Simulation Results
IV. Recursive algorithm
Fig 4.8: The standard deviation of q3.
IV. Recursive algorithm
In the third estimation, we figure out that the improvement in estimation accuracy is closely relate to k . And the reliability of 1 k depends on the accuracy of prior estimation. Now, 1 since we have more accurate result after the third estimation, we can proceed to improve the accuracy with the recursive algorithm described before.
In the beginning, we take k k2 and repeat the process similar to the 3rd estimation as follows:
k2 round(11 ) h
(4.17)
2
2 2
(2 )
* | | | | j Nk q i
i i k i i i k
G VV V V e
(4.18)
1
2
(2 ')
1
=
M j k q
j N
i i
g G g e g e
(4.19)
2 2
2 2 '
h q
k N k
(4.20)
The corresponding k should be 2
2
(1 1 ) 2
(11 ) 2 11 1023
( )
2 100 56
k round q N round N
q round
(4.21)
Fig 4.9: The standard deviation of ' and
2
' k
As depicted in Fig 4.9, we find that the performance is still bad when SNR is low, but it is improved for SNR5dB. This result is similar to the 3rd estimation.
The comparison between ('/k1) and ('/k2) is shown in Fig 4.10. Since the proposed method relies on the accuracy of prior estimation, the relationship between
) / ' ( k1
and ('/k2) is not meaningful in low SNR condition. Thus, we merely show the simulation result for SNR010dB. From Fig 4.10, the ratio of ('/k1) to ('/k2)
is pretty close to the expected value of 15
56. Recall that a wrong estimation occurs once the deviation is over 2 (0.176 )
2
1 o
N
, the correct probability of the 4th estimation is much better
than that of the 3rd.
Fig4.10: The standard deviation of
1
' k
and
2
' k
Finally, we present the correct probabilities in detecting the code phase with the recursive algorithm with n3,4,5,6 in Fig. 4.11. As shown in the figure, the correct probability is enhanced with the recursive algorithm. Since the recursive algorithm is based on the prior estimation, we can’t increase the correct probability even if we increase the number of iterations in low SNR cases. The result reveals that a more accurate estimation method is needed for the 1st estimation when SNR is low.
Fig. 4.11: The correct probabilities of recursive algorithm n=3,4,5,6
Chapter 5 Conclusions
In this thesis, we propose an effective method to achieve code synchronization. In Chapter 2, we introduce basic principle of the FFT method to achieve code synchronization. By performing FFT, the input sequence is projected to the phasor domain, where we get Xi. Then we compare the phase ofX with the local phasor Yi i. It is found that there is a regular phase difference within which the code phase q is embedded. Thus we define a factor Vi as the inner product of Xi and Yi, and it contains information of q. However, due to the inherent phase ambiguity, we could not interpret the code phase q directly from Vi. Furthermore, we employ the phasor concept and perform analysis to understand the statistic properties of these phasors.
In general, the sequence length N is a large number. It is known that the computation of FFT and IFFT is in the order of N log N, which means considerable computation is required when the sequence length is long. Therefore, we intend to reduce the computation by using the phase relationship between phasors. By the mathematic analysis, we find every term of Vi
contains information of q, but it can’t be used to get q directly due to inherent phase ambiguity. If we can solve the phase ambiguity problem, the computation caused by IFFT can be saved. Then, by observing the phase of inner product of Vi and Vi+1, we find it contains information of q without phase ambiguity in noiseless condition. However in noisy condition, the estimation will be strongly affected by noise, which leads to erroneous estimation. In order to improve it, we propose a simple solution to reduce error phase by using the Law of Large Number.
Because our estimation chooses the most possible phase among N legal phase, if the phase
deviation is larger than 2 N
, a wrong estimation occurs. From the simulation result, even a
tiny phase deviation may lead to incorrect value of q. Thus, we propose an improved method to further reduce error phase. By referring to the simple estimation before, we can get the approximated phase angel of Vi*
Vi+k. Likewise, when all the terms Vi*
Vi+k are summed, we can obtain a phase whose value is k multiple of q
N
2 , and it has the same noise variance as
Vi*Vi+1. Then the variance of error phase is divided by k so that a more accurate estimate is got.
Note that the idea of the improved method can be recursively operated to get more accurate estimation. However, the more estimation the more computation required, so the number of calculations in the recursive algorithm should depend on the SNR.
In Chapter 3 we also design a method to overcome the problem occurring when q is small. It is equivalent to shifting the estimated phase angle to a large value such that the multiplication factor k will not be too large. This method does not reduce noise variance, but requires additional computation. Thus, it is not a necessary process, which is applied only when the estimated phase is close to zero.
In chapter 4, we demonstrate our theory by simulation under different SNR conditions. By analyzing the performance in each step of estimation, we interpret the simulation results and try to overcome the difficulties when SNR is low. Since our method is based on the prior estimation, which means the error may propagate through the process, the only way to enhance the accuracy in low SNR is doing more estimations and costing more computation.
In the future, we may design a more accurate method for the 1st estimation, which will improve the performance of the recursive algorithm in low SNR conditions. Moreover, when SNR is known, we can determine the optimum k so as to have the least computation. This n would save much computation in high SNR conditions.
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