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The Exact Transition Probability and Bit Error Probability of Two-Dimensional Signaling Lei Xiao, Student Member, IEEE, and Xiaodai Dong, Member, IEEE
Abstract—A new signal-space partitioning method is proposed for the calculation of transition probabilities of arbitrary twodimensional (2-D) signaling with polygonal decision regions. This enables straightforward formulation of exact bit error probability, symbol error probability, mutual information (MI), and error probabilities of individual bits in the constellation. Previously published symbol error rate (SER) results for additive white Gaussian noise (AWGN) and various fading channels, diversity reception, and imperfect channel estimation (ICE) can all be used in conjunction with the new signal-space partitioning method to obtain the desired performance measures. Numerical results of the bit error rates (BERs) of several 8-ary and 16-ary constellations in AWGN and in Rayleigh, Ricean, and Nakagami fading channels with diversity reception are presented. Pilot-symbolassisted 2-D signaling is also studied. The proposed new approach is both exact and numerically efficient. It provides new insights into the effect of bit mapping, constellation parameter optimizations, unequal error protection (UEP) at different bit positions, and performance comparisons of different constellations. Index Terms—Bit error rate (BER), imperfect channel estimation (ICE), mutual information (MI), symbol error rate (SER), two-dimensional (2-D) signaling, unequal error protection (UEP).
I. I NTRODUCTION
I
NITIATED by Craig’s ingenious work on the calculation of the symbol error rate (SER) in additive white Gaussian noise (AWGN) channels [1], the SER of arbitrary two-dimensional (2-D) signaling in different coherent-reception scenarios has been comprehensively studied. For example, the analytical SER expressions for fading channels have been presented in [2]. Results considering diversity receptions can be found in [3] and [4]. More recently, the SERs of 2-D signaling formats in the presence of channel-estimation errors were derived in [5]–[7] for Rayleigh and Ricean fading channels, respectively. The basic idea of Craig’s method is to work with the distribution of the additive noise (expressed in polar coordinates) instead of the distribution of the decision statistic, shift the origin to the signal point considered, and divide the erroneous decision region into disjoint subregions that can be effectively described in polar form to facilitate derivation and numerical evaluation. A similar kind of general approach, however, does not exist for calculating the bit error probability of arbitrary 2-D constellations, to the best of the authors’ knowledge. In a digital
Manuscript received April 2, 2004; revised July 7, 2004 and July 8, 2004; accepted July 8, 2004. The editor coordinating the review of this paper and approving it for publication is G. Matteo Vitetta. L. Xiao is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA (e-mail:
[email protected]). X. Dong is with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC V8W 3P6, Canada (e-mail: xdong@ ece.uvic.ca). Digital Object Identifier 10.1109/TWC.2005.853821
communication system, bit error rate (BER) is the ultimate and more accurate performance measure than SER, as it tells the quality of the transmission for information source bits and, thus, includes the effect of bit-to-symbol mapping. Usually, an estimation of BER can be obtained by either adopting Gray mapping and assuming one symbol error only leads to one bit error or by employing union bound techniques. Unfortunately, the accuracy of these approximations in the low signal-to-noise ratio (SNR) region is questionable, and Gray mapping for a 2-D constellation is not always applicable when the number of nearest neighbors of a symbol is larger than the number of bits in the symbol. Other methods reported in the literature are either specific constellation [mostly rectangular-quadratic-amplitude modulation (QAM) and M -phase-shift keying (PSK)] targeted or specific bit-mapping (usually Gray mapping) oriented [8]–[13]. In [13], the authors formulated the average distance spectrum for binary reflected Gray-coded multitone PSK (MPSK). However, the exact BER analysis and calculation were not available for arbitrary 2-D constellations, such as the well-known 16-star QAM. Recently, hierarchical modulations (also known as embedded or multiresolution modulations) that find applications in digital broadcasting/multicasting channels and wireless multimedia services have been studied in [14] and [15]. These schemes encode messages or bit streams with distinct error-protection requirements to different bit positions of a nonuniformly spaced constellation. The BERs of nonuniform hierarchical M -pulse-amplitude modulation (PAM), M -PSK, and M -QAM were formulated and computed in [14] and [15]. In this paper, Craig’s wisdom is further extended to the calculation of BER for arbitrary 2-D signaling formats. The BER is acquired through transition probabilities. The erroneous decision-region partitioning in [1] for calculating SERs does not provide a solution to transition probabilities. A new partitioning method is proposed to decompose each signal space region, for a specific transition, into additions and subtractions of simple subregions, which can be efficiently evaluated. With transition probabilities available, we can compute the BER with arbitrary bit mapping, SER, mutual information (MI) of the equivalent discrete memoryless channel (DMC), and the probability of error at individual bit positions in a constellation, thus avoiding the necessity of computer simulations. This provides an analytical tool for not only the bit error-performance evaluation of 2-D signaling but also the design and analysis of unequally error-protected constellations that are not limited to nonuniform M -PAM, M -PSK, and M -QAM. The rest of this paper is organized as follows. Section II presents the relation between different performance measures and transition probabilities. The new method for calculating transition probabilities is presented in Section III. A simple way
