Integer CFO Estimation and Preamble Index Identification

Now we assume that the symbol timing and fractional CFO offset are perfect estimated and compensated. The maximum search range for integer CFO is set to ±12 ∆f , and we let the preamble index be 31. We present results for the following five joint detection methods: the correlation method, the differential method, the reduction method, the early dropping method, and the hardlimited differential method. The threshold fraction in the early dropping method is set to 1/2.

Figure 5.16 shows the simulation results on detection error probability under Vehicular A channel with FFT size 1024, where “error” means incorrect identification of the integer CFO or the preamble index or both. It is observed in the figure that the differential method gives the best performance in error probability. This is followed by the hardlimited dif-ferential method, the early dropping method, the reduction method, and the correlation

−5 0 5 10 15 20 25

Figure 5.12: Error distributions of data-aided method and modified data-aided method during normal synchronization.

method, in that order. For the early dropping method, larger window yields lower error probabilities. For the reduction method, we can find in the figure that the the performance of reduction with differential method is almost the same as that of reduction with early dropping method. We may conclude that the coarse integer CFO estimation affects the error probability significantly from this simulation result.

Figure 5.17 shows the simulation results under SUI3 channel with FFT size 1024.

Again, the differential method performs best, and the error probability for correlation method is very high. But it is observed that the probability of early dropping method with window size 20 is lower than that of hardlimited differential method under higher SNR.

Figure 5.18 shows the simulation results under SUI3 channel with FFT size 2048.

Comparing it with with FFT size 1024, we see that the curves with the correlation and reduction methods are very close, but the performance of the differential method is some-what better. This is because we do more correlations for FFT size 2048.

Now we consider the computational complexity for each method. Note that all the data we present in the following are the cases with FFT size 1024 under Vehicular A

0 5 10 15 20

8x 10⁻³ RMSE of fractional CFO synchronization under AWGN

SNR (dB)

RMSE (number of subcarrier spacings)

Initial Sync Normal Sync

Figure 5.13: RMSE of fractional CFO under AWGN channel.

channel. First of all we consider the computational complexity of the correlation method.

Since the maximum search range of integer CFO is ±12 ∆f , it needs (12/3)×2×4×(2×

12 + 1) = 800 multiplications to find the coarse integer CFO [30]. Then, if multiplication by a signed binary number is also counted a multiplication operation, the amount of real multiplications to find the preamble index is equal to 114 × (284 + 1) = 32490. In total, it needs 800 + 32490 = 33290 multiplications for the correlation method.

For the differential method and the early dropping method, we need to find the carrier-set used for the estimation and derive the differential sequence from the carrier-carrier-set first. It needs 293 × 2 × 3 + 292 × 2 = 2342 multiplications for both methods to do these. Due to the previously described preamble structure, we need to search among 1 + (12/3) × 2 = 9 different integer CFO conditions for each preamble index, for a total 114 × 9 possible CFO-preamble combinations. The amount of multiplications is equal to 2342 + 283 × 9 × 114 = 99880 for the differential method. For the early dropping method, the amount of multiplications is related to the window size, the threshold, and the iteration numbers.

We compute the number of multiplications in our simulation and average them. We can find that the amounts of multiplications are equal to 2342 + 11535 = 13877 and 2342 +

21572 = 23914, respectively, for the early dropping method with window sizes 10 and 20.

For the hardlimited differential method, we need to find the carrier-set first and it costs 293 × 2 × 3 = 1758 multiplications. Then we need 292 × 2 = 584 logic operations to derive the hardlimited differential sequence and 283 × 9 × 114 = 290358 logic operations to find the integer CFO and preamble index. Therefore, it needs 1758 multiplications and 584 + 290358 = 290942 logic operations in total.

For the reduction method, we first find the coarse integer CFO and it costs (12/3) × 2 × 4 × (2 × 12 + 1) = 800 multiplications. Then we assume the smaller search range for integer CFO is ±3 ∆f ; so we need to search among 1 + (3/3) × 2 = 3 different integer CFO conditions for each preamble index. It needs 287 × 2 × 3 + 286 × 2 = 2294 to find the carrier-set and obtain the differential sequence. The total amount of multiplications is equal to 800+2294+283×3×114 = 99880 for the reduction method with full differential search. And it costs totally 800 + 2294 + 7377 = 10471 multiplications for the reduction method with early dropping and window size 10. Note that 3812 is a average result in our simulation.

Table 5.2 lists the amounts of multiplications required for all methods under different FFT sizes. Note that the amount of multiplications for the early dropping method needs to be obtained through simulations. As we only simulate the cases with FFT sizes 1024 and 2048, the amounts of multiplications for the early dropping method with FFT sizes 128 and 512 are not available. Complexity of the other methods under FFT sizes 128 and 512 can be obtained by calculation.

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RMSE of fractional CFO synchronization under SUI3 (Initial Sync)

SNR (dB)

RMSE (number of subcarrier spacings)

0km/hr

RMSE of fractional CFO synchronization under SUI3 (Normal Sync)

SNR (dB)

RMSE (number of subcarrier spacings)

0km/hr 60km/hr 120km/hr

Figure 5.14: RMSE of fractional CFO under SUI3 channel.

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10⁻² 10⁻¹ 10⁰

Fractional CFO sync error distribution under SUI3 ,Speed=120km/hr (Initial Sync)

SNR (dB)

Probability

|error|>0.5% subcarrier spacing

|error|>1% subcarrier spacing

|error|>2% subcarrier spacing

0 5 10 15 20

10⁻³ 10⁻² 10⁻¹ 10⁰

Fractional CFO sync error distribution under SUI3 ,Speed=120km/hr (Normal Sync)

SNR (dB)

Probability

|error|>0.5% subcarrier spacing

|error|>1% subcarrier spacing

|error|>2% subcarrier spacing

Figure 5.15: Fractional CFO synchronization error distribution under different SNRs.

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Error Rate of Preamble Index/Integer CFO Estimation under Vehicular A channel

Differential Method

Figure 5.16: Error probability in either the identified preamble index or the estimated integer CFO under Vehicular A channel, where FFT size = 1024.

0 5 10 15 20

Error Rate of Preamble Index/Integer CFO Estimation under SUI3

Differential Method

Figure 5.17: Error probability in either the identified preamble index or the estimated integer CFO under SUI3 channel, where FFT size = 1024.

0 5 10 15 20

Error Rate of Preamble Index/Integer CFO Estimation under Vehicular A channel

Differential Method

Figure 5.18: Error probability in either the identified preamble index or the estimated integer CFO under Vehicular A channel, where FFT size = 2048.

Table 5.2: Computational Complexity for Integer CFO Estimation and Preamble Index Identification

Multiplications

FFT Size 128 512 1024 2048

Correlation method 5018 17216 33290 65666

Differential method 36268 146906 292700 586356

Hardlimited differential 270+ 912+ 1758+ 3462+

35998 (logic) 145994 (logic) 290942 (logic) 582894 (logic)

Reduction method 13080 50530 99880 199280

Reduction

with window size=20 N.A. N.A. 10471 13551

Early drop.

with window size=10 N.A. N.A. 13877 16117

Early drop.

with window size=20 N.A. N.A. 23914 26056

Chapter 6