CHAPTER 5 PROPOSED TIME-VARIANT CHANNEL ESTIMATION
5.4 T IME - VARIANT CHANNEL ESTIMATION BY TIME DOMAIN WLS CHANNEL ESTIMATOR
, '
k
H -1 H
a wls
'' '' ''
ph = Q Q Q Dy
(5.24)Here, we use a simple weight scheme: the weight of a pilot is set to be 1 and the weights for decisions are smaller than one and all the same. The decision weight for different modulation schemes and different ICI levels may be different and it will be determined in the simulation chapter. The comparison of the LS and WLS channel estimator will also be shown in the simulation chapter.
5.4 Time-variant channel estimation by time domain WLS channel estimator
Figure 5-6 shows the block diagram of the complete scheme for the time-variant channel estimation. As discussed in Section 4.1, we only identify the channel taps in the non-aliasing region. The detail operation is summarized in the following procedure.
Figure 5-6 Time Domain time-variant WLS channel estimation
STEP 1: We assume that the channel taps’ positions in the nonaliasing area can be
identify.STEP 2: Use the LS algorithm discussed in Section 5.2 to estimate the parameters of the
time-variant channel.STEP 3: Construct the ICI matrix M from Equation (5.7).
STEP 4: Use the zero forcing equalizer to obtain estimate transmit symbols
(
y
=Mx + w , x M y ) and make decisions as discussed in Section 4.1.3.
ˆ = −1 Using decisions as pseudo pilots; here, we let the pilot density be 1/3 as that we did in Section 4.2.STEP 5: Assume that al1 the channel taps’ positions are known (discussed in Section
4.3).The positions of the channel taps in the the nonaliasing part
LS alogorithm to estimate these taps’ start points and slopes
by pilots
Data detection
Data detection
Weighted LS to estimate these Taps’
start points and slopes by pilots and pseudo pilots
Construct the frequency domain ICI matrix
The positions of all channel taps Construct the frequency domain
ICI matrix
STEP 6: Use WLS algorithm discussed in Section 5.3 to re-estimate all the parameters
of the channel taps.STEP 7: Construct the ICI matrix M from (5.7).
STEP 8: Use the zero forcing equalizer to re-estimate transmit symbols ( x M y ).
ˆ = −1STEP 9: Go to STEP 4 if the number of re-estimation, N
re is less than a preset numberNset.
Chapter 6
Simulation Results
In this Chapter, we conduct simulation to evaluate the performance of the proposed joint time and frequency domain channel estimator in both the time-invariant and time-variant channels. In our channel model, there are 6 channel taps (paths) and the average power of these 6 taps will be determined according to the signal-to-noise ratio (SNR) used. The bit-error-rate (BER) is used as the performance index. We consider two OFDM systems; one is the standard DVB-T system and the other is an OFDM system we define. For the former system, we use two types of channels with different delay spreads.
The maximum delay (MD) for the first one is smaller than 64Ts and that for the second one is between 171Ts and 256Ts. The channel estimate for the first-type channel in the DVB-T system will not be aliased while that for the second-type channel will be aliased. We refer the first-type channel as Channel A and the second-type of channel as Channel B. The MD of the channel used for our defined system is between 43Ts and 64Ts and the corresponding channel estimate will be aliased. We refer this type of channel as Channel C. The FFT size is set as 2048 in the DVB-T system and the FFT size is set as 512 in our defined system.
Figure 6-1 shows one example of the channel. Figure 6-2 shows the variation of the 6 taps in the fading environment.
0 50 100 150 200 250 300
Figure 6-1 An example of 6-tap channel
0 2
Figure 6-2 Variation of channel taps in fading environment
6.1 Results of channel estimation in Chapter 3
Table 6-1 shows the parameters of the 6 channel taps used in simulations. The system we consider here is the 2K-mode DVB-T system and the size of the CP is 256 (1/8 of 2048).
Modulation schemes such as QPSK, 16QAM and 64QAM are used. The delays of the channel taps are randomly changed for every 7 OFDM symbols. For a block of consecutive 7 OFDM symbols, they remain the same.