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XIAO AND DONG: EXACT TRANSITION PROBABILITY AND BIT ERROR PROBABILITY OF 2-D SIGNALING
to obtain the transition probabilities of M -PSK is shown in Section IV as an example of the general approach. The extension of our results from AWGN channels to various fading channels is briefly introduced in Section V. In Section VI, the calculation of BER in Rayleigh and Ricean fading channels with imperfect channel estimation (ICE) is introduced. Numerical results and discussions are provided in Section VII, where the accuracy of the widely used Gray approximation is also scrutinized. Some conclusions are drawn in Section VIII. II. BER, SER, MI, AND T RANSITION P ROBABILITY Indicate the M -ary 2-D constellation as S = {si |si ∈ C, i = 0, 1, · · · , (M − 1)} and the bits mapped to the ith symbol si as ci . The transmitted symbol is denoted by s, and the detected symbol at the receiver is represented by sˆ. The BER of the M -ary signaling can be written as Pb =
M −1 M −1 1 d(cl , ck )P (ˆ s = sk |s = sl )P (sl ) log2 M k=0, l=0
(1)
k=l
where d(cl , ck ) represents the Hamming distance between the s = sk |s = sl ) is the transition probability vectors cl and ck , P (ˆ for the event that the received signal falls into the decision region of the symbol sk and, hence, detected as the symbol sk , while the transmitter sent the symbol sl , and P (sl ) denotes the a priori probability for the symbol sl . Since the Hamming distance between the bit mapping of different symbols and the a priori probability for each symbol are determined according to the particular modulation scheme, the BER is then a weighted sum of transition probabilities. In the same manner, the error probability of the ith bit can be formulated as Pbi =
M −1 M −1 l=0
d (cl (i), ck (i)) P (ˆ s = sk |s = sl )P (sl ). (2)
k=0, k=l
The SER of the 2-D constellation can also be written in the form of the weighted sum of transition probabilities as Ps =
M −1 M −1 l=0
Again, the MI is a weighted sum of transition probabilities. Since a digital transmission with hard decisions can be modeled as a DMC, the transition probabilities are sufficient to characterize such a channel. That is, we have complete knowledge of the channel if we know all the transition probabilities.
III. T RANSITION P ROBABILITY FOR A RBITRARY 2-D S IGNALING IN AWGN The idea of using transition probabilities to calculate BER was also adopted in [11] and [12]. However, the BER calculation employing this method was largely confined to M -PSK, because there was no general and effective way of evaluating transition probabilities. In this section, we present a new method to efficiently compute the transition probabilities for arbitrary 2-D signaling. To simplify the scenario, only AWGN channels are considered in this section. The extension to fading channels is straightforward and will be presented in Section V. In an AWGN channel, the received signal is corrupted by zero mean white Gaussian noise n with a variance of N0 /2 per dimension. Since the in-phase and quadrature component of the noise are independent, the probability density function (pdf) of the noise can be written in the form [16, eq. (6–128)]
pn (r, θ) =
P (ˆ s = sk |s = sl )P (sl ).
(3)
However, the erroneous region-partitioning method in [1] for SER involves far fewer computations. The probability that the receiver makes a decision on sk , regardless of which symbol was transmitted, is given by M −1
P (ˆ s = sk |s = sl )P (sl )
P (ˆ s = sk |s = sl ) =
θ2 ∞ PI =
P (ˆ s = sk |s = sl )P (sl ) log P (ˆ s = sk |s = sl )
l=0 k=0
−
pn (r, θ)drdθ
(7)
where the origin of the coordinates has been shifted to the transmitted signal point sl . The transition probability (7) for both the closed and open decision region shown in Figs. 1 and 2 can be calculated based on the probability that the noise superimposed on the symbol sl falls into two basic shapes, as illustrated in Fig. 3. For the type-I basic shape, the probability that the noise falls into the shadowed region can be expressed as
and the MI for this modulation scheme with hard decisions is I=
(6)
where r is the amplitude of the noise and θ the phase of the noise. For the maximum-likelihood (ML) detector, the decision region for each signal point is a polygon that may be either closed or open, as shown by the shaded area in Figs. 1 or 2. The decision region for sk is denoted as Dk . The transition probability P (ˆ s = sk |s = sl ) when the received signal is detected as sk , given that the sl sent can be written as
(4)
l=0
M −1 M −1
r r2 exp − πN0 N0
n∈Dk
k=0, k=l
P (ˆ s = sk ) =
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M −1 k=0
θ1 R(θ)
1 = 2π
θ2 θ1
P (ˆ s = sk ) log P (ˆ s = sk ). (5)
r r2 exp − drdθ πN0 N0
∆
exp −
αγs sin2 ψ dθ sin2 (|θ − θ1 | + ψ)
= G(θ1 , θ2 , ψ, α)
(8)
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Fig. 1. Typical decision region of a signal point with a closed region.
