Differentially-Encoded Turbo Coded Modulation with APP Channel Estimation

Differentially-Encoded Turbo Coded Modulation with APP Channel Estimation Sheryl Howard Dept. of Electrical & Computer Engineering University of Alberta Edmonton, AB Canada T6G 2V4 Email: sheryl@ee.ualberta.ca Christian Schlegel Dept. of Electrical & Computer Engineering University of Alberta Edmonton, AB Canada T6G 2V4 Email: schlegel@ee.ualberta.ca Abstract A simple serially concatenated turbo code using differential 8PSK encoding as the inner code and a [3,2,2] parity code as the outer code is studied. This system is decoded according to turbo principles with iterative exchange of extrinsic probabilities, without differential demodulation. Decoding over channels without prior synchronization is demonstrated to be feasible even with significant phase offset and phase noise, using a simple channel estimator that utilizes the extrinsic output symbol probabilities from the differential APP decoder. I. INTRODUCTION Coherent detection assumes perfect carrier phase knowledge at the receiver. If the received phase is rotated from the transmitted phase due to channel or local oscillator noise, the receiver is typically unable to decode correctly. Higher-order PSK constellations, while increasing spectral efficiency, are particularly sensitive to phase noise. As phase synchronization becomes more important for these higher order constellations, it conversely becomes difficult to achieve. PLLs (phase-locked loops) or Costas loops are often used for phase synchronization but result in phase ambiguities for PSK constellations. The squaring loss for higher order PSK modulation also becomes significant; for 8-PSK suppressed-carrier signalling, the squaring loss of an eighth-power-law device at =9 db is upper-bounded by -1 db [1] with respect to PLL operation on an unmodulated carrier. Additionally, at the near-capacity SNRs of turbo code operation, a loop cannot acquire phase lock quickly, if at all [2]. Wireless packet messaging is especially sensitive to phase acquisition time due to short messages. Thus we consider an alternate method of synchronizing phase. A known training sequence may be sent to estimate the channel, which incurs a rate loss. Differential encoding with differential demodulation is the classic technique for decoding without phase synchronization; however, a 3 db loss in SNR vs BER occurs for M-PSK, [3]. Differential turbo coded modulation concatenates an error-control code with a differential PSK encoder as the inner modulation code. Iterative (turbo) decoding [4], [5] is applied to decode the received signal. Differential BPSK modulation resulted in a 2.7 db loss in SNR [6] for a rate 1/2 turbo code. Differential modulation has been applied to space-time coding [7] and serial concatenation of error-control codes with a differential space-time code providing results 2.3 db from capacity in [8]. Differential QPSK modulation concatenated with a convolutional code has been decoded iteratively over multiple symbols in [9]; linear prediction results in a decoding trellis expansion states, for predictor order. Differential BPSK modulation, serially concatenated with a rate 1/2 code, is presented in [1] using discretized phase, resulting in an expanded-state decoding trellis. Similarly, channel estimation for fading channels using quantized phase in an expanded supertrellis for iterative decoding of turbo codes with QPSK modulation is considered in [11]. Rather than expand an already complex trellis, in [12] channel estimation of PSAM (pilot-symbol-assisted modulation) turbo codes is moved outside the APP decoder yet within the iterative decoding block. We adopt a similar approach regarding channel estimation within the iterative block to use APP soft information while minimizing trellis complexity, without pilot symbols. from to This paper examines an efficient yet simple serially concatenated system composed of a [3,2,2] parity check code as outer code and differential 8PSK encoding as inner code. This system is decoded iteratively by turbo principles, without differential demodulation. The differential modulation code functions as a strong inner code by itself and has a rotationally invariant trellis (to phase multiples) which is used to advantage with channel estimation. A simple channel estimation method using the extrinsic symbol probabilities from the inner APP decoder, which we term APP channel estimation, is introduced to provide phase estimation. Our channel estimation is external to the APP decoder so does not increase the complexity of the decoding trellis. Both coherent detection and decoding without channel state information (CSI) are considered. The paper is organized as follows: Section 2 describes the serially concatenated parity code/differential 8PSK system. Section 3 discusses EXIT analysis of this system. Section 4 describes a method of obtaining channel estimates when the system is decoded without CSI, termed APP channel estimation. Section 5 presents simulation results. Conclusions are discussed in Section 6.

