Iterative Detection of Unity-Rate Precoded FFH-MFSK and Irregular Variable-Length Coding

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009 3765 Correspondence Iterative Detection of Unity-Rate Precoded FFH-MFSK the demodulator [11]. The precoder imposes memory upon the chan- and Irregular Variable-Length Coding nel, thus rendering it recursive.1 Hence, SISO systems have been shown to substantially benefit from employment of rate-1 precoders Sohail Ahmed, Robert G. Maunder, Lie-Liang Yang, and Lajos Hanzo without reducing the effective throughput of the system [11]. In analogy to irregular convolutional codes (IrCCs) [9], the novel family of so-called irregular variable-length codes (IrVLCs) [12] Abstract—Iterative decoding of an irregular variable-length coding employs a number of component variable-length coding (VLC) code- (IrVLC) scheme concatenated with precoded fast frequency-hopping books having different coding rates [13], [14] to encode particular (FFH) M-ary frequency-shift keying (MFSK) is considered. We employ EXtrinsic Information Transfer (EXIT) charts to investigate the three- fractions of the input source symbol stream. With the aid of EXtrinsic stage concatenation of the FFH-MFSK demodulator, the rate-1 decoder, Information Transfer (EXIT) charts [10], the appropriate lengths of and the outer IrVLC decoder. The proposed joint source and channel these fractions may be chosen to shape the inverted EXIT curve of the coding scheme is capable of operating at low signal-to-noise ratios (SNRs) IrVLC codec to ensure that it does not cross the EXIT curve of the in Rayleigh fading channels contaminated by partial-band noise jamming (PBNJ). The IrVLC scheme is composed of a number of component inner channel codec. This way, an open EXIT chart tunnel may be variable-length coding (VLC) codebooks employing different coding rates created, even at low signal-to-noise ratio (SNR) values. In [15], we that encode particular fractions of the input source symbol stream. These proposed serial concatenation of an FFH-MFSK demodulator, a rate-1 fractions may be chosen with the aid of EXIT charts to shape the inverted decoder, and an IrVLC decoder and investigated the two-stage mutual EXIT curve of the IrVLC codec so that it can be matched with the information (MI) exchange between the rate-1 decoder and an IrVLC EXIT curve of the inner decoder. We demonstrate that using the proposed scheme, an infinitesimally low bit error ratio (BER) may be achieved at low decoder. In this paper, we extend this concept to the three-stage extrin- SNR values. sic information exchange among the demodulator, the inner decoder, Index Terms—FEC, irregular channel coding, iterative detection of and the outer decoder. By employing EXIT charts, we investigate fast frequency-hopping (FFH) M-ary frequency-shift keying (MFSK), the serial concatenation of the FFH-MFSK demodulator, the rate-1 turbo-detection, unity-rate coding, variable-length coding (VLC). decoder, and the IrVLC outer decoder to attain good performance, even at low SNR values. We contrast the two-stage iterative decoding (ID) scheme between the inner rate-1 decoder and the IrVLC outer decoder I. I NTRODUCTION to the three-stage scheme and demonstrate that the three-stage scheme Fast frequency-hopping (FFH) M -ary frequency-shift keying outperforms the two-stage scheme. The proposed system might find (MFSK) has been shown to efficiently combat partial-band noise fruitful application in FFH-MFSK-based ad hoc or cellular networks jamming (PBNJ) [1], [2]. By transmitting every symbol using multiple transmitting, for example, VLC-compressed H.264/MPEG4 video or hops, FFH systems benefit from both time and frequency diversity, audio signals. VLC compression may also provide useful data rate and which assists in mitigating the detrimental effects of PBNJ. Fur- bandwidth efficiency gains in pure data transmission by exploiting the thermore, in the FFH-MFSK receiver, a suitable diversity-combining potential latent correlation of bits. method may be invoked to further suppress the effects of PBNJ [1]. Although soft-decision decoding (SDD) of noncoherent MFSK- II. S YSTEM O VERVIEW based schemes, including slow frequency hopping (SFH), have been investigated in the published literature [3]–[8], SDD-assisted FFH In this section, we consider an IrVLC codec and an equivalent schemes have attracted little attention. The reasons for this trend regular VLC-based benchmarker in this