Power Efficient Multi-Carrier Transmission with Hard/Soft Decoding and Controlled Error Rate Houcem Gazzah and Steve McLaughlin Institute for Digital Communications The University of Edinburgh King’s Buildings, Mayfield Road, Edinburgh, EH9 3JL, UK. Email: h.gazzah,
[email protected]
Abstract— The error rate of a coded multi-carrier system can be controlled by adjusting the modulation parameters (signal constellation and level) per sub-carrier, while at the same time, minimizing the transmission power. We modify existing techniques to allow for efficient error rate control and to take the encoder/decoder into account. In particular, we study the soft decision receiver for which no adaptive allocation technique has been proposed to-date.
I. I NTRODUCTION Adaptive bit and power allocation (adaptive allocation in brief) is being proposed as a mean to combat frequency (subcarrier) selective fading. Typically, some (global) parameters (bit rate, error rate, transmission power) are fixed at some desired values while the transmission parameters (modulation size and transmission power) are adjusted to minimize some cost function(s). Hence, capacity maximization [1], rate maximization [2] and BER minimization [3], [4] approaches were proposed. In the most popular approach, the throughput and the target Bit Error Rate (BER) are fixed as a consequence of the application Quality of Service [5]. Individual (per subcarrier) bit rates and transmission powers are adjusted to minimize the overall transmission power. Optimal allocation is obtained using the Water-Pouring Algorithm (WPA) which allows the integer-value constraint on bit rates to be respected, but (sub-optimal) fast versions have been proposed as well [6], [7], [8]. While already in use in cable (xDSL) transmission, adaptive allocation is more challenging in wireless communication because of the large overhead needed to transmit the (instantaneous) Channel State Information (CSI) to the transmitter and the (bit and power) allocation information to the receiver. This requirement is less compelling in TDD systems thanks to channel reciprocity [4], [9], [8]. In this paper, we emphasis the (de)coding aspects. Apart from some fully simulation-based studies [5], [10], [7], [11], the allocation problem has been formulated as a transmission (modulation)-only problem. Hence, the encoder structure, and more importantly, the decoder nature (hard or soft) are not taken into account; while (BER) performances are more relevant when expressed in terms of the decoded bits. Existing techniques can be easily extended to Hard Decoding (HD) receivers since (de)modulation and (de)coding are conducted separately. However, the whole problem has to reformulated and solved differently when Soft Decoding (SD) is considered.
0-7803-8938-7/05/$20.00 (C) 2005 IEEE
We start by reviewing adaptive allocation for HD receivers. While this has been extensively investigated, a number of errors and imperfections still can be identified. This is illustrated by the fact that (BER-constrained) algorithms fail to reach their objective BERs. Also, they have been tested exclusively with square Quadrature Amplitude Modulations (QAMs). This was justified by the simplicity of QAMs who have independent real and imaginary components. By discarding odd bit size constellations, optimality is affected not only because of a restricted choice of signal sets, but also because the discarded cross QAMs are more power-efficient than their rectangular counter-parts [12]. This is not sensible in a power optimization approach. Furthermore, a technique is proposed [12] to achieve cross QAMs simply as square QAMs. Our main contribution is an original adaptive allocation algorithm for SD receivers. To our knowledge, adaptive allocation have been studied only for uncoded systems (or equivalently, for HD receivers) [4]. When SD receivers have been considered, allocation was applied unchanged [13] or not at all [14]. On the other hand, when bit interleaving is used, only the (clearly sub-optimum) HD receivers were considered [4]. Complexity is often cited, but difficult BER analysis of Bit Interleaving Coded Modulation (BICM) is also a reason for this state of affairs. By means of an approximated model, the SNR (per sub-carrier) can be expressed in terms of the desired BER of the (SD) decoded bits, so that the adaptation can be solved on a subcarrier-per-subcarrier basis, i.e. as simple as for the HD receiver. Minimizing the power is, then, conducted using (any version of) the WPA. II. A DAPTIVE C ODED M ULTI -C ARRIERS We consider, as depicted in Fig. 1, a set of N parallel single-input single-output ISI-free Gaussian channels. This is the case of a (perfectly synchronized) OFDM system using a sufficiently long cyclic prefix [15]. During a processing block, b coded bits are dispatched among the N sub-carriers (following some allocation procedure). The bn bits assigned to the n-th sub-carrier, are mapped into a symbol sn chosen from the 2bn -sized QAMs constellation Sbn . After amplification pn , the transmitted signal is conveyed through the n-th subchannel where it experiences an attenuation an and is affected by an AWG noise zn (with variance E |zn |2 = N0 ). The
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11- QAM b1 11 · · ·0 Bit .. . b alloc. 0110 - QAM bN I @ @ @ @ @ @
Fig. 1.