Z T II. SYSTEM DESCRIPTION Figure 1 displays the encoder. A sequence of information bits of length is encoded through the [3,2,2] parity code. The coded bits into a sequence of coded bits of length are then bitwise interleaved. These interleaved bits are mapped to 8PSK symbols, where. The 8PSK symbols serve as input to the differential encoder. Differential encoding consists of multiplying the current input Fig. 1. [3,2,2] Encoder Interleaver (11) (1) (1) (11) (1) () (11) (111) Differential Encoder Serial Turbo Encoder for Differential Turbo Coded Modulation symbol with the previously transmitted symbol "!# to obtain the current transmitted symbol$ &'!#( ). Initially, +*,-.*. The differential 8PSK symbols are then transmitted across an AWGN channel with noise variance /* in each dimension. Under coherent detection, the received symbols consist of the transmitted symbols plus complex noise with variance *, i.e., 1. We view the differential,243 encoder as an inner code of the serially concatenated system. It can be seen as a recursive non-systematic convolutional code, with a regular, fully-connected 8-state trellis. The rate of this system is 2 bits/symbol. Decoding of the serially concatenated system proceeds iteratively according to turbo decoding principles [4], [5]. No differential demodulation is used; APP decoder 1 operates on the trellis of the differential code, thus the 3 db penalty for differential demodulation is not incurred. Figure 2 displays the decoding process, with the APP channel estimation block shown in the dashed rectangle. For now, we consider coherent detection, assuming perfect CSI without channel estimation. The received channel symbols are converted into channel metrics 5 6879;: <>=? @ # BA!DC E!<F C GH I which are fed into APP decoder 1 for the differential code, along with a priori information JLKM68?= from APP decoder 2. In the first iteration, APP decoder 1 has no a priori information, and assumes uniform a priori values. [ X YX APP 1 D8PSK Soft- Output Decoder VU F W CVU F W C J K 68 O= div JDN>68 O= JDNP6R <= Interleave Combine Deinterleave Marginalize JLNP6RQS= J K 6RQS= APP 2 [3,2,2] Soft- Output Decoder Fig. 2. Serial Turbo Decoder for Differential Turbo Coded Modulation with APP Channel Estimation Using the BCJR [13] algorithm, APP decoder 1 calculates symbol probabilities on both the 8PSK symbols and the transmitted D8PSK symbols, and extrinsic 8PSK symbol probabilities JLNP6R\= are passed on to APP decoder 2. Extrinsic probabilities are first found by dividing out the _ corresponding a priori symbol probabilities J K 68 ]BA^ @ Ha` I = and then normalizing such that the extrinsic symbol probabilities sum to one. These extrinsic symbol probabilities are converted to bit probabilities through marginalization before being deinterleaved, as the interleaver works bitwise. The bit probabilities J N 68QSbc= are now deinterleaved and fed into APP decoder 2 as a priori bit probabilities JLKM6RQS=. The [3,2,2] parity code is simple enough that its APP decoder can be implemented as 6 equations, giving extrinsic probabilities that express the parity constraints as J N 68QB#)ed'=gf JDK;68Q 1hB=iJLKM6RQ jd"= 2 JLKS68Q ed'=kjdks6rq lh9= (1) and analogously for Q # mh, Q and Q. The bit probabilities are then interleaved to provide a priori bit probabilities JnKM68Q b =, which are converted back to symbol probabilities JLKS6R O= for the next iteration of APP decoder 1. The probability of a symbol is simply the normalized product of its component bit probabilities. Iterative decoding continues, with APP decoders exchanging extrinsic information until convergence is reached. III. EXIT ANALYSIS Turbo coded systems can be analyzed very elegantly by a method known as EXIT analysis [15], [16]. The reliability of the extrinsic soft information generated by each component decoder is measured by the mutual information op6rqyst= between the extrinsic information and actual symbols u associated with that soft information. Likewise, the reliability of the a priori information v into the same decoder is measured by op6rqysvo=. Plots of o<6wq$svo= versus op6rqyst=, known as extrinsic information transfer (EXIT) charts, can be used to study the convergence behavior of iterative decoding systems. Mutual information is unchanged by the