Tap Number Average Power (Lin) Average Power (dB)
1 0.9951 -0.0215
2 0.5727 0.2.421
3 0.4992 -3.0172
4 0.4273 -3.6927
5 0.3558 -4.4880
6 0.2845 -5.4595
Table 6-1 Parameters of multipath fading channel
6.1.1 Results of different interpolation methods
Figure 6-3 shows the performance comparison for different interpolation methods with Channel A. The pilot density here is 1/3 and only the channel response in the frequency domain is estimation. The modulation used is QPSK. Figure 6-4 shows the performance comparison for Channel B. Notice that the positions of the channel taps in a block of 7 OFDM symbols are the same such that the two-dimensional interpolation method can be effectively applied. In the figures, “1D linear” indicates the one-dimensional linear interpolation method, “1D cubic” the one-dimensional cubic interpolation method, “2D linear & linear” the two-dimensional linear interpolation method (linear interpolation both
interpolation method with the linearly interpolation in the temporal domain and cubic interpolation in the frequency domain, “2D cubic & cubic” the two-dimensional cubic interpolation method (cubic interpolation both in the temporal and frequency domains).
We can see from Figure 6-3 and Figure 6-4 that two-dimensional interpolation methods are better than one-dimensional interpolation methods. And the cubic interpolation is better than linear interpolation. Also, the performance of “2D linear & cubic” is similar to “2D cubic & cubic”. This is because the channel taps’ positions in a block of 7 OFDM symbols are the same and the variation of channel among the symbols is small. And since the complexity of operation of linear interpolation is lower, we can use “2D linear & cubic” as the interpolation scheme in the joint time and frequency domain channel estimation methods described in Chapter 3.
5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
BER
1D linear 1D cubic
2D linear & linear 2D linear & cubic 2D cubic & cubic Perfect CH
Figure 6-3 Comparison of different interpolation methods (Chanel A)
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
1D linear 1D cubic
2D linear & linear 2D linear & cubic 2D cubic & cubic Perfect CH
Figure 6-4 Comparison of different interpolation methods (Channel B)
6.1.2 Results of joint time and frequency domain channel estimation
Figure 6-5 and Figure 6-6 show the performance comparison for the algorithm depicted in Figure 3-9 and Figure 3-10. The modulation scheme is also QPSK. For benchmarking, the performance of the ideal channel is also shown in the Figures.
We can see from Figure 6-5 and Figure 6-6 that the performances of the joint time/frequency domain channel estimation method is good under QPSK, and the performance is improved along with the number of iteration. We also can see that the performance with 6 and 10 iteration is almost the same. This shows that 6 iterations will be sufficient. Figure 6-7 and 6-8 show the performance comparison for QPSK, 16QAM, and 64QAM. These figures show that the performance of the channel estimation is also good with 16QAM and 64QAM.
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-5 Performance of the joint time/frequency channel estimate (QPSK, Channel A)
5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-6 Performance of the joint time/frequency channel estimate (QPSK, Channel B)
5 10 15 20 25 30
Figure 6-7 Performance of the joint time/frequency channel estimate for QPSK, 16QAM, 64QAM (Channel A)
Figure 6-8 Performance of the joint time/frequency channel estimate for QPSK, 16QAM, 64QAM (Channel B)
6.2 Results of channel estimation in Chapter 4
The channel model is the same as the one shown in Table 6-1. The only difference is that the tap positions are changed for each OFDM symbol, and the delays of the channel taps in the aliasing area are set random (one or two taps). The simulation setup is also the same as that in Section 6.1. The channel estimation method is that depicted in Figure 4-5.
6.2.1 Comparison for the choice of pseudo pilots
We have discussed the choice of pseudo pilots in Section 4.3. Figure 6-9 shows the BER performance for the different choices for pseudo pilots. As defined, pseudo pilots are obtained from decisions and they can be erroneous. Figure 6-10 shows the SER of the pseudo pilots used. Figure 6-11 and 6-12 are similar to those in Figure 6-9 and 6-10 except for that Channel B is used. Here, guard band insertion is not conducted.
We can see from Figure 6-9 and Figure 6-11 that the best choice for the addition pilots is to use all detected data as the pseudo pilots. If one out three detected data is used as pseudo pilots, its performance is worse than the previous case. However, in order to compare with the performance of the conventional joint time/frequency domain channel estimator in Chapter 3, we will use the scheme with one out of three detected data as the pseudo pilots. In the simulations below, we will use the setting for simulations.
