Union Bound On The Bit Error Probability Of TCM

Union Bound On The Bit Error Probability Of TCM Coded With TAS/MRC MIMO System Subject To Weibull Fading Channels Toufik Chaayra1 , Faissal El Bouanani2 and Hussain Ben-Azza1 1

ENSAM, Moulay Ismail University, Meknes, Morocco [email protected]

2

ENSIAS, Mohammed V University, Rabat, Morocco [email protected]

1

ENSAM, Moulay Ismail University, Meknes, Morocco [email protected]

Abstract: In this work, we present a novel tight union bound (UB) for trellis-coded modulation (TCM) over multiple-input multiple-output (MIMO) system mixing transmit antenna selection (TAS) and receiver maximal-ratio combining (MRC) at the receiver subject to Weibull fading channels (WFCs). That UB is provided in product form, which makes it of great interest. The bound can be applied to whatever diversity order on both branches of transmit antenna and MRC receiver, and Weibull parameter β. For instance, simulation results show that the gain of 8-PSK TCM coded with TAS/MRC MIMO is below 0.4dB at 10−5 for various values of fading severity parameter. Simulations performed using the Monte-Carlo method show a high accuracy of the proposed upper bound. Key Words: Convolutional codes, error probability, maximal-ratio combining, pairwise error event probability (PEEP), probability density function (PDF), transmit antenna selection, trellis-coded modulation, union bound, Weibull fading. 1

Introduction

New communication services keep on demanding higher data rates and better quality of service from the digital transmission frameworks supporting them. One of the well-known systems reaching this goal is the multiple input multiple output (MIMO) when employed jointly with channel coding [1]. Among MIMO technologies, the simultaneous transmission of data streams through the multiple antennas at the transmitter and/or the receiver gives parallel radio frequency (RF) chains which come with an important cost in terms of complexity and power consumption [2], [3]. The kinds of transmission have immanent drawbacks such as inter-antenna interference and synchronization requirement [4]. Such disadvantages can be avoided using the strategy known as transmit antenna selection (TAS), therein only one transmit antenna, at a given instant, is made functional to send data according to the channel conditions. This can certainly help to reduce complexity, power consumption, and cost [5]. At the receiver, the diversity is obtained by combining independently faded copies of the transmitted signal using different combining methods. Maximal-ratio combining (MRC), known as the optimal technique of combining, consists of summing the filters outputs after being weighted by the branches fading degradation. Therein, the output signal-to-noise ratio (SNR) at the receiver is the sum of the SNRs at its branches [6]. Under Weibull fading model, the bit-error probability expressions for uncoded MIMO systems with TAS/MRC method subject to WFCs are presented in [7]. It is noteworthy that the Weibull distribution has the flexibility to describe experimental fading environment measurements for both indoor and outdoor channels. As for channel coding, the error-correcting code (ECC) is frequently used in various communication systems to mitigate a potential added noise to the transmitted data. For instance, convolutional code (CC) is the most extensively used channel code in practical wireless frameworks among the existing error control codes. This codes is produced with a different solid mathematical structure and are basically used for real-time error correction. The wide acknowledgment of CC has made many advances to extend and enhance this fundamental coding scheme. As a result, a new coding scheme, namely, trellis coded 1

modulation (TCM) has come into existence. It consists in combining both coding and modulation into a single operation [8]. TCM was originally developed for telephone uses and later received a considerable attention in mobile radios field [9]. The ability to achieve low bit error rate with an increase in the power efficiency without customary expansion of bandwidth represents one of the essential advantages of TCM. Furthermore, TCM can provide some form of time diversity when combined with sufficient depth interleaver [10]. The performance of TCM can be investigated by computing first the pairwise error event probability (PEEP) approach to upper bound the bit error rate (BER) [11]. The performance of the coded system undergoing Rayleigh or Nakagami-m fading channels was analyzed extensively in the literature [11]-[13]. Interestingly, the union bound of TCM was represented in product form, in terms of the code transfer function [11], [12]. In this paper, we extend the outcomes investigated in [12] and [14] to obtain two novel closed-form expressions for the union bound of BER in the case of TCM with M-ary modulation scheme coded with TAS/MRC MIMO system undergoing WFCs. Those bounds are obtained based on two upper bound functions of Q-function, namely Chernoff and the recent tighter one proposed in [13]. To the best of our knowledge, this topic has not been addressed before. Importantly, the bound is presented under product form allowing efficient computation of PEEP to upper bounding the BER of M-ary modulation. Motivated by this introduction, the remainder of this paper is structured as follows. In Section 2, we introduce the considered system as well as the channel model. New upper bound on PEEP for TCM with TAS/MRC MIMO operating under WFCs is derived, based on which the BER is derived and analyzed in Section 3. The numerical results and discussion are illustrated in Section 4. Lastly, a brief conclusion and some future works are pointed out in Section 5. 2