interleaving process; interleaving scrambles the symbols but leaves the first order distribution unchanged. Furthermore, the interleaver destroys any correlation between successive symbols. Using this separation assumption, the component decoder EXIT charts may be combined into a single EXIT graph which accurately describes the behavior of the iterative turbo decoding process. The outer parity decoder produces soft information x N 68QM= and x K 68QM= on the bit level, which are processed as LLRs and v. Since the inner differential code operates on 8PSK symbols, x N 68 O= and x K 6R O= must be converted from the interleaved bit probabilities xy68qmbc=. Figure 3 shows the EXIT chart for our system with the differential 8PSK curve as the inner decoder (o{z # =op6w}$b~sv#{= on the horizontal axis, o # =o<6~}tbrs # = on the vertical axis) and the [3,2,2] parity check curve as the outer decoder, with

[ [ o N swapped axes. Only the inner decoder EXIT curves depend on SNR. The significant advantage of EXIT analysis is that the turbo decoder performance near the turbo cliff region may be predicted without running simulations of the complete turbo decoder; EXIT transfer curves are obtained for each individual decoder. From Figure 3, we see that the mutual information 1.9.8.7.6.5.4.3.2.1 Differential 8PSK SNR=5 db SNR=5 db.1.2.3.4.5.6.7.8.9 1 Fig. 3. EXIT chart with trajectory for serial concatenation of [3,2,2] parity code with D8-PSK code at SNR=5 db. values for the serially concatenated system, indicated by the trajectory in blue, match well with the predicted individual decoder EXIT curves. At SNR 5 db, an open iteration channel exists and convergence occurs in 15 iterations. Each verticalhorizontal step indicates one complete iteration of decoding. Decreasing SNR values lower the differential 8PSK EXIT curve. At the turbo cliff, a narrow channel exists between the component code EXIT curves, allowing only minimal error rate improvement per iteration, resulting in a large number of iterations to reach convergence. IV. DECODING WITHOUT CHANNEL INFORMATION We now consider the case when the received channel phase is unknown and we decode without channel information. The differential outer code allows the receiver to extract soft information on the symbols and even in the absence of channel knowledge. This is achieved through the APP decoder, without differential decoding. APP decoder 1 generates extrinsic input symbol probabilies J 68 =, as well as extrinsic output symbol probabilities J 6R = which will be used to feed a channel estimator for use in the following iteration. This channel estimator should be of low complexity; thus an optimal linear estimator such as the minimum mean square error (MMSE) estimator is not feasible and a simpler filtering estimator [8] is considered below. Assuming the channel model to be 7S " < 2, where is a complex time-varying gain, we find from the first moment equation that 7 (2) o K where the expectation of the symbol, _ * J (3) is taken over the a posteriori probabilities J 68 >= generated by APP decoder 1. Normalizing eqn. 3 to lie on the unit circle gives X ;: :. Note that while h, since the APP J 68 = are not uniform due to h O "! the differential code, as can be shown by EXIT analysis. A channel estimate may be found as X Yl7B X Z (4) As the a posteriori probabilities J 6 = form the channel estimates X, we term this procedure APP channel estimation. Figure 2 shows the iterative decoding process with the APP channel estimation block enclosed in the dashed rectangle. APP decoder 1 sends its extrinsic J N 68 O= to APP decoder 2 and APP 2 generates a priori JLKS6R O= for use in the next iteration. The channel estimate X is used to calculate coherent channel metrics for APP 1 in the next iteration as J 6879;: &>= f $#& : 79 X P <;: (' (5) Each iteration improves the extrinsic values J 6R <= from APP 1, and an improved channel estimate X can be determined at each iteration. We consider a channel with time-varying phase offset and unity gain, i.e., *)P. Two different channel phase models are examined: 1) a constant phase offset, and 2) a random walk phase process. 