At the beginning, we focus on the orthogonal modulation and want to estimate the symbol ERs. For convenience, we consider a three links wireless sensor networks. Later, we will show that a wireless sensor network with more links can be decomposed into several wireless sensor networks involving three links only.
˃
Figure 5.3: A binary nonsymmetric channel with parameter e0 and e1
To derive a estimator, we first define a SMP as P(byj1 = byj2) (the probability that the hard decisions from the ith and jth links are identical). Then, by the law of total probability, we have For a three-links wireless sensor network, one has three SMPs, i.e. P(by1 = by2), P(by2 =by3), and P(by1 =by3). These three SMPs and (5.11) yield the following nonlinear
where, for i, j, k∈ {1, 2, 3} and i ̸= j ̸= k,
ai = M2pi − M, bi =−2(M − 1)(Mpi− 1), ci = (M − 1)(pi(M − 1) + (pj + pk− 1) − Mpjpk)
Hence, the procedure of the proposed ER estimator consists of two steps and is similar to that shown in Chapter 2. In the first step, we estimate {pi}3i=1 by the method of moments, we can estimate ERs by solving (5.12) and we have
bei = −bi−√
For QAM modulation scheme, we need to estimate the bit ERs{ekj}, not symbol ERs.
Hence, we need another step to transform the symbol-level decision into bit-level one, yielding a three-step estimator. Specifically, we first de-map the symbol hard decision byi
into bit decisions byik, which is often required in a typical digital link. In the second step, we estimate the bit-level SMPs P(bykj1 =byjk2) for j1 ̸= j2. We then have the the following a basic nonlinear system in a three-link sensor network ((5.12) with M = 2)
we obtain the following ER estimate via the method of moments (which can be obtained by (5.15) with M = 2)
bekj = 1 2− 1
2 vu
ut(2bP(byjk= bykj1)− 1)(2bP(byjk= byj2k)− 1)
2bP(bykj1 =bykj2)− 1 , j, j1, j2 ∈ {1, 2, 3} (5.18) For some nonbinary modulations, ekj is independent of k, hence the countings on the RHS of (5.17) for different k should be averaged to obtain an improved estimator. We summarize the complete procedure in Table 5.1. For general L > 3, we can decompose the estimation problem into several estimation subproblems involving only three links.
For instance, if L = 5, we can consider two estimation subproblems. The first subprob-lem considers the first three links while the other one involves the last three links. In this case, the ER of the third link is estimated in both subproblems and we can average these two results to get a new estimate with better performance.
Table 5.1: A blind ER estimation algorithm for high order modulation in sensor network.
Input: Received samples, y
1: Detect the signals byi and transform them into bit-level byik
2: Compute SMPs for all link pairs by (5.17).
3: Compute bei, i = 1, 2, 3 via (5.18) using the SMPs obtained in 2 Output: be1, be2 and be3.
To evaluate the performance of the blind bit-level ER estimation, we can analyze the performance on the binary symmetric channel. Although it is not binary symmetric channel for M -QAM modulation for M > 4, the results under this assumption can an approximation of the performance on the binary asymmetric channel. To derive the formula, we use the Delta method and inverse function theorem, as shown in Appendix B.5 and [33]. Based on the Delta method, the MSEE of the blind bit-level ER estimation on the binary symmetric channel is
MSEE[bekj]≈
[JCJT]
j,j
2N (5.19)
where C is the covariance matrix, J is the Jacobian matrix and [X]i,j denotes the element of the matrix X in the ith row and jth column. Unlike the binary case in Chapter 4, there is a factor 2 in the denominator. This comes from the factor that there are in-phase and quadrature-in-phase in M -QAM modulation. Since BERs of the kth bit in these two phases are identical, we have two samples to estimation the BER given a received sample.
Assuming independent links, the covariance matrix of pairwise matching indicator vector (I(ˆy1 = ˆy2), I(ˆy1 = ˆy3), I(ˆy2 = ˆy3)) can be shown to be ((B.11) or [33])
where eki can be evaluated based on (5.28) in the next section. We can derive the Jacobian matrix based on (5.18) by the definition of the Jacobian matrix and expressed it as a function of eki, i = 1, 2, 3. This procedure is quite complex. Instead, we first derive the inverse of the Jacobian matrix based on (5.16) and the inverse function theorem.
The associated inverse Jacobian matrix is
J−1 =
With (5.20)-(5.23), we can compute the MSEE on the binary symmetric channel and get the lower bound performance of the blind bit-level ER estimation on the binary asymmetric channel.
For orthogonal modulation scheme, the derivations of the MSEE formula are similar.
For covariance matrix, the formulas of pij and pijl are modified as
The other difference is the Jacobin matrix formula since the nonlinear systems (5.12)-(5.22) and (5.16) are not the same. In this case, the inverse Jacobian matrix is
J−1 =
With (5.20) and (5.24)-(5.26), the MSEE for orthogonal modulation scheme is
MSEE[bekj]≈
To further enhance the performance of the blind symbol/bit-level ER estimator, we observe that the performance of the estimator is worse with the increase of the link quality, as shown in Fig. 5.7. This observation also appears in the case with BPSK modulation in Chapter 4. In that chapter, we observe that the estimator of the SMPs is a compare-and-count process (5.17) which is similar to that used in simulation-based ER estimations and IS techniques can be applied. It motives us to inject the noise into the received samples to alter their statistics and perform the estimation based on them.
Because the binary symmetric model is similar to the case in Chapter 4 with binary modulation, MSEE can be reduced possibly by injecting noise. To achieve the better performance, we propose a noise-enhanced ER estimations similar to the estimation proposed in 4 with some modifications.
The procedure of the proposed noise-enhanced ER estimations involves four steps for M -QAM modulation scheme. First, we inject noise into the received signal before we