Approximations of the packet error rate under quasi-static fading in direct and relayed links
- Paul Ferrand^{1}Email author,
- Jean-Marie Gorce^{1} and
- Claire Goursaud^{1}
https://doi.org/10.1186/s13638-014-0239-4
© Ferrand et al.; licensee Springer. 2015
Received: 24 June 2014
Accepted: 30 December 2014
Published: 25 January 2015
Abstract
The packet error rate (PER) is a metric of choice to compute the practical performance of communication systems experiencing block fading, e.g., fading processes whose coherence time is relatively slow when compared to the symbol transmission rate. For these types of channels, we derive a closed-form asymptotic expression which approximates the value of the PER for high signal-to-noise ratio (SNR). We also provide another approximation based on a unit-step formulation of the symbol error rate (SER). We show that the two approximations are related and may be derived from one another, thereby allowing us to obtain closed-form approximations of the block fading PER in both coded and uncoded systems. We then show how these approximations may be used in practice, through the derivation of a packet error outage (PEO) metric covering the case where the links experience shadowing on top of block fading, as well as asymptotically optimal power allocations in relay channels under a block fading hypothesis.
Keywords
1 Introduction
Most of the existing literature on performance evaluation of fading channels is concentrated on the symbol error rate of the links. A good review of these results as well as an interesting framework for the evaluation of symbol error rates in fading channels is available in the book of Simon and Alouini [1]. These results are focused on the symbol error rate of fading channels; when the fading is relatively fast compared to the symbol transmission duration, with proper interleaving, one can extend them to packet error rates [2]. On the other hand, when the fading is much slower than the packet transmission time, one has to consider that most symbols in the packet will experience the same fading state - a model known as block fading or quasi-static fading. A metric of choice for performance evaluation of this system models is the outage capacity [3-5]. Being reliant on infinitely long random codes, these results provide a theoretical bound on the performance of transmission schemes over block fading channels, but they fail to capture the behavior of practical systems, which is the foremost motivation for the current work.
The results presented here root themselves in the work of Wang and Giannakis [6] who presented an asymptotic approximation of symbol error rates in fading channels using a Taylor expansion limited to the first term. This approach is well suited to channels whose probability density function is approximately polynomial near zero, but fails for certain models of fading such as log-normal shadowing. Wang and Giannakis’ approach has been extended recently by Xi et al. for the packet error rate of block fading channels [7]. Alternatively, in the case of block fading Rayleigh channels - a common model - Liu et al. derived a tight approximation of the PER of uncoded packet transmissions [8].
These asymptotic approximations have a number of useful applications. They are in general well suited to the study of relay networks and have been extended to general amplify-and-forward relays by Ribeiro et al. [9], who showed that in that case, the optimal selection criterion for relays is to maximize the harmonic mean of the source-relay and relay-destination links. Liu et al. [10] further extended these results and produced a comprehensive treatment of the end-to-end symbol error rates in relay channel for both amplify-and-forward and decode-and-forward protocols, including asymptotic approximations, and for a variety of modulation schemes (see in particular [10], Ch.5.). In a similar manner to the approach we present here, Annavajjala et al. treated the asymptotical outage probability of direct links as well as relayed amplify-and-forward and decode-and-forward protocols [11]. A similar work has been treated by [12] in the context of fully cooperative relay channels under a block fading model.
The work presented in this paper extends the one of Xi et al. in [7]. We strengthen their results by giving a closed-form approximation of the asymptotic coding gain rather than a numerical evaluation, for usual forms of bit error rate (BER) expressions used in realistic cases. We then express that a unit step approximation can be derived for the packet error rate of block fading channels that closely matches the numerical computation of the packet error rate in a tractable closed-form expression. Such an approximation has been studied in the context of long block codes [13,14] as well as [15]. We survey these results and show in fact that both approximations depend on the same parameters, thereby allowing to easily compute one given the other. We illustrate both these approaches through the approximations of uncoded and coded packet error rates in various fading channel models. Using the fact that the asymptotic formulation is invertible with respect to the mean SNR, we present a packet error outage metric, with applications to channels where links are subject to both fading and log-normal shadowing effects simultaneously. We finally use the asymptotic approximation to derive the optimal power allocation for different relaying protocols, where we derive complementary results to [10-12] for relay channels where full cooperation is not technically achievable.
