This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
A Novel Non-Parametric Iterative Soft Bit Error Rate Estimation Technique for Digital Communications Systems ‡
Samir Saoudi‡ , Tarik Ait-Idir‡ , and Yukou Mochida Signal and Communications Department, Institut Telecom / Telecom Bretegne/LabSticc, and Universite Europeenne de Bretagne (UeB) Brest, France Communications Systems Department, INPT, Rabat, Morocco Orange-Labs, Tokyo, Japan Email:
[email protected]
Abstract—This paper addresses the problem of unsupervised soft BER estimation for digital communications systems, where no prior knowledge about transmitted information bits is available. We show that the problem of BER estimation is equivalent to estimating the conditional probability density functions (pdf)s of soft channel/receiver outputs. We also propose a non parametric Gaussian Kernel-based pdf estimation technique. Then, we introduce an iterative stochastic expectation maximization (EM) algorithm for the estimation of both a priori and a posteriori probabilities of transmitted information bits, and the classification of soft observations according to transmitted bit values. These inputs serve in the iterative procedure used for the estimation of conditional pdfs. We analyze the performance of the proposed BER estimator and show that is asymptotically unbiased and pointwise consistent. We also provide numerical results in the case of CDMA systems and show that attractive performance is achieved compared with conventional Monte Carlo (MC)-aided techniques.
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I. I NTRODUCTION Monte Carlo (MC) techniques are generally used to evaluate the bit or block error rate (BER or BLER) of digital communication systems. In [1], a tutorial exposition of different MC-aided techniques has been provided, with particular reference to four methods: i) Modified Monte Carlo simulation (i.e., importance sampling); ii) extreme value theory; iii) tail extrapolation; and iv) quasi-analytical method. The modified Monte Carlo technique is achieved by importance sampling which means that important events, i.e., errors, are artificially generated by biasing the noise process. At the end of the simulation, the error count must be properly unbiased [2]. The extreme value theory [3], assumes that the probability density function (pdf) can be approximated by exponential functions. The tail extrapolation method, which is a subclass of the extreme value theory technique, is based on the assumption that the tail region of the pdf can be described by a generalized exponential class. The quasi-analytical method combines noiseless simulation with analytical representation of the noise. Except for the MC method, all these techniques assume perfect knowledge of the type and/or the form of noise statistics. In [4], high order statistics (HOS) of the bit log-likelihood-ratio (LLR) are used to evaluate the error rate performance of turbo-like codes. In this case, the characteristic function of the bit LLR is estimated using its first cumulants, i.e., moments, and the pdf is deduced with the aid of the inverse Fourier transform (IFT). In [5], a BER estimation technique where the transmitter sends a fixed information bit value has been proposed. At the receiver side, the BER is computed by estimating the pdf of received soft channel/receiver outputs. This technique is called soft BER estimation since the BER is estimated using soft channel observations/outputs without requiring hard decisions about information bits 1 . Unfortunately, all these methods assume that the estimator perfectly knows transmitted data. In many practical communication systems, the BER is required to be on-line estimated in order to perform system-level functions such as scheduling, resource allocation, power control, or link adaptation where the transmission scheme is adapted to the channel conditions. Under this framework, the BER estimation problem becomes very challenging because of the following main reasons: 1 In soft BER estimation, soft observations are used instead of hard decisions. This provides reliable BER estimates and reduces the number of required samples/observations compared with hard decision-based BER estimation techniques [5].
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In MC-based techniques, the unknown transmitted information bit values are required for computing the BER estimate, while in practical communications systems the BER estimation should be performed in an unsupervised fashion because the estimator has no information about transmitted data. Most of practical communication channels are quasi-static block fading, i.e., constant over a certain block duration, and randomly changes from block to block. Therefore, in order to provide the transmitter with a reliable BER information feedback for a given block, i.e., channel state information (CSI), only the soft observations corresponding to the actual block have to be used for estimating the instantaneous BER. In the case of MC-based techniques, this results in unreliable or even wrong BER estimates because the number of observations/bits is not sufficient. The knowledge of the transmitter scheme, channel and interference models, and receiver technique greatly impacts the reliability of the BER estimate. In other words, it is generally quite hard to derive analytical expressions of the bit error probability (BEP) when the system suffers from interference and/or non-linear receiver techniques (e.g. iterative interference cancellation receivers) are used.