1) For a constant phase offset, *). The individual APP channel estimates X + provide phase estimates X, which + are averaged to obtain a constant phase estimate X. Figures IV and IV show simulation results of the channel phase estimation for a constant phase offset vs. iterations. A phase offset of Md(, rads can be compensated for with our APP channel estimation method in 1 iterations at SNR 4.8 db; a phase offset of requires 25 iterations. A feature of the differential 8-PSK trellis works to our advantage in this estimation process, that is, the rotational invariance of the differential 8-PSK trellis to multiples of rads phase offset. In the absence of noise, a channel phase offset of rotates the symbols in the transmitted sequence and thus cyclically permutes each state in the traversed state sequence. This results in incorrectly decoded transmitted symbols but correctly decoded 8-PSK symbols. The system can thus coherently decode any channel phase rotation of an integer multiple of, without absolute CSI. Any phase rotation only needs to be corrected through channel phase estimation to the closest integer multiple of rads with our system. A phase rotation of modulo rads will be the most difficult to estimate, as it lies halfway between two valid phase values. Forcing the differential trellis beginning and end states to be state will cause endpoint errors for a phase offset of multiple. The rest of the trellis shifts to a rotated sequence,

[ but those points are pegged at state. Therefore, we use a floating trellis, where both beginning and end states are assumed unknown and set to uniform probabilities. 2) The random + walk phase process is a Markov process described by + + '!# 2, where 2 is the channel phase offset at symbol interval, is a zero-mean Gaussian distributed random phase with variance, and is a constant channel phase rotation. Initial channel estimates are found as per equation 4. These initial estimates are then filtered through a moving average filter with exponential decay, corrected for lag, to obtain 6id L= * '! X. The filtered channel estimates are used to calculate improved channel metrics in the next iteration. Estimated Phase in Degrees 12 1 8 6 4 2 Fig. 4. 2 15 1 5 SNR=4.8 db dbd Md(, rads 5 1 15 2 25 3 35 4 45 5 Iterations Channel Phase Estimation vs. Iterations, offset Estimated Phase in Degrees Fig. 5. rads 5 1 15 2 25 3 35 4 45 5 Iterations SNR=4.8 db Channel Phase Estimation vs. Iterations, "! offset The rotational invariance of the differential 8-PSK trellis to multiples of phase rotation is displayed in Figure 6, which shows the random walk channel phase at top and the APP channel phase estimate beneath. The phase estimate slips twice to a phase rotated by rads from the actual channel phase. However, there are no decoding errors, even at the phase discontinuities, due to the rotationally invariant trellis. Phase in Rads 1.8.6.4.2 -.2 -.4 -.6 -.8 random walk phase estimated phase random walk channel phase estimated channel phase decoding errors -1 1 2 3 4 5 6 8PSK symbols Fig. 6. Random walk channel phase model and estimated phase with # phase slips, decoded without errors. V. SIMULATION RESULTS Simulation results are provided for the [3,2,2] parity code/differential 8PSK modulation under coherent operation and using channel estimation when the channel phase is unknown. Two different channel phase models are simulated: a constant phase offset and a random walk phase process that varies with each symbol. Figure 7 shows results for the [3,2,2] parity outer code with differential 8PSK modulation as inner code, with coherent decoding. A block length of 1 information bits is used. 8- PSK capacity at a rate of 2 bits/symbol is at $ $*O 2.9 db. An 8-PSK mapping described in [18], designed to increase the minimum distance of this system, which is 1.172 for natural mapping, is used. Natural mapping provides a.2 db advantage in turbo cliff onset, at the cost of a higher error floor [18]. Bit Error Probability (BER) 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 5 iterations 1 iterations 2 iterations 5 iterations 5 iters 2 iters 1 iters 5 iters 3 4 5 6 7 8dB Fig. 7. Performance of the serially concatenated D8-PSK system with outer [3,2,2] parity code. '&

