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Maximum Free Distance Binary to -ary Convolutional Codes for Pseudo Chaotic Type Time Hopping PPM Impulse Radio UWB S. Villarreal-Reyes, Student Member, IEEE, and R. M. Edwards, Member, IEEE
Abstract—The topic of spectral line suppression is of major interest in the design of pulse position modulated time hopping impulse radio ultra wideband communications systems. Previously pseudo-chaotic time hopping (PCTH) has been introduced to eliminate spectral lines and improving bit error rates (BER). In this -ary maximum free distance convolutional letter, new binary to codes are introduced showing similar power spectral density characteristics as those obtained with PCTH and better BER for hard and soft decision decoding.
convolutional encoders for rates of 1, 1/2, and 1/3, with similar spectral characteristics to those obtained with PCTH for 16-ary, 32-ary, 64-ary, and 128-ary OPPM TH–IR UWB systems, are introduced. Note for these encoders the rate is given as a function of the number of channel symbols produced for each input bit [3].
Index Terms—Convolutional Codes, power spectral density (PSD), pulse position modulated (PPM), time hopping impulse radio (TH–IR), ultra wideband (UWB).
This work follows the basic model introduced in [1] and [2] where the PCTH encoder—Gray mapper is replaced by a binary to -ary convolutional encoder followed by OPPM. It is assumed that the encoder rate is 1 where is the number of -ary symbols produced per input bit. Then, the transmitted signal can be described by
I. INTRODUCTION
II. SYSTEM MODEL
I
N ONE of its early forms, time hopping impulse radio (TH–IR) ultra wideband (UWB) systems used pulse position modulation (PPM) for the transmission of information. The method has low cost and is simple to implement but leads to “spectral lines” within its power band. Spectral lines in PPM TH–IR UWB systems can be reduced using pseudo chaotic time hopping (PCTH) [1], which can be interpreted as the concatenation of a convolutional encoder (which resembles -ary the dynamics of Bernoulli shift and tent maps) with PPM based UWB. PCTH can be seen as a set of rate 1 binary to -ary convolutional encoders with constraint length , used in conjunction with -ary orthogonal PPM (OPPM) UWB. PCTH’s useful spectral line reduction characteristics can be explained by its Markov chain (MC) model, [1], [2], therefore there exists the possibility of finding binary to -ary convolutional encoders with better bit error rate (BER) performance and similar spectral characteristics to those obtained with PCTH. Encoders for binary to -ary orthogonal signaling have been reported in [3]–[5] for 4-ary, 8-ary and 16-ary orthogonal alphabets. Application of superorthogonal and punctured convolutional encoders to -ary OPPM UWB systems was proposed in [6] but without considering the spectral characteristics of the signal. These encoders were neither designed for -ary orthogonal alphabets nor interpreted as binary to -ary encoders. In this letter new maximum free distance (MFD) binary to -ary
Manuscript received July 25, 2006; revised December 4, 2006. The authors are with the Department of Electronic and Electrical Engineering, Loughborough University, Loughborough LE11 3TU, U.K. (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LMWC.2007.892935
(1) where accounts for the transmitted pulse, is the mean is the PPM modulation index repetition time between pulses, is the th -ary symbol produced by the encoder and due to the th data input bit. The received signal can be modeled as
(2) is the received pulse, is the delay, and is where and are known at the additive noise. It is assumed that is AWGN for comparison purposes. Furreceiver and that thermore, to guarantee orthogonallity and good spectral characteristics the transmitted and received pulse durations are set to and is set to . be less than III. DISCRETE POWER SPECTRAL DENSITY AND CODE SEARCH PROCEDURE Convolutional encoders can be modeled as Melay finite state machines (FSM), [7]. This model can be converted to an equivalent Moore FSM which can be used to obtain a Markov chain (MC) model of the encoder when its input consist on a sequence of independent uniformly distributed symbols as in PCTH, [2]. The advantage of the Moore-MC model is that the output depends exclusively on the MC states and therefore results such as [7, p. 220] and [8] can be used for power spectral density (PSD) evaluation of the “convolutional driven” -ary PPM TH–IR
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VILLARREAL-REYES AND EDWARDS: MAXIMUM FREE DISTANCE BINARY TO
UWB signal. The power spectral density discrete component is then given by [2], [7], [8]
-ary CONVOLUTIONAL CODES
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TABLE I BEST BINARY TO -ARY ENCODERS WITH THE FIRST EIGHT ELEMENTS OF ITS INFORMATION WEIGHT SPECTRUM