role. We refer to these schemes might be the difficulty to derive soft information from the received as the IrVLC- and VLC-FFH-MFSK arrangements, respectively. The signal, as well as the challenge of the subsequent exploitation of schematic that is common to both of these schemes is shown in the soft information forwarded by the soft-input–soft-output (SISO) Fig. 1. We consider K = 16-ary source symbol values that have the decoder to the demodulator. By contrast, a number of useful tools probabilities of occurrence that result from the Lloyd–Max (LM) have been employed for the analysis, as well as for the performance quantization [15], [16] of independent Laplacian distributed source enhancement, of iteratively decoded coherently modulated schemes samples. In the transmitter of Fig. 1, the source symbol frame s to be [9]–[11]. A powerful technique to improve the iterative gain of iter- transmitted is partitioned into J 4-bit source symbols corresponding atively decoded schemes is constituted by precoders, which improves to K = 16-ary values of {sj }Jj=1 ∈ [1, . . . , K]. These 4-bit source the extrinsic information exchange between the channel decoder and symbols are then decomposed into N component streams {sn }N n=1 to be protected by N different rate IrVLC component codes, where we opted for N = 16 in the case of the IrVLC-FFH-MFSK scheme and N = 1 in the case of the VLC-based benchmarker scheme. The Manuscript received January 27, 2008; revised October 5, 2008, number of symbols in the source symbol frame s that are decomposed December 8, 2008, and January 14, 2009. First published January 27, 2009; current version published August 14, 2009. This work was supported by the into the source symbol frame component sn is specified as Jn , where Engineering and Physical Sciences Research Council, U.K., and the European we have J1 = J in the case of the VLC-based scheme. By contrast, in Union under the auspices of the OPTIMIX projects and of the Higher Education the case of the IrVLC-based scheme, the specific values of {Jn }N n=1 Commission, Pakistan. The review of this paper was coordinated by Prof. J. Wu. may be chosen to shape the inverted EXIT curve of the IrVLC codec The authors are with the School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: sa03r@ ecs.soton.ac.uk; [email protected]; [email protected]; [email protected]. 1 Recursivity in this context implies that the channel has an infinite impulse ac.uk). Digital Object Identifier 10.1109/TVT.2009.2013989 response (IIR). 0018-9545/$26.00 © 2009 IEEE Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on September 30, 2009 at 03:38 from IEEE Xplore. Restrictions apply. 3766 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009 Fig. 1. Schematic of the IrVLC- and VLC-based schemes, employing three-stage serial concatenation of the demodulator, the rate-1 decoder, and the outer VLC/IrVLC decoder. In the IrVLC-coded scheme, N = 16, whereas in the VLC-coded scheme, N = 1. so that it does not cross the EXIT curve of the precoder, as detailed in generated by a pseudonoise (PN) generator, where L is the number of Section IV.2 frequency hops per symbol. Ts by Th = Ts /L. Each of the N source symbol frame components {sn }N n=1 is VLC The channel is assumed to be a frequency-flat Rayleigh fading encoded using the corresponding codebook from the set of N VLC medium for each of the transmitted frequencies. Furthermore, we codebooks {VLCn }N N Nn=1 , having a range of coding rates {Rn }n=1 ∈ assume that the separation between the adjacent frequencies is Rh = [0, 1], satisfying n=1 αn Rn = R, where αn = Jn /J is the partic- 1/Th , which also represents the bandwidth occupied by a single FFH- ular fraction of the source symbol frame coded by the nth subcode, MFSK tone and is higher than the coherence bandwidth of the channel. and R is the average code rate of the VLC or the IrVLC scheme. Therefore, all FFH tones conveying the same symbol experience inde- The N VLC codewords that represent the Jn source symbols in the pendent fading. The transmitted signal is also corrupted by additive source symbol frame component sn are concatenated to provide the white Gaussian noise and a PBNJ signal [17] having single-sided transmission frame component un . Owing to the variable length of power spectral densities of N0 and NJ , respectively. We assume that the VLC codewords, the number of bits comprised by each transmis- the PBNJ signal jams a fraction 0 ≤ ρ ≤ 1 of