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers Perfect CH
Figure 6-9 BER comparison of different pseudo pilot selection schemes (without guard band insertion, Channel A)
5 10 15 20 25 30
10-3 10-2 10-1 100
SNR(dB)
Symbol Error Rate
Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers
Figure 6-10 SER of pseudo pilots for different pseudo pilot selection schemes (without guard band insertion, Channel A)
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers Perfect CH
Figure 6-11 BER comparison of different pseudo pilot selection schemes (without guard band insertion, Channel B)
5 10 15 20 25 30
10-2 10-1 100
SNR(dB)
Symbol Error Rate
Sample every 1 subcarrier Sample every 3 subcarriers Sample every 4 subcarriers Sample every 6 subcarriers
Figure 6-12 SER of pseudo pilots for different pseudo pilot selection schemes (without guard band insertion, Channel B)
guard band insertion. We can see that under the same number of iterations, the performance with the guard band insertion is much better than that without. Notice that we conduct the guard band insertion only when the channel response is completely estimated.
5 10 15 20 25 30
Nre=0 & no insertion Nre=0 & with insertion Nre=1 & no insertion Nre=1 & with insertion Nre=6 & no insertion Nre=6 & with insertion Nre=10 & no insertion Nre=10 & with insertion Perfect CH
Figure 6-13 Performance comparison for the channel estimator with/without guard band insertion (Channel A)
6.2.2 The Results of iterative channel estimation with pilots and pseudo pilots
Figure 6-14 and Figure 6-15 show the performance of the proposed estimator for different number of iterations. The modulation scheme is QPSK. Here, guard band insertion is conducted and the data in the original pilots are used in the SIC-LS algorithm. In other words, the pseudo pilots are only used to help locate channel taps. As we can see, the performance is satisfactory when the number of iteration is 6. Also, its performance is almost as good as that when the number of iteration is 10.
5 10 15 20 25 30 10-4
10-3 10-2 10-1
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-14 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses pilots, Channel A)
5 10 15 20 25 30
10-4 10-3 10-2 10-1
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-15 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses pilots, Channel B)
method with QPSK, 16QAM, 64QAM modulation. The number of iteration is set as 6. As we can see, for each case the performance is almost as good as the perfect channel.
5 10 15 20 25 30
Figure 6-16 Performance comparison for the channel estimator with QPSK, 16QAM, and 64QAM (SIC-LS uses pilots, Channel A)
5 10 15 20 25 30
Figure 6-17 Performance comparison for the channel estimator with QPSK, 16QAM, and 64QAM (SIC-LS uses pilots, Channel B)
Figure 6-18 and Figure 6-19 show the performance of the proposed estimator when
both pilots and pseudo pilots are used in the SIC-LS algorithm. The performances with one, 6 and 10 iterations are almost the same and it is also the same with that of the perfect channel. We also see that the performance is similar to that in Figure 6-14 and Figure 6-15.
5 10 15 20 25 30
Figure 6-18 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses original and pseudo pilots, Channel A)
5 10 15 20 25 30
Figure 6-19 Performance comparison for the channel estimator with different numbers of
Figure 6-20 and Figure 6-21 show the performance of the proposed estimator when all the detected data are taken as pseudo pilots. Compared to Figure 6-18 and Figure 6-19 (only one out of three is used as a pseudo pilot), the performance here is better. In Figure 6-20, the performance is still satisfactory even when there is no iteration. In Figure 6-21, only one iteration is sufficient to obtain good performance.
5 10 15 20 25 30
10-4 10-3 10-2 10-1
SNR(dB)
BER
Nre=0 Nre=1 Nre=3 Nre=6 Nre=10 Perfect CH
Figure 6-20 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses original pilots and all decisions, Channel A)
5 10 15 20 25 30 10-4
10-3 10-2 10-1
SNR(dB)
BER
Nre=0 Nre=1 Nre=3 Nre=6 Nre=10 Perfect CH
Figure 6-21 Performance comparison for the channel estimator with different numbers of iteration (SIC-LS uses original pilots and all decisions, Channel B)
6.3 Results of WLS algorithm
In this section, we report simulation results for the system we have defined. As mentioned, the number of subcarriers and the FFT size is 512 and the pilot density is 1/12. The CP size here is 64 (1/8 of 512) and the guard band size is also 64. The number of pilots here is 39.
Modulation schemes QPSK, 16QAM and 64QAM are used in our simulations. The delays of the channel taps are also set random for different OFDM symbols.
Figure 6-22 shows the performance of the proposed channel estimation (without weighting). The modulation scheme is QPSK. Here one out of three decisions is used as a pseudo pilot, and the SIC-LS method uses pilots only. As we can see, the performance is far from satisfactory even the number of iteration is large. This is because the number of pilots is not large enough such that the SIC-LS method cannot function properly.
