Pairwise Error Probability of Turbo Codes over Joint ... - IEEE Xplore

Globecom 2014 - Wireless Communications Symposium

Pairwise Error Probability of Turbo Codes over Joint Fading and Two-Path Shadowing Channels I. Dey and G. G. Messier

S. Magierowski

Electrical and Computer Engineering University of Calgary Calgary, Canada Email: [email protected], [email protected]

Electrical Engineering and Computer Science York University Ontario, Canada Email: [email protected]

Abstract—The performance of Turbo coded Binary Phase Shift Keying (BPSK) over the Joint Fading and Two-path Shadowing (JFTS) environment is analyzed. Exact analytical expression for the Pairwise Error Probability (PEP) of Turbo codes over fully interleaved JFTS channels is presented using the probability distribution of the squared independent and identically distributed JFTS random variables. Finally the Turbo coded Average Bit Error Rate (ABER) of a wireless communication system is simulated over JFTS fading / shadowing channels and compared with the derived analytical result.

I. I NTRODUCTION An appropriate composite fading / shadowing channel model that characterizes the transition from local small scale fading to global shadowing statistics for users confined to small coverage areas in a large office environment is proposed in [1]. A joint distribution called the Joint Fading and Twopath Shadowing (JFTS) distribution that combines Rician fading and the two wave with diffuse power (TWDP) shadowing model is shown to fit the measurement data. Expressions for the joint moments and amount of fading (AF) of the JFTS distribution are derived in [2]. The work in [1] and [2] is extended in [3] and [4] by deriving the expressions for the Cumulative Distribution Function (CDF) of the JFTS distribution and the Average Bit Error Rate (ABER) of Binary Phase Shift Keying (BPSK) over a JFTS channel respectively. A reduction in ABER can be obtained by the application of forward error correction (FEC) codes with iterative or non-iterative detection techniques in presence or absence of channel interleavers and de-interleavers. For any fixed set of JFTS parameters, the application of non-iterative FEC coding is shown to offer a maximum of around 10 dB improvement over an uncoded BPSK modulated wireless communication system in [4]. Further improvement can be achieved through the application of iterative FEC coding techniques like Turbo [5], Repeat and Accumulate (RA) [6] and LDPC [7] codes, of which Turbo codes offer impressive performances over fading channel conditions. However, performance evaluation in terms of deriving closed-form ABER expressions for Turbo coding is exhaustive and laborious. Hence, the bulk of existing literature [8], [9] resort to bounding techniques based on computationally complex maximum likelihood (ML) decoding, most of which uses the tight upper bounds provided by [9] for the

978-1-4799-3512-3/14/$31.00 ©2014 IEEE

Additive White Gaussian Noise (AWGN) and Rician fading channel cases. As an alternative, the work introduced in [10] uses a probability distribution to derive the pairwise error probability for performance analysis of turbo codes over Rayleigh fading channels. In particular, [10] showed that this approach led to computationally efficient results. A similar approach is also used in [11] and [12] to derive exact and efficient expressions for pairwise error probability (PEP) over fully interleaved Rician and Nakagami-m fading channels respectively. The derived expressions for PEPs in [10]–[12] are also approximated to yield tight upper bounds which stay close to the exact expressions. The primary contribution of this paper is to analyze the ABER performance of Turbo coded Binary Phase Shift Keying (BPSK) over fully interleaved JFTS block fading / shadowing channels using the same approach as in [10]. In order to do that, a closed-form expression for the PDF of the sum of the squared independent and identically distributed (i.i.d) JFTS random variables is derived, which in turn is used to deduce the exact expression for PEP. Finally the analytical results are compared with the simulation ABER results in order to verify the validity of the derived expression. The rest of the paper is organized as follows. Section II introduces the channel model and the performance analysis is presented in Section III. Numerical results and discussion are given in Section IV, while concluding remarks are provided in Section V. II. C HANNEL M ODEL √ Let sk ∈ {± Es } represent a BPSK symbol amplitude that is transmitted over a composite slow shadowed and flat faded wireless communication channel with JFTS statistics, where k is an integer symbol index. With appropriate sampling and perfect coherent demodulation, the discrete representation of the demodulator output can be expressed as, yk = αk sk + nk

