3.3 Simulation and Discussion
3.3.1 Monte Carlo simulation
m m
I
1 Q
-0 tan
. (3.20)
2) Initial value of SNR
After the initial phase is determined, let for 0 I(θ) and Q(θ) in Eq. (3.14).
Then Eq. (3.14) is degenerated to be a function of . Subsequently, the initial SNR can be determined by searching the minimum value with respect to in Eq. (3.14). The one-dimensional search technique, the Golden section method [40], is used to find the initial SNR, i.e. 0. Note that the accuracy of 0 is affected by the variance of 0. Hence a loose stop criterion is used in the Golden section method to find the coarse 0, and the computation time can be conserved.
3.3 Simulation and Discussion
3.3.1 Monte Carlo simulation
The Monte Carlo simulations are used to verify the joint phase and SNR estimation method, wherein each case of different SNR and N is tested for 1000 trials. Let
MHz 157 .
1
fc and fs 4.096MHz in the simulation, then p4096 according to Eq.
(2.3). Assume ˆ is the estimated phase of the j-th trial, and j is the true phase. The root mean-squared error (RMSE) of phase estimation is defined by
2
For SNR estimation, the RMSE is normalized to clarify the performance, which is denoted by
2
In the nonlinear least-square algorithm of Eq. (3.18), the stop criterion is given by 0001
.
0
. In addition, for using the Golden section method to determine the initial SNR, a coarse resolution of 0.5 dB is used for the stop criterion. The one-bit ADC is commonly used in low-power satellite applications. For potential applications on GNSS receivers, the proposed method will be simulated for SNR as low as –30 dB. On the other hand, to illustrate wide applicable range of our method, SNR up to 15 dB is also considered.
The performance of phase and SNR estimation are illustrated in Fig. 3.1 and Fig. 3.2, respectively. Note that the RMSE of SNR is normalized and denoted by percentages. In Fig.
3.1, the RMSE of the phase is improved with increasing SNR and N . The phase RMSE’s are below one degree when SNR is higher than –12 dB and –22 dB for N 105 and
106
N cases, respectively. Moreover, the accuracy of 0.1 degrees phase RMSE is achieved when SNR is higher than 12dB and –2dB in each case. According to Fig. 3.2, the SNR estimation performs well in moderate SNR. Accurate SNR information can be used to learn the achieved accuracy of phase estimation in this range. Specifically, the nRMSE is less than 10% for 21dBSNR13dB when N 105 and for 30dBSNR14dB when
Fig. 3.1. RMSE of phase estimation.
Fig. 3.2. Normalized RMSE of SNR estimation.
Note that the nRMSE attains the lower bound when SNR is around 4 dB, and increases in higher SNR. This phenomenon is investigated by plotting the mean I-Q outputs for
dB 15 SNR dB
0 as shown in Fig. 3.3. When SNR increases, the distance between neighboring curves becomes smaller and the distinction between them is little. Specifically, for SNR4dB (dotted lines), the separations between the curves are recognizable.
However, the distinction between curves becomes closer when SNR4dB (solid lines).
Consequently, when slight variance occurs in the obtained I-Q correlation outputs for SNR higher than 4 dB, the obtained Im and Qm will be fitted to I(θ) and Q(θ) with significant SNR error in the cost function of Eq. (3.14). Therefore, the performance of SNR estimation becomes degraded. Especially, when SNR10dB , the curves are nearly overlapped, and the error in SNR estimation increases rapidly.
Fig. 3.3. Mean I-Q correlation outputs for SNR between 0 to 15dB in 1dB step.
Recalling that the variance of I-Q correlation outputs is approximately inversely proportional to N , the standard deviation (STD) will decrease with N . Hence, the performance of the nonlinear least-square algorithm can be improved with N . This is consistent with the simulation results shown in Fig. 3.1 and Fig. 3.2. The results suggest that
N can be increased to compensate for the loss of amplitude information due to one-bit quantization and the desired accuracy can be achieved.
3.3.2 Range of applications
GNSS receiver: For applications using carrier phase in GNSS receivers, the signal bandwidth is assumed to be 2 MHz concerning the coarse/acquisition (C/A) code. According to the measured results of [46], when the elevation angles between 20° and 90° are of interest, SNR from –30dB to –10 dB will be considered for different applications. According to the simulation results of N 106, the phase RMSE’s of our method are 2.5545 and 0.25203 degrees for SNR of –30dB and –10 dB, respectively. For L1 carrier (1575.42 MHz), these errors correspond to 1.35 mm and 0.13 mm in length, and 1.73 mm to 0.17 mm for L2 carrier (1227.6 MHz). Such performances are close to those of high-quality receivers [44, 46]. Note that C/A code synchronization is assumed to be achieved before carrier phase estimation. In addition, the nRMSE of SNR is less than 9% when N 106 and SNR is between –30dB and –10 dB. Regarding the potential application of POD in low Earth orbit satellites (LEO), the accuracy of the orbit will be influenced by the satellite center variation, the attitude error, and the antenna phase variation, which are at the centimeter level [47]. Our achieved phase RMSE is much smaller than these variations. In addition, the phase RMSE is also insignificant with respect to the ultimate accuracy after the orbit determination algorithm reported in [47-49]. Hence, even though the proposed one-bit processing method is adopted, the reported accuracy of POD in literatures can still be achieved. Furthermore, the quality of data can be controlled by the estimated SNR. When the quality of data deteriorates owing to
the low elevation angle, the cycle slip or the multipath, the unfavorable low-SNR can be detected immediately. These data can be omitted to enhance the accuracy of POD.