This paper was mainly motivated by the above considerations. We assume that the estimator has no knowledge either about transmitted data, or the transmitter/receiver scheme and the communication model. We focus on the problem of BER estimation in a completely unsupervised 2 fashion. To the best of the authors knowledge, this issue has not been addressed before in the literature. We first provide a problem formulation where we show that BER estimation is equivalent to estimating the pdfs of conditional soft observations corresponding to transmitted information bits. Then, we introduce a non parametric Gaussian Kernel-based pdf estimation technique where no analytical model of soft observations is assumed. We study the asymptotic behavior of the resulting BER estimator and show that is unbiased and point-wise consistent. As pdf estimation requires the knowledge of both a priori probabilities of information bits and the classification of soft observations according to the +1 and −1 values of transmitted bits, we introduce an iterative stochastic expectation maximization (EM) algorithm for iteratively computing these parameters. We analyze the performance of the proposed unsupervised technique in the case of a multiuser code division multiple access (CDMA) system with conventional single-user detection. Performance comparison shows that the proposed estimator clearly outperforms supervised MC-aided estimation. Interestingly, reliable BER estimates are achieved even in the high signal to noise ratio (SNR) and using only a few number of soft observations. Throughout the paper, the following notation is used. The cardinality N of set C is denoted N = |C|. When X is a random variable, E [X] and V ar [X] denote the mathematical expectation and variance of X, respectively. When f has a second derivative function, (f ) (x) and (f ) (x) denote its first and second derivative at point x, respectively. sgn (.) denotes the sign of the argument, and ln(.) is the natural log 2 The aim of unsupervised estimation is to estimate the BER based only on soft observations that serve for computing hard decisions about information bits without requiring any information about transmitted data and transceiver scheme/model.
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function. P [.] is the probability of a given event, and superscript denotes the transpose operator. The remainder of the paper is organized as follows. In Section II, we describe the considered framework and detail soft output pdf and BER estimation; Section III details the stochastic EM iterative algorithm; In Section IV, we study the asymptotic behavior of the BER estimator and provide numerical results; The paper is concluded in Section V. The proofs of the results are provided in the Appendices A–E. II. S YSTEM F RAMEWORK , O UTPUT PDF
AND
BER E STIMATION
A. System Framework We consider a digital communication system where a sequence of N independent and identically distributed (i.i.d) information bits (bi )1≤i≤N ∈ {+1, −1} is transmitted using a specific transmission scheme. At the receiver side, we assume the presence of soft values corresponding to transmitted bits. Let X denote the random variable (RV) corresponding to the receiver soft outputs, i.e., observations, and (Xi )1≤i≤N be the sequence of the realizations of X where each Xi corresponds to the decision statistic that serves for the computation of hard decision ˆbi about transmitted information bit bi as ˆbi = sgn (Xi ). Note that Xi may contain any type of interference such as co-channel interference (CCI), intersymbol interference (ISI). We assume no knowledge either about the transmission scheme, or about the channel model and the receiver technique. Our main purpose is to estimate in an unsupervised fashion the BER at the output of the receiver. Let π+ and π− denote the probability that the transmitted bit bi is equal to +1 and −1, respectively, i.e., π+ P [bi = +1] , (1) π− P [bi = −1] , where π+ + π− = 1. Note that π+ and π− are not known at the receiver. The soft outputs (Xi )1≤i≤N are random variables having the same pdf fX (x). The BEP is then given by pe
=
π+
0
b
−∞
fX+ (x) dx + π−
b
+∞ 0
b
fX− (x) dx,
(2)
where hN+ (respectively, hN− ) is a smoothing parameter which depends on the number of observed samples, i.e., N+ (respectively, N− ). Function K(·) is any pdf (called the Kernel) assumed to be an even and regular (i.e., square integrated) function with zero mean and unit variance. In the following, we provide equations only in the case of class C+ as those corresponding to C− can easily be obtained by replacing “+” by “−”. The choice of the optimal smoothing parameter is very critical since it impacts the accuracy of the pdf estimate. In [10], [11], it has been shown that if hN+ tends towards 0 when N+ tends towards +∞, the b+ estimator fˆX,N (x) is asymptotically unbiased. It has also been shown + that if hN+ → 0 and N+ hN+ → +∞ when N+ → +∞, then the MSE of the Kernel estimator tends towards zero. The optimal integrated mean squared error (IMSE)-based smoothing parameter h∗N+ is given as [5], − 1 1 1 5 b h∗N+ = N+ − 5 J fX+ (M (K))+ 5 ,
+∞ 2 b = K (x) dx, and J fX+ where M (K) = −∞
2 +∞ b+ fX (x) dx is the integrated square second derivative −∞ b
of the conditional pdf fX+ . As it can be seen from (6), the computation of h∗N+ unfortunately b depends on the unknown pdf fX+ . We therefore suggest to use the Gaussian Kernel whose parameter M (K) is given as [5] 1 M (K) = √ , 2 π
fX (x)
=
b π+ fX+
(x) +
b π− fX−
(x) .