the total spread-spectrum sion frame component un will typically vary from frame to frame. To bandwidth Wss , as discussed in [15]. facilitate VLC decoding of each transmission frame component un , The receiver schematic is also shown in Fig. 1, where the frequency it is necessary to explicitly convey the exact number of bits In to the dehopper, which is identical to and aligned with the frequency hopper receiver with the aid of side information. This may be achieved using of the transmitter, despreads the received signal using the knowledge of a low-rate block code or repetition code, for example. For the sake of the transmitter’s unique FFH address. The demodulator is composed of avoiding obfuscating details, this is not explicitly shown in Fig. 1. M branches, each corresponding to a single MFSK tone and consisting The N transmission frame components {un }N n=1 encoded by the of a square-law detector [17] and a diversity combiner, which performs different IrVLC component codes are concatenated at the transmitter, clipping followed by linear combining of the signals received in all as hops. The process of clipping may be expressed by [1] shown N in Fig. 1. The resultant transmission frame u has a length of n=1 n I bits. Following the bit interleaver, the binary transmission C, if Uml ≥ C  frame u is precoded, and the precoded bits are converted to M -ary f (Uml ) = Uml , otherwise symbols [7], which are transmitted by the FFH-MFSK modulator of Fig. 1. To further elaborate, in the FFH-MFSK transmitter, the M -ary m = 0, 1, . . . , M − 1; l = 0, 1, . . . , L − 1 (1) symbols are mapped to M frequency tones [17]. The particular MFSK tone chosen for transmission modulates a carrier generated by a where Uml represents the square-law detector’s output corresponding frequency synthesizer, which is controlled by the L-tuple FFH address to the mth tone in the lth hop, and C represents an appropriately chosen clipping threshold. The decision variable recorded after clipped L−1 combining is given by Zm = l=0 f (Uml ), m = 0, 1, . . . , M − 1. 2 In this context, the authors would like to gratefully acknowledge the helpful Using the diversity combiner outputs, the corresponding symbol prob- comment of the anonymous reviewer that by designing the IrVLC code for the abilities and log-likelihood ratios (LLRs) are computed, as explained sake of approaching the capacity, it may seem that we have to surrender some of in the following section, which are then fed to the rate-1 decoder. the flexibility in designing the VLC code to match the source statistics, which would potentially reduce the achievable compression ratio. We note, however, that this potential problem was circumvented by our two-stage design, where in III. I TERATIVE D ECODING the first stage, we designed the VLC codebook for K = 16-ary source symbol values generated from LM quantized independent Laplacian distributed source In this section, we discuss how soft information is derived from the samples. In the second stage, the resultant stream of 4-bit symbols was then decomposed into N substreams encoded by the N IrVLC component codes. channel’s output observations and how ID is carried out by exchanging The IrVLC source-code design is, hence, decoupled from the near-capacity extrinsic information between the demodulator, the rate-1 decoder, and EXIT-chart matching. the outer decoder, as shown in Fig. 1. Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on September 30, 2009 at 03:38 from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009 3767 To compute the LLRs, we need the probability that the mth symbol or FSK tone was transmitted, m = 0, . . . , M − 1, given that Z = [Z0 , Z1 , . . . , ZM −1 ] is received, which represents the set of M outputs of the diversity combiners of Fig. 1. This probability is given by p(Z|m)P (m) P (m|Z) = (2) p(Z) where p(Z|m) is the probability density function (pdf) of the received signal Z, given that the mth symbol is transmitted. Furthermore, P (m) is the a priori probability of the symbol m, whereas p(Z) = M −1 m=0 p(Z|m)P (m) is the pdf of the received signal set Z, which is the same for all m. Moreover, for equiprobable symbols, we have P (m) = 1/M . Hence, the pdf p(Z|m) uniquely and unambiguously describes the statistics required for estimating the probability P (m|Z). For independent fading of all tones, the pdf p(Z|m) is given by M −1  p(Z|m) = fZm (xm |m) fZn (xn |m) (3) n=0, n=m where fZn (xn |m) represents