System And Channel Model

We consider an (Lt , Lr ) TAS/MRC system subject to flat WFCs, employing Lt and Lr transmit and receive antennas, respectively. The antenna maximizing the received SNR is chosen at the transmitter, and all the Lr received signals are used to combine through MRC receiver. At the transmitter, the communication system is composed of an encoder (e.g., convolutional code, TCM, turbo code), interleaver, and an M-ary modulator. The encoder operates by accepting a data bits source of K symbols to form a codeword of M encoded symbols, S = (sh )1≤h≤M . Therefore, the associated code rate is Rc = K/M . Moreover, after selecting the best ith transmit antenna, the hth symbol received on the jth receive branch can be expressed as p (h) (h) (h) yi,j = Es ai,j sh + zi,j , h = 1, .., M, (1) i = 1, .., Lt , j = 1, .., Lr , where   (h) • ai,j h≤M are the complex fading amplitudes associated to the channels linking the ith transmit i≤Lt j≤Lr antenna Ti

with the jth receive one Rj . For the sake of simplicity, we assume perfect interleaving and independent and identically distributed (i.i.d) WFCs is considered, • Es denotes the average energy per received symbol,   (h) • zi,j h≤M represent the associated additive white Gaussian noise (AWGN) samples, assumed to i≤Lt j≤Lr

be i.i.d. with zero mean and noise variance of the power spectral density equals N0 .

2

3

Pairwise Error Event Probability

In this section, two upper bounds of PEEP based on two bounds of Q-functions are investigated. The PEEP is defined as the conditional probability of decoding a codeword Sb instead of the original sent one, namely S, given a fading gains vector A. That is, the PEEP can be written in product form as [11, Eq. (23)] M  Y b P S −→ S | A = Kc W (sh , sbh ) ,



(2)

h=1

where W (sh , sbh ) denotes the error weight profile between sbh and sh [15, Eq. (4.15)], and Kc is a constant independent from the error sequence. The expression of PEEP in (2) is used to determinate the union bound on BER. In particular, the conditional PEEP for MRC diversity given A can be written as [12, Eq. (3)].   Lr  M X 2 2    X p p (h) (h) (h) (h) b P S −→ S|A =P (3) yi,j − Es ai,j sh − yi,j − Es ai,j sbh ≥ 0 |A  , h=1 j=1

In what follows, we combine TAS with MRC diversity and extend the practice of [11] and [12] to the WFCs. In what follows, let’s define • for a real x and a positive integer n, the function  n , dn (x) = Γ 1 + x

(4)

where Γ(.) denotes the Gamma function [16, Eq. (8.310-1)], (h)

(h)

• Zij = |aij | as the magnitude of the fading complex amplitude, assumed to be normalized, i.e.    (h) 2 E Zij = 1, where E[.] denotes the expectation operator. The shape parameter β is considered the same for all channels, • γ ij is the average SNR, associated to the link Ti − Rj , assumed also to be constant ( i.e., γ ij = γ). The instantaneous SNR at the input of the receiver is given by   (h) (h) 2 γij = γ ij Zij .

(5)

For an (1, Lr ) MRC system with a single ith transmit antenna and Lr receive antennas operating over P r (h) (h) WFCs, the PDF of the instantaneous SNR at the MRC output, i.e., γi = L j=1 γij , can be approximated by [6] (γΨ)Φ exp (−γΨ) , (6) fγ (h) (γ) ≈ γΓ(Φ) i where Φ = K (β) Lr , Ψ = and K (β) =

K (β) , γ

d22 (β) . d4 (β) − d22 (β) 3

(7)

(8)

(h)

Now, γi

(h)

are rearranged in ascending order of magnitude, denoting by γ(l) , 1 ≤ l ≤ Lt . The PDF of

(h)

SNR γ(Lt ) at the receiver output can be approximated by [14, Eq. (16)] ∞

Lt Ψ X fγ (h) (γ) ≈ ∆Lt −1,k (Ψγ)ΦLt +k−1 e−γΨLt Γ(Φ) (Lt )

(9)

 Γ(Φ+1) Pk   ∆r,k = k q=1 (q(r + 1) − k) bq ∆r,k−q , r ∆r,0 = Γ (Φ + 1) ,  1  bq = Γ(Φ+q+1) , q ≥ 1.