(3)
B. Output PDF Estimation b
b
Both conditional pdfs fX+ (x) and fX− (x) depend on the communication channel model and receiver scheme. Therefore, it is extremely difficult or even infeasible to find out the exact parametric model of these distributions. In this subsection, we introduce a Kernel-based method [7], b b [8], [9] to estimate both pdfs fX+ (x) and fX− (x). The proposed technique is non parametric, and only requires soft outputs for each class. Let us assume that we can classify the set C = {X1 , · · · , XN } into two classes (i.e., partitions) C+ and C− , where C+ (respectively, C− ) contains the observed received soft output Xi such that the corresponding transmitted bit is bi = +1 (respectively, bi = −1). In Section III, we will introduce a stochastic EM-based algorithm that allows us to classify the sets C+ and C− . Let N+ (respectively, N− ) denote the cardinality of C+ (respectively, C− ), i.e., N+ = |C+ |, and N− = |C− |. Using the Kernel technique, it follows that each conditional distribution can be estimated using the elements of sets C+ and C− as, 1 x − Xi b+ , (4) (x) = K fˆX,N + N+ hN+ X ∈C hN+ i
b− fˆX,N −
1 (x) = N− hN−
+
Xi ∈C−
K
x − Xi hN−
,
(5)
(7)
b
while J(fX+ ) can be expressed in the case of a Gaussian conditional b 2 ), as distribution, i.e., fX+ ∼ N (m+ , σ+ 3 b J(fX+ ) = √ 5 , 8 πσ+
(8)
2 where m+ , and σ+ are the mean and variance of subset C+ , respectively. Details regarding the derivation of (8) are not provided because of space limitation. It follows from (6) that the optimal smoothing parameter in the case of a Gaussian Kernel, and a Gaussian conditional pdf is
b
where fX+ (·) (respectively, fX− (·)) is the conditional pdf of X such that bi = +1 (respectively, bi = −1). fX (x) is a mixture of the two b b conditional pdfs fX+ (x) and fX− (x) and can therefore be written as
(6)
h∗N+ =
4 3N+
1 5
σ+ .
(9)
C. BER Estimation As it can be seen from the generic expression of the BEP pe in (2), the b b BER can be estimated using the two conditional pdfs fX+ (·), fX− (·), and the a priori probabilities π+ and π− . Note that all these parameters are unknown, and are required to be estimated before computing the BER. The b b estimation of conditional pdfs fX+ (·) and fX− (·) can be performed based on observations X1 , · · · , XN , where only those corresponding to transmitted information bit value +1 (respectively, −1) are used to estimate b b fX+ (·) (respectively, fX− (·)). Unfortunately, the problem of classifying soft outputs X1 , · · · , XN according to transmitted information bit values is itself a detection problem. The unsupervised BER estimator we propose in this paper allows us to classify soft outputs X1 , · · · , XN and estimate b b both the conditional pdfs fX+ (·) and fX− (·) and a priori probabilities π+ and π− in an iterative fashion using a posteriori probability (APP) values P [bi = +1 | Xi ] and P [bi = −1 | Xi ] ∀ i = 1, · · · , N . Let T denote the number of iterations (index t = 1, · · · , T ). At each iteration t, all APP values are computed using a priori probabilities and pdf estimates obtained at the previous iteration t − 1. The resulting APPs allow us to update the estimates of a priori probabilities and classify soft outputs X1 , · · · , XN into two classes according to transmitted information bit values. The two conditional pdfs are then re-estimated. This process is repeated T times, and the BER is estimated with the aid of parameters estimated at the last iteration T . Note that, at the first iteration, since no prior information is available, the classification of soft outputs X1 , · · · , XN can be performed with respect to the sign of each
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observation Xi , while the initial values of a priori probabilities π+ and π− can be derived using this initial classification. In the following, we derive the expression of the BER estimate assuming Gaussian Kernel-based pdf estimator. Let θ π+ , N+ , hN+ , π− , N− , hN− . The value of θ is unknown and is iteratively estimated by iteratively classifying soft outputs X1 , · · · , XN into two classes. At the last iteration T , a reliable estimate of θ is reached and the BER is computed using the obtained estimate. Let us recall the expression of the BEP (2). Replacing the two conditional pdfs by their Gaussian Kernel-based estimates (4) and (5), and given the value of θ, the BER estimate is simply computed as
pˆe,N
=
π− π+ Xi Xi + , (10) Q Q − N+ i∈C hN+ N− i∈C hN− +
−
where Q(·) denotes the complementary unit cumulative Gaussian dis+∞ 1 √ tribution, i.e., Q(x) = x exp (−t2 /2)dt. Details regarding the 2π derivation of (10) are omitted because of space limitation.
