Iterative ML Estimator – Data Channel

To overcome the problem of bias in the C/N0 estimate for weak signals, iterative

approaches can be used (Li et at 2002). In this work, the Newton-Raphson method is used to solve Eq. (3.43), the ML equation for amplitude. Initial estimates for the iterative procedure are obtained using the estimates given by the approximate solution described above. Let 𝑙𝑙_𝑑𝑑(𝑃𝑃) be

Figure 3-5: The mean of tanh(2αym/A) evaluated as a function of C/N0 with

𝑙𝑙𝑑𝑑(𝑃𝑃) = 𝑃𝑃 −_{𝑁𝑁 � �𝑠𝑠}1 𝑚𝑚tanh �2𝛼𝛼_{𝑃𝑃 𝑠𝑠}𝑚𝑚�� 𝑚𝑚

. (3.49)

The roots of 𝑙𝑙_𝑑𝑑(𝑃𝑃) correspond to the ML estimate of 𝐿𝐿. In this iterative method, 𝛼𝛼 is considered independent of 𝐿𝐿 and thus is treated as a constant in the partial derivative 𝜕𝜕𝑙𝑙𝑑𝑑/𝜕𝜕𝑃𝑃. An optimization with respect to 𝐿𝐿 is performed to reduce the dimensionality of

the search.

The update for 𝐿𝐿̂_𝑅𝑅+1 is obtained as 𝐿𝐿̂𝑅𝑅+1 = 𝐿𝐿̂𝑅𝑅−_{𝜕𝜕𝑙𝑙}𝑙𝑙𝑑𝑑�𝐿𝐿̂𝑅𝑅�

𝑑𝑑(𝑃𝑃)

𝜕𝜕𝑃𝑃 �_{𝐿𝐿�𝑅𝑅} (3.50)

where 𝐿𝐿̂_𝑅𝑅 is the amplitude estimate at the 𝑅𝑅-th iteration and the partial derivative is given by 𝜕𝜕𝑙𝑙𝑑𝑑(𝑃𝑃) 𝜕𝜕𝑃𝑃 = 1 + 1 𝑁𝑁 2𝛼𝛼 𝑃𝑃2� � 𝑠𝑠_𝑚𝑚2 cosh2�2𝛼𝛼_{𝑃𝑃 𝑠𝑠𝑚𝑚}�� 𝑚𝑚 . (3.51)

Once 𝐿𝐿̂_𝑅𝑅+1 is obtained, 𝜎𝜎�_𝑛𝑛2_𝑅𝑅+1 is updated using Eq. (3.45), which is then used to compute 𝛼𝛼�𝑅𝑅+1. The initial estimate, 𝐿𝐿̂1, is obtained using the approximate version of the ML

estimator, as given by Eq. (3.47). The above implementation actually searches for a root of the equation 𝑙𝑙_{𝑀𝑀𝐿𝐿,𝑑𝑑}(𝑃𝑃) = 0, given by

𝑙𝑙𝑀𝑀𝐿𝐿,𝑑𝑑(𝑃𝑃) = 𝑃𝑃 −_{𝑁𝑁 � �𝑠𝑠}1 𝑚𝑚𝑃𝑃𝑃𝑃𝑛𝑛ℎ � _𝑃𝑃 𝑃𝑃 𝑑𝑑 − 𝑃𝑃2𝑠𝑠𝑚𝑚�� 𝑚𝑚 (3.52) where 𝑃𝑃𝑑𝑑 =_{𝑁𝑁 � 𝑥𝑥}1 𝑛𝑛2 𝑛𝑛 . (3.53)

Figure 3-6 shows the improvement in performance obtained by using the iterative method to compute the C/N0 as compared to the MLE with approximation. The plot is

obtained with 10 iterations per estimate with a C/N0 output rate of 500 ms (corresponding

𝑇𝑇𝑐𝑐𝑐𝑐ℎ = 1 𝑚𝑚𝐿𝐿, 𝐾𝐾 = 20, 𝑁𝑁 = 500). A total of 104 estimates are averaged for each

considered C/N0. As shown in Figure 3-6, the ML estimator with approximation is

considerably biased for C/N0 values less than 20 dB-Hz, and hence does not parallel the

CRLB bound for variance. The iterative method is significantly less biased than the approximate MLE.

Although the iterative method involves computing the tanh⁡(. ) and cosh2(. ) functions, this can be performed in offline software receivers where computational complexity is not of significant concern. This is particularly helpful in determining the

Figure 3-6: Performance analysis of iterative MLE against MLE with approximation using data channel only

performance of algorithms proposed for weak signal environments, where offline tests can be carried out. Another option is to use look-up tables for real-time applications.

Since the initial estimate for the data channel only C/N0 estimator is obtained

from the MLE with approximation, this might lead to poor initialization of the iterative algorithm. Further, as shown in Figure 3-7, the roots of 𝑙𝑙_{𝑀𝑀𝐿𝐿,𝑑𝑑}(𝑃𝑃) includes {±𝐿𝐿, 0} even under noiseless conditions. Hence, depending on the initial estimates used for the iterative procedure, the iteration can converge to any one of the three roots. Ignoring the possibility of negative initialization for the amplitude estimate, the iteration can converge either close to the original amplitude (𝑃𝑃 = +𝐿𝐿) or the origin (𝑃𝑃 = 0). There are also some cases where the noise level is too high and the root 𝑃𝑃 = +𝐿𝐿 vanishes under such low SNR conditions. When the iteration converges to 𝑃𝑃 = 0, the C/N0 estimate becomes −∞.

This condition is referred to as divergence. Convergence is declared when the iterative algorithm settles on a non-zero root (𝑃𝑃 > 0).

Figure 3-7: Plot of gML,d(a) for a C/N0 of 15 dB-Hz under two different

conditions (i) Convergence and (ii) Divergence. The reference curve corresponds to the noise-less condition. Original amplitude (A) is set as 10.

Hence, in order to analyse the convergence percentage, a simple detector is utilized. For each C/N0 estimate, the algorithm is allowed to run for 100 iterations to

check for divergence. Since the maximum variance of the estimator can be of the order of 102 _{(corresponding to a C/N}

0 of 5 dB-Hz), which translates to 1𝜎𝜎_{𝑅𝑅𝐿𝐿𝑃𝑃} = ±10 𝑑𝑑𝐵𝐵,

convergence is declared when both of the following conditions are met:

i. | Final Estimate – Initial Estimate from ML with approximation| < 30 dB

ii. |Final Estimate – True C/N0| < 10 dB

The first condition is a rough check designed for early detection of divergence. The true C/N0 is the C/N0 used to simulate the input vectors and would not be available under live

signal conditions. The results shown in Figure 3-8 are averaged across 104 such estimates.

The convergence percentage of the iterative algorithm drops slowly below 20 dB- Hz, but is approximately 70% or greater for C/N0 values which are of interest to weak

Figure 3-8: Convergence percentage for the iterative ML estimator using data channel only

signal positioning users (10 dB-Hz and above). This problem can be minimized by making use of the presence of the pilot channel (i.e., pilot channel ML estimates can be used as initial C/N0 estimates).