(10)

k=0

where

To obtain [11, Eq. (30)], let’s define in like manner `h =

Es |sh −b sh | 2 , δh 4N0

=

`h `h 1+ ΨL

 (h) and ωh = γ(Lt ) 1 +

`h ΨLt



.

t

Now, substituting (9) into [11, Eq. (28)], yields the average PEEP for M-ary modulation     Z ∞ Z ∞ sX   Y X 1 1 erfc  δh ωh  exp  P S −→ Sb = ·· δh ωh   ΦLt 2 `h 0 0 h∈η 1 + h∈η h∈η ΨLt ×g(ω1 ) · · · g(ωM )dω1 · · · dωM ,

(11)

where η = {h : sh 6= sbh } , Lη = |η| represents the number of the set’s elements η, erfc(.) denotes the complementary error function [17, Eq. (7.1.2)], and ∞

g(x) =

Lt Ψ X ∆Lt −1,k ΦLt +k−1 −xΨLt , e  k (Ψx) Γ(Φ) `h k=0 1 + ΨLt

(12)

√ Let’s define δmin = min {δh , h ∈ η} . Taking into consideration that erfc ( x) ex is monotonically decreasing for all x ≥ 0 and that X X δh ωh ≥ δmin ωh , (13) h∈η

h∈η

the PEEP can be upper bounded by   A Y `h −ΦLt b P S −→ S ≤ 1+ , 2 ΨLt 

(14)

h∈η

where Z A =

∞

Z ..

0

0

∞

    s X X Y ωh  exp δmin ωh  g(ωh )dωh , erfc  δmin h∈η

h∈η

(15)

h∈η

In the following, the integral in (15) is simplified using two approaches resulting in two bounds on the PEEP.

4

3.1

Proposed Bound

Theorem 1 The BER of TCM coded with TAS/MRC MIMO system undergoing WFCs can be upper bounded by    Lη ∞ ∞ ΦL +k ΦL +k Y X Ψ t ∆L −1,k 1 Lt   Y X Ψ t ∆Lt −1,k t Pb ≤ 3  k Bk (δmin ) +  k Bk (0) 8K Γ(Φ) `h `h h∈η k=0 h∈η k=0 1 + ΨLt 1 + ΨLt   Y `h −ΦLt 1+ × . (16) ΨLt h∈η

Proof. Incorporating the relationship between the Q-function and the erfc function [18, Eq. (1)] into (15), alongside with the recent tighter upper bound of Gaussian Q-function [13, Eq. (9)], the integral (15) can be upper bounded by     Z ∞ Z X Y 1 ∞ 3 exp −δmin .. ωh  + 1 A ≤ g(ωh )dωh , (17) 4 0 0 h∈η

h∈η

Now, replacing (12) into (17), yields    Lη ∞ ∞ ΦL +k Y X ΨΦLt +k ∆L −1,k 1 Lt  Y X Ψ t ∆Lt −1,k  t A≤ 3 k Bk (δmin ) + k Bk (0) ,   4 Γ(Φ) `h `h h∈η k=0 h∈η k=0 1 + ΨL 1 + ΨL t t

(18)

where for any y ∈ R+ , Z Bk (y) =

∞

xΦLt +k−1 e−(ΨLt +y)x dx =

0

Γ(ΦLt + k) (ΨLt + y)ΦLt +k

,

Therefore, substituting (18) into (14), the PEEP can be simplified as    Lη ∞ ∞   ΦL +k ΦL +k Y X Ψ t ∆L −1,k 1 Lt  Y X Ψ t ∆Lt −1,k  t P S −→ Sb ≤ 3 k Bk (δmin ) + k Bk (0)   8 Γ(Φ) `h `h h∈η k=0 h∈η k=0 1 + ΨL 1 + ΨL t t −ΦLt Y `h × . (19) 1+ ΨLt h∈η

Using [15, Eq. (5.95)], the BER can be upper bounded as given in (16), which concludes the proof of Theorem 1.