the pdf of the nth diversity combiner output, n = 0, 1, . . . , M − 1, given that the mth symbol is transmit- Fig. 2. EXIT characteristics of the demodulator, the rate-1 decoder, and ted. To derive the pdfs of the diversity combiner outputs, we first the combined module in Rayleigh fading channels contaminated by PBNJ, consider the pdfs of the square-law detector outputs before diversity assuming Eb /N0 = 6 dB, Eb /NJ = 10 dB, and ρ = 0.1. combining, as shown in Fig. 1. Assuming that the mth symbol is transmitted in the lth hop, it can readily be shown that for independent Rayleigh fading, the pdf of the square-law detector output Uml , as Upon inserting (8) in (2), we can derive the corresponding symbol shown in Fig. 1, may be expressed as [15], [17] probabilities. The resultant bit probabilities can be derived from the 1 −ym symbol probabilities, assuming the bits-to-symbol mapping of [7]. fUml (ym |m) = e 1+γh , ym ≥ 0 (4) Finally, the LLRs can be computed using the bit probabilities [10]. 1 + γh Note that earlier in this paper, we have derived the soft information where γh = bREb /(N0 L) is the SNR per hop, Eb is the transmitted from the channel’s output observations, assuming a somewhat simplis- energy per bit, and b = log2 M is the number of bits per symbol. tic but tractable interference-free channel. Moreover, although clipped From (4), using the characteristic function (CF) approach [17],3 we combining is employed, our analysis assumes linear combining, i.e., can derive the pdf of the linear combiner’s output Zm , as shown in without clipping. This assumption has been stipulated for the sake of Fig. 1, which can be expressed as further simplifying the analysis and is supported by the observation that clipping is an operation performed to reduce effects of PBNJ. We xL−1 m will demonstrate in Section IV that valuable performance improve- fZm (xm |m) = e−xm /(1+γh ) . (5) (1 + γh )L Γ(L) ments can be achieved using this suboptimal soft information. Following the derivation of the soft information from the received Similarly, for all the nonsignal tones, n = 0, 1, . . . , M − 1, n = m, signal, two types of serial concatenation schemes are possible. In the we have first case, the a posteriori probability (APP) SISO rate-1 decoder and xL−1 n the outer decoder perform ID [15], [18]. We refer to this configuration fZn (xn |m) = e−xn . (6) Γ(L) of the ID as the two-stage scheme. Alternatively, the system may be modified so that the FFH-MFSK demodulator, the rate-1 inner Inserting (5) and (6) in (3) and after further simplifications, we have decoder, and the IrVLC outer decoder exchange their extrinsic MI,  M −1    as shown in Fig. 1. We refer to this arrangement as the three-stage 1 1  xm γh scheme, which requires an additional interleaver between the precoder p(Z|m) = xL−1 n e−xn exp . (1+γh )L ΓM (L) 1+γh and the FFH-MFSK modulator of Fig. 1. n=0 (7) In [15], we demonstrated that the FFH-MFSK demodulator yields low-gradient EXIT curves for all values of the modulation order M . We can see in (7) that all the terms except the last exponential When the SNR is sufficiently high, the EXIT curves can be shifted term are common for any of the mth symbol, m = 0, 1, . . . , M − 1. upward in the EXIT plane, and hence, an arbitrarily low bit error ratio Since the computation of the LLRs requires the logarithm of the bit (BER) may be achieved. However, this would be achieved at the cost probabilities, we consider the common terms as a normalization factor of having a large area between the demodulator’s and the decoder’s and express the normalized probability p(Z|m) as EXIT curves, implying that the scheme operates far from capacity. By   contrast, it was shown in [15] that the precoder renders the channel to xm γh appear recursive [11], i.e., of IIR, and hence results in steeper EXIT p(Z|m) = exp . (8) 1 + γh curves than the stand-alone demodulator. Furthermore, as the pre- coder’s memory is increased, the EXIT curves become steeper. More- 3 The CF is the Fourier transform of the pdf, and the fact that the CF of a over, in contrast to the demodulator, the rate-1 decoder’s EXIT curves sum of random variables is the product of their individual CFs [17] has been do indeed reach the (Ie , Ia ) = (1, 1) point, implying that the precoder exploited here. allows the ID to converge to an arbitrarily low BER. Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on September 30, 2009 at 03:38 from IEEE Xplore. Restrictions apply. 