guidelines for channel code design in quasi-static ... - Semantic Scholar
GUIDELINES FOR CHANNEL CODE DESIGN IN QUASI-STATIC FADING CHANNELS Meritxell Lamarca*
Hanqing Lou and Javier Garcia-Frias*
Dept. of Signal Theory and Communications Technical University of Catalonia Barcelona (Spain) [email protected]
Dept. of Electrical and Computer Engineering University of Delaware Newark, DE 19716 {lou,jgarcia}@ee.udel.edu
ABSTRACT* In this paper we provide guidelines for the design of good binary error correcting codes in quasi-static fading channels, using established design rules in AWGN channels as a starting point. The proposed analysis is based on the Gaussian assumption of demodulator log-likelihood ratios. This assumption allows to decouple the influence of the convergence threshold, slope in the BER waterfall region and error floor of the channel code, so that these parameters can be analyzed separately. Our analysis evidences that, contrary to what happens in AWGN channels, the design of good low rate codes in quasi-static fading channels is much simpler than those of standard rates (.3 to .8 ).
I. INTRODUCTION Quasi-static fading channels have a constant gain throughout the transmission of one codeword and, therefore, from the point of view of the error correcting code (ECC) they behave as an additive white Gaussian noise (AWGN) channel with a random instantaneous SNR that follows certain statistics (e.g. a Rayleigh probability density function). In spite of that, the design of ECCs for the AWGN and the quasi-static fading channels have distinctive features (e.g. see [1, section VI]), since the channel outage is the dominant impairment in quasi-static fading channels [2], whereas it does not appear in the AWGN channel. This different behavior is evidenced in figures 1 and 2. These figures depict the performance of two channel codes with rates .113 and .52 (see code parameters in Table 1) in two different scenarios. Figure 1 shows the Frame Error Rate (FER) of both codes when BPSK modulation is used in an AWGN channel. The parameter SNRmin in the figure stands for the minimum SNR required by an ideal code of the same rate to achieve FER=0 (i.e. the SNR for which the channel capacity1 equals the de* This work has been partially supported by the European Commission (FEDER) and Spanish Government under project TIC2003-05482 and by the National Science Foundation under PECASE/CAREER Award 0093215. 1 With an abuse of notation, throughout the paper we denote as channel capacity of the different transmission schemes the mutual information achieved by each one of them under the constraint of equally likely alphabet symbols.
sired rate). Note that in terms of threshold and error floor, the rate .113 code performance is clearly worse than that of the rate .52 code, evidencing the difficulties encountered in the design of very extreme rates (rates close to 0 or 1) in the AWGN channel. Figure 2 depicts the performance of the same codes when they are applied to a bit-interleaved coded modulation (BICM) [3] with 16QAM Gray system operating in a quasistatic Rayleigh fading and compares it with the channel outage probability for the same rate, defined as: r
Poutage (r ) = ∫ f C fad BICM (C )dC 0
being r the rate and fC fad BICM (·) the probability density function of the channel capacity when 16QAM with BICM is applied in the quasi-static Rayleigh fading channel. Note that both codes have very similar behavior for the whole range of SNR’s but the roles have been exchanged with respect to the AWGN case: none of the codes has an error floor2 and, for FER=10-1, the gap to channel outage (i.e., the gap with respect to the performance of an ideal code) is 0.5 and 0.75 dB for the rate .113 and .52 codes, respectively (i.e., contrary to the AWGN case, the gap is smaller for the code with rate .113). Although not included in the paper due to space constraints, the same behavior has been observed for these codes when they are employed in quasi-static fading channels using other transmission schemes such as 16QAM with multi-level codes (MLC) [4] or BPSK. Hence, the comparison of figures 1 and 2 evidences that, in spite of the similarities between the quasi-static fading channel and the AWGN channel, the design of channel codes for both scenarios should follow different criteria. In this paper we provide an analysis on the performance of ECCs in the quasi-static fading channel that explains this behavior. The analysis is then used to derive some guidelines that can be employed for the design of ECC’s in quasi-static fading channels taking as a starting point the simpler and better characterized design rules existing for the AWGN channel. Due to space constraints, the proposed analysis is only described in the context of BICM, but has also been 2
An error floor will eventually arise at lower FER values. However, in practical applications the range of values of interest is around .1 and .01, i.e. 10% and 1% outage, so it would not be relevant.