a1 z1 ?- x? g- +g y1
6g s1 x I @ @
6g sN x 6
@
III. A DAPTIVE B IT AND P OWER A LLOCATION We denote the target BER by BERdec at the decoder output. We show that this target BER directly implies fixed values for the receiver (sub-carrier) SNR. How these are determined is detailed in Sec. III-A and Sec. III-B for HD and SD receivers, respectively. Given the CSI, the allocation problem can be formulated as looking for b1 , · · · , bN that minimize [7] N SNRn n=1 bn = b , subject to bn ≤ bmax |an |2
aN zN g- +? - x? g- yN
CSI pN · · · p1 ? Power alloc. Adaptive algorithm
n=1,···,N
where 2bmax stands for the largest affordable QAM. This reduces the bit and power allocation problem to a power allocation problem, which can be solved using the WPA.
Adaptive Multi-carrier transmission.
A. Hard Decision Receiver
n-th sub-channel output is given by
BERdec can be translated into a BER at the decoder input (noted BERcod ) by means of some function
yn = pn an sn + zn , n = 1, · · · , N.
BERdec = fHD (BERcod ) .
BICM was introduced to deal with burst errors that result from deep channel fades. Because (de)encoder and (de)modulator are now separated (by the interleaver), the channel output has no relevance for the decoder and so, cannot be used, as such. Soft metrics were proposed to substitute for the (unavailable) exact bits likelihood [16]. The performance of this (sub-optimal) decoder can be further enhanced using iterative (Turbo) decoding [17]. This, however, requires highly complex soft output decoders. Application of BICM techniques to adaptive multi-carrier systems is straightforward [13], since time diversity is now replaced by frequency diversity. We assume a convolutional encoder and, for simplicity, a Viterbi decoder with hard output, as depicted in Fig. 2. This excludes iterative decoding in the SD receiver. However, the allocation algorithm for SD receivers proposed in Sec. III-B remains applicable to any SD architecture. Also, for simplicity, we do not explicitly mention (bit) interleaving. Apart from using the appropriate bit (metrics) (de)interleaver, interleaving does not affect the decoder structure. Notice that bit interleaving is implicitly assumed since coded bits are assumed to be independent, which is not true unless (ideal) bit interleaving is used. Let c1n , · · · , cbnn be the bits mapped into the symbol sn . We let Sbin be the set of those signal in Sbn whose labels have a bit ”0” at the i-th position. Prior to decoding, coded bits can only be assumed i.i.d. the coded bit and i |c = 0 , are given soft metrics, an approximation of P y n n by 21−bn s∈S i P (yn |sn = s) [18]. In contrast to BICM,
fHD can be approximated by the union bound [19] d≥df ed Pd where df is the free distance of the code and d d d (1 − BERcod )d−d . d odd d =(d+1)/2 Cd BERcod def d d d d−d Pd = d =d/2+1 Cd BERcod (1 − BERcod ) d/2 d/2 +(1/2)Cd [BERcod (1 − BERcod )] . d even In practice, the sums above reduce to finite sums and numerical values of ed are given in tables as in [19]. Not only is the union bound not invertible, it is also a loose one [20]. Alternatively, simulation results can be used to predict the decoder (error) −1 means using performance [20]. In both cases, referring to fHD some look-up tables. −1 The (overall) error rate BERcod = fHD (BERdec ) is (closely) attained if the BER at each sub-carrier is also equal to BERcod [6], [3]. Hence, a second step is to determine the required SNR (say SNRn ) at the input of the n-th de-mapper so that the BER of the demapped symbols equals BERcod . This is expressed by the (modulation dependent) functions def
BERcod = fbn (SNRn ). In the existing algorithms [9], [8], fbn is approximated by the Symbol Error Rate (SER) which, for square QAMs, is given by [21, (5.2-80)]. First, the SER is a loose upper-bound of the BER, even for Gray labeling. Second, this restricts
bn
21−bn is not a constant. However, it can still be ignored since the resulting bit weights are path-independent. M (cin = def 0) = − log P yn |cin = 0 (1/N0 ) mins∈Sbi |yn −an s|2 , n and its counter-part M (cin = 1) do not verify, in general, M (cin = 0) + M (cin = 0) = 1 [16]. Hence, both (0 and 1) metrics need to be computed.