(1)

where nk is the i.i.d. complex AWGN component with zero mean and power spectral density, N0 /2. The amplitude, αk denotes the JFTS distributed composite fading / shadowing envelope. Since, we are assuming that the communication

3995

Globecom 2014 - Wireless Communications Symposium

channel is fully interleaved, the αk ’s are independent. For our analysis, we will be using an Rc -rate Turbo code with an input size of L bits and an output encoded stream of N bits. The first order statistics of the channel fading stochastic process, αk , can be represented by the random variable A, which has a Probability Density Function (PDF) given by,  r  4 2K α e(−K−Sh ) X bi D α ; Sh , ∆Mi fA (α) = P1 P2 P1 P2 i=1   m X w~ r~ 2 (2P1 − 1) α2 · exp − (2) |r~ | 2P1 2P2 r~2

Using the infinite series expansion of the modified Bessel P+∞ /2)2ξ function of the first kind, I0 (f ) = ξ=0 (f(ξ!) 2 , (4) can be expressed as, fZ (z) =

where Hm−1 (·) is the Hermite polynomial with roots r~ for ~ = 1, 2, . . . , m and bi = ai I0 (1) where a1 = 751/17280, a2 = 3577/17280, a3 = 49/640 and a4 = 2989/17280 for i = 1, 2, 3, 4 respectively. The fading parameter, K is the Rician K-factor given by K = Specular Power / Diffused Power. The value of the shadowing parameter, Sh , is set based on the range of shadowing values experienced by a user while traveling through different shadowing clusters. The ∆-parameter is the shape parameter of the shadowing distribution and represents the transition from one scattering cluster to the next one. The parameters P1 and P2 are the mean-squared voltages of the diffused and the shadowed components respectively. As α1 , α2 , . . . ατ are modeled as independent JFTS random variables, each of the ατ2 terms will have a PDF of, r  4 X bi 2Kxτ fX (xτ ) = D ; Sh , ∆Mi P P P1 P2 i=1 1 2   m X xτ (3) · R exp − K − Sh − 2P2 r~2 ~=1

where R = (w~ /|r~ |) exp (r~ 2 (2P1 − 1)/2P1 ) and xτ = ατ2 . The expression of the PDF in (3) can be derived using the procedures for deducing bi-variate and joint distributions followed in [13]. we define PNext PT another random variable Z, T such that Z = τ =1 ατ2 = τ =1 xτ . The PDF of Z in turn can be calculated as, X r  T 4 X bi 2Kxτ fZ (z) = D ; Sh , ∆Mi P P P1 P2 τ =1 i=1 1 2   m X xτ · R exp − K − Sh − (4) 2P2 r~2

(5)

~=1

where, r

r KSh KSh B1 = 2 (1 − ∆Mi ), B2 = 2 (1 + ∆Mi ) P1 P2 P1 P2 B3 = exp (Sh ∆Mi ), B4 = exp (−Sh ∆Mi )

~=1

p where D(γ; η,p δ) = 1/2[exp (ηδ)I0 (γ 2η(1 − δ)) + exp (−ηδ)I0 (γ 2η(1 + δ))], Mi = cos((i − 1)π/7), I0 is the zeroth order modified Bessel function of the first kind and m is the approximation index. For our analysis, we have chosen m = 20, as is done in [4]. The multiplier w~ denotes the associated weights and is given by, √ 2m−1 m! π w~ = [m Hm−1 (r~ )]2

+∞ 2ξ X B3 B2ξ bi 1 + B4 B2 2P1 P2 (ξ!)2 i=1 ξ=0   m X z · R z ξT exp − K − Sh − 2P2 r~2 4 X

In the next section, we will be using (5) to derive the PEP expression for a Turbo coded BPSK over a fully interleaved JFTS faded / shadowed communication channel. III. P ERFORMANCE A NALYSIS Assuming the all-zero codeword and ML decoding at the receiver, the upper bound on the word error probability of Turbo coding can be expressed as [5], Pword ≤

N X

C(δ) P2 (δ)

(6)