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(3) where is the number of states in the MC, are the MC is the time interval between MC steady state probabilities, is the Fourier transform of the output state transitions and . In our case and signal when the MC is in state (3) becomes
(4) where is the Fourier transform of the pulse shape over the is the th -ary symbol produced when the channel and encoder is in state . From (4), it can be seen that by proper design of the convolutional encoder the spectral lines in the power spectral density can be reduced. Using (4), a search procedure for codes with similar spectral line suppression characteristics as PCTH and maximum -ary free distance was performed. Our code search procedure is similar to that described in [3], [4], with the addition that before computing the encoder’s free distance its Moore—MC model and the corresponding stationary probabilities are obtained as in [2]. With this information encoders showing inferior spectral line reduction capabilities than PCTH were discarded, [2]. In order to provide rate adaptability the rate 1/2 and 1/3 encoders were found through nested search. Table I presents the code search results for 16-ary to 128-ary OPPM TH–IR UWB. In order to guarantee all possible -ary symbols are used, the minimum constraint length for each case , [2]. For OPPM the -ary free distance, was set to , must be considered rather than the binary free distance, [2]–[5]. All of the encoders shown in Table I have -ary maximum free distance. The results in Table I are interpreted as follows: for each -ary alphabet and constraint length, , the first line gives the and the first eight composet of generators (in octal form), nents of the information weight spectrum (IWS) for rate 1 encoders. The second and third lines give the second and third sets of generators that must be added to form the rate 1/2 and 1/3 enand IWS. For coders with their respective only one set of generators was listed (rate 1/2 and 1/3 maximum free distance codes can easily be obtained by repeating the generators set). Note for each set of generators the binary output must be converted to the corresponding -ary symbol as in as in [4]. Optionally an equivalent representation over GF [3] can be obtained by following [5]. The binary to 16-ary encoders presented here have better information weight spectrum than the encoders reported in [3]. When the PCTH scheme is interpreted as a set of rate 1 binary to -ary convolutional encoders, the set of generators needed for -ary PCTH can be obtained (in octal form) as (5)
Fig. 1. Power spectral densities obtained when using (a) the 32-ary PCTH encoder and (b) the rate 1, 7, binary to 32-ary convolutional encoder given 8 ns and 0.25 ns. The third in Table I. The system parameters are derivative Gaussian pulse is used with 0.24 ns. In order to be able to compare the PSD’s continuous and discrete components a double simulation-estimation procedure was performed as described in [9].
K=
T = T
T =
is the encoder’s constraint length. In [1], where the interpretation of PCTH as a set of rate 1 binary to -ary encoders is not addressed. Instead, it is considered that the PCTH encoders have a binary rate of 1 . Note how in PCTH the number of generators (outputs) is equal to , and itself depends on the number of PPM positions . In contrast, the encoders reported in Table I do not follow a specific construction . rule and the constraint length, , can be larger than Furthermore, the encoders in Table I have larger -ary free dis], better information weight spectrum tance [for and provide rate adaptability from 1 to 1/3 by adding (or removing) different sets of generators to the rate 1 base set. Fig. 1 shows power spectral densities obtained when using 7, binary to 32-ary encoder given in Table I and the rate 1,
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binary Hamming distance (this is the reason why the BER in the PCTH scheme is improved for hard Viterbi decoding upon the introduction of Gray labeling). Fig. 2 shows the BER performance for several of the rate 1 binary to 32-ary encoders given in Table I and 32-ary PCTH. The plots in Fig. 2 were obtained by simulation for AWGN channel. It is readily seen that the BER can be improved with the use of the encoders reported in Table I while preserving the spectral line reduction characteristics obtained with PCTH. Furthermore, a significant gain over uncoded binary PPM (BPPM) can be achieved even with hard decision decoding. As the number of information bits per signal is the same for BPPM and a rate 1 binary to -ary convolutionally coded system, BPPM was chosen as reference. V. CONCLUSION Fig. 2. BER versus bit energy to noise ratio for several of the rate 1 binary to 32-ary encoders presented in Table I and 32-ary PCTH. The plots were obtained by simulation for AWGN channel. The constraint length of each code is indicated in the legend where HD stands for hard Viterbi decoding and SD stands for soft Viterbi decoding. The curves for BPPM and 32-ary PPM are plotted as a reference.
