International Symposium on Information Theory and its Applications, ISITA2004 Parma, Italy, October 10–13, 2004
Tight Bounds and Accurate Approximations for the BER of DQPSK Transmission from Novel Bounds on the Marcum Q-Function Gianluigi FERRARI† and Giovanni E. CORAZZA‡ †
Dipartimento di Ingegneria Informazione (DII) Universit`a di Parma, 43100 PARMA, ITALY E-mail:
[email protected]
‡
Dipartimento di Elettronica, Informatica e Sistemistica (DEIS) Universit`a di Bologna 40136 BOLOGNA, ITALY E-mail:
[email protected]
Abstract In this paper, we apply recently proposed bounds for the Marcum Q-function in order to derive novel tight bounds for the bit error rate (BER) in the case of differential quaternary phase shift keying (DQPSK) transmission, with Gray coding, over an additive white Gaussian noise (AWGN) channel. We then propose simple, yet very accurate, approximations for the BER. The computational complexity of the final approximate expression is significantly lower than that of the exact expression, but the entailed degradation is minimal. 1. Introduction The focus of this paper is on the performance analysis of differential quaternary phase shift keying (DQPSK) transmission with Gray coding over the additive white Gaussian noise (AWGN) channel. This modulation format, due to its robustness to phase uncertainties, is applied in many practical communication systems [1]. The bit error rate (BER) in the case of Gray coding can be written in terms of the Marcum Q-function and the zeroth order modified Bessel function of the first kind, I0 (x). In turn, the Marcum Q-function can also be expressed as an integral involving I0 (x), which is itself defined by an integral expression. Calculation of the generalized Marcum Q-function of order M , and particularly the popular case (M = 1) indicated as Marcum Q-function, Q(a, b), is important in many problems of signal detection [1–3]. Several algorithms have been devised in order to numerically evaluate this function [4– 6], but it is sometimes useful to have simple bounds, in order to gain immediate physical insight and avoid tedious and difficult numerical computations. Various bounds have been proposed, based on suitable representations of the Marcum Q-function and generalized Marcum Q-function [3, 7–9]. Recently, novel and very tight bounds on the Marcum Qfunction were proposed [10]. In this paper, by applying the bounds on the Marcum
Q-function proposed in [10], we show that it is possible to find very tight bounds and accurate, yet simple, approximate expressions for the BER of DQPSK with Gray coding, which completely avoid the computation of the Marcum Q-function. In particular, the final derived approximate expression entails only the computation of the exponential function, but the performance degradation is negligible at all BER values of interest. This paper is structured as follows. In Section 2, the reader is provided with preliminaries on the Marcum QFunction, and upper and lower bounds for this function, introduced in [10], are recalled. Based on these bounds, in Section 3 upper and lower bounds for the BER expression of DQPSK with Gray coding are presented. In Section 4, from the proposed bounds, accurate approximations for the BER expression of DQPSK with Gray coding are obtained. Finally, Section 5 concludes the paper. 2. Preliminaries on the Marcum Q-Function The Marcum Q-function is widely used for performance analysis of digital communication systems [1]. The definition of this function is the following: ¶ µ 2 w∞ x + a2 I0 (ax) dx. (1) Q(a, b) , x exp − 2 b
It is known that the computation of the integral in (1) may lead to numerical problems, due to the semi-infinite integration region. For this reason, in [7, 11] the authors propose alternative integral expressions for the Marcum Q-function with integration over a finite interval. In both cases, they consider suitable integral expressions for I0 (x). Instead of transforming the integration region of the integral in (1), upper and lower bounds for I0 (x), extremely tight over the integration region, are introduced in [10]. Based on these bounds, tight bounds on the Marcum Q-function are derived for both cases with b ≥ a and b < a, respectively. As it will be clear in the remainder of this paper, the case with b ≥ a is of interest for the analysis of the BER
expression of DQPSK with Gray coding. In the following, we summarize the derivation of the upper and lower bounds for the Marcum Q-function proposed in [10].
UB
2.1. Upper Bounds A known upper bound for the zeroth order modified Bessel function of the first kind is the following [1]: I0 (ax) ≤ exp(ax)
∀x ≥ 0.