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y1
yN
SNR1- Soft metrics computation BERdec.. P/S - VA . SNRN- Soft metrics computation Fig. 2.
Soft-decision receiver.
assigned bits to have even values. Consequently, no only does the algorithm in [9] fail to reach the optimal power allocation but it also fails to reach the target BER. This last fact is not contradicted by the reported simulations. This drawback can be easily corrected by using analytic approximate BER expressions from the existing literature, which, furthermore, are invertible and accurate for a large range of BER values. For (Gray labeled) square QAMs, we have [22, (12)]
3 2 2bn /2 − 1 erfc BERcod SNRn . bn 2bn /2 2(2bn − 1) For cross QAMs, we can use the Smith’s approximation [12]
Neigh 1 BERcod Gp erfc (1) D SNRn , bn 2 where Gp stands for the Gray penalty (Gray labeling is not possible for cross QAMs), Neigh stands for the average bn number of nearest neighbors √ and D equals 48/(2 31 − 32) if bn ≥ 5 [23, (5)], or 1/(3+ 3) if bn = 3. For BPSK signaling, (1) is the exact BER where Gp , D, Neigh and bn all equal 1. For cross 8-QAM and cross 32-QAM, respectively, Gp equals 11/8 and 7/6; and Neigh equals 3 and 13/4 [23]. B. Soft Decision Receiver
b1
{
SNRcod .. SNRcod - BPSK+AWGN -
- BPSK+AWGN . .. .
bN
- BPSK+AWGN .
{ - BPSK+AWGN
Fig. 3. system.
..
P/S - VA BERdec-
-
SNRcod -
An equivalent model for the SD-decoded adaptive multi-carrier
We continue to refer to fn (s) as the BER obtained at the output of an n-QAM demapper when its input has an SNR equal to s. A BICM system can be modeled as a set of parallel binary channels connected to the encoder by a random switch [18], as depicted in Fig. 3. There, while the modulation is not adaptive, the sub-carriers noise powers depends on the actual (adaptive) bit and power allocation. Intuitively, for optimal bit and power allocation, the equivalent model should exhibit the same noise energy (or equivalently, the same SNR, denoted by SNRcod in Fig. 3) across sub-carriers, i.e. a model that is not sub-carrier selective. Hence, the adaptive multi-carrier system can be modeled as a BPSK transmission over b parallel identical AWGN channels characterized by the same energy per (coded) bit SNRcod (the random switch plays no role here). SNRcod expresses the quality of the soft input to the Viterbi decoder. It, hence, can be linked to BERdec by means of some function BERdec = fSD (SNRcod ) .