can express Q θ(t) as N (t) = E ln P [Xi , bi ] | X1 , · · · , XN , Q θ =
N
i=1
(t) (t) b ρi+ ln π+ fˆ +
i=1
(t) (t) b + ρi− ln π− fˆ − (t)
A. Estimation Step
P [bi = −1 | Xi , θ(t−1) ],
(11)
of unobserved information bits bi , for i = 1, · · · , N , conditioned on observations X1 , · · · , XN and the estimate θ(t−1) of θ obtained at the maximization step of previous iteration t − 1. Using simple mathematical manipulations, we show that the likelihood probabilities of bit bi at iteration t can be computed as, ⎧ (t−1) ˆb+ π+ (Xi ) f ⎪ (t−1) ⎪ X,N ⎪ (t) + ⎪ ⎪ ρi+ = (t−1) b+ , ⎪ (t−1) ˆb− ⎪ π+ (Xi )+π− (Xi ) fˆ f ⎨ (t−1) (t−1) X,N X,N + − (12) (t−1) ˆb− π− (Xi ) f ⎪ ⎪ (t−1) ⎪ (t) X,N ⎪ − ⎪ = (t−1) b+ . ρi ⎪ (t−1) ˆb− ⎪ π+ (Xi )+π− (Xi ) fˆ f ⎩ − (t−1) (t−1) X,N +
X,N −
Note that for the sake of notation simplicity, the iteration index is dropped b b from pdf estimates fˆ + (t−1) (Xi ) and fˆ − (t−1) (Xi ). X,N+
X,N−
B. Maximization Step At iteration t, the maximization step allows us to compute the estimate θ based on conditional probabilities obtained at the estimation step of (t) is obtained by maximizing the the same iteration t. The estimate θ (t) of the log-likelihood of the joint event conditional expectation Q θ at iteration t. Assuming independent soft observations X1 , · · · , XN , we (t)
.
(13)
(t)
By invoking the fact that π+ +π− = 1, the maximization of Q (θ) leads to the new estimates of a priori probabilities at iteration t, (t) N (t) ρi+ , π+ = N1 (14) i=1 (t) (t) N π− = N1 i=1 ρi− . The proof of (14) follows using Lagrangian techniques. The details are omitted because of space constraint. C. Classification Step
X,N+
In this section, we introduce a stochastic EM-based algorithm to iteratively classify soft outputs X1 , · · · , XN into the two classes C+ and C− , and estimate θ. The EM algorithm was first introduced by Dempster et. al. in [12]. It iteratively computes, with the aid of the estimation and maximization steps, maximum likelihood (ML) estimates when the observations can be viewed as incomplete data that contain missing information (i.e., the unknown data we have to decide about). Stochastic EM presents a generalization of EM techniques by introducing a random rule in the EM algorithm. Some applications of stochastic EM can be found in [13], [14]. In our case, as we want to estimate the BER in an unsupervised fashion, the incomplete data correspond to observations X1 , · · · , XN , while the missing data are transmitted information bits b1 , · · · , bN .
(t)
(t) (Xi )
X,N−
(t)
ρi−
(Xi )
(t)
(t)
(t)
The remaining parameters N+ , hN+ , N− , hN− and the two condib b tional pdf estimates fˆ + (t) (.) and fˆ − (t) (.) depend on the outcome
III. S TOCHASTIC EM-BASED PARAMETER E STIMATION
In the estimation step of iteration t, we estimate the APPs (t) ρi+ P [bi = +1 | Xi , θ(t−1) ],
(t) X,N+
X,N−
of the classification procedure of subsets C+ and C− at iteration t. Therefore, this procedure should be carefully performed since it greatly impacts the reliability of both pdf estimates and a posteriori probabilities in subsequent iterations, and consequently the accuracy of the b b BER estimate. Given estimates θ(t−1) , fˆ + (t−1) (.), and fˆ − (t−1) (.) X,N+
X,N−
available at iteration t, we can classify soft outputs X1 , · · · , XN according to joint probabilities P [Xi , bi = +1], and P [Xi , bi = −1]. Us(t) (t) t ing Bayes rule, we can obtain subsets C+ and C− at iteration (t)
as C+ =
(t−1) ˆb+
Xi : π+
f
(t−1)
X,N+
(t−1) ˆb−
(Xi ) ≥ π−
f
(t−1)
X,N−
(Xi ) , and