shown to provide insightful results for coding design in layered architectures such as MLC or V-BLAST [5] in multiple antenna (MIMO) systems. Note that although BICM in quasi-static fading does not benefit from the code diversity that improves the performance in fast fading channels, it is still meaningful to use it in this scenario, since the capacity losses of this scheme in AWGN channel and quasi-static fading channels are negligible when Gray labeling is employed [3]. Thus, BICM can be used as an illustrative example that is easier to describe than other schemes such as, for instance, MLC. II. PROBLEM STATEMENT A. Signal model Figure 3a depicts the block diagram of the system under analysis: a binary ECC is used in combination with a multilevel modulation to transmit data through a Rayleigh fading channel. Channel state information is perfectly known at the receiver, but it is not available at the transmitter site. We focus in a BICM scheme: the interleaved coded bits are mapped to constellation symbols and, at the receiver, the log-likelihood ratios (LLR) on each coded bit are computed without demapper iterations and delivered to the channel decoder. We consider a non-frequency selective channel where the coherence time is greater than the codeword length, so that the fading gain h remains constant within one codeword transmission and, therefore, the propagation channel can be accurately modeled as a quasi-static fading channel. We will denote as SNRinst=|h|2/σ2 the instantaneous SNR associated to a certain channel realization and as SNR its mean value Eh{|h|2}/σ2. Figure 3b shows the block diagram of the AWGN channel with BPSK modulation that will be used as the starting point for the design of ECCs for quasi-static fading channels. Note that from the point of view of the ECC, the AWGN channel with BPSK modulation and the quasi-static fading channel with BICM behave as binary input channels whose inputs are the coded bits and whose outputs are the LLR’s on those bits (see [3] for the model of this equivalent channel in BICM). The respective capacities of those equivalent binary input channels determine the performance of the ECC code in each scenario. Once more, if MLC or MIMO systems were used an equivalent binary input channel with different LLR statistics would be obtained. The proposed analysis is based on the mapping between the equivalent binary channels, as shown next. B. ECC performance characterization The theoretical limit in AWGN channels is defined by the Shannon capacity, while the performance of a given turbolike ECC in this channel can be described by the FER plot, which typically has a shape similar to the one depicted in figure 4, being SNRmin defined as in section I and figure 1. Hence, the departure of a practical ECC from an ideal code
can be approximately described by the parameters indicated in the figure as SNR gap, Waterfall slope, FER floor and Error floor slope. The theoretical limit in quasi-static fading channels is defined by the ε-outage capacity for a certain SNR: the maximum transmission rate that this channel can support with a probability of error of ε. The FER performance of an ECC in the quasi-static fading channel depends on the outage probability of the channel, as well as on the characteristics of the ECC itself. Thus, the usefulness of the parameters SNR gap, Waterfall slope, Error floor slope employed to characterize the code performance in the AWGN case must be reviewed in this case. For example, the waterfall slope of the FER vs. SNR plot of an ideal code would not be infinite but it would rather coincide with the slope of the channel outage probability for the same code rate. In the following sections we propose a method that can be used to predict the relationship between the performance of a certain code in the AWGN channel and the quasi-static fading channel. Specifically, we discuss what the characteristics (parameters SNR gap, Waterfall slope, and Error floor slope) of a code designed for the AWGN case should be if such a code is to present good performance in quasi-static fading channels. III. FER PREDICTION FOR BICM AND QUASISTATIC FADING CHANNELS We propose a simple method to predict the FER performance of an ECC in the quasi-static fading channel based on the knowledge of the FER performance of this code in the AWGN channel. The proposed method relies on the assumption that the log-likelihood ratios (LLR’s) of the coded bits are Gaussian. This assumption is strictly true for BPSK and QPSK modulation in the AWGN channel and is only an approximation for other modulations such as BICM or MLC. Nevertheless, it has been shown to be useful for the performance analysis of BICM and MLC schemes [6]. A. FER in AWGN channels with BPSK modulation The FER performance of an ECC operating in a quasi-static fading channel with BPSK can be easily predicted if the channel statistics and the FER plot for the AWGN channel are known, as shown next. Consider an ECC whose performance in the AWGN channel with BPSK is described by the function GAWGN BPSK (·) FERAWGN BPSK= GAWGN BPSK (SNR). Equivalently, the code performance can also be described by taking the channel capacity of the equivalent binary-input channel (see figure 3) rather than the SNR as the function parameter. Indeed, the channel capacity for BPSK modulation in AWGN [7] performs a nonlinear mapping of the [-∞,+∞] range of SNR’s into the [0,1] range of capacities CAWGN BPSK=MIAWGN BPSK (SNR),
and it is depicted in figure 5. Since there is a one to one mapping between the channel capacity C and the SNR, we can write FERAWGN BPSK = GAWGN BPSK (MI-1AWGN BPSK (CAWGN BPSK)) = T AWGN BPSK (C AWGN BPSK), where MI-1AWGN BPSK (·) stands for the inverse function of MIAWGN BPSK(·) and T AWGN BPSK= GAWGN BPSK (MI-1AWGN BPSK (·)). As an example, figure 6 depicts the FERAWGN BPSK vs CAWGN BPSK plot of the rate r=.113 code, obtained by combining the FER to SNR plot in figure 1 with the SNR to capacity mapping in figure 5. B. FER in quasi-static fading channels with BPSK modulation In the case of a quasi-static fading channel, the propagation channel for a certain channel realization is equivalent to an AWGN channel with SNR equal to SNRinst. Hence, the FER for that channel realization is given by GAWGN BPSK (SNRinst) and, therefore, the average FER for the quasi-static fading channel can be calculated as
{
}
h
2
σ2
f h (h )dh
{
(
)}
= ∫ TAWGN BPSK (C )f Cfad BPSK (C )dC
C. FER in AWGN channels with BICM modulation The FER performance of a certain ECC in the AWGN channel using BPSK modulation can be used to estimate the performance of the ECC for any other transmission scheme where the bit LLR’s at the decoder input can be accurately modeled as iid Gaussian variables. If this condition holds, the FER performance of the new transmission scheme (BICM) will be the same as the one obtained using a BPSK signal that yields the same Gaussian statistics at the equivalent channel output. Thus, if the bit LLR’s are Gaussian and we define the equivalent LLR signal to noise ratio as SNReq=E2{LLR}/Var{LLR}, then CAWGN BICM= CAWGN BPSKeq= MIAWGN BPSK (SNReq) Therefore, we conclude that the FER of the ECC code in the new transmission scheme (BICM) can be expressed by combining the expression of the equivalent binary channel capacity with that of the FER of the ECC in the AWGN-BPSK configuration: FER AWGN BICM ≅ FER AWGN BPSKeq
(
)
(
I.
Obtain the p.d.f. of the fading gain h (typically |h| follows a Rayleigh distribution).
II.
Obtain the capacity vs SNR plot of the desired scheme (e.g. BICM scheme): CAWGN BICM=MIAWGN BICM (SNR).