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-22 Comparison of the proposed channel estimator (no weighting, SIC-LS uses pilots, Channel C)
Figure 6-23 shows the performance of the proposed channel estimation (without
weighting) for the QPSK, 16QAM, 64QAM modulation. Different from that in Figure 6-22, the SIC-LS method now uses the original and pseudo pilots. It is apparent that the
performance has been improved. However, for 16QAM and 64QAM, the performance in high SNR areas seems less satisfactory. Unlike the DVB-T system, the number of pseudo pilots is not large. As a result, erroneous decisions will affect the estimation performance more seriously.
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
BPSK BPSK perfect QPSK QPSK perfect 16QAM 16QAM perfect 64QAM 64QAM perfect
Figure 6-23 Performance of the proposed channel estimator (no weighting, QPSK, 16QAM, and 64QAM, Channel C)
6.3.1 Results of the weighted LS algorithm by pilots and pseudo pilots
Figure 6-24 shows the performance of the proposed channel estimation method with the WLS algorithm. The channel estimation method is that depicted in Figure 4-9. Here, the number of iteration is set as 6, one out of 3 decisions is used as a pseudo pilot, and the modulation scheme is BPSK. Here, we apply the first weighting scheme as described in Section 4.4. The weight of each decision regions is shown in Table 6-2 and the weights of pilots are all 1. In the table,
x
represents the estimated transmitted signal. We have tried some combinations with different regions and different weights. The weight values in Table 6-2 are obtained by trial-and-error. As we can see, the performance in Figure 6-24 is somewhat degraded in high SNR region.Region
Table 6-2 The weights of the first weighting method
5 10 15 20 25 30
Figure 6-24 Performance of proposed channel estimation method with WLS (the first weighting method, Channel C)
Figure 6-25 shows the performance of the proposed channel estimation method with the WLS algorithm. Here, the second weighting method is applied. The weight is determined by the SNR of each subcarrier as discussed in Section 4.4. Table 6-3 shows the weights. In the table,
x
represents the SNR(dB) of each subcarrier and the weight for a pilot is 1. The weight values in Table 6-3 are also obtained with trial-and-error. As we can see, the performance is improved with the second weighting scheme.Region x≤ −5 − < ≤5 x 0 0< ≤x 5 5< <x 10 10 x≤
Weight 0.05 0.1 0.5 0.7 1
Table 6-3 The weights of the second weighting method
5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
BER
The first weighting method The second weighting method Perfect CH
Figure 6-25 Performance of proposed channel estimation method with WLS (the second weighting method, Channel C)
We now conduct simulations to evaluate the performance of the proposed estimator with the WLS algorithm for QPSK, 16QAM, 64QAM. The weights for the WLS algorithm are shown in Table 6-4, Table 6-5 and Table 6-6. Similar to the previous cases, the weights of pilots are all 1.
Region x≤0 0< ≤x 5 5< ≤x 8 8< <x 10 10 x≤
Weight 0.1 0.35 0.7 0.8 1
Table 6-4 Weights used for QPSK
Region x≤0 0< ≤x 10 10< ≤x 15 15< <x 20 20 x≤
Weight 0.05 0.3 0.85 0.9 1
Table 6-5 Weights used for 16QAM
Regio n
0
x≤ 0< ≤x 10 10< ≤x 15 15< ≤x 20 20< ≤x 25 25< <x 30 30 x≤
Weight 0.05 0.3 0.4 0.45 0.55 0.6 0.8
Table 6-6 Weights used for 64QAM
Figure 6-26 shows the performance comparison. From the figure, we see that the performance of the proposed WLS channel estimator for BPSK and QPSK is very close to the optimum, that for 16QAM is improved (compared with the one without weighting), and that for 64QAM is not improved (compared with the one without weighting).
5 10 15 20 25 30
Figure 6-26 Performance of proposed channel estimation method with WLS for BPSK, QPSK,
16QAM, 64QAM (Channel C)
Figure 6-27 shows the performance comparison for the proposed channel estimators with LS and WLS methods for BPSK, QPSK, 16QAM, and 64QAM.
5 10 15 20 25 30
Figure 6-27 Performance comparison for proposed channel estimators with LS and WLS methods (BPSK, QPSK, 16QAM, 64QAM, Channel C)
In order to further improve the performance of the proposed channel estimator with the WLS method, we use all the decisions as the pseudo pilots. Figure 6-28 and Figure 6-29 show the performance of the proposed channel estimator with the WLS method for 16QAM and 64QAM. The weights used in the simulation f or16QAM and 64QAM are the same as those in Table 6-5 and 6-6. We can see that the performance is almost as good as the optimum.