(t) = C¯+ . Unfortunately, when the information bit values +1 and −1 are not equiprobable, this classification procedure will prevent some erroneous soft outputs (i.e., those positive soft outputs corresponding to transmitted information bit value −1 and vise versa) from being exchanged in the course of iterations between subsets C+ and C− . In the following, we introduce a stochastic EM-based algorithm that randomly performs the classification of soft outputs X1 , · · · , XN using APP values. This method relaxes the condition on the exchange of soft outputs between the two subsets C+ and C− . The stochastic EM technique uses a random Bayesian rule. At iteration (t) (t) t, it combines likelihood probabilities (15) and realizations U1 , · · · , UN of a uniform random variable U defined over the interval [0, 1]. The classification of soft outputs is performed as follows,
(t) (t) (t) , C+ = Xi : ρi+ ≥ Ui (15) (t) (t) ¯ C− = C+ . (t) (t) (t) (t) Parameters N+ and N− are simply obtained as N+ = C+ , and (t) (t) (t) (t) N− = C− . The optimal smoothing parameters hN+ and hN− are computed either with the aid of (9) assuming Gaussian conditional pdfs, or b b by iteratively using the exact expressions of J(fX+ ) and J(fX− ) followed by (9). The two conditional pdfs are then estimated using (4) and (5). Note (t) (t) (t) that σ+ and σ− correspond to the standard-deviation of subsets C+ and (t) C− at iteration t. The proposed stochastic EM-based unsupervised BER estimation algorithm can now be summarized as in Table I. (t) C−
IV. P ERFORMANCE E VALUATION A. Analysis of the BER Estimator In this subsection, we study the MSE-based convergence of the proposed iterative BER estimator. The first theorem shows that the proposed estimator is asymptotically unbiased.
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Table I S UMMARY OF THE S TOCHASTIC EM-BASED U NSUPERVISED BER E STIMATION A LGORITHM
1.
2.
Initialization (t = 1) 1.1. Classify soft outputs X1 , · · · , XN according to, (1) (1) i.e., C+ = {Xi : Xi ≥ 0}, and C+ = {Xi : Xi < 0}. (1) (1) (1) (1) 1.2. Deduce N+ and N− , i.e., N+ =| C+ | and (1) (1) N− =| C− |. (1) (1) 1.3. Compute standard-deviations σ+ and σ− (1) (1) corresponding to C+ and C− , respectively. (1) (1) 1.4. Compute the smoothing parameters hN+ and hN− according to (9). (1) (1) 1.5. Deduce a priori probabilities as π+ = N+ /N (1) (1) and π− = N− /N . 1.6. Compute the two conditional pdfs. Parameter Estimation at Iteration t 2.1. Estimation Step (t) (t) 2.1.1. Estimate APPs ρi+ and ρi− using (12). 2.2. Maximization Step (t) (t) 2.2.1. Compute a priori probabilities π+ and π− using (14). 2.3. Classification Step (t) (t) 2.3.1. Classify subsets C+ and C− using APP values (t) (t) ρi+ and ρi− according to (15). 2.3.2. 2.3.3. 2.3.4.
(t)
(t)
(t)
(t)
N+ =| C+ | and N− =| C− |.
(t)
(t)
Compute the standard-deviations σ+ and σ− (t) (t) corresponding to C+ and C− , resp. (t) Compute the smoothing parameters hN+ and
Then, by noting that pˆe,N is asymptotically unbiased, i.e., (E [ˆ pe,N ] − 0 pe ) → 0, (see Theorem 1), and that the variance of pˆe,N tends towards pe,N − pe ])2 = 0. as N → +∞ (see Theorem 2), we get limN →+∞ E (ˆ B. Numerical Results To evaluate the performance of the proposed unsupervised and non parametric BER estimator, we consider the framework of a synchronous CDMA system with two users having binary phase-shift keying (BPSK) and operating over an additive white Gaussian noise (AWGN) channel. We restrict ourselves to the conventional single user CDMA detector. Performance assessment in the case of advanced signaling/receivers is not reported in this paper due to space limitation and is left for future contributions. With respect to the considered framework, the received LSF × 1 chiplevel signal vector at discrete time instant i can be expressed as (1)
(1)
(t)
Xi
hN
according to (9). 2.3.5. Compute the two conditional pdfs. Parameter Estimation Compute the BER estimate. −
3.