Fig. 3. Two basic shapes that act as components to form a decision region. (a) Type I. (b) Type II.
Fig. 2. Typical decision region of a signal point with an open region.
and the function R(θ) is R(θ) =
x sin ψ sin (|θ − θ1 | + ψ)
(9)
√ where x = αγs , γs is the average received SNR, and √ α denotes the distance x when the average energy of the constellation is normalized to one. Geometric parameters θ1 , θ2 , and ψ are labeled in Fig. 3(a). The reason for using the absolute value (|θ − θ1 |) in the above definition of G(θ1 , θ2 , ψ, α) will be obvious when the type-II basic shape is discussed, although it has no effect on the PI calculation. Fig. 3(b) shows the type-II basic shape that has only one boundary connected to the symbol sl (the origin of the new coordinates). We always denote the angle of the boundary that passes the origin as θ1 . Parameter θ2 is the angle of the line
parallel to the other boundary. Then, it is not guaranteed that θ1 will always be smaller than θ2 , which is the case for the type-I basic shape. Fig. 3(b) is such an example. Therefore, |θ − θ1 | is needed in G(θ1 , θ2 , ψ, α) as θ might be smaller than θ1 . The probability that the noise falls into the type-II basic shape is given by θ2 2 1 sin ψ αγ s exp − 2 PII = dθ sin (|θ − θ1 | + ψ) 2π θ1
= |G(θ1 , θ2 , ψ, α)|
(10)
where the absolute value outside the integral is necessary to guarantee that the probability is always positive, whether θ1 > θ2 or θ1 < θ2 . Later, it will be seen that such a definition of θ1 and the possible negative result of the integral in (10) facilitates the probability computation in the open-region case. Both PI and PII are now expressed by the G function, given the geometric parameters of the two basic shapes defined in Fig. 3.
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Note that the angle ψ ∈ (0, π) always. The G function is a single integral with finite integration limits and a well-behaved integrand and can be evaluated efficiently with high accuracy. Next, we show that the decision region Dk ’s in Figs. 1 and 2 can be decomposed into the summation and subtraction of the two types of basic shapes. A. Closed-Region Case Without loss of generality, Fig. 1 is used as an example to illustrate the partition when sk has a closed decision region Dk . The polygonal decision region can be divided into triangles and quadrangles by straight lines from the signal point sl to the vertices of the polygonal decision region’s boundaries. The decision region Dk is then composed of disjoint slices labeled as 1, 2, and 3 in Fig. 1. For convenience, we denote them as Dk1 , Dk2 , and Dk3 , respectively. Thus, (7) can be rewritten as P (ˆ s = sk |s = sl ) =
+
1 n∈Dk
+
2 n∈Dk
pn (r, θ)drdθ.
(11)
3 n∈Dk
One useful observation is that slice Dk1 can be obtained as subregion EAIF subtracting subregion EABF , slice Dk2 as subregion F IDG subtracting subregion F BJG, and slice Dk3 as subregion GDCH subtracting subregion GJCH. All these subregions are type-I shapes, and therefore, the probability that the received signal falls into each subregion is given by (8). Then, the probability for the ith slice Dki is Pci =
pn (r, θ)drdθ i n∈Dk
= G θ1i , θ2i , ψi,1 , αi,1 − G θ1i , θ2i , ψi,2 , αi,2 (12) where all the geometric parameters are shown in Fig. 1 and √ αi,j = xi,j when the average energy of the M -ary constellation is one. The transition probability for a closed decision region is given by P (ˆ s = sk |s = sl ) =
N
Pci
i=1
=
N i i G θ1 , θ2 , ψi,1 , αi,1 i=1
− G θ1i , θ2i , ψi,2 , αi,2
(13)
where N is the total number of slices as a result of the closed decision-region Dk partitioning. B. Open-Region Case When the erroneous symbol sk has an open decision region, the situation is more complicated. For an arbitrary open de-
Fig. 4. Values of (b1 , b2 , b3 ) in (14) for six possible positions of sl . Subregion 1 is always the region formed by boundary AB and the line connecting sl and B, subregion 2 is bounded by the lines connecting sl and B, sl , and C, and boundary BC, and subregion 3 is bounded by boundary DC and the line from sl to C.