δ=1

where C(δ) is the number of codewords and P2 (δ) is the probability of incorrectly decoding a codeword with weight δ. In order to eliminate a computationally exhaustive search,an average bound can be constructed using the average weight distribution over all possible interleavers. In this case, the average weight distribution can be written as, L   X L C(δ) = p(δ|l) (7) l l=1
where Ll is the number of input words with Hamming weight l and p(δ|l) is the probability that an input word with Hamming weight l can be encoded to a codeword with Hamming weight δ. Using (6) and (7), we will be deriving the average upper bound for word and bit error rates in Subsection III-A and the PEP of Turbo coded BPSK over fully interleaved JFTS channels with perfect channel state information (CSI) in Subsection III-B respectively. A. Average Upper Bound The average upper bound for word error rate over any fading, shadowing or composite fading / shadowing communication channel can be given by,

~=1

3996

P word ≤

N X

C(δ) P2 (δ)

δ=δmin L   X L p(δ|l) P2 (δ) l δ=δmin l=1 L   X   L ≤ Eδ|l P2 (δ) l

≤

N X

l=1

(8)

Globecom 2014 - Wireless Communications Symposium

where δmin is the minimum Hamming weight of the generated  codewords and Eδ|l · is the expectation with respect to p(δ|l). Consequently, the average upper bound for bit error rate can be expressed as,

The inner integral in (13) can be solved by using the infinite series representation of [15] as, ζ2 Es /N0

Z Iι =

z ξδ exp

0

  L X   l L ≤ Eδ|l P2 (δ) . L l

P bit

=

(9)

l=1

This average bound can be used for computing p(δ|l) by utilizing the state transition matrix of the recursive systematic convolutional (RSC) encoders proposed in [14]. Subsequently, using p(δ|l), the PEP P2 (δ) of Turbo coded BPSK over a JFTS fading / shadowing channel will be formulated in the following subsection.

·

# "v u δ u Es X t 2 αk P (c0 , cj |A) = Q N0

(10)

∞

Z Io =

e 0

·

P2 (δ) = E Q

Es Z N0

0

(12)

4 m +∞ 2ξ X X bi e(−K−Sh ) X B3 B2ξ 1 + B4 B2 R 2P1 P2 (ξ!)2 i=1 ξ=0

~=1

·

e 0

ζ 2 /2

0

ζ2 Es /N0

ξδ

z exp



r

ξδ

z exp



  z − dz dζ 2P2 r~2

ξδ X π ξδ! − 2 2v!(Es /N0 )v v=0

4 m +∞ 2ξ X X bi e(−K−Sh ) X B3 B2ξ 1 + B4 B2 R 2P1 P2 (ξ!)2 i=1 ~=1 ξ=0 r ξδ X ξδ! π ξδ! · − 2 ξδ+1 (1/2P2 r~ ) 2 2v!(Es /N0 )v v=0

Γ(v + 1/2) (1/2P2 r~2 )ξδ−v+1 (1/2 + N0 /2Es P2 r~2 )v+1/2 (16)

Finally (16) will be used to evaluate ABER performance of Turbo coded BPSK over JFTS fading / shadowing wireless communication channels in the following section and the analytical results will be compared with the simulated ABER performance. It is to be noted that for simulation we will be using an iterative suboptimal soft-decision decoder where each constituent RSC is decoded separately. The constituent decoders exchange bit-likelihood information iteratively using a Log-MAP decoding algorithm [16]. Though this algorithm does not provide optimal ML decoding, it is shown to perform within 0.7 dB of the Shannon limit on the AWGN channel for an ABER of 10−5 [17]. IV. N UMERICAL R ESULTS AND D ISCUSSION

Changing the order of integration, we can write (12) as,

Z

ζ2 Es /N0

0

·

(11)

m +∞ 4 2ξ X X B3 B2ξ bi e(−K−Sh ) X 1 + B4 B2 R 2P1 P2 (ξ!)2 i=1 ~=1 ξ=0     Z ∞ Z ∞ z ζ 2 /2 ξδ · e dζ z exp − dz. q Es 2P2 r~2 0 N z

∞

Z

Γ(v + 1/2) (15) (1/2P2 r~2 )ξδ−v+1 (1/2 + N0 /2Es P2 r~2 )v+1/2

P2 (δ) =

#)