III. Combine the results in I, II to obtain the p.d.f. of the instantaneous capacity Cfad BICM =MIAWGN BICM (|h|2/σ2)
IV. Assume that the LLR statistics at the decoder input in the BICM quasi-static fading channel with instantaneous gain h are the same as in an AWGN channel with BPSK modulation and capacity Cfad BICM . Based on this assumption, compute the average FER for BICM in the quasi-static fading channel as
{
(
)}
FER fad BICM = E Cfad BICM TAWGN BPSK C fad BICM =
(1)
Notice that the influence of the different scenario parameters is reflected in (1). For instance:
where fh(h) is the probability density function (p.d.f.) of h. Equivalently, the average FER can also be described in terms of the instantaneous capacity of the fading channel, Cfad BPSK=MIAWGN BPSK (SNRinst ), as FER fad BPSK = E Cfad BPSK TAWGN BPSK C fad BPSK
We now exploit the mapping described in the previous subsections to estimate the FER performance of a BICM scheme using a certain ECC in a quasi-static fading channel:
= ∫ TAWGN BPSK (C )f Cfad BICM (C)dC
FER fad BPSK = E h G AWGN BPSK (SNR inst ) = ∫ G AWGN BPSK h
D. FER in quasi-static fading channels with BICM modulation
≅ TAWGN BPSK C AWGN BPSKeq = TAWGN BPSK C AWGN BICM
)
• The MI function performs a nonlinear mapping from the infinite range of SNR to the [0,1] range of capacities. Hence, the same SNR intervals may correspond to different values of capacity intervals and, therefore, the same SNR gap may be mapped to different capacity margins, depending on the channel code rate. This explains the different behavior in the quasistatic fading channel of codes with different rates and similar performance in the AWGN channel. • According to (1), the average FER in the quasi-static fading channel is obtained as the FER for each instantaneous capacity weighted by the probability of occurrence of that capacity. Thus, the same FER values in the AWGN channel might yield different FER values in the quasi-static fading channel depending on the fading statistics. Figures 6 to 8 illustrate this procedure: figure 7 depicts the p.d.f. of the instantaneous capacity for BICM with 16QAMGray constellation. When figures 6 and 7 are combined following (1), the prediction of the average FER for BICM in the quasi-static fading channel depicted in figure 8 is obtained for the rate r=.113 code. Figure 8 illustrates the precision of the proposed approach for FER prediction. It compares the true FER performance of the codes in Table 1 in the BICM-quasi static fading scenario with the predicted value obtained from the application of (1). Note that the analysis is not exact but it is quite accurate for all SNR’s.
IV. CODE DESIGN CRITERIA FOR QUASI-STATIC FADING CHANNELS The example introduced at the end of the previous section has evidenced that the mapping in equation (1) provides quite accurate FER estimates in the quasi-static fading channel based on the availability of the simulated FER of the same code in the AWGN channel. Comparing the performance of the rate .113 and .52 codes in AWGN and quasi-static fading (figures 1 and 2), it is clear that a poor performance in the AWGN channel in terms of SNR gap, Waterfall slope and Error floor slope (see figure 4) does not necessarily translate into a bad performance in the quasi-static fading channel. For example, the error floor of the r=.113 code that can be observed in the AWGN case has no impact in performance for the considered range of FER (up to 10-4) in quasi-static fading. This occurs even though the AWGN performance plot starts to flatten for FER’s in the range of 10-2-10-3. Although the behaviors in the two scenarios (AWGN and quasi-static fading) are very different, (1) indicates the type of relationship existing between them. In this section we aim at characterizing the correspondence between the FER performance in the BPSK-AWGN channel and in the BPSK-quasi-static fading set up. Once this characterization is clarified, it will be possible to use the wealth of knowledge available for the design of codes in the BPSKAWGN channel to build good channel codes for the quasistatic fading channel. In order to do so, we propose to apply the analysis described in section III to a hypothetical ECC such that its FER performance in the AWGN channel follows the plot in figure 4. In this way, by modifying the synthetically built FER plot at our will we can analyze the impact of the design parameters in AWGN (SNR gap, Waterfall slope, FER floor and Error floor slope) on the performance in a BICM quasi-static fading scenario. Table 2 lists the features of the synthetic FER plots in AWGN that were used in the analysis. Codes I-III have no SNR gap and infinite waterfall slope, whereas they have different error floor behavior. Besides, code IV has a non-infinite waterfall slope and code V has, additionally a SNR gap to capacity in the convergence threshold. The performance of these hypothetical codes in the BICM quasi-static fading setup predicted from the analysis in the previous section is depicted in figure 9. The comparison in these figures is further complemented by the results shown in table 3, which lists the SNR gap in the AWGN channel that results in a SNR loss of 2 or 4 dB for FER=10-1 with respect to channel outage in the quasi-static fading channel. Note that for the same SNR loss in the quasi-static fading channel, rate .113 codes can have larger SNR gaps in the AWGN channel. According to these results the following conclusions can be drawn:
• The existence of an error floor in the AWGN case only degrades FER performance in the BICM quasistatic fading set-up when it starts at FER values close to the target outage probability and has a small slope. Otherwise, it does not affect performance. • The parameter in the AWGN case that has the strongest influence in the performance of the BICM quasistatic fading set-up is the convergence threshold, followed by the slope of the FER performance in the waterfall region. • The design of medium and high rate codes for the BICM quasi-static fading set-up should be made more carefully than low rate codes. The reason is that the same SNR gap or the same FER slope in the waterfall region for the AWGN case result, for the BICM quasi-static fading scenario, in performance losses that increase with the code rate. V.
CONCLUSIONS
In this paper we have presented a simple approach to the analysis of FER performance of ECC’s in quasi-static fading channels. This analysis provides some guidelines for code design based on the performance of the ECC in AWGN. For example, it evidences that in quasi-static fading environments the design of low rate codes is simpler than codes of medium and high rate ones, as opposed to AWGN channels. VI.
