Phase Offset Estimation using Enhanced Turbo Decoders Bartosz Mielczarek and Arne Svensson Communication Systems Group Department of Signals and Systems Chalmers University of Technology SE-412 96 Göteborg, Sweden PH: +46 31 772 1763 FAX: +46 31 772 1748 Abstract - This paper discusses a realistic turbo coding system with the signal phase which has not been perfectly estimated. We propose improved decoding algorithms for the situations when the residual phase error can be modelled by the Gaussian proba- bility distribution and a Markov chain, a model which can be used in many actual phase estimators. It is shown that increasing the state space of the decoders can decrease the bit error probability. I. INTRODUCTION One of the most important factors determining the efficien- cy of a wireless system is the power required for successful transmission of data. The required power needed to reliably transmit a signal can be reduced by using special coding tech- niques. One of the latest and the most prominent of such tech- niques is turbo coding with a performance which is almost able to reach the Shannon channel capacity limit [1]. The problem lies, however, in the practical applications of the turbo codes. Unfortunately, reducing the transmitted power makes it more difficult to estimate the channel and properly synchronize the phase of the incoming signal. In our paper we propose algorithms aiming at improving performance of turbo-coded systems with non-perfect phase offset estimation. II. PHASE SYNCHRONIZATION In the majority of digital wireless communication systems, the incoming HF signal needs to be downconverted to lower frequency ([3]). The good frequency and phase synchroniza- tion is therefore essential for the reliability of wireless systems. There exist a number of different techniques estimating carrier parameters (such as Phase Locked Loops, Costas loops etc.) but none of these algorithms succeed to provide perfect estimation of the carrier signal. One of the reasons for such performance loss is that synchronization of phase is done be- fore the decoding process and cannot use the code properties to improve its accuracy. This is due to the fact that most decoders will not work without a proper phase estimation and must rely on some initial estimates of the signal phase. If, on the other hand, the phase estimator/decoder knows the structure of the data signal, it can use this knowledge in joint phase and data estimation and improve the system’s performance. We will use this approach for the algorithms presented in this paper. III. SYSTEM MODEL The system analysed in this paper is presented in Fig. 1. A typical turbo encoder ([1],[2]) of rate 1/3 generates codewords consisting of systematic bits and the parity bits , . The stream of code bits is BPSK-modulated and trans- mitted over an AWGN channel as real-valued signal samples . The encoded signal suffers from a phase noise process (which can be a result of a fading process or oscillator instabil- ity) and is corrupted by the white, Gaussian noise. The incom- ing distorted signal is fed to the phase synchronizer, which produces estimates of the phase noise process . After adjust- ing the phase error, the signal is decoded (the decoded data se- quence can then be used to refine the phase estimation process but this problem is not addressed in this paper). Formally, the signal after the AWGN channel, phase esti- mation and receiver matched filtering (the timing recovery is assumed to be perfect) can be expressed as , (1) where is the complex received signal (which can be the sys- tematic bit , the first parity bit or the second parity bit ) and is the white, additive, Gaussian complex noise with . is the residual phase offset estima- tion error ; its statistics are discussed in the next section. Note that the amplitude of the signal is assumed to be constant, i.e. the fading is compensated by a perfect power control. The extension of the channel model to the non-com- pensated fading channels is relatively straightforward and will not be discussed in this paper. IV. RESIDUAL PHASE ERROR MODELLING The residual phase error can be modelled as Gaussian distributed, with known variance and zero mean for non- biased phase estimators (which are the most common solutions [3]). Such an assumption is rather popular in the existing liter- ature and seems to be quite realistic since typical synchronizers produce an error distribution with a similar shape and known variance (for example, the Tikhonov distribution after the PLL loop, see [5]). x k s x k p 1 , x k p 2 , c k θ k θ ˆ k y k e j φ k c k n k + = y k y k s y k p 1 , y k p 2 , n k E n k 2 [ ] N 0 = φ k φ k θ ˆ k θ k – = φ k σ φ 2