Pδ 2 where Z = k=1 αk and α1 , α2 , . . . , αδ are i.i.d JFTS random variables. The PDF of Z will be given by (5). By putting (5) in (11), we obtain the expression for PEP as,

Z

ζ 2 /2

where Γ(·) is the upper incomplete Gamma function. Substituting (15) back in (12), the final expression for PEP of Turbo coded BPSK over a JFTS faded / shadowed communication channel can be obtained as,

where cj differs from c0 in δ bit positions with a known fading vector A indexed by 1, 2, . . . , δ and Q(¯ x) is the Gaussian QR∞ 2 4 function defined as, Q(¯ x) = √12π x¯ eζ /2 dζ. As mentioned in Subsection III-A, the PEP can be deduced by taking the expectation over (10) as, ( "r

(14)

Putting (14) back in (13) we can solve the outer integral as,

k=1

P2 (δ) =

(ζ 2 N0 /Es )v . v! (1/2P2 r~2 )ξδ−v+1

ξδ! = (1/2P2 r~2 )ξδ+1

Assuming perfect CSI at the receiver, the conditional PEP of decoding a codeword c0 into a codeword cj over a fading / shadowing channel can be given by,

 z dz 2P2 r~2   ζ 2 N0 − exp − 2Es P2 r~2 −

ξδ X ξδ! v=0

B. Pairwise Error Probability

P2 (δ) =

ξδ! (1/2P2 r~2 )ξδ+1



  z − dz dζ. 2P2 r~2 (13)

In this section, the derived analytical expression for PEP of coherently detected BPSK with perfect CSI at the receiver is numerically evaluated and plotted as functions of the parameters of the communication channel model with iterative Turbo coding. The analytical results for average upper bound for bit error rate are compared with simulation results in order to verify the validity of the derived expression. In order to demonstrate the effect of the channel model on the system

3997

Globecom 2014 - Wireless Communications Symposium

Fig. 1. Analytical and simulated average bit error rates (ABER) of 1/2-rate TC coded BPSK with code length, N = 1024 over fully interleaved JFTS faded / shadowed communication link, where the curves are generated by varying all the JFTS parameters, K, Sh and ∆ simultaneously.

performance, the parameters of the JFTS distribution are also varied. The wireless communication channel between the transmitter and the receiver is assumed to be suffering from AWGN and the composite fading / shadowing envelope is JFTS distributed, where the communication channel is fully interleaved. All the analytical and simulation results are evaluated using a single input single output (SISO) system and are averaged over 100 independent random channel realizations. The ABER results are plotted as functions of the average received SNR per bit, Eb /N0 in dBs. The results in Fig. 1 are generated by varying the JFTS parameters K, Sh and ∆. The values for each set of parameters are chosen from the ranges of their numerical values proposed in [1], depending on the relative position of the mobile LAN user and the access point. Fig. 1 compares the simulated ABER with the analytical average upper bound for bit error rate, where the exact PEP is calculated using (16). For this set of results, the 1/2-rate Turbo coding with a memory of 4 is used with generators (23, 35) and a frame size of N = 2×512 coded bits. It is evident from Fig. 1 that the analytical average upper bound for bit error rate tightly approximates the simulated ABER for a broad range of JFTS channel parameter values. However, a minor difference exists between the simulated and the predicted results for lower ABERs. The reason behind this can be attributed to the fact that the analytical bound is truncated after codewords with distances δ > 8. The first set of curves in Fig. 1 are generated for K = 10 dB, Sh = 10.5 dB and ∆ = 0.8, representing a condition where both the user and the access point are located in the same room. As the user moves to a different room separated by one set of wall or partition from that of the access point, system performance degrades, as exhibited by the second set of curves in Fig. 1. In order to achieve an ABER of 10−4 ,

Fig. 2. Comparative simulated average bit error rates (ABER) of BPSK with iterative coding technique like Turbo Coding (1/2-rate, N = 1024), noniterative coding technique like Convolutional Coding (1/2-rate, [7, 5] octal) with Hard-Decision Decoding and uncoded BPSK over JFTS block fading / shadowing communication link, where the curves are generated by varying the K-parameter of the JFTS distribution, with fixed Sh = 2 dB and fixed ∆ = 0.3.