REFERENCES
[1] J. Hu and S.L. Miller, “Performance analysis of convollutionally coded systems over quasi-static fading channels”, IEEE Trans. on Wireless Communications, pp. 789795, April 2006 [2] D.Tse, P.Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. [3] G. Taricco, G. Caire, and E. Biglieri, “Bit-interleaved coded modulation”, IEEE Trans. on Information Theory, pp.927-946, May 1998. [4] H.Imai and S.Hirakawa, “A new multilevel coding method using error correcting codes”, IEEE Trans. on Information Theory, pp.371-377, May 1977. [5] P.W.Wolniansky, G.J.Foschini, G.D.Golden, and R.A. Valenzuela, “V-BLAST: an architecture for realizing very high data rates over the rich-scattering wireless channel”, Proc.of URSI Intl. Symp. on Signals, Systems, and Electronics (ISSSE ’98), pp.295–300, Pisa(Italy), Sept.1998. [6] A.Guillen, A.Martinez, G.Caire, “Error probability of bit-interleaved coded modulation using the Gaussian approximation”, Proc. of Conf. On Inform. Sciences (CISS’04), Princeton (USA), March 2004.
[7] S.Benedetto, E.Biglieri, Principles of digital transmssion with wireless applications, Kluwer Academic 1999.
0
10
r=.113 r=.52
[8] J. Garcia-Frias and W. Zhong, “Approaching near Shannon performance by iterative decoding of linear codes with low-density generator matrix”, IEEE Communications Letters, pp.266-268, June 2003.
−1
FER
10
−2
10
Table 1. Code parameters. Rate .113
Rate .52
Overall codeword length 4444 bits 2 serially concatenated LDGM codes [8] with individual rates .834 and .1355
−3
10
Overall codeword length 4444 bits 2 serially concatenated LDGM codes [8] with individual rates .959 and .543
0
SNRfloor
FERfloor
1
1.5
2
2.5
3
3.5
4
SNR−SNRmin (dB)
Figure 1. AWGN performance of the codes with rate .113 and .52 described in Table 1.
Table 2. Features of the synthetic FER plots in an AWGN channel. Code SNRgap
0.5
0
Error floor decay
10
Channel outage in BICM quasi−static fading Code performance in BICM quasi−static fading
I
0 dB
0 dB
10-2
constant error floor
II
0 dB
0 dB
10-2
1 order of magnitude/20dB
0 dB
-2
1 order of magnitude/6dB
-3
constant error floor
-3
constant error floor
Rate r=.52 −1
0 dB
III
0 dB
IV
3 dB
3 dB
10 10 10
Rate r=.113 FER
V
3 dB
10
−2
10
Table 3. SNRgap in AWGN channel vs SNR loss with respect to channel outage in the BICM-quasi-static fading environment at FER=10-1 SNR Loss (FER=10-1) in quasi-static fading
SNRgap in BPSK-AWGN
−3
10
Rate r=.113
Rate r=.52
2 dB
1.7 dB
1.2 dB
4 dB
3.2 dB
2.3 dB
−4
10
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
SNR (dB)
Figure 2. FER performance of the codes with rates .113 and .52 in a BICM-quasi-static fading scenario. The corresponding channel outage probability is also depicted. wgn: N(0,σ2 )
(a)
ECC encoder
bi
Interleaver
Cons tellation Mapping
Transmitter
h De mapper Quasi-s tatic fading channel
Deinterleaver
LLR(b i)
ECC Decoder
LLR(b i)
ECC Decoder
Receive r
Equivalent binary-input channel
wgn: N(0,σ2 )
(b)
ECC encoder
bi
BPSK Mapping Transmitter
De mapper AW GN channel
Receive r
Equivalent binary-input channel
Figure 3. (a) Block diagram of the BICM system in a quasi-static fading channel; (b) AWGN–BPSK model used as a starting point for the proposed analysis..
0
10
FER
True value with BICM 16QAM−Gray quasi−static fading Prediction with BPSK AWGN channel
Convergence Threshold
1
Rate r=.52 −1
W aterfall reg ion waterfa ll slope Error floor Floor slope SNRgap
SNR floor
Rate r=.113 FER
FER floor
10
−2
10
SNR-SNRm in
Figure 4. Typical behavior of the FER vs SNR curve for a turbo-like ECC in the AWGN channel.
−3
10
1 0.9
BICM, 16QAM−Gray BPSK
CAWGN BPSK , CAWGN BICM
0.8
−4
10
2.5
5
7.5
10
12.5
15
0.7
17.5
20
22.5
25
27.5
30
SNR (dB)
0.6
Figure 8. Comparison of the FER performance of the codes with rates .113 and .52 in a BICM-quasi-static fading scenario with their estimated values based on the application of (1).
0.5 0.4 0.3 0.2 0.1 0 −10
−5
0
5 SNRAWGN
10
15
20 0
10
Figure 5. Capacity of the equivalent binary-input channel for the AWGN channel with BPSK and with 16QAM-Gray BICM. −1
10 0
(a)
AWGN BPSK
FER = T
FER
−2
10
(C
AWGN BPSK
)
10
−2
10
Outage r=.113 Code I r=.113 Code II r=.113 Code III r=.113 Outage r=.52 Code I r=.52 Code II r=.52 Code III r=.52
−4
10
−3
10 −6
10
−4
10
2.5
−8
10
0
0.2
0.4
C
0.6
0.8
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
SNR (dB)
1
AWGN BPSK 0
10
Figure 6. FER plot of the r=.113 code for the AWGN channel depicted in figure 1 with the horizontal axis rescaled in terms of CAWGN BPSK .
−1
10
2
SNR=2.5dB SNR=7.5dB SNR=10dB SNR=12dB SNR=15dB
1
10
(b)
f(C)
0
10
FER
10
−2
10
Outage r=.113 Code IV r=.113 Code V r=.113 Outage r=.52 Code IV r=.52 Code V r=.52
−3
10
−1
10
−4
10
10
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
SNR (dB)
−2
0
0.2
0.4
0.6
Cfad BICM
0.8
1
Figure 7. Probability density function of the instantaneous capacity of BICM-16QAM Gray in the quasi-static fading channel for several SNR values.