3768 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009 Fig. 3. Inverted VLC EXIT curves and rate-1 decoder EXIT curves in uncorrelated Rayleigh fading channels, assuming Eb /NJ = 10 dB and ρ = 0.1. Let us now employ the EXIT chart technique to investigate the MFSK modulation order of M = 16, and FFH diversity order of L = 3. three-stage receiver scheme, namely, the FFH-MFSK demodulator, Finally, we assume having the optimum clipping thresholds defined in the inner rate-1 decoder, and the outer IrVLC decoder. Ideally, this (1) for all simulations [1]. Usually, the most suitable clipping levels would require a 3-D EXIT chart depicting the evolution of the MI at for various values of the system parameters can be determined in an the output of the three concatenated components. However, a simpler empirical manner, e.g., by plotting the system BER against the clipping and almost equally effective method of investigating this three-way level and choosing the clipping threshold that yields the lowest BER. MI exchange was proposed in [19], which implies treating the FFH- In Fig. 2, we compare the EXIT characteristics of the three types MFSK demodulator and the rate-1 inner decoder as a single module, of inner modules considered, when the channel was also contaminated although they also exchange extrinsic information in a single step. by PBNJ. The inverted EXIT curve of a half-rate recursive systematic This effectively allows us to analyze the three-stage concatenation convolutional decoder characterized by the octal generator polynomial as a two-stage one using 2-D EXIT charts. We will refer to this of (7,5) is also shown. We observe that the combined module has a su- module as the combined module. All MI measurements were made perior EXIT curve, and although its EXIT curve has a similar gradient using the histogram-based approximation of the true distribution [10]. as that of the rate-1 decoder, its EXIT curve emerges from a higher Note that in Fig. 1, Λ(·) denotes the LLRs of the bits concerned, point in the EXIT plane at a zero abscissa value, indicating that the where the superscript i indicates the inner decoder (or demodulator), combined module would allow satisfactory communication at lower whereas o corresponds to the outer decoder. Additionally, a subscript SNR values. The superiority of the combined module over the stand- denotes the dedicated role of the LLRs, with a, p, and e indicating alone rate-1 decoder stems from the fact that when employing the a priori, a posteriori, and extrinsic information, respectively. More- combined module, MI is exchanged between the two SISO modules, over, unless otherwise stated, we employ the following parameter i.e., the demodulator and the rate-1 decoder, thus enabling optimum values: source symbol frame length J = 70 000, code rate R = 0.5, exploitation of the soft information. Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on September 30, 2009 at 03:38 from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009 3769 Fig. 4. BER versus Eb /NJ performance of the two- and three-stage VLC- and IrVLC-based schemes in jammed uncorrelated Rayleigh fading channels, assuming Eb /N0 = 10 dB and ρ = 0.5. Note that our combined module invokes a single iteration of MI [15]. Fig. 3 also shows the inverted EXIT curve of the IrVLC scheme. exchange between the demodulator and the rate-1 decoder. We have This was obtained as the appropriately weighted superposition of the also investigated the use of more than one iterations between these N = 16 component VLC codebooks’ inverted EXIT curves, where the two blocks, but it was found that virtually no additional benefits weight applied to the inverted EXIT curve of the component VLC were achieved to justify its added complexity. Hence, when the three- codebook VLCn is proportional to the specific number of source stage scheme of Fig. 1 is employed, each ID iteration involves one symbols employed for encoding Jn [9]. Using the approach of [9], the iteration between the demodulator and the rate-1 decoder, followed values of {Jn }N n=1 given in Fig. 3 were designed so that the IrVLC by a single iteration between the rate-1 inner decoder and the outer coding rate matches that of our regular VLC scheme, namely, 0.5. decoder. Moreover, for the three-stage scheme, we invoke a precoder Furthermore, we ensured that the inverted IrVLC EXIT curve did not of memory 1, whereas for the two-stage scheme, we employ a precoder cross