Simulation results for APP channel estimation with a constant channel phase offset of Md(, rads are shown in Figure 8. Performance degrades somewhat as phase offset approaches rads. Random walk phase results with channel estimation are provided in Figure 9 with =.99 and. Near- BER 1 3 1 4 1 5 1 6 1 7 h 1 5 iterations 1 1 1 1 iterations 2 iterations 5 iterations 5 coherent decoding 1 2 5 iterations 1 iters 2 iters coherent results 3 4 5 6 7 8dB Fig. 8. BER results without CSI, phase offset= rads, using APP channel estimation BER 1 1 1 1 2 1 3 1 4 1 iters 1 5 25 iters 1 6 1 iterations 5 iters 25 iterations 5 iterations coherent & 1 7 coherent decoding results 3 4 5 6 7 db Fig. 9. BER results without CSI, random walk phase process, using APP channel estimation coherent performance is achieved without CSI using APP channel estimation. Use of training symbols would improve performance further. VI. CONCLUSIONS It was shown that a simple serial concatenated system composed of a [3,2,2] parity code as outer code and differential & 8PSK modulation as inner code provides very good results when iteratively decoded. Higher order modulation provides good spectral efficiency, while the differential code has a rotationally invariant trellis which aids in channel estimation. Near-coherent performance without CSI is achieved with a simple channel estimation technique using the extrinsic information from the inner APP decoder, without pilot symbols or differential demodulation. This system is easily encoded and decoded, and could be used with packet transmission, where short messages increase the need for phase offset immunity. ACKNOWLEDGMENT Many thanks to Lovisa Björklund for generating the data for the EXIT chart, and to Alex Grant for helpful discussions. REFERENCES [1] S.A. Butman, J.R. Lesh, The Effects of Bandpass Limiters on n-phase Tracking Systems, IEEE Trans. on Inform. Theory, vol. 25, June 1977, pp. 569-576. [2] H. Meyr and G. Ascheid, Synchronization in Digital Communications, Vol. 1, Wiley Series in Telecommunications, 199. [3] J.G. Proakis, Digital Communications, 3rd edition, McGraw-Hill, 1995. [4] C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit errorcorrecting coding and decoding: Turbo codes, Proceedings of the IEEE International Conference on Communications, Geneva, Switzerland, 1993, pp. 164-17. [5] S. Benedetto, D. Divsalar, G. Montorsi and F. Pollara, Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding, IEEE Trans. on Inform. Theory, 44(3), May 1998, pp. 99-926. [6] E.K. Hall and S.G. Wilson, Turbo codes for noncoherent channels, Proc. GLOBECOM 97, Nov. 1997, pp. 66-7. [7] B.L. Hughes, Differential space-time modulation, IEEE Trans. on Inform. Theory, 46(7), Nov. 2, pp. 2567-2578. [8] C. Schlegel and A. Grant, Concatenated space-time coding, Proc. Symposium on Personal, Indoor and Mobile Radio Commun., PIMRC 21, pp. 139-143, Sept. 3-Oct. 3, San Diego, CA. [9] P. Hoeher and J. Lodge, Turbo DPSK : Iterative differential PSK demodulation and channel decoding, IEEE Trans. on Commun., 47(6), June 1999, pp. 837-843. [1] M. Peleg, S. Shamai and S. Galán, Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel, IEE Proc. Commun., vol. 147, Apr. 2, pp. 87-95. [11] C. Komninakis and R. Wesel, Joint Iterative Channel Estimation and Decoding in Flat Correlated Rayleigh Fading, IEEE J. Sel. Areas Commun., vol. 19, No. 9, Sept. 21, pp. 176-1717. [12] M.C. Valenti and B.D. Woerner, Iterative channel estimation and decoding of pilot symbol assisted turbo codes over flat-fading channels, IEEE J. Sel. Areas Commun., vol. 9, Sept. 21, pp. 1691-176. [13] L.R. Bahl, J. Cocke, F. Jelinek and J. Raviv, Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate, IEEE Trans. on Inform. Theory, vol. 2, Mar. 1974, pp. 284-287. [14] C. Schlegel, Trellis Coding, IEEE Press, Piscataway, NJ, 1997. [15] S. ten Brink, Convergence behavior of iteratively decoded parallel concatenated codes, IEEE Trans. on Commun., 49(1), Oct. 21, pp. 1627-1737. [16] S. ten Brink, Design of serially concatenated codes based on iterative decoding convergence, in 2nd International Symposium on Turbo Codes and Related Topics, Brest, France, 2. [17] A. Grant and C. Schlegel, Differential turbo space-time coding, Proc. IEEE Information Theory Workshop 21, pp. 12-122, Cairns, Sept. 21. [18] S. Howard, C. Schlegel, L. Pérez, F. Jiang, Differential Turbo Coded Modulation over Unsynchronized Channels, Proc. of IASTED 3rd Int. Conf. on Wireless and Optical Communications, Banff, Alberta, Canada, 22, pp. 96-11.