Dynamic Loop Bandwidth Control

Beside the phase detector design, the approach using dynamic loop bandwidth control is an-other feasible way [52] to provide large acquisition range and good tracking stability. Fig.2.20 illustrates the structure of the proposed dynamic loop bandwidth control. Unlike the scheme proposed by Ke et al. [52], the proposed scheme comprises a lock indicator and a lookup table of the LF parameters. In mathematical, the loop parameters k_i = f_i(p) and k_p= f_p(p) are both function of the lock value p.

Lock Indication

The lock detector is an indicator of the locking quality and used to monitor and control the synchronizer and equalizer. Lee et al. [58] propose a simplification of the detector in [69] for QPSK constellation, and it is give by

p = E n

(√

2 + 1)y_min− y_max o

, (2.96)

where y_max= max {|y_r|, |y_i|} and y_min = min {|y_r|, |y_i|}. However, this indicator can not provide precise accuracy for high-order QAMs.

Based on the concept that y with larger |y| provides better reliability of the angle information, we modify the lock indicator(2.96) by a conditional expectation as

p = E { (√

2 + 1)y_min− y_max||y| > T_p} . (2.97) We only count the lock indication value when |y| is larger than a predefined threshold T_p.

The selection of T_p depends on the conditions including the mean behavior of p, its variance and the percentages of the useful samples. Figs.2.21 shows the experimental results of the different thresholds. We try the thresholds from 0 to 1.4 with 0.1 step-size. Part(a) demonstrates the mean curve of the lock value. As shown, the larger thresholds yield larger peak values.

Part(b) is the variance of the lock value when θ = 0, which presents the reliability of the estimated lock value. Part(c) is the used percentages of the total samples; straightforwardly, the lock indicator with larger threshold can use fewer samples and the part(c) results supports this inference.

To choice the threshold, we should take into consideration of the decision among the peak value, the variance and useful samples. In this case, T_p = 1.3 is a considerable selection because of its large peak value, small variance and acceptable sample utility.

Dynamic Loop Bandwidth Control

The proposed dynamic loop bandwidth control is achieved via the changes of the (k_i, k_p) accord-ing to the lock value p. We consider a simple multi-step function to implement this function set.

−0.1 −0.05 0 0.05 0.1 0.15

−1

−0.5 0 0.5 1

Phase offset: θ/(2π)

Average lock value

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Threshold

Variance of lock value

(b)

−0.20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Threshold

Usage of samples

(c)

Figure 2.21: Selection of the threshold (T ) of proposed lock detector in 64QAM system with 18dB SNR. (a) Mean behavior of the lock value versus θ. (b) Variance of the lock value when θ = 0 versus T . (c) Useful percentages of the samples versus T .

Let L be the total stages of the multi-step function. The control maintains L step thresholds P and L sets of (k_i, k_p). When p value locates at the region P (l) =< p < P (l + 1), l-th parameter pair: (k_i, k_p)_l is the output. To design the coefficient pairs, we fix the damping ratio to 0.707, and decide nature frequency ω(l) in each stage.

Now, we give a design example in the 64QAM system. The function with four steps is used.

For convenience, we assume P (1) = −∞; thus, the step thresholds are P = [−∞ 0.25 0.5 0.7] and the nature frequency are ω = [0.05 0.01 0.003 0.001] which yield k_p = [0.0707 0.0141 0.0042 0.0014]

and k_i = [2.41 0.0993 0.009 0.001] × 10⁻³ accordingly. The threshold of lock indication value estimator is 1.3. We use the exponential average filter with 0.99 forgetting factor to smooth the estimation of p.

We now present the simulation studies for the proposed algorithms. The experimental set-tings are given by: SNR = 15 dB, 64QAM system, f = 0.11 normalized to sampling rate. We test four cases as follows:

• case 1, the CRL using RCPD without dynamic loop bandwidth control;

• case 2, the CRL using HPD without dynamic loop bandwidth control;

• case 3, the CRL using RCPD with dynamic loop bandwidth control;

• case 4, the CRL using HPD with dynamic loop bandwidth control.

Figs. 2.22, 2.23, 2.24 and 2.25 illustrate the simulation results of the case 1 to case 4 sequentially. First, when comparing the steady-state performances of all, the steady-state MSE of the frequency estimations are given by σ²_f = 6.52 × 10⁻⁴ for case 1, σ_f² = 1.52 × 10⁻⁵ for case 2, σ²_f = 5.06 × 10⁻¹¹ for case 3, and σ_f² = 6.55 × 10⁻¹²for case 4.

It shows the case 4 is superior than others. Considering the comparison of the HPD and RCPD, the tracking stability of HPD is much better than the RCPD as compared in case 1 and case 2, and the HPD can provide much faster stabilized transition from initial step to final step as compared in case 3 and case 4. Considering the ability of the dynamic loop bandwidth control, no matter case 3 or case 4, the CRL with dynamic loop bandwidth control has more better stability than that without dynamic loop bandwidth control in case 1 and case 2, which is demonstrated in the trajectory of the frequency estimation and its steady-state MSE. Thus, we get a brief conclusion: the CRL using HPD with dynamic loop bandwidth control provides the superiority in both acquisition ability and tracking stability.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.05 0.1

Iterations Est. freq.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

−0.1 0 0.1 0.2 0.3 0.4

Iterations Lock value

Figure 2.22: Simulated results of Case 1, the CRL using RCPD without dynamic loop bandwidth control. At top: estimated frequency normalized to sampling frequency. At bottom: lock indication value.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.05 0.1

Iterations Est. freq.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

−0.2 0 0.2 0.4 0.6

Iterations Lock value

Figure 2.23: Simulated results of Case 2, the CRL using HPD without dynamic loop bandwidth control. At top: estimated frequency normalized to sampling frequency. At bottom: lock indication value.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.05 0.1

Iterations Est. freq.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.2 0.4 0.6 0.8

Iterations Lock value

Figure 2.24: Simulated results of Case 3, the CRL using RCPD with dynamic loop bandwidth control. At top: estimated frequency normalized to sampling frequency. At bottom: lock indication value.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0

0.05 0.1

Iterations Est. freq.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

−0.2 0 0.2 0.4 0.6 0.8

Iterations Lock value

Figure 2.25: Simulated results of Case 4, the CRL using HPD with dynamic loop bandwidth control. At top: estimated frequency normalized to sampling frequency. At bottom: lock indication value.