1.5 Thesis Organization And Studied Algorithms
2.1.5 Optimization of Multi-resolution Algorithm
Two issues should be considered in the realization of the proposed algorithms. First, the used orders of the correlations affect both the complexity and the accuracy. Secondly, the ratio of the orders of the correlations between two stages will constrain the correctness of the unwrap process. The proposed design flow first selects the last stage correlation order, which would optimize the estimation accuracy. Then, we decide the numbers of the stages and the ratios of the orders between any two successive stages. Therefore, there are two questions for the design flow. First is what the optimal correlation order is; the second is the selection of the best ratio that minimizes the unwrap error.
Selection of Optimal Correlation Order
To answer first question, we consider the performance of the estimator using k-th correlation.
It has two cases: k < T /2 and K > T /2.
The solutions are given by
ko= T 6
2 − 5ρ ±p
4 + 28ρ + ρ2
2 − ρ . (2.36)
If ρ is small, we can get the approximate optimal value of k as k1o≈ T
3, (2.37)
and the optimal MSE is given by J1o = Fs2
which is roughly 12.5% larger than the CRB.
If K > T2, similar derivation gives the results that k2o≈ 2T
Therefore, k(L) = L3 or 2L3 at final stage is recommend. The choice of L3 or 2L3 is the trade-off between the operations of the correlator and the length of the buffer size. When T3 is used, it requires 2T3 operations of correlator and T3 buffer sizes; on contrast, if k(L) = T3, it only requires
T3 operations of correlator but 2T3 buffer sizes.
Selection of Ratio of Orders between Two Stages
The selection of the ratio of orders between two stages depends on the correctness of unwrapping operation. Let ˜Ωok(l) = ˜Ωk(l) − ²k(l) and Ωok(l − 1) = Ωk(l − 1) − ²k(l−1) be the true values of
If ξ(l) has a Gaussian distribution, which condition holds when the noise is Gaussian and T is large, the probability of the incorrect unwrapping at lth stage is given by
Pe(l) = 2 where Φ(z) is the Q function of Gaussian distribution and σξ2(l) is the variance of ξ(l). The full error probability is given by
Pe=
Moreover, we have Pe(1) = 0 since the first order delayed correlation is used at first stage.
There are some comments on the derivation. First, the incorrect unwrap occurs if one error at any stage happens at least, so the overall error probability is approximately the summation of the error probabilities at each stage. Besides, The Q function is dramatic increasing with its argument; therefore, the error probability is minimized by the selecting those rls that have balanced σξ2(l)s. Therefore, a sub-optimal heuristic choice is considered to use “near-constant”
ratios.
To give more insights of the unwrap error probability, we need to analyze the variance of
Therefore, the variance of ξ(l) is given by
σξ2(l) = C(l)ρ + D(l)ρ2 (2.49)
Then, when k(l) > T2 and k(l − 1) < T2, similar derivation yields the same formulation of σ2ξ(l) in (2.49), but its coefficient C(l) is different. There are two cases in this condition. When T ≤ k(l − 1) + k(l), we have
−10 −5 0 5 10−3
10−2 10−1 100 101 102 103
SNR (1/ρ) in dB
MSE
sim., actual noise sim.,imaginary part noise analy.
Figure 2.3: The simulated and analytic variance of the unwrap error σ2ξ for T = 100 and different rl.
If the SNR ρ is high, we can omit the ρ2 in (2.49); thus, ratio setup mainly depends on the C(l). The minimization of C(l) yields minimization of σξ2(l) as well as Pe(l). But, when ρ is low, such as ρ > 1, the parameter D(l) becomes the dominated term and the ratio setup in this situation mostly relies on D(l) rather than C(l).
Verification of Unwrap Error Probability
We first compare the analytical derivation and the simulated result of σ2ξ. We fix the first stage as k(1) = 1, and test different second stage order as follows k(2) = 2, 4, 8 and 16 for T = 100.