2 System model
Probability density functions for the fading models considered in this paper
p.d.f. of γ =| h [ m ]| ^{ 2 } | Parameters | |
---|---|---|
Rayleigh model | \(\displaystyle \frac {1}{\bar {\gamma }} \exp \left (- \frac {\gamma }{\bar {\gamma }} \right) \) | \(\bar {\gamma }\) |
Rice model | \(\displaystyle \frac {(1+K)e^{-K}}{\bar {\gamma }}\exp \left (-\frac {\gamma (1+K)}{\bar {\gamma }}\right)I_{0}\left (2 \sqrt {K(K+1)\frac {\gamma }{\bar {\gamma }}} \right)\) | \(K,\bar {\gamma }\) |
Nakagami model | \(\displaystyle \frac {m^{m} \gamma ^{m - 1}}{\left (\bar {\gamma } \right)^{m} \Gamma (m)} \exp \left (-\cfrac {m \gamma }{\bar {\gamma }} \right) \) | \(m, \bar {\gamma }\) |
3 Approximations of the PER in direct links
The first function is the theoretical symbol error rate of a binary PSK (BPSK) modulation when ν=1 and k=2 and an approximation for higher order PSK modulations and quadrature amplitude modulations (QAM) [1]. Both parameters depend on the constellation size, codeword mapping and geometry; specific values for the parameters may be found in [1]. The second function in (7) is classically used to fit bit or symbol error rates in realistic systems.
3.1 Asymptotic approximations
Parameters a and t for channels of interest
Model | t | a |
---|---|---|
Rayleigh | 0 | 1 |
Rice | 0 | (1+K)e ^{−K } |
Nakagami | m−1 | m ^{ m }/Γ(m) |
where G _{ c } is termed the coding gain and G _{ d } is the diversity gain. This result has been stated without proof in [12] to the block PER (5), replacing the term p _{ s }(γ) in (8) by p _{ p }(γ) from (4). We provide here a thorough statement and proof of this extension.
Theorem 1.
and parameters a and t depend on the fading distribution f _{ γ }(γ) and are listed in Table 2 for common fading models.
Proof.
The complete proof is detailed in Appendix Appendix 1 Proof of Theorem 1. A key difference between our hypotheses and the ones of [6] is that they require the p.d.f. of β to have the asymptotic form a β ^{ t }+o(β ^{ t+ε }), whereas the above proof and the conditions in [12] are looser and require a O(β ^{ t+ε }) term. The proof of [6] actually does not use the o(β ^{ t+ε }) term but rather a o(β ^{ t }) term, which may mean that the o(β ^{ t+ε }) is a typographic error. Some parts of the proof are actually made simpler by using a O(·) term since we can swap the O(·) terms and integrations. The conclusions are the same, since O(β ^{ t+ε })⊂o(β ^{ t }) for ε>0. We added a condition p _{ p }(γ)∈O(γ ^{−(t+1+ε)}) compared to the statement in [12], but as far as we could tell, it is needed for the proof to be complete. In essence, it is close to the discussion of [6]; it ensures that \(p_{p}(\beta \bar {\gamma })\) tends to a dirac function as \(\bar {\gamma } \to \infty \). The examples given by [6] all have an exponential decrease, as do most bit or error rates in practice, and thus this additional condition is verified for packet error rates in most cases of interest.
Using the relation between the Q function and the incomplete gamma function ([17], Eq.7.11.2), we can express (14) and thus an upper bound on the coding gain \(G_{c}^{\mathrm {(block,th)}}\) of theoretical PSK modulations in block fading channels using the following proposition, with N the block size and a,t from Table 2.
Proposition 1.
Proof.