3.2

Chernoff Bound

Theorem 2 The BER of Chernoff bound for TCM coded with TAS/MRC MIMO system undergoing WFCs can be upper bounded by Pb ≤

    ∞ 1 Lt Lη Y X Γ(ΦLt + k)∆Lt −1,k Y `h −ΦLt . 1 +  k 2K Γ(Φ) ΨLt `h ΦLt +k h∈η k=0 1 + h∈η Lt ΨLt 5

(20)

Proof. Using the Chernoff bound that is well-known upper bound of Q-function 2

Q(x) ≤ 0.5e−0.5x , the integral (15) can be upper bounded as follows Z ∞ Z A ≤ .. 0

∞

0

Y

g(ωh )dωh ,

(21)

(22)

h∈η

Now, substituting (12) in (22), we obtain 

Lt A≤ Γ(Φ)

Lη Y X ∞ Γ(ΦLt + k)∆Lt −1,k ,  k ` ΦLt +k h∈η k=0 1 + h L t ΨLt

(23)

Therefore, substituting (23) into (14), the PEEP of Chernoff bound can be upper bounded as   ∞  1  L Lη Y X Γ(ΦLt + k)∆Lt −1,k Y `h −ΦLt t b P S −→ S ≤ 1+ . k  2 Γ(Φ) ΨLt `h ΦLt +k h∈η h∈η k=0 1 + Lt ΨLt 

(24)

Using [15, Eq. (5.95)], the BER can be upper bounded as given in (20), which concludes the result of the Theorem 2. Remark 1 It is noteworthy that the bound (16) is tighter than (20) as the new recent bound on Qfunction [13, Eq. (9)] is tighter than the Chernoff bound. 4

Simulation And Results Bound

For illustration, the proposed union bound was evaluated for a code rate 2/3 and the 8-PSK 8-states TCM for TAS/MRC MIMO system subject to WFCs. Fig. 1 depicts the performance of BER versus the average SNR per information bit γ¯b = Eb /N0 in dB. We observe that the new bound is closer to the simulation curve than the one based on Chernoff bound, for diversity orders Lt × Lr (2 × 2, 3 × 3 and 4 × 4) and the shape parameter β = 1.5. In addition, it is obvious that the bound gain increases with the diversity orders namely, Lt and Lr . Fig. 2 compares the new bound expression of BER for 8-PSK TCM coded with double-branches transmitter, double-branches MRC receiver, β = 1.2, and β = 2.2, plotted from (16). It can be also noticed that the greater β and γ¯b , the smaller the BER. Fig. 3 plots the proposed bound given in (16) and simulation of BER versus β for γ¯b = 0.4dB and various values of Lt × Lr MIMO. The main remark is that the bounds are more tighter for 2 × 3 MIMO and 3 × 2 MIMO than other MIMO schemes. Additionally, the greater the fading severity β, the smaller the BER, and consequently the system becomes more reliable. Fig. 4 depicts the simulated and the analytical expression based on the new bound and Chernoff bound of Q-function, given in (16) and (20), respectively. The curves are plotted for β = 3, two values of Lt (Lt = 2 and Lt = 4) and two values of γ¯b (¯ γb = 2.5dB and γ¯b = 4.5dB). Again, we can observe that the analytical and the simulated curves are close considerably compared to the one based on Chernoff bound. Besides, obviously, based on new upper bound of Q-function, the greater is Lt , the smaller is the BER. In addition, it can be seen that increasing the receiver antennas number decreases considerably the BER, which shows that diversity in MIMO system can effectively combat fading.

6

* 0.010

*

*

*

*

MIMO(2,2)

*

*

*

*

0.001 BER

* *

*

* *

10-4

*

*

10-5

*

*

Simulation Chernoff bound Proposed upper bound, Eq. (20)

2

4

*

*

*

10-6

*

*

*

MIMO(4,4)

MIMO(3,3)

*

*

*

* *

*

6

8

10

12

γ b [dB] Figure 1: The BER of the 8-PSK 8-states TCM with TAS/MRC MIMO system over WFCs for different diversity orders, and β = 1.5.

**

*

BER

0.010

* *

* *

* *

0.001

* *

* β=3

* 10-5

*

*

10-4

*

*

Simulation Chernoff bound Proposed upper bound, Eq. (20)

5

6

7

β = 1.2

8

9

* 10

* 11

12

γ b [dB] Figure 2: BER of 8-PSK 8-states TCM with TAS/MRC for 2 × 2 MIMO scheme over WFCs and various values of β.

7

0.100

MIMO(2,3)

0.010 BER

MIMO(3,2) MIMO(3,4)

0.001

MIMO(4,3)

Simulation Proposed upper bound, Eq. (20)

10-4 1.0

1.5

2.0

2.5

3.0

3.5

4.0

β Figure 3: BER of 8-PSK TCM coded with TAS/MRC under several MIMO schemes over WFCs and γ¯b = 0.4dB.