The variance of imagine part of rRx(1) − Rx(k(2)), denoted as σξ20, is also provided in this study when f = 0. Fig.2.3 illustrates the comparisons. We can find that the σ2ξ0 perfectly matches the analytical results in any cases of any SNRs, which demonstrates the correctness of the analysis.
However, the actual σ2ξ approaches the analytical results only if the SNR is high. This is because of the mismatch of the linear approximation of the phase noise in the analysis.
Next, consider the scenario of the constant ratio settings. We use the pairs of different stage orders as follows: (k(1), k(2)) = (1, 3),(k(1), k(2)) = (3, 9) and (k(1), k(2)) = (9, 27), but constant ratio r = 3 with T = 100. As illustrated in Fig.2.4, the difference of the variances σ2ξ with different simulation settings are not significant. Thus, the heuristic near constant ratio assignment is reasonable in the system design.
We now consider a study of two stages approach to verify the derivation of the error prob-ability. The parameters are given as follows: T ∈ [100 200 400], k(1) = 1, k(2) = 66. The frequency used in the testing is f = 0.2Fs. From numerical calculation, the ratio r2 = 66 and the variance of ξ(2) is σ2ξ(2) = 0.4346ρ + 22.0147ρ2 for T = 100, σξ2(2) = 0.1088ρ + 10.9485ρ2 for T = 200 and σξ2(2) = 0.027ρ + 5.4601ρ2 for T = 400. Fig.2.5 illustrates the comparisons of the error probabilities of the setups. As shown, the analytic results almost perfectly match the simulated ones.
−5 −4 −3 −2 −1 0 1 2 3 4 5
Figure 2.4: The simulated and analytic variance of the unwrap error σ2ξ for T = 100 and constant ratio rl= 3.
The choice of total stages relies on the length of observations and longer one requires more stages. Fortunately, the number of stages has a log relationship with the total length in the near constant ratio scheme. The stage ratio depends on both the length of observation and the target operation environment. For example, as the cases given above, the stage ratios of the cases all are 66. If Pe = 10−4 is the near error-free condition, the threshold is roughly 8 dB for T = 100, 6.5 dB for T = 200 and less than 5 dB for T = 400.
Design Example and Performance Study
Now, we present a design example of the parameter setup when T = 200. First of all, we set final stage order as k = d2T3 e = 134. Then, we consider the designs using 3 stages, 4 stages and 5 stages. The constant ratio is equal to rl=√
134 = 11.52 for L = 3, rl= 1341/3= 5.12 for L = 4, and rl= 1341/4= 3.402 for L = 5. After theoretical error probability test, we set k = [1 12 134]
for L = 3, k = [1 5 27 134] for L = 4 and k = [1 4 13 40 134] for L = 5, which are the optimal settings of each design. The required complexity in the estimation of the correlations, which is the major cost of the correlation-based estimators, is given
N =
Therefore, the complexities of the setups are given as follows: 453 for L = 3, 633 for L = 4, and 808 for L = 5.
Fig.2.6 shows the simulated and the theoretical results. As shown, all of the settings can approach the bound at the SNR higher than 0 dB. The threshold in 3 stage approach is -1 dB, in 4 stage approach is -4 dB and in 5 stage approach is -5 dB, which are related to the unwrap error probability. As shown in Fig.2.7, the analytical error probabilities versus different settings demonstrate the thresholds of the SNR. For example, in 5 stages approach, the error probability
−8 −6 −4 −2 0 2 4 6 8 10−4
10−3 10−2 10−1 100
SNR(1/ρ) (dB)
Pe
Sim.,T=100 Sim.,T=200 Sim.,T=400 Analy.,T=100 Analy.,T=200 Analy.,T=400
Figure 2.5: The simulated and analytic probabilities of the unwrapping error.
is less than 10−4 when SNR is larger than -5.5 dB which is close to the SNR threshold in 5 stages approach. The same result holds on the thresholds and the analytical unwrap error probabilities for 3 stages approach and 4 stages approach.