its major disadvantages. We compare the (log-)MAP, max-log-MAP and SOVA from a theoretical point of view to illuminate their commonalities and differences. As a practical example forming the basis for simulations, we consider Turbo decoding, where recursive systematic convolutional component codes are decoded with the three algorithms, and we also demonstrate the practical suitability of the log-MAP by including quantization effects. The SOVA is, at 10/sup -4/, approximately 0.7 dB inferior to the (log-)MAP, the max-log-MAP lying roughly in between. We also present some complexity comparisons and conclude that the three algorithms increase in complexity in the order of their optimality.

Two-dimensional pilot-symbol-aided channel estimation by Wiener filtering

The potential of pilot-symbol-aided channel estimation in two dimensions are explored. In order to procure this goal, the discrete shift-variant 2-D Wiener filter is derived and analyzed given an arbitrary sampling grid, an arbitrary (but possibly optimized) selection of observations, and the possibility of model mismatch. Filtering in two dimensions is revealed to outperform filtering in just one dimension with respect to overhead and mean-square error performance. However, two cascaded orthogonal 1-D filters are simpler to implement and shown to be virtually as good as true 2-D filters.

Optimal and sub-optimal maximum a posteriori algorithms suitable for turbo decoding

For estimating the states or outputs of a Markov process, the symbol-by-symbol maximum a posteriori (MAP) algorithm is optimal. However, this algorithm, even in its recursive form, poses technical difficulties because of numerical representation problems, the necessity of non-linear functions and a high number of additions and multiplications. MAP like algorithms operating in the logarithmic domain presented in the past solve the numerical problem and reduce the computational complexity, but are suboptimal especially at low SNR (a common example is the Max-Log-MAP because of its use of the max function). A further simplification yields the soft-output Viterbi algorithm (SOVA). In this paper, we present a Log-MAP algorithm that avoids the approximations in the Max-Log-MAP algorithm and hence is equivalent to the true MAP, but without its major disadvantages. We compare the (Log-)MAP, Max-Log-MAP and SOVA from a theoretical point of view to illuminate their commonalities and differences. As a practical example, we consider Turbo decoding, and we also demonstrate the practical suitability of the Log-MAP by including quantization effects. The SOVA is, at 10−4, approximately 0.7 dB inferior to the (Log-)MAP, the Max-Log-MAP lying roughly in between. The channel capacities of the three algorithms -when employed in a Turbo decoder- are evaluated numerically.

Bandwidth-efficient turbo trellis-coded modulation using punctured component codes

We present a bandwidth-efficient channel coding scheme that has an overall structure similar to binary turbo codes, but employs trellis-coded modulation (TCM) codes (including multidimensional codes) as component codes. The combination of turbo codes with powerful bandwidth-efficient component codes leads to a straightforward encoder structure, and allows iterative decoding in analogy to the binary turbo decoder. However, certain special conditions may need to be met at the encoder, and the iterative decoder needs to be adapted to the decoding of the component TCM codes. The scheme has been investigated for 8-PSK, 16-QAM, and 64-QAM modulation schemes with varying overall bandwidth efficiencies. A simple code choice based on the minimal distance of the punctured component code has also been performed. The interset distances of the partitioning tree can be used to fix the number of coded and uncoded bits. We derive the symbol-by-symbol MAP component decoder operating in the log domain, and apply methods of reducing decoder complexity. Simulation results are presented and compare the scheme with traditional TCM as well as turbo codes with Gray mapping. The results show that the novel scheme is very powerful, yet of modest complexity since simple component codes are used.

Illuminating the structure of code and decoder of parallel concatenated recursive systematic (turbo) codes

A coding scheme (turbo codes) was proposed, that achieves almost reliable data communication at signal-to-noise ratios very close to the Shannon-limit. We show that the associated iterative decoder can be formulated in a simpler fashion by passing information from one decoder to the next using log-likelihood ratios as opposed to channel values that need to be normalized. Also no heuristically determined correction parameters are necessary for stable decoding. In addition, we can reduce the average number of iterations needed for the same BER performance by determining when further iterations achieve no more benefit. Furthermore, it seems that the trellis-termination problem appears non-trivial and we give a pragmatic suboptimal solution. We investigate different block sizes and also a hybrid scheme that performs extremely well with less computations. A drawback of the codes has been discovered: the BER curves show a flattening at higher signal-to-noise ratios, this is due to the small minimum distance of the whole code. By analyzing the interleaver used in the encoder we can calculate approximations to the BER at high SNRs. Finally, by careful interleaver manipulation the minimum distance of the code can be increased and the error-coefficient for the remaining small distance events can be further reduced. Furthermore, we have investigated the influence of the interleaver length on the SNR needed to achieve a certain BER. Simulations confirm both the analytical approximation to the BER as well as the method for interleaver design which yields a marked improvement at higher SNR.