Figure 9. Prediction of the performance of the synthetic codes in the quasi-static fading with BICM: (a) Codes I-III, (b) Codes IV-V.
Hanqing Lou and Javier Garcia-Frias*
Dept. of Signal Theory and Communications Technical University of Catalonia Barcelona (Spain) [email protected]
Dept. of Electrical and Computer Engineering University of Delaware Newark, DE 19716 {lou,jgarcia}@ee.udel.edu
ABSTRACT* In this paper we provide guidelines for the design of good binary error correcting codes in quasi-static fading channels, using established design rules in AWGN channels as a starting point. The proposed analysis is based on the Gaussian assumption of demodulator log-likelihood ratios. This assumption allows to decouple the influence of the convergence threshold, slope in the BER waterfall region and error floor of the channel code, so that these parameters can be analyzed separately. Our analysis evidences that, contrary to what happens in AWGN channels, the design of good low rate codes in quasi-static fading channels is much simpler than those of standard rates (.3 to .8 ).
I. INTRODUCTION Quasi-static fading channels have a constant gain throughout the transmission of one codeword and, therefore, from the point of view of the error correcting code (ECC) they behave as an additive white Gaussian noise (AWGN) channel with a random instantaneous SNR that follows certain statistics (e.g. a Rayleigh probability density function). In spite of that, the design of ECCs for the AWGN and the quasi-static fading channels have distinctive features (e.g. see [1, section VI]), since the channel outage is the dominant impairment in quasi-static fading channels [2], whereas it does not appear in the AWGN channel. This different behavior is evidenced in figures 1 and 2. These figures depict the performance of two channel codes with rates .113 and .52 (see code parameters in Table 1) in two different scenarios. Figure 1 shows the Frame Error Rate (FER) of both codes when BPSK modulation is used in an AWGN channel. The parameter SNRmin in the figure stands for the minimum SNR required by an ideal code of the same rate to achieve FER=0 (i.e. the SNR for which the channel capacity1 equals the de* This work has been partially supported by the European Commission (FEDER) and Spanish Government under project TIC2003-05482 and by the National Science Foundation under PECASE/CAREER Award 0093215. 1 With an abuse of notation, throughout the paper we denote as channel capacity of the different transmission schemes the mutual information achieved by each one of them under the constraint of equally likely alphabet symbols.
sired rate). Note that in terms of threshold and error floor, the rate .113 code performance is clearly worse than that of the rate .52 code, evidencing the difficulties encountered in the design of very extreme rates (rates close to 0 or 1) in the AWGN channel. Figure 2 depicts the performance of the same codes when they are applied to a bit-interleaved coded modulation (BICM) [3] with 16QAM Gray system operating in a quasistatic Rayleigh fading and compares it with the channel outage probability for the same rate, defined as: r
Poutage (r ) = ∫ f C fad BICM (C )dC 0
being r the rate and fC fad BICM (·) the probability density function of the channel capacity when 16QAM with BICM is applied in the quasi-static Rayleigh fading channel. Note that both codes have very similar behavior for the whole range of SNR’s but the roles have been exchanged with respect to the AWGN case: none of the codes has an error floor2 and, for FER=10-1, the gap to channel outage (i.e., the gap with respect to the performance of an ideal code) is 0.5 and 0.75 dB for the rate .113 and .52 codes, respectively (i.e., contrary to the AWGN case, the gap is smaller for the code with rate .113). Although not included in the paper due to space constraints, the same behavior has been observed for these codes when they are employed in quasi-static fading channels using other transmission schemes such as 16QAM with multi-level codes (MLC) [4] or BPSK. Hence, the comparison of figures 1 and 2 evidences that, in spite of the similarities between the quasi-static fading channel and the AWGN channel, the design of channel codes for both scenarios should follow different criteria. In this paper we provide an analysis on the performance of ECCs in the quasi-static fading channel that explains this behavior. The analysis is then used to derive some guidelines that can be employed for the design of ECC’s in quasi-static fading channels taking as a starting point the simpler and better characterized design rules existing for the AWGN channel. Due to space constraints, the proposed analysis is only described in the context of BICM, but has also been 2
An error floor will eventually arise at lower FER values. However, in practical applications the range of values of interest is around .1 and .01, i.e. 10% and 1% outage, so it would not be relevant.