b
b
Theorem 1. Assume that the two conditional pdfs, fX+ and fX− , have second derivative functions, that hN+ → 0 and hN− → 0 as N → +∞. Then the soft BER estimator pˆe,N given by (10) is asymptotically unbiased, i.e., pe,N ] = pe . lim E [ˆ N →+∞
Proof: See Appendix A for the proof. The following theorem shows that the variance of the proposed estimator also tends towards zero. b
b
Theorem 2. Assume that the two conditional pdfs, fX+ and fX− , have second derivative functions, that hN+ → 0 and hN− → 0 as N → +∞. Then the variance of the soft BER estimator pˆe,N tends towards zero as N tends towards +∞, i.e., pe,N − E[ˆ pe,N ])2 = 0. lim E (ˆ N →+∞
Proof: See Appendix B for the proof. Using Theorems 1 and 2, it is easy to show that the proposed estimator is point-wise consistent, i.e., the MSE tends towards zero as the number of samples N tends towards +∞. This result can be formulated in the following corollary: b
b
Corollary 3. Assume that the two conditional pdfs, fX+ and fX− , have second derivative functions, that hN+ → 0 and hN− → 0 as N → +∞. Then the MSE of the soft BER estimator pˆe,N tends towards zero as N tends towards +∞, i.e., pe,N − pe ])2 = 0. lim E (ˆ N →+∞
Proof: is straightforward. First of all, we this corollary The proof 2of pe,N ]])2 + (E[ˆ pe,N − E[ˆ pe,N ] − pe ])2 . write E (ˆ pe,N − pe ]) = E (ˆ
(2)
(16) ri = A1 bi s1 + A2 bi s2 + ni ,
√ LSF is where LSF denotes the spreading factor, and sk ∈ ±1/ SF the spreading code corresponding to user k. Ak is the amplitude of user (k) k = 1, 2, bi is the information bit value ∈ {±1} of user k at time instant i, and ni ∈ CLSF is the
temporally and spatially white Gaussian noise, i.e., ni ∼ N 0, σ 2 ISF . The a priori probabilities of information (k) bits are assumed identical for both users, i.e., π+ = P bi = +1 , and (k) π− = P bi = −1 ∀k, i. The decision statistic that serves for detecting user 1 at time instant i (1) is Xi = s 1 ri [6] and is given as, (1)
= A1 bi
(2)
(1)
+ A2 bi ρ + n ˜i ,
(17)
where ρ is the normalized cross-correlation between the two spreading (1) ˜ i is the noise at the output of the single codes s1 and s2 , and n
Gaussian (1) user detector, i.e., n ˜ i ∼ N 0, σ 2 . The decision about information bit (1) (1) bi corresponds to the sign of decision statistic Xi . Note that the soft (1) output Xi in (17) contains a mixture of a Gaussian noise and a RV whose pdf is unknown at the receiver (because a priori probabilities π+ and π− are unknown). Using (17), we can easily show that the BEP for user 1 is 2 A1 − A2 ρ A1 + A2 ρ 2 + π+ . (18) + π− Q pe1 = 2π+ π− Q σ σ In s1
the =
following, we use the two spreading codes 1 √ + + + + − − − , and s = 2 7 1 √ − − + + − − − , where the cross-correlation 7 is ρ = 0.4286. We consider the case where the two users have equal powers A1 = A2 = 1. The SNR at the output of the MF of each user is therefore SNR = 1/2σ 2 . Note that under these assumptions, soft (1) (1) decisions X1 , · · · , XN will be corrupted by severe MAI. This is extremely challenging for the proposed BER estimator since parameter estimation (such as a priori probabilities) will be performed in the presence of strong interference. In all simulations, we consider T = 6 iterations for the stochastic EM-based parameter estimation while at each iteration t = 1, · · · , 6, two iterations are used for computing the optimal smoothing parameters h∗N+ and h∗N− as mentioned in Subsection III-C. Note that for all the scenarios we consider in the following, we assume that the proposed unsupervised and non parametric BER estimator has no prior knowledge either about a priori probabilities π+ and π− or (1) (1) the classification of soft outputs X1 , · · · , XN . Because of space limitation, we only consider the case of equiprobable information bits i.e., (π+ = π− = 1/2). The number of soft outputs that serve for estimating the BER is N = 104 observations. In Fig. 1.a and Fig. 1.b we present both the theoretical and the estimated conditional pdfs (at the last iteration T = 6) for SNR = 0dB and 10dB, respectively. We observe that the proposed BER estimator provides accurate estimates of conditional pdfs. Note that
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
V. C ONCLUSIONS
1 Estimated pdf (right) Estimated pdf (right)
0.9
Estimated pdf (left)
Estimated pdf (left)
1.2
Theoretical pdf (left)
Theoretical pdf (left)
1 Conditional pdfs
0.7 Conditional pdfs
Theoretical pdf (right)
Theoretical pdf (right)
0.8
0.6 0.5 0.4 0.3
0.8
0.6
0.4
0.2 0.2 0.1 0 −5
−4
−3
−2
−1
0 x
1
2
3
4
0 −5
5
−4
−3
−2
−1
(a)
0 x
1
2
3
4
5
(b)
Figure 1. Estimated conditional pdfs for (a) SNR = 0dB, and (b) SNR = 10dB.
