Iterative Bi-directional Kalman-DFE Equalizer for the High Data Rate HF Waveforms in the HF Channel Mahmoud A. Elgenedy Comm. System Design Varkon Semiconductors Cairo, Egypt melgenedy@varkonsemi.com Essam Sourour Dept. of Electrical Engineering Faculty of Engineering, Alexandria University Alexandria, Egypt sourour@ieee.org Magdy Fikri Dept. of Electrical Engineering Faculty of Engineering, Cairo University Cairo, Egypt magdi.fikri@gmail.com Abstract— In this paper, we present the Kalman decision feedback equalizer (Kalman-DFE) as a solution for the high data rate HF waveforms. We show that the basic performance for the Kalman-DFE is far from the standard requirements of the HF and the performance breaks down for the higher constellation. As a result, we add two major enhancements for the Kalman- DFE: the bi-directional and iterative structures. The bi- directional structure successfully prevents the performance break down at the higher constellation and enhances the overall performance. The iterative structure then enhances the performance significantly (about 6dB gain at 1e-5 BER for 64QAM after three iterations). We show that the proposed equalizer in the final form achieves significant performance enhancements over the normal one. Moreover, the proposed iterative Kalman-DFE is much simpler than turbo equalizer, especially for the fractional-spaced model, for two reasons: first, the direct adaptation structure used in Kalman filter is much simpler than the indirect adaptation structure usually used with the turbo structure (it does not require matrix inversion). Secondly, the iterative structure (exchanges hard information) is simpler than turbo (exchanges soft information) as the later requires a lot of changes in both equalizer and decoder to support soft information. We perform the simulations on the MIL-STD-188-110 (Appendix C) waveforms, transmitted over an ITU-R poor channel (commonly used channel to test HF modem). Keywords— HF modems, Kalman equalizer, DFE, Iterative equalization. I. INTRODUCTION Transmission in the high frequency band (HF, 3 to 30MHz) has a lot of difficulties and challenges due to severe channel fading, especially at high latitudes, which impedes the mitigation (equalization) process. The task of equalizer becomes more difficult when high data rates (high constellation orders) are assumed. Various types of Kalman-DFE in HF channel for a medium data rate are proposed by many researches [1-4]. Eleftheriou and Falconer in [1] propose both LMS (the least mean square) and FRLS (the fast recursive least squares) adaptation algorithms with periodic restart. They indicate that FRLS adaptation yields a superior performance to LMS in rapid fading conditions. A more numerically stable algorithm for fixed point implementation is the square root Kalman, presented by HSU in [4]. Bi-directional DFE processing has been previously proposed in [5-7]. The time-reversal equalization mode improves decision feedback equalizer (DFE) with a small number of forward filter taps to perform equally well for both minimum-phase and maximum-phase channel characteristics. It converts the non minimum-phase to the minimum-phase condition by the time-reversal operation. The mean-squared error (MSE) between the input and output of the DFE decision device is used as the criterion to choose between the results of forward and reverse processing in [6] and [7]. The MSE criterion is applied to an entire frame of data. In contrast to the preceding global MSE criterion, a local MAP decision between two candidate sequences is used in [5], since each decision is based only on a window around the bit of interest. In this paper, we use the global MSE as selection criteria. Most of equalizer errors result from the bad reliability of hard decisions of equalizer output, which affect the adaptation process. The main idea behind iterative or turbo structures is to exchange the information (hard or soft information) between the decoder and the equalizer iteratively (through an interleaver). A returned data from decoder are much more reliable than equalizer output hard decisions and are improved with iterations. The main difference between iterative and turbo (in this paper) is the type of data exchanged. If hard information is used, we call this iterative, but the famous expression (turbo) is used with soft information. 978-1-4673-2821-0/13/$31.00 ©2013 IEEE In the case of the iterative structure, we use the same equalizers and decoders as conventional receivers. The main difference as explained previously is to use the data returned from decoder instead of hard decisions of equalizer output. Returned data from decoder should first pass on the interleaver then the encoder and finally to the modulator. The iterative structure enhances the estimated error signal used for equalizer adaptation. In case of Decision feedback equalizers, the use of good quality decoder output data instead of equalizer output decisions enhances the ISI cancellation. The structure of turbo equalizer is quite similar to the iterative structure except exchanging soft values (LLR) instead of hard decisions. Exchanging soft values requires some modifications for both equalizer and decoder, as both of them should accept soft information at its input and provide soft information as output (SISO modules). Also, the modulator block is replaced by another block which accepts input soft values and provides both mean and variance for encoded symbols. Finally, turbo equalizer structure can be viewed as serially concatenated SISO modules. Linear MMSE with soft ISI cancellation in a turbo equalization structure (assuming symbol-spaced model) is proposed by Roald Otnes in [8] for the HF medium data rate standard, and he extends his results for high data rates in [9]. In this paper, we first investigate the performance of basic Kalman-DFE with detailed tuning for all equalizer parameters. The bi-directional structure is then tested and successfully prevents the performance break down at the higher constellation. Finally, the iterative structure achieves a significant performance enhancement (about 6 dB gain at 1e-5 BER for the 64QAM using three iterations). The rest of the paper is organized as follows. The next section presents the system model. In Section III, a study for the normal Kalman-DFE performance is introduced. Section IV presents a trial of testing the Kalman-DFE in the backward direction. In Section V, we present the bi-directional structure. Finally, Section VI shows the final form of the iterative bi- directional Kalman-DFE structure and the final BER performance results. II. SYSTEM MODEL A. Transmitter The transmitter structure for high data rate serial tone HF is defined in (Annex B of STANAG 4539 and Appendix C of MIL-STD-188-110B). The transmitter starts with the blocking operation which is responsible of dividing the input raw bits into blocks of size dependent on the rate and the interleaver length. Each code block shall be interleaved within a single interleaver block of the same size. The full-tail-biting and puncturing techniques shall be used with a rate of 1/2 convolutional code to produce a rate of 3/4 block code that is the same length as the interleaver. The interleaver output is then scrambled and input to modulation block. The modulation used (QPSK, 8PSK, 16QAM, 32QAM and 64QAM) shall depend upon the data rate (3200, 4800, 6400, 8000 and 9600 bps). An 8PSK training symbols (initial preamble, mini-probes and reinserted preamble) are then inserted within the data frames (256 symbols per frame) through the framing process. Finally, the output frames are filtered using SRRC filter (roll-off 0.25 – filter length 12 symbol time) to get the transmitted data samples . B. Channel Model In an ionospheric HF communication system, the transmitter and receiver are not moving (or moving slowly relative to the wavelength), while the radio waves are being reflected by a large number of randomly moving ions. This suggests that the Doppler shift has a Gaussian distribution, as was verified experimentally by Watterson in [10]. A Gaussian Doppler spectrum can be written as                     (1) where   is the mean value and    is the variance of the Doppler shift. For a Gaussian fading spectrum, the Doppler spread   is by convention defined as twice the standard deviation of the Doppler shift (    ). Using this definition and assuming that   is zero, (1) becomes,                   (2) C. Receiver Model The receiver operations start with the SRRC filter matched to the TX (could be removed if fractional equalizer is used) followed by the equalization process then the reverse operations for the transmitter blocks. The decoder is a Viterbi decoder supports tail biting initialization. The receiver assumes perfect time and frequency synchronization. III. F ORWARD KALMAN FILTER First, we introduce the normal operation for the Kalman- DFE which is the case of adapting in the forward direction. Fig. 1 shows the equalizer structure, where [n] is the equalizer input (received data),    and    are the equalizer filter coefficients with lengths   forward and   in feedback,  is the equalizer soft output,  is the hard decision output symbol, and  is the corresponding error signal. The following equations describe the Kalman algorithm as derived in [11], Define the vector  with the total length of ,      , contains the data received following to current symbol and past decisions with lengths     respectively:                            Also define a vector of equalizer forward and backward filter coefficients;            The equalizer soft output is       . (3) Now, define the Kalman gain vector as follows                               (4) where  represents a weighting factor and    is the inverse of the correlation matrix D N [n] of the received signal R N [n] (measurement),                    (5) In the current notation, we define      as the inverse of the received signal (measurement) correlation matrix as derived in [11]; however, the more popular definition for the    matrix is the correlation matrix of the estimated signal. The      matrix can be calculated recursively as follows                                (6) Finally, update the filter coefficients                    (7) or                        (8) The initial value for the   matrix is     , where  is a small positive constant whose value is small for high SNR and large for low SNR and  is the identity matrix. Also, the initial value for filter taps is all zeros except the position of the output which may start with one. Note that previous equations and equalizer