shown to provide insightful results for coding design in layered architectures such as MLC or V-BLAST [5] in multiple antenna (MIMO) systems. Note that although BICM in quasi-static fading does not benefit from the code diversity that improves the performance in fast fading channels, it is still meaningful to use it in this scenario, since the capacity losses of this scheme in AWGN channel and quasi-static fading channels are negligible when Gray labeling is employed [3]. Thus, BICM can be used as an illustrative example that is easier to describe than other schemes such as, for instance, MLC. II. PROBLEM STATEMENT A. Signal model Figure 3a depicts the block diagram of the system under analysis: a binary ECC is used in combination with a multilevel modulation to transmit data through a Rayleigh fading channel. Channel state information is perfectly known at the receiver, but it is not available at the transmitter site. We focus in a BICM scheme: the interleaved coded bits are mapped to constellation symbols and, at the receiver, the log-likelihood ratios (LLR) on each coded bit are computed without demapper iterations and delivered to the channel decoder. We consider a non-frequency selective channel where the coherence time is greater than the codeword length, so that the fading gain h remains constant within one codeword transmission and, therefore, the propagation channel can be accurately modeled as a quasi-static fading channel. We will denote as SNRinst=|h|2/σ2 the instantaneous SNR associated to a certain channel realization and as SNR its mean value Eh{|h|2}/σ2. Figure 3b shows the block diagram of the AWGN channel with BPSK modulation that will be used as the starting point for the design of ECCs for quasi-static fading channels. Note that from the point of view of the ECC, the AWGN channel with BPSK modulation and the quasi-static fading channel with BICM behave as binary input channels whose inputs are the coded bits and whose outputs are the LLR’s on those bits (see [3] for the model of this equivalent channel in BICM). The respective capacities of those equivalent binary input channels determine the performance of the ECC code in each scenario. Once more, if MLC or MIMO systems were used an equivalent binary input channel with different LLR statistics would be obtained. The proposed analysis is based on the mapping between the equivalent binary channels, as shown next. B. ECC performance characterization The theoretical limit in AWGN channels is defined by the Shannon capacity, while the performance of a given turbolike ECC in this channel can be described by the FER plot, which typically has a shape similar to the one depicted in figure 4, being SNRmin defined as in section I and figure 1. Hence, the departure of a practical ECC from an ideal code
can be approximately described by the parameters indicated in the figure as SNR gap, Waterfall slope, FER floor and Error floor slope. The theoretical limit in quasi-static fading channels is defined by the ε-outage capacity for a certain SNR: the maximum transmission rate that this channel can support with a probability of error of ε. The FER performance of an ECC in the quasi-static fading channel depends on the outage probability of the channel, as well as on the characteristics of the ECC itself. Thus, the usefulness of the parameters SNR gap, Waterfall slope, Error floor slope employed to characterize the code performance in the AWGN case must be reviewed in this case. For example, the waterfall slope of the FER vs. SNR plot of an ideal code would not be infinite but it would rather coincide with the slope of the channel outage probability for the same code rate. In the following sections we propose a method that can be used to predict the relationship between the performance of a certain code in the AWGN channel and the quasi-static fading channel. Specifically, we discuss what the characteristics (parameters SNR gap, Waterfall slope, and Error floor slope) of a code designed for the AWGN case should be if such a code is to present good performance in quasi-static fading channels. III. FER PREDICTION FOR BICM AND QUASISTATIC FADING CHANNELS We propose a simple method to predict the FER performance of an ECC in the quasi-static fading channel based on the knowledge of the FER performance of this code in the AWGN channel. The proposed method relies on the assumption that the log-likelihood ratios (LLR’s) of the coded bits are Gaussian. This assumption is strictly true for BPSK and QPSK modulation in the AWGN channel and is only an approximation for other modulations such as BICM or MLC. Nevertheless, it has been shown to be useful for the performance analysis of BICM and MLC schemes [6]. A. FER in AWGN channels with BPSK modulation The FER performance of an ECC operating in a quasi-static fading channel with BPSK can be easily predicted if the channel statistics and the FER plot for the AWGN channel are known, as shown next. Consider an ECC whose performance in the AWGN channel with BPSK is described by the function GAWGN BPSK (·) FERAWGN BPSK= GAWGN BPSK (SNR). Equivalently, the code performance can also be described by taking the channel capacity of the equivalent binary-input channel (see figure 3) rather than the SNR as the function parameter. Indeed, the channel capacity for BPSK modulation in AWGN [7] performs a nonlinear mapping of the [-∞,+∞] range of SNR’s into the [0,1] range of capacities CAWGN BPSK=MIAWGN BPSK (SNR),
and it is depicted in figure 5. Since there is a one to one mapping between the channel capacity C and the SNR, we can write FERAWGN BPSK = GAWGN BPSK (MI-1AWGN BPSK (CAWGN BPSK)) = T AWGN BPSK (C AWGN BPSK), where MI-1AWGN BPSK (·) stands for the inverse function of MIAWGN BPSK(·) and T AWGN BPSK= GAWGN BPSK (MI-1AWGN BPSK (·)). As an example, figure 6 depicts the FERAWGN BPSK vs CAWGN BPSK plot of the rate r=.113 code, obtained by combining the FER to SNR plot in figure 1 with the SNR to capacity mapping in figure 5. B. FER in quasi-static fading channels with BPSK modulation In the case of a quasi-static fading channel, the propagation channel for a certain channel realization is equivalent to an AWGN channel with SNR equal to SNRinst. Hence, the FER for that channel realization is given by GAWGN BPSK (SNRinst) and, therefore, the average FER for the quasi-static fading channel can be calculated as
{
}
h
2
σ2
f h (h )dh
{
(
)}
= ∫ TAWGN BPSK (C )f Cfad BPSK (C )dC
C. FER in AWGN channels with BICM modulation The FER performance of a certain ECC in the AWGN channel using BPSK modulation can be used to estimate the performance of the ECC for any other transmission scheme where the bit LLR’s at the decoder input can be accurately modeled as iid Gaussian variables. If this condition holds, the FER performance of the new transmission scheme (BICM) will be the same as the one obtained using a BPSK signal that yields the same Gaussian statistics at the equivalent channel output. Thus, if the bit LLR’s are Gaussian and we define the equivalent LLR signal to noise ratio as SNReq=E2{LLR}/Var{LLR}, then CAWGN BICM= CAWGN BPSKeq= MIAWGN BPSK (SNReq) Therefore, we conclude that the FER of the ECC code in the new transmission scheme (BICM) can be expressed by combining the expression of the equivalent binary channel capacity with that of the FER of the ECC in the AWGN-BPSK configuration: FER AWGN BICM ≅ FER AWGN BPSKeq
(
)
(
I.
Obtain the p.d.f. of the fading gain h (typically |h| follows a Rayleigh distribution).
II.
Obtain the capacity vs SNR plot of the desired scheme (e.g. BICM scheme): CAWGN BICM=MIAWGN BICM (SNR).