In this paper, we considered the problem of unsupervised BER estimation for digital communication systems. We proposed a BER estimation algorithm where only soft observations that serve for computing hard decisions about information bits are used to estimate the BER, and no prior knowledge about transmitted information bits is required. First of all, we provided a formulation of the problem where we showed that BER estimation is equivalent to the estimation of conditional pdfs of soft observations. We then proposed a BER computation technique based on pdf estimation. Then, we introduced an iterative stochastic EM technique to compute the parameters that serve for the estimation of conditional pdfs based only of soft observations. Finally, we analysed the theoretical performance of the estimator, and showed that it is asymptotically unbiased, and point-wise consistent. We also provided numerical performance in the framework of CDMA systems to corroborate the theoretical analysis. Interestingly, we showed that when classical MC methods fail to perform BER estimation in the region of high SNR, the proposed estimator provides reliable estimates using only few soft observations.
0
0
10
10
A PPENDIX A P ROOF OF T HEOREM 1 −2
10
Let us recall that the proposed soft BER estimator is given by
−1
10
BER
1
BER
1
pˆe,N where,
−2
10
B+
−6
10 Classical Monte Carlo Theoretical BER −3
10
0
−8
Unsupervised Soft BER 1
2
3
10 4
5 SNR1
(a)
6
7
8
9
10
=
π+ B+ + π− B− ,
(A.1)
−4
10
0
=
Classical Monte Carlo theoretical BER Unsupervised Soft BER 2
4
6
0 −∞
8
10
12
14
16
B−
SNR
=
1
(b)
Figure 2. BER performance comparison when π+ = π− = 1/2 for (a) N = 104 , and (b) N = 103 soft observations.
for low SNR (Fig. 1.a), the variance of the MAI plus thermal noise is very high, and therefore the set of soft outputs has a large definition support. This means that in order to achieve smooth pdf estimates, a large number of observations is required. This explains the oscillatory behavior we observe around +1 and −1 for low SNR = 0dB. When the SNR is increased to SNR = 10dB (Fig. 1.b), no oscillations are observed since the MAI plus noise variance is reduced and the definition support of soft observations becomes very tight. In Fig. 2, we provide the BER estimation performance. From Fig. 2.a, we notice that the proposed unsupervised and non parametric BER estimator offers the same performance as the MC-aided method where the estimator has perfect knowledge about the transmitted information bits, i.e., perfect classification of soft (1) (1) outputs X1 , · · · , XN . Note that when the proposed estimator knows the transmitted information bits, then its supervised version corresponds only to the computation of the smoothing parameters followed by the BER estimation according to (10) and the iterative stochastic EM algorithm is not required. We now focus on the behavior of the proposed BER estimator at the very high SNR region where it is difficult to achieve a reliable BER estimate when using MC-aided techniques with a limited number of soft observations. In Fig. 2.b, we report the BER estimation performance using only N = 103 soft observations. The proposed technique provides reliable BER estimates (with respect to the theoretical curve) for SNR values up to SNR = 16dB, while the MC technique fails to do so and stops at SNR = 8dB because of the very limited number of transmitted information bits. This presents a major strength of the proposed BER estimator, and is very promising for many practical systems where it is required to estimate the BER of the communication link in a real-time fashion based only on a few data frames.
1 N+ hN+
+∞ 0
K
Xi ∈C+
1 N− hN−
x − Xi hN+
K
Xi ∈C−
x − Xi hN−
dx,
(A.2)
dx.