UBS
(2)
≤
r µ · ¸ ¶ b−a (b − a)2 π erfc √ exp − +a 2 2 2 (3)
where x 2 w erfc(x) , √ exp(−t2 ) dt. π
(4)
h i 2 exp − (b−a) 2´o ³ pπ b−a √ +a 2 erfc 2 b b−a
i h 2 exp − (b−a) 2
UBC
³ 2 2´ exp − a +b 2 ³ I0 (ab) ´ pπ √ +a 8 erfc b−a 2
UBMG
h i 2 exp − (b−a) 2
Based on this inequality, it is easy to derive the following upper bound for the Marcum Q-function: Q(a, b)
n
I0 (ab) exp(ab)
Table 1: Upper bounds for the Marcum Q-function in the region b ≥ a.
0
The upper bound (3) can be significantly improved by observing that one is interested in bounding I0 (ax) only in the interval [b, ∞). Therefore, one simply needs to find a tighter approximation in this interval. Since I0 (x) − I1 (x) d ex = ex >0 dx I0 (x) I02 (x)
∀x ≥ 0
(5)
it follows that I0 (x) ≤ ex
I0 (b) exp(b)
∀x ≥ b.
(6)
In order to find a lower bound for the Marcum Q-function, the following inequality on the modified Bessel function I0 (x) can be verified: I0 (x) ≥
I0 (b)b ex eb x
∀x ≥ b.
(8)
The above bound is tight and valid exclusively for x > b. Based on this bound, one can derive the following lower bound on the Marcum Q-function:
r ¶ µ π I0 (ab)b b−a The following improved upper bound can therefore be ob√ . (9) Q(a, b) ≥ erfc 2 eab 2 tained by multiplying the right-hand side of (3) by I0 (ab)/ exp(ab), obtaining This lower bound will be referred to as LB. In Table 2, the ½ · ¸ I0 (ab) (b − a)2 new lower bound is shown and compared to bounds proQ(a, b) ≤ exp − exp(ab) 2 posed in [7–9] and indicated as LBS, LBC and LBMG, rer µ ¶¾ spectively. π b−a erfc √ +a . (7) 2 2 The upper bound in (7) is referred to as UB. In Table 1, the expression of the new upper bound is shown, together with bounds previously appeared in the literature. More precisely the bound indicated as UBS is proposed in [7], the bound UBC is proposed in [8] and UBMG is proposed in [9]. 2.2. Lower Bounds
3. Bounds for the BER of DQPSK with Gray Coding In the case of DQPSK transmission with Gray coding over an AWGN channel, the BER has the following expression [1]: µ 2 ¶ a + b2 1 BER = Q(a, b) − I0 (ab) exp − 2 2
(10)
I0 (ab) b p π exp(ab) 2 erfc
LB
b b+a
LBS
³
b−a √ 2
´
BERUB
h i 2 exp − (b+a) 2
LBMG
h
exp − (b+a) 2
+a
Table 2: Lower bounds for the Marcum Q-function in the region b ≥ a. where a
,
b
,
v à r ! u u t2γb 1 − 1 2 v à r ! u u t2γb 1 + 1 2
1 2 I0 (ab) exp
BERUBC
i 2
³
´
³ 2 2´h b exp − a +b exp(ab) 2 b−a ¤ 1 − 2 I0 (ab)
BERUBS
³ 2 2´ I0 (ab) exp − a +b 2
LBC
hp
a π √ erfc b−a 2 exp(ab) ³ 2 2 ´i 2 + 21 exp − a +b 2
I0 (ab)
BERUBMG
pπ
³
8 erfc
2
2
− a +b ³ 2´ b−a √ 2
´
³ 2 2´ exp − a +b [exp(ab) 2 ¤ − 12 I0 (ab)
Table 3: Upper bounds for the BER of DQPSK with Gray coding. (11)
(12)
and γb is the bit signal-to-noise ratio (SNR). Since b > a, by applying the bounds for the Marcum Q-function introduced in Section 2, it is possible to derive novel bounds for the BER expression in (10). We distinguish between upper and lower bounds: in each case, we also consider the application of the other bounds for the Marcum Q-function proposed in the literature and indicated in Table 1 and Table 2, respectively. 3.1. Upper Bounds Using the upper bound (7) for the Marcum Q-function, it is possible to derive the following upper bound for the BER in the case of DQPSK with Gray coding: ·r ¶ µ π a b−a BER ≤ I0 (ab) erfc √ 2 exp(ab) 2 µ 2 ¶¸ a + b2 1 + exp − . (13) 2 2 The upper bound (13) will be referred to as BERUB . In Table 3, the expression of the new upper bound is shown, together with BER bounds derived by using the upper bounds for the Marcum Q-function given in Table 1. The upper bounds on the BER for DQPSK transmission with Gray coding over an AWGN channel, summarized in