the rate-1 decoder’s EXIT curve at Eb /N0 = 6.6 dB. As shown of memory 3, since using these precoder memories, attractive EXIT in Fig. 3, the presence of the resultant open EXIT chart tunnel implies characteristics are achieved. that an infinitesimally low BER may be achieved by the IrVLC-FFH- Since N separate VLC encoders are employed in the IrVLC-FFH- MFSK scheme for Eb /N0 values above 6.6 dB. By contrast, having an MFSK transmitter, N separate VLC decoders are employed in the open EXIT chart tunnel is not afforded for Eb /N0 values below 6.9 dB corresponding receiver seen in Fig. 1. The a priori LLRs Λoa (u) are in the case of the benchmarker VLC-based scheme, which is identical decomposed into N components, as shown in Fig. 1. This is achieved to the VLC codebook VLC10 of the IrVLC scheme. Analogous to the with the aid of the explicit side information that we assume for con- IrVLC design of Fig. 3, we have designed IrVLC codes for both the veying the number of bits In in each transmission frame component two- and three-stage ID schemes, assuming various jamming scenarios un . Each of the N VLC decoders is provided with the a priori LLR in Rayleigh fading channels. subframe Λoa (un ), and in response, it generates the a posteriori LLR Let us now focus our attention on the BER performance of the subframe Λop (un ), n ∈ [1, . . . , N ]. These a posteriori LLR subframes proposed system in the context of the two- and three-stage schemes are concatenated to provide the a posteriori LLR frame Λop (u), as considered. Note that the BER depicted in Figs. 4 and 5 corresponds shown in Fig. 1. During the final decoding iteration, N bit-based MAP to the encoded bits of the transmission frame u and the corresponding VLC sequence estimation processes are invoked instead of single- received frame u ˜. class APP SISO VLC decoding, as shown in Fig. 1. In this case, each In Fig. 4, we provide the BER versus Eb /NJ performance com- transmission frame component un is estimated from the corresponding parison of the two- and three-stage schemes, assuming Eb /N0 = a priori LLR frame component Λoa (un ). The resultant transmission 10 dB and ρ = 0.5. We observe that both the VLC- and IrVLC-based frame component estimates u ˜ n of Fig. 1 may be concatenated to schemes result in superior performance compared to the system oper- provide the transmission frame estimate u ˜ , as explained in [15]. ating without the precoder, which encounters an error floor. We also note that the three-stage IrVLC scheme yields a further improvement IV. S YSTEM P ARAMETER D ESIGN AND R ESULTS of nearly 3 dB over the two-stage IrVLC scheme. This performance gain confirms the EXIT chart prediction of Fig. 2. Finally, we note In Fig. 3, we provide the inverted EXIT curves that characterize in Fig. 4 that the IrVLC-based three-stage scheme outperforms the the bit-based APP SISO VLC decoding of the aforementioned VLC corresponding VLC-based scheme by approximately 1.1 dB. codebooks, together with the rate-1 decoder’s EXIT curves at Eb /N0 An increased complexity is imposed by the increased number of values of 6.6 and 6.9 dB. All the EXIT curves were generated using decoding iterations, which is a natural consequence of operating at uncorrelated Gaussian distributed a priori LLRs, based on the assump- lower SNRs. This is particularly true in the case of the IrVLC scheme, tion that the transmission frame’s bits have equiprobable logical values where typically a narrower EXIT tunnel exists between the EXIT Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on September 30, 2009 at 03:38 from IEEE Xplore. Restrictions apply. 3770 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009 Fig. 5. BER versus Eb /N0 performance of the two- and three-stage IrVLC-based schemes in jammed uncorrelated Rayleigh fading channels for various numbers of decoding iterations, assuming Eb /NJ = 10 dB and ρ = 0.1. curves of the outer and inner decoders, as shown in Fig. 3. In Fig. 4, [4] K. Cheun and W. E. Stark, “Performance of robust metrics with convo- up to 80 iterations are needed for achieving convergence to the point lutional coding and diversity in FHSS systems under partial-band noise jamming,” IEEE Trans. Commun., vol. 41, no. 1, pp. 200–209, Jan. 1993. of (Ie , Ia ) = (1, 1), which is associated with an infinitesimally low [5] M. C. Valenti and S. Cheng, “Iterative demodulation and decoding of BER, when employing the proposed IrVLC-based scheme. The effect turbo-coded