Similarly as for fHD , the union bound [19] or simulated performances [20] can be used. The former √ is given by BERdec ≤ e P where P = (1/2)erfc d SNRcod . d d d d≥df Rather than the virtual SNRcod relative to each coded bit, we want to express the required actual SNR per subcarrier SNRn . To establish a relationship between SNRcod and SNRn , we imagine that the i-th virtual binary channel is completed by a slicer. The set of the bn virtual channels conveyed by √ the n-th subcarrier exhibit the same BER given by (1/2)erfc SNRcod . Estimates of the same coded bits can also be obtained by demapping the subcarrier output and the associated BER is given by fbn (SNRn ). We make the assumption that both situations are so that, for √ equivalent, n = 1, · · · , N we have (1/2)erfc SNRcod = fbn (SNRn ) or also
1 SNRn = fb−1 erfc SNR . cod n 2 Strictly speaking, the assumtion that led to the above approximation is wrong. Since it refers to hard decoding, we expect it to be a pessimistic one. We now describe the WPAbased bit and power allocation for the SD receiver. √ def 1) Initialization : Let α = (1/2)erfc SNRcod . For n = 1, · · · , N , let bn = 0 and ∆n = f1−1 (α) /|an |2 . 2) Bit allocation : Repeat b times a) n0 = argminn=1,···,N ∆n . b) Increment bn0 by 1. c) if bn0 < bmax, −1 then ∆n0 = fb−1 (α) − f (α) /|an |2 , bn 0 n0 +1 otherwise, ∆n0 = A, where A is an arbitrarily large constant. 3) Amplification : Compute, for n = 1, · · · , N , pn = SNRn N0 /Ebn /|an |. Notice that the noise level affects the amplification step only. Depending on the (sub-)channel condition, the allocation may N fail to assign all bits ( n=1 bn < b) because of the limitation on the size of the QAM constellations. IV. S IMULATIONS The transmitter of Fig. 1 is simulated with a 1/2-rate encoder (generator matrix 57 in octal) generating b = 768 bits dispatched among N = 256 sub-carriers. After bit and power allocation, BPSK, QPSK, cross 8-QAM, 16-QAM, cross 32QAM and/or 64-QAM (or no modulation at all if no bits are assigned to the sub-carrier) are conducted. Constellations are normalized to have an energy per coded bit equal to 1. The sub-carriers (single-tap) responses, a1 , · · · , aN , are generated as (normalized) independent random Rayleigh variables. This corresponds to a non light-of-sight propagation, a model suitable for reception in dense urban area and indoor environments [15]. Monte Carlo runs are repeated until 100 erroneous information bits are counted. Performances of the Viterbi decoder are predicted using the union bound or simulations results summarized in table I.
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BER
0.8 10−2
10−3
10−4
2 10−5
0.0504 0.8 10−3
0.0264 10−4
0.0126 10−5
0.0074
1.2686
1.6131
2.0106
−1
10
Target BER =0.8 10−2 −3 Target BER =10 Target BER =10−4 Target BER =2 10−5
BSC channel [20, Fig. 7] cross-over probability BER AWGN channel [20, Fig. 5]
Eb /N0 per coded bit
−2
10
Measured BER
TABLE I
P ERFORMANCE OF THE HARD / SOFT INPUT V ITERBI DECODER .
−4
10
−5
10
8
8.5
9
10 10.5 9.5 Eb/N0 per coded bit [dB]
11
11.5
12
Fig. 5. HD receiver. Performance of the Viterbi decoder is predicted
by the union bound (dashed line) and simulations (solid line). −3
10
Target BER =10−3 Target BER =0.8 10−3 Target BER =10−4 Target BER =2 10−5 −5 Target BER =10
−4
10 Measured BER
A first series of tests, presented in Fig. 4, aims at verifying the revisited HD adaptive allocation technique in the same conditions as previous published works, i.e. with no coding/decoding. The target/measured BER is associated with the (hard) output demodulator. Another reason for first assuming an uncoded system is to assess the expected negative consequences of the loose union bound used to evaluate the decoder performance. Fig. 4 shows a perfect match between target and measured BER for BER values as low as 10−8 . Looseness of the union bound is well reflected in Fig. 5 where very low target BERs are attained only approximately. Some enhancement is obtained using simulated performance from [20]. In Fig. 6, the performance of the proposed allocation algorithm for SD receivers is presented and compared to that of the existing algorithm (for HD receivers). Obviously, the allocation algorithm for HD receivers has no relevance when a SD receiver is used, contrarily to the proposed algorithm which successfully approaches the target BER. A lower (than the target) BER is even achieved, which is a consequence of the pessimistic approximations made in Sec. III-B. For a given target BER, the power savings are important, as shown by Fig. 7(a). They are approximately 2 dB, which is the usual gain from using a soft decoder [21]. W.r.t. the BERs actually met, the power savings are only marginal, as shown by Fig. 7(b), but one should keep in mind that these (measured) BERs can be predicted only if the proposed algorithm is used.