The proof is given in Appendix Appendix 2 Proof of Proposition 1.
A similar bound for symbol error rates of the form (7) may be derived in a simple way along the line of the proof of Proposition 1, and we have the following proposition:
Proposition 2.
3.2 Unit step approximations
We can see that both channels have a similar behavior for low values of \(\bar {\gamma }\), but the Rician fading model generates an asymptote in \(O(\bar {\gamma }^{-1})\), whereas the Nakagami fading model’s asymptote is \(O(\bar {\gamma }^{-m})\) (see Tables 2 and (8)). Using the simple asymptote formulations, it is possible to extract the crossing between the asymptotes and therefore define a piecewise function approximation of the PER of a Rician block fading channel better at medium SNR. However, this method is not entirely satisfactory. We show in this section that another form of approximation provides closer results in this case, and that this approximation is actually based on the exact same coefficients as the asymptotic approximation described above.
This approximation is mathematically valid whenever the c.d.f. exists and can thus provide a way to treat cases where the asymptotic approximation is looser than expected, e.g., for the Rician fading model. It can also be of use when the asymptotic approximation does not exist, which is the case for the log-normal fading model whose p.d.f. behaves exponentially near 0 (recall the conditions in [6], Sec. II).
Unit step approximations of the block fading PER for usual channel models
Model | Block PER approximate |
---|---|
Rayleigh | \(\displaystyle 1- \exp \left (-\frac {\gamma _{\text {th}}}{\bar {\gamma }} \right)\) |
Rice | \(\displaystyle 1- Q_{1}\left (\sqrt {2K},\sqrt {2(K+1)\frac {\gamma _{\text {th}}}{\bar {\gamma }}}\right)\) |
Nakagami | \(\displaystyle \frac {1}{\Gamma (m)} \gamma \left (m,\frac {m \gamma _{\text {th}}}{\bar {\gamma }}\right)\) |
Log-normal | \(\displaystyle Q \left (\frac {\bar {\gamma } - 10 \log _{10} \left (\gamma _{\text {th}}\right)}{\sigma _{S}} \right)\) |
A similar argument may be extended to the criterion of [14]; if (21) is not verified, then both integrals in (19) and (20) are effectively unbounded. Equation (21) thus provides a necessary condition for minimizing either the absolute or relative error over the SNR range, but a priori not a sufficient one. For fading channels with an polynomial asymptotic expansion in 0 [6], only one threshold will verify (21).
Theorem 2.
Remark 1.
The following corollary derives from the proof of Theorem 2 and indicates that the choice of threshold for the unit-step approximation is the best choice when considering the relative and absolute error criterions.
Corollary 1.
For fading channels whose p.d.f. verify the conditions in Theorem 2, choosing γ _{0} as(23) minimizes both the absolute and relative error criterions (19) and (20).
Proof.
Both proofs are detailed in Appendix Appendix 3 Proofs of Theorem 2 and Corollary 1.
4 Applications of the PER approximations
These approximations have a number of practical applications. They are easy to evaluate and, in some cases, invertible with respect to \(\bar {\gamma }\). In this section, we present two of these applications; we define a packet error outage metric for the case where \(\bar {\gamma }\) is only known at the transmitter through its statistical distribution. In a second part, we survey and illustrate how the asymptotic approximations may be used to compute optimal power allocations in multi-user channels.
4.1 Packet error outage
4.2 End-to-end PER of relay channels
As represented on Figure 6, the path loss is different for the three links in the relay channel and captured through the terms s _{1},s _{2}, and s _{3}. We aim at allocating a global power P _{tot} between the source and the relay - or equivalently a global normalized transmit SNR \(\bar {\gamma }_{\text {tot}} = P_{\text {tot}}/N_{0}\). With δ∈[ 0,1], the power sharing term, we define \(\bar {\gamma }_{1} = s_{1} \delta \bar {\gamma }_{\text {tot}}\), \(\bar {\gamma }_{2} = s_{2} \delta \bar {\gamma }_{\text {tot}}\), and \(\bar {\gamma }_{3} = s_{3} (1-\delta) \bar {\gamma }_{\text {tot}}\) as the SNR of the source-destination, source-relay, and relay-destination links, respectively.