cision region with more than three boundaries, as shown in Fig. 2, we can always use a straight line (BD) connecting the first and the last intersection of the decision boundaries to partition Dk into two disjoint regions. One is an open region Dko (ABDE), which has three boundaries and two intersections, and the other is a closed region Dkc (BCDB), which can be solved employing the method presented in Section III-A. Therefore, any open decision regions are formed by either two boundaries (such as M -PSK excluding BPSK), three boundaries, or three boundaries plus a closed region. Thus, we only need to solve the probability for an open region with no more than three boundaries. The partition of the three-boundary open region is done by drawing two straight lines from the signal point sl to the two intersections of the decision boundaries, as illustrated in Figs. 2 and 5. Regardless of the relative position of the symbol sl to the open decision region of sk , the transition probability is always given by the summations and subtractions of the probabilities for the subregions formed by the partition. For example, the open region ABDE in Fig. 2 can be expressed as the sum of type-II subregion ABG and type-I subregion GBDF minus type-II shape F DE. The probability that the received signal falls into the open region ABDE can be expressed as Po = b1 G θ11 , θ21 , ψ1 , α1 + b2 G θ12 , θ22 , ψ2 , α1 + b3 G θ13 , θ23 , ψ3 , α2
(14)
where (b1 , b2 , b3 ) = (+1, +1, −1) in this example. The relative position of the sl to the boundaries of Dk determines the values of b1 , b2 , and b3 in (14), with six different scenarios illustrated in Fig. 4. The need to determine six different sl positions relative to the three-boundary open decision region complicates the transition probability calculation. We observe that the absolute
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Equations (12), (15), and (16) together constitute our new approach to the transition probability calculation, which can be summarized as P (ˆ s = sk |s = sl ) =
i ∈D Dk k
Po +
Pci ,
i i ∈D c Pc , Dk k
when Dk is closed when Dk is open
(17)
for l = k. P (ˆ s = sk |s = sk ) can be obtained by subtracting s = sl |s = sk ) from unity. k=l P (ˆ
IV. M -PSK: A N E XAMPLE
Fig. 5. New treatment of the three line open subregion without using (b1 , b2 , b3 ).
operation over the G function in (14) eliminates the sign resulting from the integral in G, while (b1 , b2 , b3 ) are introduced later on to add signs before the sum. It is natural to ask whether it is possible to take advantage of the sign in G(θ1 , θ2 , ψ, α) to circumvent the need for classifying the relative position of sl to determine (b1 , b2 , b3 ). As a matter of fact, a single universal formula for a three-boundary open region is available for any sl locations. The probability of the three-boundary open region can be expressed without using (b1 , b2 , b3 ) as Po = |G(Θ1 , Θ2 , ψ2 , α1 ) − G(Θ1 , Θ3 , ψ1 , α1 ) + G(Θ2 , Θ4 , ψ3 , α2 )|
(15)
√ where Θ1 , Θ3 , and α1 = x1 , as shown in Figs. 2 and 5, should be the angle of the line originating from the signal point sl to one of the boundary intersections, the angle of the semi-infinite boundary that passes this intersection, and the normalized distance between the signal point sl and the same intersection, respectively. Same rules apply for Θ2 , Θ4 , and √ α2 = x2 for the other intersection. Though not shown here, (15) is valid for all the relative positions of sl . For open regions having two lines as boundaries (for example, M -PSK shown in Fig. 6), we can deem them as a special case of the three-boundary region, where the two intersections of boundaries overlap, and therefore, (15) can be applied. Note that the two lines connecting the signal point and the respective intersection (now, there is only one intersection) are the same line, i.e., Θ1 = Θ2 and α1 = α2 (x1 = x2 ) in Fig. 5, if the three-boundary open region is degenerated into a twoboundary region. In this case, the first term in (15) is zero because the lower and the upper limits of the integral in G are identical. Then, the transition probability for a two-boundary open region is given by Po,2 = |G(Θ1 , Θ3 , ψ1 , α1 ) − G(Θ1 , Θ4 , ψ3 , α1 )| .
(16)
To better illustrate the new method, M -PSK is used as an example for the transition-probabilities calculation. Fig. 6 shows the constellation and decision boundaries of 8-PSK. The constellation points for M -PSK are
2πn sn = exp j M
,
n = 0, 1, . . . , (M − 1).
(18)
The decision region for sk , in polar coordinates, is given by (2k + 1)π (2k − 1)π