−3
10
−5
10
−6
10
8
8.5
9
9.5 10 10.5 Eb/N0 per coded bit [dB]
11
11.5
12
Fig. 6. SD receiver. Adaptive allocation assumes HD (dashed line) and SD (solid line) receiver. Performance of the Viterbi decoder is predicted using simulations.
−2
10
10−2 10−3 10−4 −5 10 10−6 10−7 −8 10
−3
10
V. C ONCLUSION Two adaptive allocation techniques were developed for hard and soft decision decoding receivers, respectively, with the objective of achieving a desired BER and minimizing the transmission power. While the former follows from straightforward extensions of an existing technique, the latter is an original one and proves to be efficient in achieving the objectives stated above. Simulations of the uncoded system show the error rate to be perfectly controllable. However, difficult error analysis of the Viterbi decoder translates into the BER objective being only approximated. This suggests that enhancements of the proposed algorithm are possible and should be investigated.
−4
Measured BER
10
−5
10
−6
10
−7
10
−8
10
8
Fig. 4.
8.5
9
9.5 10 10.5 Eb/N0 per coded bit [dB]
11
11.5
12
Uncoded system. The legend shows the target BER.
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−3
10
−4
BER
10
−5
10
−6
10
4.5
5
5.5
6
7 7.5 6.5 8 Transmission SNR per coded bit [dB]
8.5
9
9.5
Fig. 7. SD receiver. Power requirements. Adaptive allocation assumes HD (dashed line) and SD (solid line) receiver. Target BER (resp. BER measured at Eb /N0 = 10 dB per coded bit) is presented by ’o’ (resp. ’x’). Performance of the Viterbi decoder is predicted using simulations.
ACKNOWLEDGMENT
[12] J. G. Smith, “Odd-Bit Quadrature Amplitude-Shift Keying,” IEEE Trans. Commun., vol. 23, pp. 385-389, Mar. 1975. [13] H. Dai and H. V. Poor, “Turbo Multiuser Detection for Coded DMT VDSL Systems,” IEEE J. Select. Areas Commun., vol. 20, pp. 351-362, Feb. 2002. [14] M. R. G. Butler and I. B. Collings, “A Zero-Forcing Approximate LogLikelihood Receiver for MIMO Bit-Interleaved Coded Modulation,” IEEE Commun. Lett., vol. 8, pp. 105-107, Feb. 2004. [15] Y. L. Guan, Z. Song and K. Zhang, “Modeling of Frequency-Domain Fading Characteristics for OFDM Systems,” in Proc. ICICSP, 2001. [16] E. Zehavi, “8-PSK Trellis Codes for a Rayleigh Channel,” IEEE Trans. Commun., vol. 40, pp. 873-884, May 1992. [17] X. Li and J. A. Ritcey, “Trellis-Coded Modulation with Bit Interleaving and Iterative Decoding,” IEEE J. Select. Areas Commun., vol. 17, pp. 715-724, Apr. 1999. [18] G. Caire, G. Taricco and E. Biglieri, “Bit-Interleaved Coded Modulation,” IEEE Trans. Inform. Theory, vol. 44, pp. 927-946, May 1998. [19] A. G. Burr, “Bounds and Approximations for the Bit Error Probability of Convolutional Codes,” Electronics Letters, vol. 29, pp. 1287-1288, July 1993. [20] J. A. Heller and I. M. Jacobs, “Viterbi Decoding for Staellite and Space Communication,” IEEE Trans. Commun. Technol., vol. COM-19, pp. 835848, Oct. 1971. [21] J. G. Proakis, Digital Communications, 4-th Edition, McGraw-Hill, 2001. [22] K. Cho, D. Yoon, W. Jeong and M. Kavehrad, “BER Analysis of Arbitrary Rectangular QAM,” in Proc. Asilomar Conf. Signals, Systems and Computers, 2001, pp. 1056-1059. [23] P. K. Vitthaladevuni and M.-S. Alouini, “Exact BER Computation for Cross QAM Constellations,” IEEE Trans. Wireless Commun., accepted.