M -ary noncoherent orthogonal modulation,” IEEE J. Sel. of reducing the number of iterations on both the two- and three- Areas Commun., vol. 23, no. 9, pp. 1739–1747, Sep. 2005. stage schemes is depicted in Fig. 5, where the BER versus Eb /N0 [6] Q. Zhang and T. Le-Ngoc, “Turbo product codes for FH-SS with partial- performance of the IrVLC schemes considered is shown. We observe band interference,” IEEE Trans. Wireless Commun., vol. 1, no. 3, pp. 513– 520, Jul. 2002. that the number of decoding iterations can significantly be reduced by [7] U. C. Fiebig, “Soft-decision and erasure decoding in fast frequency- increasing the Eb /N0 value by as little as 0.2–0.4 dB. hopping systems with convolutional, turbo, and Reed–Solomon codes,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1646–1654, Nov. 1999. [8] D. Park and B. G. Lee, “Iterative decoding in convolutionally and turbo V. C ONCLUSION coded MFSK/FH-SSMA systems,” in Proc. IEEE ICC, Jun. 2001, vol. 9, pp. 2784–2788. We have investigated the serial concatenation of IrVLC coding with [9] M. Tüchler and J. Hagenauer, “EXIT charts of irregular codes,” in Proc. an FFH-MFSK modem operating in a Rayleigh fading channel when Conf. Inf. Sci. Syst., Princeton, NJ, Mar. 2002, pp. 748–753. the transmitted signal was corrupted by PBNJ. We employed 2-D [10] S. ten Brink, “Convergence of iterative decoding,” IEEE Trans. Commun., EXIT charts to investigate the three-stage concatenation of the FFH- vol. 49, no. 10, pp. 1727–1737, Oct. 2001. [11] R. Y. S. Tee, S. X. Ng, and L. Hanzo, “Precoder-aided iterative detection MFSK demodulator, the rate-1 decoder, and the outer IrVLC decoder. assisted multilevel coding and three-dimensional EXIT-chart analysis,” in Furthermore, the IrVLC code was designed in such a way that the Proc. IEEE WCNC, Apr. 2006, vol. 3, pp. 1322–1326. inverted EXIT curve of the IrVLC decoder matches the EXIT curve [12] R. G. Maunder, J. Wang, S. X. Ng, L.-L. Yang, and L. Hanzo, “Iteratively of the inner decoder. This way, an open EXIT chart tunnel may be cre- decoded irregular variable length coding and trellis coded modulation,” in ated, even at low SNR values, providing source-correlation-dependent Proc. IEEE Workshop Signal Process. Syst., Shanghai, China, Oct. 2007, pp. 222–227. additional performance gains of up to 1.1 dB over the VLC-based [13] V. Buttigieg and P. G. Farrell, “Variable-length error-correcting codes,” scheme. Consequently, we noted that the precoder-aided schemes yield Proc. Inst. Elect. Eng.—Commun., vol. 147, no. 4, pp. 211–215, Eb /N0 gains in excess of 7 dB over the system dispensing with the Aug. 2000. precoder, which suffers from an error floor, when jamming is severe. [14] R. Bauer and J. Hagenauer, “Iterative source/channel-decoding using reversible variable length codes,” in Proc. Data Compression Conf., Moreover, we demonstrated that the three-stage concatenated scheme Snowbird, UT, 2000, pp. 93–102. yields approximately 3-dB better performance compared with the two- [15] S. Ahmed, R. G. Maunder, L. L. Yang, S. X. Ng, and L. Hanzo, “Joint stage arrangement. source coding, unity rate precoding and FFH-MFSK modulation using iteratively decoded irregular variable length coding,” in Proc. 66th IEEE VTC—Fall, Sep./Oct. 2007, pp. 1042–1046. R EFERENCES [16] S. Lloyd, “Least squares quantization in PCM,” IEEE Trans. Inf. Theory, [1] K. C. Teh, A. C. Kot, and K. H. Li, “FFT-based clipper receiver for vol. IT-28, no. 2, pp. 129–137, Mar. 1982. fast frequency-hopping spread-spectrum system,” in Proc. IEEE Symp. [17] J. G. Proakis, Digital Communications. Singapore: McGraw-Hill, 2001. Circuits Syst., May/Jun. 1998, vol. 4, pp. 305–308. [18] V. B. Balakirsky, “Joint source-channel coding with variable length [2] X. F. Wu, C. M. Zhao, X. H. You, and S. Q. Li, “Robust diversity-combing codes,” in Proc. IEEE Int. Symp. Inf. Theory, Ulm, Germany, Jun. 1997, receivers for LDPC coded FFH-SS with partial-band interference,” IEEE p. 419. Commun. Lett., vol. 11, no. 7, pp. 613–615, Jul. 2007. [19] J. Wang, S. X. Ng, A. Wolfgang, L.-L. Yang, S. Chen, and L. Hanzo, [3] P. C. P. Liang and W. E. Stark, “Algorithm for joint decoding of turbo “Near-capacity three-stage MMSE turbo equalization using irregular con- codes and M -ary orthogonal modulation,” in Proc. IEEE Int. Symp. Inf. volutional codes,” in Proc. Int. Symp. Turbo Codes, Munich, Germany, Theory, Jun. 2000, p. 191. Apr. 2006. 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