BER

0.100*

* * 0.010*

* *

0.001

* *

L t = 2 , γ b = 4.5 dB

* *

* *

*

10-4

* *

L t = 4 , γ b = 2.5 dB L t = 4 , γ b = 4.5 dB

10-6 10-7

* 1

*

*

* * * *

Simulation Chernoff bound Proposed upper bound, Eq. (20)

2

*

*

*

10-5

L t = 2 , γ b = 2.5 dB

3

* 4

5

6

Lr Figure 4: BER vs. number of receive antennas Lr for β = 3, and various values of Lt and γ¯b .

8

5

Conclusion

In this paper, two union bounds on the bit error probability for MIMO systems based on trellis coded modulation and the TAS/MRC combining technique, operating under Weibull multipath fading models are derived. Numerical and simulation results show that the proposed bound based on the recent bound of Q-function is close to simulation results for moderate diversity orders. Furthermore, the new bound, expressed in closed-form, is simple to evaluate. In upcoming work, we intend to investigate, based on these results, the performance of the considered system subject to correlated generalized fading channels. References [1] M.-K. Simon, M.-S. Alouini. Digital Communication over Fading Channels. John Wiley & Sons, Hoboken, 2005. [2] E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor. MIMO Wireless Communications. Cambridge, UK: Cambridge University Press, 2007. [3] S. Sanayei, and A. Nosratinia. Antenna selection in MIMO systems. IEEE Commun. Mag., 42(10):6873, October 2004. [4] J. Jeganathan, A. Ghrayeb, L. Szczecinski, and A. Ceron. Space shift keying modulation for MIMO channels. IEEE Trans. Wirel. Commun., 8(7):3692-3703, July 2009. [5] A. F. Molisch and M. Z. Win. MIMO systems with antenna selection. IEEE Microw. Mag., 5(1):46-56, March 2004. [6] F. El Bouanani and H. Ben-Azza. Efficient Performance Evaluation for EGC, MRC and SC Receivers over Weibull Multipath Fading Channel. In 2015 International Conference on Cognitive Radio Oriented Wireless Networks (CROWNCOM) Proceedings, pages 346-357. Springer, 21-23 April 2015. [7] P. L. Yeoh, M. Elkashlan, N. Yang, D. B. da Costa, and T. Q. Duong. MIMO Multi-Relay Networks with TAS/MRC and TAS/SC in Weibull Fading Channels. In 2012 23rd International Symposium on Personal, Indoor and Mobile Radio Communications Proceedings, pages 2314-2318. IEEE, Sydney, Australia, 9-12 September 2012. [8] G. Ungerboeck. Channel coding with multilevel/phase signals. IEEE Trans. Inform. Theory, 28(1):5567, January 1982. [9] S. H. Jamali and T. Le-Ngoc. A new 4-state 8PSK TCM scheme for fast fading, shadowed mobile radio channels. IEEE Trans. Veh. Technol., 40(1):216-222, February 1991. [10] G. Femenias and I. Furi´ o. Analysis of switched diversity TCM-MPSK systems on Nakagami fading channels. IEEE Trans. Veh. Technol., 46(1):102-107, February 1997. [11] S. Al-Semari and T. Fuja. Performance analysis of coherent TCM with diversity reception in slow Rayleigh fading. IEEE Trans. Veh. Technol., 48(1):198-212, January 1999. [12] S. A. Zummo. Union Bounds on the Bit Error Probability of Coded MRC in Nakagami-m Fading. IEEE Communications Letters, 10(11):769-771, November 2006. [13] Y. Mouchtak and F. El Bouanani. New tighter upper bounds on the performance of convolutional code over exponentially correlated Rayleigh fading channel. In 2016 International Wireless Communications and Mobile Computing Conference (IWCMC) Proceedings, pages 863-868. IEEE, Paphos, September 2016. 9

[14] T. Chaayra, F. El Bouanani and H. Ben-azza. Performance Analysis of TAS/MRC based MIMO systems over Weibull Fading Channels. In 2016 International Conference on Advanced Communication Systems and Information Security (ACOSIS) Proceedings, pages 1-6. IEEE, Marrakesh, October 2016. [15] S. H. Jamali and T. Le-Ngoc. Coded-Modulation Techniques for Fading Channels. Boston: Kluwer Academic Publisher, 1994. [16] S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products. Academic Press, 6th, 2000. [17] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formula graphs and mathematical tables. National Bureau of Standards. Applied Mathematical Series, June 1964. [18] M. Simon and D. Divsalar. Some new twists to problems involving the Gaussian probability integral. IEEE Trans. Commun., 46(1):200-210, February 1998.

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