structure consider the case of the symbol-spaced model. ]N[ a +nr ][ nr ba , a N− a 1N a +− a 0 a 1 b 2 b b N b ∑ ∑ ][ ˆ nx ][ ~ nx ]N[ ~ b −nx Fig.1. Kalman-DFE equalizer. Start considering the case of reference-mode, where the perfect data are fed back instead of the hard decisions. It is an important starting point for testing the adaptive equalizer for two reasons. First, examine the equalizer performance limitation (get the best performance). Second, find an easy way to set optimum values for equalizer parameters. In the following, we tune the equalizer parameters in the reference-mode and show the final simulation results. A. Number of forward and backward coefficients     As indicated in [1], number of equalizer taps is the major source of performance limitation for the directly adapted equalizers. Choosing the optimum number of taps for both forward and backward filters is a major problem in the finite length equalizers [13], [14] and there is no closed form for these lengths. In a classic design of the adaptive equalizer, the filter lengths are usually fixed to some compromise values. Other approaches use exhaustive search algorithms which increase the complexity with a great amount. Suboptimum search algorithms are introduced by [13]. In our simulation, we choose to fix the forward and backward number of taps to a suboptimum value, getting it from the simulation. The main parameter that controls the number of filter taps is the channel delay spread. Simulation results show that the forward number of taps   should not be less than  and the performance is improved with increasing that number but with limits. However, it is sufficient to put the backward   taps equals to  (famous choice as indicated in [13] and [8]). B. Tap of decision (decision delay) In finite length adaptive equalizers, decision delay choice is also an interesting problem, especially for short filter lengths [13-16], and to get an optimum choice, we should study it together with filter lengths [13, 14, 16]. Also, this is not the scope of this paper and then we use a fixed suboptimum choice for the decision delay. For the case of forward direction adaptation model we found that the last tap is the correct choice in most cases (it could be seen as collecting symbol power from the successive symbols in the forward filter and subtracting the ISI from the preceding symbols in the feedback filter). C. Periodic restart We choose to periodically restart the filter taps and all related variables each training sequence. Our simulation shows a slight change in the performance (loss) if we continue without restarting, as the mini-probe length (31 symbols) is sufficient in all cases to achieve initial convergence. D. Adaptation rate Adaptation rate depends mainly on the channel variation speed (fading rate) and the transmission data rate. The adaptation rate should be increased (w decreased) with fading rate increase. Fig. 2 shows the final results after optimizing the equalizer parameters in reference-mode for all supported rates. Fig.2. Forward Kalman-DFE, reference-mode, 72 frame interleaver size,   =11,   =5, Doppler=1 Hz, delay spread=2 msec, adaptation rate = [0.99- 0.98-0.97-0.96-0.93] for [QPSK-8PSK-16QAM-32QAM-64QAM]. Now, let us test the more realistic scenario by replacing the perfect data in the feedback by the hard decision one. Simulation results show that Kalman-DFE adapting in the forward direction can not cope with channel parameters (Poor channel) for all supported rates. Fig. 3 shows the BER curves for all rates after reoptimizing the equalizer parameters (some changes are needed in the realistic scenario like the adaptation rate). We note that only QPSK and 8PSK can work properly. IV. B ACKWARD KALMAN FILTER In the current standard specification, there is a similarity for training sequences around the data frame and hence we can adapt the Kalman filter in the reverse direction. Fig. 4 describes the behavior of the backward equalizer (Fig. 4b) compared to the forward one (Fig. 4a) assuming 3-path symbol spaced channel and filter lengths are   =3 and   =2. The most important parameter that we have to change in the backward case is the tap of decision (decision delay). Fig.3. Forward Kalman-DFE performance, decision-directed mode, all data rates, 72 frame interleaver size,   =11,   =5, Doppler = 1 Hz, delay spread = 2 msec. adaptation rate = [0.99-0.97-0.93-0.86-0.72] for [QPSK- 8PSK-16QAM-32QAM-64QAM]. ]0[ 0 xh ]1[ 0 xh + ]0[ 1 xh ]2[ 0 xh + ]1[ 1 xh + ]0[ 2 xh ]3[ 0 xh + ]2[ 1 xh ]1[ 2 xh + ][ 0 nxh ]1[ 1 −nxh ]2[ 2 −nxh + + ]1[ 0 +nxh ][ 1 nxh + ]1[ 2 −nxh + ]2[ 0 +nxh + ]1[ 1 +nxh ][ 2 nxh + 2− a 1− a 0 a 1 b 2 b ][nr ]1[ +nr ]2[ +nr ]1[ ~ −nx ]2[ ~ −nx Fig.4a. Forward Kalman-DFE (adaptation behavior), delay of decision is the last tap, 3-paths symbol-spaced channel,   =3,   =2. ]0[ 0 xh ]1[ 0 xh + ]0[ 1 xh ]2[ 0 xh + ]1[ 1 xh + ]0[ 2 xh ]3[ 0 xh + ]2[ 1 xh ]1[ 2 xh + ][ 0 nxh ]1[ 1 −nxh ]2[ 2 −nxh + + ]1[ 0 +nxh ][ 1 nxh + ]1[ 2 −nxh + ]2[ 0 +nxh + ]1[ 1 +nxh ][ 2 nxh + 2− a 1− a 0 a ][nr ]1[ +nr ]2[ +nr 1 b 2 b ]1[ ~ +nx ]2[ ~ +nx Fig.4b. Backward Kalman-DFE (adaptation behavior), 3-path symbol- spaced channel,   =3,   =2. As shown in Fig. 4b, the tap of decision should be transferred to the first tap (it could be seen as collecting symbol power from the