The work reported in this paper has formed part of the Wireless Enablers area of the Core 3 Research Programme of the Virtual Centre of Excellence in Mobile & Personal Communications, Mobile VCE, www.mobilevce.com, whose funding support, including that of EPSRC, is gratefully acknowledged. Fully detailed technical reports on this research are available to Industrial Members of Mobile VCE. R EFERENCES [1] P. S. Chow, J. M. Cioffi and J. A. C. Bingham, “A Practical Discrete Multitone Transceiver Loading Algorithm for Data Transmission over Spectrally Shaped Channels,” IEEE Trans. Commun., vol. 43, pp. 773775, Feb. 1995. [2] B. S. Krongold, K. Ramchandran and D. L. Jones, “Computationally Efficient Optimal Power Allocation Algorithms for Multicarrier Communication Systems,” IEEE Trans. Commun., vol. 48, pp. 23-27, Jan. 2000. [3] R. F. H. Fischer and J. B. Huber, “A New Loading Algorithm for Discrete Multitone Transmission,” in Proc. IEEE GLOBECOM, 1996, pp. 724-728. [4] T. Hunziker and D. Dahlhaus, “Optimal Power Allocation for OFDM Systems with Ideal Bit-Interleaving and Hard-Decision Decoding,” in Proc. IEEE ICC, 2003, pp. 3392-3397. [5] H. Rohling and R. Gr¨ unheid, “Adaptive Coding and Modulation in an OFDM-TDMA Communication System,” in Proc. IEEE VTC, 1998, pp. 773-776. [6] L. Piazzo, “Fast algorithm for power and bit allocation in OFDM systems,” Electronics Letters, vol. 35, pp. 2173-2174, Dec. 1999. [7] Sai Kit Lai, R. S. Cheng, K. B. Letaief and R. D. Murch, “Adaptive Trellis Coded MQAM and Power Optimization for OFDM Transmission,” in Proc. IEEE VTC, 1999, pp. 290-294. [8] C. Mutti, D. Dahlhaus, T. Hunziker and M. Foresti, “Bit and Power Loading Procedures for OFDM Systems with Bit-Interleaved Coded Modulation,” in Proc. ICT, 2003, pp. 1422-1427. [9] C. Y. Wong, R. S. Cheng, K. B. Letaief and R. D. Murch, “Multiuser OFDM with Adaptive Subcarrier, Bit, and Power Allocation,” IEEE J. Select. Areas Commun., vol. 17, pp. 1747-1758, Oct. 1999. [10] H. Rohling and R. Gr¨ unheid, “Performance of an OFDM-TDMA Mobile Communication System,” in Proc. IEEE VTC, 1996, pp. 1589-1593. [11] Kai-Kit Wong, Sai-Kit Lai, R. S-K Cheng, Khaled Ben Letaief and Ross D. Murch, “Adaptive Spatial-Subcarrier Trellis Coded MQAM and Power Optimization for OFDM Transmission,” in Proc. IEEE VTC, 2000, pp. 2049-2053.
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