When all the links have a similar path loss, the asymptotically optimal allocation gives marginal benefits only for the second cooperation mode. On the other hand, as seen on Figure 8, when the S→D link is of lower quality, using a relay provides a large gain even if the S→R link is also weak. Furthermore, if we compare the performances of the second and third model, we can see a large performance discrepancy when an equal power allocation is used. However, when an asymptotically optimal power allocation is used, both models show similar performances while the third model is more complex to implement in practice. Further analyses have shown that this fact is conditioned on the quality of the S→R link; when its quality is low, as in Figure 8, the performance of both models will be close, whereas the third model shows performance gains when the S→R link is of superior quality.
5 Conclusions
In this paper, we studied the PER of communication systems subject to block fading effects. We derived a closed form upper bound on the coding gain, leading to asymptotic approximations similar to those of [6] for fast fading channels. We then studied unit-step approximations of the PER and showed that the approximation can be quite close on the whole SNR range if the threshold of the unit step is chosen wisely. For fading models behaving polynomially near 0, we showed that the optimal threshold of the unit-step approximation, w.r.t. both absolute and relative error criterions, and the coding gain of the asymptotic approximation are directly related and may be deduced from one another. This allows a simple treatment of both coded and uncoded transmission schemes. We then applied these results to two practical use cases. By defining a packet error outage metric, we showed how to use the asymptotic approximation to derive a performance evaluation of systems subject to both fading and shadowing effects simultaneously. Finally, in the context of cooperative communications, we derived asymptotically optimal power allocations for relay channels which were showed to provide gains on the whole SNR range.
6 Appendices
6.1 Appendix 1 Proof of Theorem 1
Now since p _{ p }(γ) is bounded and continuous, the integral on the right-hand side is well defined. The original integral thus has a bounded value, and the proof is complete.
6.2 Appendix 2 Proof of Proposition 1
To integrate the second term in (14), we will make use of the following lemma.
Lemma 1.
Proof.
Identifying −x ^{ t+s } e ^{−x }=Γ ^{′}(t+s+1,x) completes the proof.
Since \((2\sqrt {\pi })^{-1}\Gamma (1/2,k\gamma ^{*}/2) = 1/N\) by definition of γ ^{∗}, the first two terms cancel out. Reinjecting the remaining term in (8) gives the proposition.
6.3 Appendix 3 Proofs of Theorem 2 and Corollary 1
On the other hand, from Theorem 1, we know the asymptotic expansion of the left-hand side of (22). Identifying the value of γ _{0} in (35) and (36) completes the proof.
The above assertion is verified if and only if \(\bar {p}_{p}(\bar {\gamma }) \sim F(\gamma _{0}, \bar {\gamma })\). Through the proof of Proposition 2, we can see that for fading channels verifying the conditions, there is only one choice of γ _{0} for the functions to be asymptotically equivalent, and the necessary condition is thus sufficient in that case. For the relative error criterion, the derivation is even more direct since (37) is readily verified if \(\bar {p}_{p}(\bar {\gamma }) \sim F(\gamma _{0}, \bar {\gamma })\) without changing the assertion.
6.4 Appendix 4 Asymptotically optimal power allocation for the relay channel
For every model considered, we can readily see that the asymptotically optimal power allocation is only a function of s _{2} and s _{3} and the coding gains, but is not related to the quality of the S→D link, nor is it related to the power to allocate \(\bar {\gamma }_{\text {tot}}\). The actual end-to-end performance is in fact dependent on both s _{1} and \(\bar {\gamma }_{\text {tot}}\), but not the asymptotically optimal power allocation.
Declarations
Acknowledgements
The authors would like to thank P. Mary, for helpful discussions and comments on the initial version of this paper, as well as the anonymous reviewers for their comments and suggestions which greatly improved the quality of the manuscript.
Authors’ Affiliations
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