Table 3, are shown, as functions of the bit SNR γb , in Figure 1. For comparison, the exact BER is also shown. It is immediately recognized that the proposed bound BERUB is very tight, and outperforms significantly the other bounds. 3.2. Lower Bounds Using the lower bound (9) for the Marcum Q-function, it is possible to derive the following lower bound for the BER in the case of DQPSK with Gray coding: BER
≥
·r
µ ¶ b b−a π erfc √ I0 (ab) 2 exp(ab) 2 µ 2 ¶¸ a + b2 1 − exp − . (14) 2 2
The lower bound (14) will be referred to as BERLB . In Table 4, the expression of the new lower bound is shown, together with the bounds derived by using the lower bounds on the Marcum Q-function given in Table 2. The lower bound BERLB is shown, as a function of the bit SNR γb , in Figure 2. For comparison, the lower bound BERLBC is also shown, together with the exact BER. The remaining lower bounds indicated in Table 4, i.e., BERLBS and BERLBMG , are not shown in Figure 2, since they are negative and approximately equal to zero—this is expected, since the corresponding lower bounds on the Marcum Qfunction, from which they are derived, are loose [10].
0
10
-1
³ 2 2´h b exp − a +b 2 b+a¤exp(−ab) − 12 I0 (ab)
UBS
BER -3
10
-4
10
BERLBC
-5
10 0
2
4
6 γb [dB]
8
10
12
Figure 1: Upper bounds for the BER of QDPSK transmission, with Gray coding, over an AWGN channel. For comparison, the exact BER is also shown.
BERLBMG
0
´
BERLBS
b π √ erfc b−a 2 exp(ab) ³ 2 2 ´i 2 − 21 exp − a +b 2
-2
BER UBMG BER UBC BER UB BER Exact
³
I0 (ab)
10
10
hp
BERLB
1 2
³ 2 2´ I0 (ab) exp − a +b 2
³ 2 2´ [exp(−ab) exp − a +b 2 ¤ 1 − 2 I0 (ab)
10
Table 4: Lower bounds for the BER of DQPSK with Gray coding.
Exact LB BER LBC BER
-1
10
-2
BER
10
-3
10
-4
10
-5
10 0
2
4
6 γb [dB]
8
10
12
Figure 2: Lower bounds BERLB and BERLBC for the BER of QDPSK transmission, with Gray coding, over an AWGN channel. For comparison, the exact BER is also shown. 4. Approximate BER Expressions for DQPSK with Gray Coding Based on the tight bounds (13) and (14) and because of their striking formal symmetry, it is intuitive to derive an approximate expression for the BER by simply considering the arithmetic average between these two bounds. This observation leads to the following expression: r µ ¶ b−a π I0 (ab) (a + b) erfc √ BERapprox , . (15) 8 exp(ab) 2 In order to estimate the tightness of the proposed approximate expression (15), it is expedient to define an error and a percentage relative error as follows: ∆
,
²
,
(BERapprox − BER) 100 × (∆/BER).
(16) (17)
In Table 5, the exact BER, the approximate expression (15), ∆ and ² are shown for increasing values of the SNR. It is immediate to notice that the numerical difference between the exact BER (10) and the approximate expression (15) is basically negligible (² < 1%) for all BER values of interest (say, BER< 10−2 ). According to [12], for large values of their arguments, it is possible to consider exp(x) I0 (x) ' √ 2πx
(18)
and erfc(x) '
exp(−x2 ) √ . πx
(19)
Applying the approximations (18) and (19) in (15), one obtains the following approximate expression (with increasing accuracy for larger SNR values): · ¸ (b − a)2 1 b+a exp − BER0approx , √ 2 8πab b − a √ h i √ 2+1 1 = p √ √ exp −(2 − 2)γb 8π 2 γb 0.40 ' √ exp (−0.56 γb ) (20) γb The exact BER and the approximate expression (20) are compared, as functions of the SNR, in Figure 3. As one can see, the simple approximation (20), which entails a very low computational complexity, is very accurate for any BER value of interest.