III. Combine the results in I, II to obtain the p.d.f. of the instantaneous capacity Cfad BICM =MIAWGN BICM (|h|2/σ2)
IV. Assume that the LLR statistics at the decoder input in the BICM quasi-static fading channel with instantaneous gain h are the same as in an AWGN channel with BPSK modulation and capacity Cfad BICM . Based on this assumption, compute the average FER for BICM in the quasi-static fading channel as
{
(
)}
FER fad BICM = E Cfad BICM TAWGN BPSK C fad BICM =
(1)
Notice that the influence of the different scenario parameters is reflected in (1). For instance:
where fh(h) is the probability density function (p.d.f.) of h. Equivalently, the average FER can also be described in terms of the instantaneous capacity of the fading channel, Cfad BPSK=MIAWGN BPSK (SNRinst ), as FER fad BPSK = E Cfad BPSK TAWGN BPSK C fad BPSK
We now exploit the mapping described in the previous subsections to estimate the FER performance of a BICM scheme using a certain ECC in a quasi-static fading channel:
= ∫ TAWGN BPSK (C )f Cfad BICM (C)dC
FER fad BPSK = E h G AWGN BPSK (SNR inst ) = ∫ G AWGN BPSK h
D. FER in quasi-static fading channels with BICM modulation
≅ TAWGN BPSK C AWGN BPSKeq = TAWGN BPSK C AWGN BICM
)
• The MI function performs a nonlinear mapping from the infinite range of SNR to the [0,1] range of capacities. Hence, the same SNR intervals may correspond to different values of capacity intervals and, therefore, the same SNR gap may be mapped to different capacity margins, depending on the channel code rate. This explains the different behavior in the quasistatic fading channel of codes with different rates and similar performance in the AWGN channel. • According to (1), the average FER in the quasi-static fading channel is obtained as the FER for each instantaneous capacity weighted by the probability of occurrence of that capacity. Thus, the same FER values in the AWGN channel might yield different FER values in the quasi-static fading channel depending on the fading statistics. Figures 6 to 8 illustrate this procedure: figure 7 depicts the p.d.f. of the instantaneous capacity for BICM with 16QAMGray constellation. When figures 6 and 7 are combined following (1), the prediction of the average FER for BICM in the quasi-static fading channel depicted in figure 8 is obtained for the rate r=.113 code. Figure 8 illustrates the precision of the proposed approach for FER prediction. It compares the true FER performance of the codes in Table 1 in the BICM-quasi static fading scenario with the predicted value obtained from the application of (1). Note that the analysis is not exact but it is quite accurate for all SNR’s.
IV. CODE DESIGN CRITERIA FOR QUASI-STATIC FADING CHANNELS The example introduced at the end of the previous section has evidenced that the mapping in equation (1) provides quite accurate FER estimates in the quasi-static fading channel based on the availability of the simulated FER of the same code in the AWGN channel. Comparing the performance of the rate .113 and .52 codes in AWGN and quasi-static fading (figures 1 and 2), it is clear that a poor performance in the AWGN channel in terms of SNR gap, Waterfall slope and Error floor slope (see figure 4) does not necessarily translate into a bad performance in the quasi-static fading channel. For example, the error floor of the r=.113 code that can be observed in the AWGN case has no impact in performance for the considered range of FER (up to 10-4) in quasi-static fading. This occurs even though the AWGN performance plot starts to flatten for FER’s in the range of 10-2-10-3. Although the behaviors in the two scenarios (AWGN and quasi-static fading) are very different, (1) indicates the type of relationship existing between them. In this section we aim at characterizing the correspondence between the FER performance in the BPSK-AWGN channel and in the BPSK-quasi-static fading set up. Once this characterization is clarified, it will be possible to use the wealth of knowledge available for the design of codes in the BPSKAWGN channel to build good channel codes for the quasistatic fading channel. In order to do so, we propose to apply the analysis described in section III to a hypothetical ECC such that its FER performance in the AWGN channel follows the plot in figure 4. In this way, by modifying the synthetically built FER plot at our will we can analyze the impact of the design parameters in AWGN (SNR gap, Waterfall slope, FER floor and Error floor slope) on the performance in a BICM quasi-static fading scenario. Table 2 lists the features of the synthetic FER plots in AWGN that were used in the analysis. Codes I-III have no SNR gap and infinite waterfall slope, whereas they have different error floor behavior. Besides, code IV has a non-infinite waterfall slope and code V has, additionally a SNR gap to capacity in the convergence threshold. The performance of these hypothetical codes in the BICM quasi-static fading setup predicted from the analysis in the previous section is depicted in figure 9. The comparison in these figures is further complemented by the results shown in table 3, which lists the SNR gap in the AWGN channel that results in a SNR loss of 2 or 4 dB for FER=10-1 with respect to channel outage in the quasi-static fading channel. Note that for the same SNR loss in the quasi-static fading channel, rate .113 codes can have larger SNR gaps in the AWGN channel. According to these results the following conclusions can be drawn:
• The existence of an error floor in the AWGN case only degrades FER performance in the BICM quasistatic fading set-up when it starts at FER values close to the target outage probability and has a small slope. Otherwise, it does not affect performance. • The parameter in the AWGN case that has the strongest influence in the performance of the BICM quasistatic fading set-up is the convergence threshold, followed by the slope of the FER performance in the waterfall region. • The design of medium and high rate codes for the BICM quasi-static fading set-up should be made more carefully than low rate codes. The reason is that the same SNR gap or the same FER slope in the waterfall region for the AWGN case result, for the BICM quasi-static fading scenario, in performance losses that increase with the code rate. V.
CONCLUSIONS
In this paper we have presented a simple approach to the analysis of FER performance of ECC’s in quasi-static fading channels. This analysis provides some guidelines for code design based on the performance of the ECC in AWGN. For example, it evidences that in quasi-static fading environments the design of low rate codes is simpler than codes of medium and high rate ones, as opposed to AWGN channels. VI.