successive symbols in the forward filter and subtracting the ISI of these successive symbols in the feedback filter). Fig. 5 shows the performance for the backward Kalman-DFE assuming reference-mode. We notice that the performance is almost the same as forward adaptation but they are independent as will be shown. V. B I-DIRECTIONAL KALMAN FILTER An efficient structure for the equalizer can be achieved by combining the two previous structures (forward and backward) [5-7]. In this case, the equalizer is running in the two directions (for each data frame) then chooses the best result by comparing the mean square error between the two directions (MSE is measured between soft equalizer output and corresponding hard decisions). The new structure shown in Fig. 6 introduces a real diversity in the results as we found that convergence may be achieved in a direction while the other is failed. The reference-mode simulation results for the bi-directional equalizer for all data rates are shown in Fig. 7 and they show a significant improvement over one directional (forward or backward) equalizer. Fig.5. Backward Kalman-DFE, reference-mode, 72 frame interleaver size,   =11,   =5, Doppler = 1 Hz, delay spread = 2 msec. Fig.6. Bi-directional Kalman-DFE structure. Fig.7. Bi-directional Kalman-DFE, reference-mode, 72 frame interleaver size,   =11,   =5, Doppler = 1 Hz, delay spread = 2 msec. Fig. 8 shows the simulation results for decision-directed bi- directional equalizer. Although we have a noticeable enhancement in the performance compared to one directional structure, unfortunately, the performance is still far from the standard expectations. VI. I TERATIVE STRUCTURE Previous tests and results show that using the hard decision data in the feedback (real decision-directed mode) degrades the performance (from the optimum reference-mode) with a big margin. Iterative equalization through the decoder aims to enhance these hard decisions by feeding back the decoder enhanced results. There is no doubt that iterative structure will increase the complexity and latency of the system as each iteration should wait for the entire block of data (interleaver block) to be deinterleaved and decoded then interleave and encode the decoder output again to input to the equalizer. System model for one direction is shown in Fig. 9. Fig. 10 shows the BER for the 64QAM constellation with the equalizer iteration increase. Simulation results show a great enhancement, but we notice that zeroth iteration always limits the performance, i.e., if the BER of the zeroth iteration is less than 10 -3 , a great enhancement can be achieved with iterations; otherwise, no improvement can be achieved. The BER curves for all data rates for the iterative structure of the bi-directional Kalman-DFE are shown in Fig. 11. Fig.8. Bi-directional Kalman-DFE, decision-directed, 72 frame interleaver size,   =11,   =5, Doppler=1 Hz, delay spread=2 msec. Fig.9. Iterative Kalman-DFE structure (one direction). Fig.10. Iterative Kalman-DFE with iterations, decision-directed, 64QAM, 72 frame interleaver size,   =11,   =5, adaptation rate=0.91 for first iteration and 0.93 for other iterations, Doppler=1Hz, delay spread=2 msec. Fig.11. Iterative Bi-directional Kalman-DFE, decision-directed, 72 frame interleaver size,   =11,   =5, Doppler = 1 Hz, delay spread = 2 msec. VII. CONCLUSIONS In this paper, we treated the ITU-R Poor channel for the HF high data rates using directly adapted fractional spaced Kalman-DFE with two major enhancements in the structure. The bi-directional structure successfully prevents the performance break down at the higher constellation and enhances the overall performance for all rates. In the bi- directional structure, a real diversity is achieved when adapting the equalizer in the reverse direction. The iterative structure then enhances the performance significantly (about 6dB gain at 1e-5 BER for 64QAM using three iterations). The iterative structure enhances the reliability for the feedback data by replacing the hard decisions with the decoder feedback data. We can notice that the proposed solution is much simpler than turbo equalizers as it uses direct adaptation structure (eliminates the use of matrix inversion) and also uses the less complicated iterative technique to enhance the reliability of the feedback data. Finally, both structures (bi-directional and iterative) can be used also with indirect adaptive equalizers (adaptation through channel estimation) and should improve its performance. A CKNOWLEDGMENT Some parts of this research were a part of a project in MTSE Company. We are greatly thankful for the CEO Essam El Aasar and the department director Ehab Samy. R EFERENCES [1] E. Eleftheriou and D. D. 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Abstract— In this paper, we present the Kalman decision feedback equalizer (Kalman-DFE) as a solution for the high data rate HF waveforms. We show that the basic performance for the Kalman-DFE. Iterative Bi-directional Kalman-DFE Equalizer for the High Data Rate HF Waveforms in the HF Channel Mahmoud A. Elgenedy Comm. System Design. depends mainly on the channel variation speed (fading rate) and the transmission data rate. The adaptation rate should be increased (w decreased) with fading rate increase. Fig. 2 shows the final