(A.3)
i is identical for all Xi ∈ C+ and can be By noting that E K x−X hN +
evaluated only for a particular Xj ∈ C+ , we have, +∞ 0 1 x−u b E [B+ ] = fX+ (u)du dx.(A.4) K hN+ −∞ hN+ −∞ Using the following change of variable t = E [B+ ]
=
0 −∞
+∞ −∞
x−u hN+
, we get,
b K (t) fX+ (x − hN+ t)dt dx. (A.5)
b fX+
is assumed to have a second derivative pdf function, we can write As b a Taylor series expansion of fX+ as follows, h2N+ t2 b+ b b b fX fX+ (x − hN+ t) = fX+ (x) − hN+ t fX+ (x) + (x) 3 3 2 + O hN+ t . (A.6) Then, combining (A.5) and (A.6) we get, 0 +∞ b b fX+ (x) K (t) dt − fX+ (x)hN+ × E [B+ ] = −∞ −∞ +∞ +∞ h2N+ b+ 2 fX tK (t) dt + (x) t K (t) dt dx + O(h3N ). 2 −∞ −∞ (A.7) As +∞ and unit variance Gaussian +∞ 2Kernel, we get +∞K (.) is a zero mean K (t) dt = 1, tK (t) dt = 0, and t K (t) dt = 1. −∞ −∞ −∞ 0 b+ b Exploiting these properties and −∞ fX (x)dx = fX+ (0), (A.7) can be expressed as, 0 h2N+ b+ b fX fX+ (x)dx + (0) + O(h3N+ ). (A.8) E [B+ ] = 2 −∞
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings
As hN+ → 0 when N → +∞, we therefore get, limN →+∞ E [B+ ] = 0 b fX+ (x)dx. Similarly, we can show that limN →+∞ E [B− ] = −∞ +∞ b− fX (x)dx. Therefore, we get, 0 pe,N ] lim E [ˆ
N →+∞
=
π+
=
pe .
0
b
−∞
fX+ (x)dx + π−
+∞ 0
b
fX− (x)dx (A.9)
A PPENDIX B P ROOF OF T HEOREM 2 First, recall that the proposed soft BER estimator is given by, =
pˆe,N
π+ B+ + π− B− ,
(B.1)
where, B+ =
B− =
1 N+ hN+ 1 N− hN−
Xi ∈C+
K −∞
Xi ∈C−
0
+∞
K 0
x − Xi hN+
x − Xi hN−
Ai+
=
K −∞
Ai−
0
=
x − Xi hN+
+∞
K
(B.2)
dx.
(B.3)
0
dx, where Xi ∈ C+ ,
(B.4)
=
dx, where Xi ∈ C− .
(B.5)
From (B.2) and (A.8), we get, E Ai+ = hN+
t∈R
0
b
−∞
K (t) fX+
thN x+ √+ 2
dt dx.
(B.11)
b
2 2 pe+ (1 − pe+ ) hN+ b+ 1 1 f = V ar A + (0) − p e i + X + N+ h2N+ N+ N+ 4 4 2 hN+
1 b fX+ (0) + − O h5N+ . (B.14) 4N+ N+ As hN+ → 0 and N+ ≈ π+ N when N → +∞, it fol
limN →+∞
V ar A2 i− N− h2 N−
V ar A2 i N+ h2 N
+
+
= 0. Similarly, we can show that
= 0. By recalling the expression in (B.6), we finally
get, limN →+∞ V ar [ˆ pe,N ] = 0.
2 2 π+ V ar(B+ ) + π− V ar(B− ), 2 2 π− π+ V ar(Ai+ ) + V ar(Ai− ). (B.6) 2 N+ hN+ N− h2N−
=
it follows
As fX+ is assumed to have a second derivative pdf, a Taylor series b expansion of fX+ can be written as, thN thN b b b fX+ x + √ + = fX+ (x) + √ + fX+ (x) 2 2 t2 h2N+ b+
fX (x) + O t3 h3N+ . (B.12) + 4 From (B.11), and (B.12), and exploiting the fact that K (.) is a zero-mean and unit variance Gaussian Kernel, we get h2N+ b+
2 2 fX (B.13) (0) + O h5N+ . E Ai+ = hN+ pe+ + 4
As X1 , · · · , XN are independent, the variance of pˆe,N is V ar [ˆ pe,N ]
E A2i+ = h2N+
lows that, limN →+∞
x − Xi hN−
that
u−x√ , hN+ / 2
Using (B.7) and (B.13), we obtain dx,
Let us introduce the following quantities,
Using (B.10) and the following change of variable, t =
h3N b b fX+ (x)dx + + fX+ (0)+hN+ O(h3N+ ). 2 −∞ ! "# $ 0
pe+
(B.7) Now, expression of (B.6), we must calculate to determine the analytical E A2i+ . E A2i− and E Ai− can be similarly deduced. Using (B.4), we get, E A2i+ = 0 0 y − Xi x − Xi E K dx dy | Xi ∈ C+ . K hN+ hN+ −∞ −∞
(B.8)
We can easily show that for the chosen Gaussian Kernel, we have, Xi − x+y x − Xi y − Xi x−y 2 √ √ . K K =K K hN+ hN+ hN+ / 2 2hN+ (B.9) Using (B.8), (B.9), and the following change of variable, (v, w) = x+y ( 2 , x − y), and by noting that K(·) is a pdf (and therefore K(w)dw = 1), we get R 0 √ 2 Xi − v √ E Ai+ = E 2hN+ K dv | Xi ∈ C+ . (B.10) hN+ / 2 −∞
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