Beacon receiver: For the tri-band radio beacon (150 MHz, 400 MHz and 1067 MHz) signals now on board of several LEO’s, the received power on ground is at least –140 dBm [50].
Assume the noise floor is –173 dBm/Hz, and the bandwidth is 20 KHz concerning the Doppler shift [51]. The minimum input SNR of the beacon receiver is then –10 dB. Suppose the LEO is at an altitude of 450 Km above the sea level. The variation in signal strength is approximately 15 dB. Hence, SNR’s between –10 dB and 5 dB are considered at the receiver.
When N 105, the corresponding phase RMSE of the proposed method is from 0.80765 to 0.16384 degrees for SNR between –10 dB and 5 dB. Meanwhile, the nRMSE of SNR is less than 3%. For measuring the TEC of the ionosphere in the beacon receiver, the relationship between the accumulated carrier phase and the TEC is given by [51-52]
NT
f2 2 . The term related to travelling distance in Eq. (3.23) can be eliminated by means of a differential phase technique. The phase difference measured on the frequency fr is then denoted by
When the proposed method with N 105 and SNR 10 dB is applied to TEC measurements, according to Eq. (3.24), the resulting measurement error is 8.75104 TECU (TECU: 1016electrons/m2 ). The achieved accuracy is sufficient for science requirement, i.e. 0.003 TECU [53].
From the above discussion, the use of one-bit ADC cooperated with the proposed joint phase and SNR estimation can achieve high accuracy comparable to conventional approaches.
Since one-bit signal processing is simple, fast and without AGC, the proposed approach is feasible for various high precision applications.
3.4 Summary
In this chapter, the method utilizing the nonlinear least-square algorithm to accurately estimate both the phase and SNR of sinusoidal carriers is proposed. From simulation results, the phase RMSE decreases with SNR, as shown in Fig. 3.1. The phase RMSE is less than 0.1 degrees when SNR is higher than 12dB and –2dB in the cases of N 105 and N 106, respectively. In addition, SNR estimation performs well in the middle range, as shown in Fig.
3.2. In particular, the nRMSE is less than 1% for N 106 and SNR between –11dB and 9dB. The nRMSE increases in high SNR region because of the tiny distinction between I-Q correlation outputs as shown in Fig. 3.3. Furthermore, since the STD of the I-Q correlation outputs decreases with N , the accuracy of the estimated phase and SNR can be improved by increasing the number of samples, as verified in Fig. 3.1 and Fig. 3.2. Finally, potential applications for the proposed efficient one-bit processing method in GNSS and beacon receivers have been illustrated with respect to the high accuracy of the phase estimation.
Chapter 4
Code Phase Coherence Acquisition Method
In this chapter, a computationally-efficient code phase acquisition method, termed the Phase Coherence Acquisition (PCA), is proposed. The method requires much less computation than the FFT-based acquisition to search for the cross-correlation peak between two PN sequences. This superiority becomes evident when the sequence length N is very large such that the FFT-based approach is difficult to be implemented. For instance, the acquisition of precision code (P code) with extremely long length in the GNSS system would be one of the potential applications. We describe the motivation behind our work first. We then develop our approach in the noiseless case and provide the essential idea in our development. To achieve noise-robustness, we incorporate a novel segmentation scheme and propose the PCA method. Simulation results are provided to verify the analysis and demonstrate the performance of the proposed method. Finally, the computations involved in PCA and the FFT-based method are discussed. Note that the PN sequence with length of 220 is used to demonstrate the two-layer PCA, which is a good compromise between the 1 simulation burden and performance illustration.
4.1 Motivation
The convolution theorem states that under general conditions the Fourier transform of a convolution between two sequences is the pointwise product of the Fourier transforms of these two sequences. The theorem can be represented by
[ ] [ ]
[ ] [ ]
F x n y n F x n F y n (4.1) where F denotes Fourier transform.
By applying the inverse Fourier transform F1, we have
[ ] [ ] 1 [ ] [ ]
x n y n F F x n F y n . (4.2)
In many applications, the code phase search between two sequences is usually implemented by FFT and its inverse due to the efficient computation compared with the exhaustive direct serial search method. The computation of FFT of N points involves complex multiplications and additions of order Nlog2 N . Due to the diverse need for applications and the increasing complexity of modern algorithms, a more computationally efficient method is needed when the length of a processed sequence becomes so large that implementation using the FFT method becomes difficult. Our proposal for the code phase acquisition that involves much less computation is developed as follows.