5 10 15 20 25 30 10-3
10-2 10-1 100
SNR(dB)
BER
16QAM original 16QAM weighted 16QAM perfect CH
Figure 6-28 Performance of proposed channel estimators with WLS method for 16QAM (all decisions are used, Channel C)
5 10 15 20 25 30
10-3 10-2 10-1 100
SNR(dB)
BER
64QAM Original 64QAM Weighted 64QAM Perfect CH
Figure 6-29 Performance of proposed channel estimators with WLS method for 64QAM (all decisions are used, Channel C)
6.3.2 The Weighted LS channel estimation with different aliasing power
In this Section, we would like to test how strong the aliasing power we can tolerate for the proposed channel estimator with the WLS method in the DVB-T system. The parameters of the DVB-T system are the same as those in Section 6.2. For the WLS method, all decisions are used as pilots. The power profiles of the first 4 taps are the same as those in Table 6-1. We let the last two taps be located in the aliasing area. Table 6-7, Table 6-8, Table 6-9 are the channel power profiles used for QPSK, 16QAM, and 64QAM. For the QPSK scheme, the total power is 4.1955 (2.4882+1.7073) and the maximum power we can use for the last two taps is 1.7073 (40.7% of the total power). For the 16QAM scheme, the total power is 3.8399 (2.4882+1.3517) and the maximum power we can use for the last two taps is 1.3517 (35.2% of the total power). Finally, for the 64QAM scheme, the total power is 3.413 (2.4882+0.9248) and the maximum power we can use for the last two taps is 0.9248 (27.1% of the total power). Figure 6-30, Figure 6-31 and Figure 6-32 show the simulation results.
Aliasing or not
Tap
Number Average Power (Lin) Total Power(Lin) 1 0.9951
(0.593 of the total power)
5 0.8539 Aliasing
6 0.8534
1.7073
(0.407 of the total power) Table 6-7 The channel taps power for QPSK scheme
Aliasing or not
Tap
Number Average Power (Lin) Total Power(Lin) 1 0.9951
(0.648 of the total power)
5 0.7116 Aliasing
6 0.6401
1.3517
(0.352 of the total power) Table 6-8 The channel taps power for 16QAM scheme
Aliasing or not
Tap
Number Average Power (Lin) Total Power(Lin) 1 0.9951
(0.729 of the total power)
5 0.4981 Aliasing
6 0.4267
0.9248
(0.271 of the total power) Table 6-9 The channel taps power for 64QAM scheme
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-30 Performance of proposed channel estimators with WLS method for QPSK (all decisions are used, DVB-T, Channel B)
5 10 15 20 25 30
10-3 10-2 10-1 100
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-31 Performance of proposed channel estimators with WLS method for 16QAM (all decisions are used, DVB-T, Channel B)
5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
Nre=0 Nre=1 Nre=6 Nre=10 Perfect CH
Figure 6-32 Performance of proposed channel estimators with WLS method for 64QAM (all decisions are used, DVB-T, Channel B)
6.4 Results of proposed time-variant channel estimation in Chapter 5
In this section, we evaluate the performances of the proposed time-variant channel estimation method. The simulations are conducted for the 2K-mode DVB-T system with the CP size of 256 (1/8 of 2048). Also, modulation schemes QPSK, 16QAM and 64QAM are used. The delays of the channel taps are still randomly set (with one tap or two taps in the aliasing area), and the tap positions are changed for different OFDM symbols. In this section, we will consider two cases; one is for the normalized Doppler frequency of 0.0244, and the other is for the normalized Doppler frequency of 0.1016. For all simulations, we assume that the tap positions are known. Also, one out of three decisions is used as a pseudo pilot.
6.4.1 Results of proposed time-variant channel estimation with normalized Doppler frequency of 0.0244
The channel power profile is shown in Table 6-10. The time-variant channel is generated with Jake’s model and the normalized Doppler frequency is set to 0.0244.
Tap Number Average Power (Lin) Average Power (dB)
1 0.953 -0.2091
2 1.086 0.3583
3 0.9873 -0.0555
4 0.6999 -1.5496
5 0.5139 -2.8912
6 0.3728 -4.2852
Table 6-10 Channel tap power profile
Figure 6-33 and 6-34 show the performance of the proposed time-variant channel
see that the performance is slightly worse than the optimum.
5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
BER
Estimated the taps in the nonaliasing area by pilots Estimated all taps by pilots
Perfect CH
Figure 6-33 Performance of proposed time-variant channel estimator for QPSK (SIC-LS uses
Figure 6-33 Performance of proposed time-variant channel estimator for QPSK (SIC-LS uses