Pilot-symbol-aided channel estimation in time and frequency

The potentials of pilot-symbol-aided channel estimation in two dimensions are explored for mobile radio and broadcasting applications. In order to procure this goal, the discrete shift-variant 2-D Wiener filter is analyzed given an arbitrary sampling grid, an arbitrary (but possibly optimized) selection of observations, and the possibility of model mismatch. Filtering in two dimensions is revealed to outperform filtering in just one dimension with respect to overhead, mean-square error performance and latency. Conceptually, the discrete shiftvariant 2-D Wiener filter is the optimal linear estimator for the given problem, however, two cascaded orthogonal 1-D filters are simpler to implement and virtually as good as true 2-D filters. Analytical results are presented, verified by Monte-Carlo simulations.

Analysis of the effects of phase-noise in orthogonal frequency division multiplex (OFDM) systems

In OFDM transmission schemes, phase-noise from oscillator instabilities in the receiver is a potentially serious problem, especially when bandwidth efficient, high order signal constellations are employed. The paper analyses the two effects of phase-noise: inter-carrier interference (ICI) and a phase error common to all OFDM sub-carriers. Through numerical integration, the ICI power can be evaluated and is shown as a function of the number of OFDM sub-carriers and various parameters of the phase-noise model. Increasing the number of sub-carriers causes an increase in the ICI power, which our analysis indeed shows to become a potential problem, since it can lead to a BER floor. The analysis allows the design of low-cost tuners through specifying the required phase-noise characteristics. A similar technique is applied to calculate the variance of the common phase error. After showing that the common phase error is essentially uncorrelated from symbol to symbol, we propose a simple feed-forward correction technique based on pilot cells, that dramatically reduces the degradation due to phase-noise. This is confirmed by BER simulations of a coded OFDM scheme (proposed for terrestrial transmission of digital television) with 64 QAM.

Improved decoding with the SOVA in a parallel concatenated (Turbo-code) scheme

We show that the soft output of the soft-output Viterbi-decoder (SOVA) suffers from two distortions. Firstly, for bad channels the reliability information of the decoder output is too optimistic. The output can be considered as being multiplied by a factor, that depends on the current bit-error-rate (BER). To become closer to the true log-likelihood ratio the output has to be normalized. Secondly, the soft-output of the SOVA-when used in Turbo decoding-is effected by a correlation between the so called extrinsic and intrinsic information. Since the extrinsic information is fed forward to be the a-priori information in the next decoding stage and is treated as being uncorrelated with the systematic-information, a correcting term has to be introduced in order to compensate this correlation. We show how these distortions can be corrected and we apply the correcting measures on the SOVA that is used as component decoder of a Turbo-code scheme. Finally, we provide simulation results that illustrate the gain due to normalizing the extrinsic information and exploiting the correlation in the soft output.

Analysis of the loss of orthogonality through Doppler spread in OFDM systems

The paper quantifies the significant effects of Doppler spread in a time variant mobile radio channel on the performance of OFDM multi-carrier systems. A suitable model for OFDM is introduced and serves as the basis for the derivation of the inter-carrier interference (ICI) due to the loss of sub-carrier orthogonality. Several Doppler spectra are analyzed and compared, such as the classical and uniform distributions, as well as the case of two distinct values for the Doppler frequency. We can thus employ the computed variance of the ICI process, together with the channel noise, in system analyses and designs, and avoid lengthy time-domain simulations. We also analyze optimal receiver frequency synchronisation in the case of Doppler spreads.

Integration of foot-mounted inertial sensors into a Bayesian location estimation framework

An algorithm for integrating foot-mounted inertial sensors into a Bayesian location estimation framework is presented. The proposed integration scheme is based on a cascaded estimation architecture. A lower Kalman filter is used to estimate the step-wise change of position and direction of the foot. These estimates are used in turn as measurements in an upper particle filter, which is able to incorporate nonlinear map-matching techniques. Experimental data is used to verify the proposed algorithm.