the corresponding 32-ary PCTH encoder. It is readily seen that the spectral line suppression capabilities of both systems are the same. IV. BIT ERROR RATE PERFORMANCE Due to orthogonal signaling is used, the received signal in each -ary symbol interval can be expressed in vector form as [7] (6) where are zero mean mutually independent is the pulse energy. Then, Gaussian random variables and correlators (the actual number the receiver can use a set of needed can be reduced) followed by -ary Viterbi decoding (VD). Soft or hard decision Viterbi decoding can be implemented depending on the way the correlators’ output is fed to the decoder. If the correlators’ outputs are fed to the decoder as a measure of the Euclidean distance between the received -ary orthogonal vector (6) and each one of the possible signals, then soft Viterbi decoding can be performed by using these outputs to form the branch metrics in a similar fashion to that described in [6]. If a decision based on the pulse position ( -ary symbol) is performed before decoding, then hard Viterbi decoding must be implemented. Branch metrics must be calculated based on the -ary Hamming distance. A 1 is added to the corresponding branch metric if the hard symbol given by the pulse position demodulator is different to the corresponding -ary symbol in the branch. This hard metric is different to the one proposed in PCTH which is based in the
In this letter, new binary to -ary convolutional encoders for use with orthogonal PPM TH–IR UWB systems were introduced. The new encoders have convenient spectral line suppression characteristics which are similar to the ones obtained with PCTH. With the new encoders an improvement on the BER over PCTH can be obtained with a minimum increase in the system complexity. Furthermore, it was shown that by computing the branch metrics based on the -ary Hamming distance for hard Viterbi decoding a significant gain over binary PPM can be achieved. This characteristic is important if such encoders are to be used in low rate low power systems where soft decision decoding may not be feasible. REFERENCES [1] G. M. Maggio, N. Rulkov, and L. Reggiani, “Pseudo-chaotic time hopping for UWB impulse radio,” IEEE Trans. Circuits Syst. I, vol. 48, no. 12, pp. 1424–1434, Dec. 2001. [2] S. Villarreal-Reyes and R. M. Edwards, “New convolutional encoders for spectral line suppression in pulse position modulated time hopping impulse radio ultra wideband systems,” in Proc. IET Sem. Ultra Wideband Syst., Technol. Appl., Apr. 2006, pp. 206–210. [3] W. E. Ryan and S. G. Wilson, “Two classes of convolutional codes over GF(q) for q-ary orthogonal signaling,” IEEE Trans. Commun., vol. 39, no. 1, pp. 30–40, Jan. 1991. [4] J. J. Chang, D. J. Hwang, and M. C. Lin, “Some extended results on the search for good convolutional codes,” IEEE Trans. Inform. Theory, vol. 43, no. 6, pp. 1682–1697, Sep. 1997. [5] S. L. Miller, “Design and analysis of trellis codes for orthogonal signal sets,” IEEE Trans. Commun., vol. 43, no. 2–4, pp. 821–827, Feb./Mar./ Apr. 1995. [6] A. R. Forouzan and M. Abtahi, “Application of convolutional error correcting codes in ultrawideband M-ary PPM signaling,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 8, pp. 308–310, Aug. 2003. [7] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [8] S. Villarreal-Reyes and R. M. Edwards, “Power spectral density of convolutionally encoded biorthogonal pulse position modulated time hopping UWB signals,” in Proc. IEEE Global Telecommun. Conf. (Globecom’06), Nov. 2006, [CD ROM]. [9] S. Villarreal-Reyes, R. M. Edwards, and J. C. Vardaxoglou, “On the use of simulation-DFT based analysis for spectral estimation of PPM TH–IR UWB signals,” in Proc. Veh. Technol. Conf. (VTC’05), May 2005, pp. 1370–1374.