the average SNR per bit requirement increases by around 1.5 dB. The average SNR requirement increases even further by around 10 dB if the user and the access point are separated by two or three sets of partitions. This happens due to the lack of strong specular components and the presence of at least two scattering clusters between the transmitter and the receiver, which jointly deteriorates the overall system performance. Fig. 2 shows that the performance improves at higher SNR if non-iterative FEC coding like Convolutional Coding (CC) is employed at the transmit side. Hard-decision decoding (HDD) using Viterbi decoding (VD) with no trace back memory is applied at the receiver side. Both in the case of uncoded and convolutional coded BPSK, performance deteriorates with the decrease in K-factor. As K decreases, the power contributed by the strong specular component decreases in comparison to that contributed by the diffused and the scattered components resulting in the degradation of the overall system performance. This difference in performance is obliterated over the application of 1/2-rate Turbo coding with a memory of 4 and a frame size of N = 2 × 512 coded bits (δ = 5). At the receiver Log-MAP is used for decoding with at most 25 iterations. The fact that Turbo coding is able to eliminate the performance reduction experienced by a 3 dB drop in K-factor is quite surprising if compared with the results for simple Rician fading channels. The JFTS fading distribution combines Rician fading with that TWDP shadowing model. This means that, in addition to K-factor, the shadowing factor Sh , and the shape parameter, ∆, also affect the channel behavior. For this set of analyses, Sh = 2 dB and fixed ∆ = 0.3. A high Sh factor corresponds to a low severity in shadowing.

3998

Globecom 2014 - Wireless Communications Symposium

Fig. 3. SNR requirement for a (23, 35) Turbo coded BPSK to achieve an ABER of P bit = 10−4 for varying K-factors of the JFTS distribution with fixed Sh = 2 dB and fixed ∆ = 0.3, where the curves are generated for different codeword distances, δ.

A low ∆ represents a scenario where only one scattering cluster dominates instead of two thereby also resulting in low shadowing severity. Hence, with a high Sh and low ∆ coupled with high K-factors, the communication channel approaches the ”no fading” scenario. This means that changing the Kfactor from 5 dB to 8 dB does not increase fading enough to cause a meaningful performance degradation when Turbo codes are used. To expand on this point, it is possible to further improve performance over a range of K-factors by increasing the codeword distance, δ. Fig. 3 exhibits the SNR requirement to achieve an ABER of P bit = 10−4 for different K-factors but same Sh and ∆ as in Fig. 2. Since we are assuming the channel to be fully interleaved, all the fading amplitudes are independent of each other. In that case, the codeword distance δ corresponds to the diversity order of the system. As a result with the increase in diversity order, the overall performance improves for a high K-factor even in presence of a high Sh and low ∆. The effect of the variation in Sh -parameter on the performance of uncoded BPSK is preserved even when TC is employed, as can be seen in Fig. 4. A rate 1/3 turbo code with a memory of 2, code structure of (1, 7/5, 7/5), an input block size of L bits and an output encoded stream of N = 3(L + 2) bits is used. With both encoders terminating in the zero state, ABER performance is analyzed for two selected codes with message block length of L = 1000 bits. The performance of both coded and uncoded BPSK degrades with the decrease in Sh -parameter while the employment of Turbo codes offers an average coding gain of 11 dB at an ABER of P bit = 10−3 . A large Sh -factor represents large variations in the main wave amplitudes contributed by each scattering neighborhood resulting in approximately equable number of high and low

Fig. 4. Comparative simulated average word error rates (ABER) of BPSK with Turbo Coding (1/3-rate, N = 3006) and uncoded BPSK over JFTS block fading / shadowing communication link, where the curves are generated by varying the Sh -parameter of the JFTS distribution, with fixed K = 6 dB and fixed ∆ = 0.8.