0
BER
BERapprox
∆
²
0 1 2 3 4 5 6 7 8 9 10 11 12
0.16 0.13 9.9E-2 7.2E-2 4.8E-2 3.04E-2 1.72E-2 8.5E-3 3.64E-3 1.267E-3 3.432E-4 6.790E-5 9.053E-6
0.17 0.14 0.10 7.3E-2 4.9E-2 3.07E-2 1.73E-2 8.6E-3 3.6E-3 1.269E-3 3.436E-4 6.795E-5 9.057E-6
0.01 6.3E-2 3.4E-3 1.6E-3 7.2E-4 2.9E-4 1.0E-4 3.5E-5 9.8E-6 2.3E-6 4.0E-7 5.2E-8 4.6E-9
6.2% 4.8% 3.4% 2.3% 1.5% 0.95% 0.62% 0.40% 0.27% 0.18% 0.12% 0.077% 0.05%
Table 5: Comparison, for increasing SNR, of the exact BER, the approximate BER expression (15), their difference and the relative percentage error for DQPSK transmission over an AWGN channel
5. Conclusions In this paper, by using recently proposed bounds for the Marcum Q-function, we have derived novel tight bounds for the BER of DQPSK with Gray coding. These bounds have been shown to outperform other bounds on the BER derived from existing bounds on the Marcum Q-function. Due to the symmetry of the obtained bounds, it was natural to use their arithmetic average as an approximate BER expression, from which a further simple approximate expression, characterized by a very low computational complexity, has been derived. It has been shown that the latter proposed approximate expression is extremely accurate for all BER values of interest. References [1] J. G. Proakis, Digital Communications, 4th Edition. New York: McGraw-Hill, 2001. [2] M. K. Simon and M.-S. Alouini, “A unified approach to the probability of error for noncoherent and differentially coherent modulations over generalized fading channels,” IEEE Trans. Commun., vol. 46, pp. 1625– 1638, December 1998. [3] ——, Digital Communication over Generalized Fading Channels: a Unified Approach to the Performance Analysis. John Wiley & Sons, 2000. [4] S. Parl, “A new method of calculating the generalized Q function,” IEEE Trans. Inform. Theory, vol. 26, pp. 121–124, January 1980.
10
-1
10
-2
10
BER
γb [dB]
-3
10
-4
10
-5
10 0
2
4
6 γb [dB]
8
10
12
Figure 3: Comparison between the approximate expression (20) and the exact expression for the BER of DQPSK transmission with Gray coding over an AWGN channel [5] P. E. Cantrell and A. K. Ojha, “Comparison of generalized Q-function algorithms,” IEEE Trans. Inform. Theory, vol. 33, pp. 591–596, July 1987. [6] C. W. Helstrom, “Computing the generalized Marcum Q-function,” IEEE Trans. Inform. Theory, vol. 38, pp. 1422–1428, July 1992. [7] M. K. Simon, “A new twist on the Marcum Q-function and its application,” IEEE Commun. Lett., vol. 2, pp. 39–41, February 1998. [8] M. Chiani, “Integral representation and bounds for Marcum Q-function,” IEE Electronics Letters, vol. 35, pp. 445–446, 18th March 1999. [9] M. K. Simon and M.-S. Alouini, “Exponential-type bounds on the generalized Marcum Q-function with application to error probability analysis over fading channels,” IEEE Trans. Commun., vol. 48, pp. 359– 366, March 2000. [10] G. E. Corazza and G. Ferrari, “New bounds on the Marcum-Q function,” IEEE Trans. Inform. Theory, vol. 48, no. 11, pp. 3003–3008, November 2002. [11] X. Chen, “Iterative data detection: Complexity reduction and applications,” Ph.D. dissertation, University of Southern California, Los Angeles, CA, December 1999. [12] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions. Dover, 1972.