REFERENCES
[1] J. Hu and S.L. Miller, “Performance analysis of convollutionally coded systems over quasi-static fading channels”, IEEE Trans. on Wireless Communications, pp. 789795, April 2006 [2] D.Tse, P.Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. [3] G. Taricco, G. Caire, and E. Biglieri, “Bit-interleaved coded modulation”, IEEE Trans. on Information Theory, pp.927-946, May 1998. [4] H.Imai and S.Hirakawa, “A new multilevel coding method using error correcting codes”, IEEE Trans. on Information Theory, pp.371-377, May 1977. [5] P.W.Wolniansky, G.J.Foschini, G.D.Golden, and R.A. Valenzuela, “V-BLAST: an architecture for realizing very high data rates over the rich-scattering wireless channel”, Proc.of URSI Intl. Symp. on Signals, Systems, and Electronics (ISSSE ’98), pp.295–300, Pisa(Italy), Sept.1998. [6] A.Guillen, A.Martinez, G.Caire, “Error probability of bit-interleaved coded modulation using the Gaussian approximation”, Proc. of Conf. On Inform. Sciences (CISS’04), Princeton (USA), March 2004.
[7] S.Benedetto, E.Biglieri, Principles of digital transmssion with wireless applications, Kluwer Academic 1999.
0
10
r=.113 r=.52
[8] J. Garcia-Frias and W. Zhong, “Approaching near Shannon performance by iterative decoding of linear codes with low-density generator matrix”, IEEE Communications Letters, pp.266-268, June 2003.
−1
FER
10
−2
10
Table 1. Code parameters. Rate .113
Rate .52
Overall codeword length 4444 bits 2 serially concatenated LDGM codes [8] with individual rates .834 and .1355
−3
10
Overall codeword length 4444 bits 2 serially concatenated LDGM codes [8] with individual rates .959 and .543
0
SNRfloor
FERfloor
1
1.5
2
2.5
3
3.5
4
SNR−SNRmin (dB)
Figure 1. AWGN performance of the codes with rate .113 and .52 described in Table 1.
Table 2. Features of the synthetic FER plots in an AWGN channel. Code SNRgap
0.5
0
Error floor decay
10
Channel outage in BICM quasi−static fading Code performance in BICM quasi−static fading
I
0 dB
0 dB
10-2
constant error floor
II
0 dB
0 dB
10-2
1 order of magnitude/20dB
0 dB
-2
1 order of magnitude/6dB
-3
constant error floor
-3
constant error floor
Rate r=.52 −1
0 dB
III
0 dB
IV
3 dB
3 dB
10 10 10
Rate r=.113 FER
V
3 dB
10
−2
10
Table 3. SNRgap in AWGN channel vs SNR loss with respect to channel outage in the BICM-quasi-static fading environment at FER=10-1 SNR Loss (FER=10-1) in quasi-static fading
SNRgap in BPSK-AWGN
−3
10
Rate r=.113
Rate r=.52
2 dB
1.7 dB
1.2 dB
4 dB
3.2 dB
2.3 dB
−4
10
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
SNR (dB)
Figure 2. FER performance of the codes with rates .113 and .52 in a BICM-quasi-static fading scenario. The corresponding channel outage probability is also depicted. wgn: N(0,σ2 )
(a)
ECC encoder
bi
Interleaver
Cons tellation Mapping
Transmitter
h De mapper Quasi-s tatic fading channel
Deinterleaver
LLR(b i)
ECC Decoder
LLR(b i)
ECC Decoder
Receive r
Equivalent binary-input channel
wgn: N(0,σ2 )
(b)
ECC encoder
bi
BPSK Mapping Transmitter
De mapper AW GN channel
Receive r
Equivalent binary-input channel
Figure 3. (a) Block diagram of the BICM system in a quasi-static fading channel; (b) AWGN–BPSK model used as a starting point for the proposed analysis..
0
10
FER
True value with BICM 16QAM−Gray quasi−static fading Prediction with BPSK AWGN channel
Convergence Threshold
1
Rate r=.52 −1
W aterfall reg ion waterfa ll slope Error floor Floor slope SNRgap
SNR floor
Rate r=.113 FER
FER floor
10
−2
10
SNR-SNRm in
Figure 4. Typical behavior of the FER vs SNR curve for a turbo-like ECC in the AWGN channel.
−3
10
1 0.9
BICM, 16QAM−Gray BPSK
CAWGN BPSK , CAWGN BICM
0.8
−4
10
2.5
5
7.5
10
12.5
15
0.7
17.5
20
22.5
25
27.5
30
SNR (dB)
0.6
Figure 8. Comparison of the FER performance of the codes with rates .113 and .52 in a BICM-quasi-static fading scenario with their estimated values based on the application of (1).
0.5 0.4 0.3 0.2 0.1 0 −10
−5
0
5 SNRAWGN
10
15
20 0
10
Figure 5. Capacity of the equivalent binary-input channel for the AWGN channel with BPSK and with 16QAM-Gray BICM. −1
10 0
(a)
AWGN BPSK
FER = T
FER
−2
10
(C
AWGN BPSK
)
10
−2
10
Outage r=.113 Code I r=.113 Code II r=.113 Code III r=.113 Outage r=.52 Code I r=.52 Code II r=.52 Code III r=.52
−4
10
−3
10 −6
10
−4
10
2.5
−8
10
0
0.2
0.4
C
0.6
0.8
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
SNR (dB)
1
AWGN BPSK 0
10
Figure 6. FER plot of the r=.113 code for the AWGN channel depicted in figure 1 with the horizontal axis rescaled in terms of CAWGN BPSK .
−1
10
2
SNR=2.5dB SNR=7.5dB SNR=10dB SNR=12dB SNR=15dB
1
10
(b)
f(C)
0
10
FER
10
−2
10
Outage r=.113 Code IV r=.113 Code V r=.113 Outage r=.52 Code IV r=.52 Code V r=.52
−3
10
−1
10
−4
10
10
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
27.5
30
SNR (dB)
−2
0
0.2
0.4
0.6
Cfad BICM
0.8
1
Figure 7. Probability density function of the instantaneous capacity of BICM-16QAM Gray in the quasi-static fading channel for several SNR values.
Figure 9. Prediction of the performance of the synthetic codes in the quasi-static fading with BICM: (a) Codes I-III, (b) Codes IV-V.