discrete shadowing values contributing to a lower severity in shadowing. While a low Sh factor represents a high severity in shadowing as a very small range of discrete shadowing values are encountered repeatedly. V. C ONCLUSION The performance of Turbo coding over the JFTS fading / shadowing communication link is analyzed in this paper. The analytical expression for the exact PEP of Turbo codes over fully interleaved JFTS faded / shadowed channels is derived using the PDF of the sum of the squared i.i.d JFTS random variables. The ABER performances of Turbo coded BPSK are simulated assuming coherent detection and perfect CSI at the receiver side. Simulated performances fall within 0.1 dB of SNR difference with that of the analytical upper bound of ABER over JFTS communication channels. R EFERENCES [1] I. Dey, G. G. Messier and S. Magierowski, “Joint Fading and Shadowing Model for Large Office Indoor WLAN Environments,” in IEEE Trans. Antenna Propag., vol. 62, no. 4, pp. 1–14, 2014. [2] I. Dey, G. G. Messier and S. Magierowski, “Fading Statistics for the Joint Fading and Two Path Shadowing Channel,” accepted to the IEEE Wireless Commun. Letters, on Mar. 10, 2014. [3] I. Dey, G. G. Messier and S. Magierowski, “The Cumulative Distribution Function for the Joint Fading and Two Path Shadowing Channel : Expression and Application,” submitted to Proc. IEEE Veh. Tech. Conf. VTC Fall 2014. [4] I. Dey, G. G. Messier and S. Magierowski, “Performance Analysis of BPSK over Joint Fading and Two-Path Shadowing Channels,” submitted to Proc. IEEE Veh. Tech. Conf. VTC Fall 2014. [5] E. K. Hall and S. G. Wilson, “Design and analysis of turbo codes on Rayleigh fading channels,” in IEEE J. Sel. Areas Communs.,, pp. 160– 174, Feb. 1998. [6] H. Jin, A. Khandekar, and R. McEliece, “Irregular repeat- accumulate codes,” in Proc. 2nd International Symposium on Turbo Codes, pp. 1–8, 2000.

3999

Globecom 2014 - Wireless Communications Symposium

[7] R. G. Gallager, “Low Density Parity Check Codes,” Cambridge, MA: MIT Press, 1963. [8] S. A. Zummo, P. Yeh and W. E. Stark, “A union bound on the error probability of binary codes over block-fading channels,” in IEEE Trans. Veh. Tech., vol. 54, no. 6, pp. 2085–2093, 2005. [9] I. Sason and S. Shamai, “On improved bounds on the decoding error probability of block codes over interleaved fading channels, with applications to Turbo-like codes,” in IEEE Trans. Inf. Theory, vol. 47, no. 6, pp. 2275–2299, 2001. [10] E. A. Ince, N. S. Kambo and S. A. Ali, “Efficient expression and bound for pairwise error probability in Rayleigh fading channels, with application to union bounds for turbo codes,” in IEEE Commun. Lett., vol. 9, no. 1, pp. 25–27, Jan. 2005. [11] S. A. Ali and E. A. Ince, “Closed form expression and improved bound on pairwise error probability for performance analysis of turbo codes over Rician fading channels,” in IEEE Commun. Lett., vol. 10, no. 8, pp. 599–601, Aug. 2006. [12] E. A. Ince, N. S. Kambo and S. A. Ali, “Exact expression and tight bound on pairwise error probability for performance analysis of turbo codes over Nakagami-m fading channel,” in IEEE Commun. Lett., vol. 11, no. 5, pp. 399–401, May 2007. [13] A. Papoulis and S. U. Pillai, “Probability, Random Variables and Stochastic Processes,” Fourth Edition. Mcgraw Hill, 2002. [14] S. Benedetto and G. Montorsi, “Unveiling turbo codes : Some results on parallel concatenated coding schemes,” in IEEE Trans. Inf. Theory, vol. 42, pp. 409–429, Mar. 1996. [15] I. S. Gradshteyn and I. M. Ryhik, “tables of Integrals, Series and Products,” Sixth Edition. San Diego: Academic Press, 2000. [16] P. Robertson, E. Villebrun, and P. Hoeher, “A comparison of optimal and sub-optimal map decoding algorithms operating in the log domain,” in Proc. IEEE Int. Conf. Commun. (ICC), pp. 1009–1013, 1995. [17] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding,” in Proc. IEEE Int. Conf. Commun. (ICC), pp. 1064–1070, 1993.

4000