Simulation Results and Analysis

For verification of the simulation results, note that the theoretical MSE for linear interpola-tion in AWGN is given by

MSE = E[|h − ˆh|²]

0 2 4 6 8 10 12 14

Figure 4.5: The SER curve for uncoded QPSK resulting from simulation matches the theo-retical one.

Table 4.6: MSE of Eight Data Subcarriers at Antennas 0 and 1 Carrier Index, Antenna 0 1,8 2,7 3,6 4,5

Carrier Index, Antenna 1 2,7 1,8 4,5 3,6 MSE ⁴¹₈₁σ₀^{2 29}₈₁σ₀^{2 5}₉σ₀^{2 25}₈₁σ²₀

Note also that, the MSE is different at different data subcarriers. The MSE at the eight data subcarriers in a tile are listed in Table 4.6.

The theoretical MSE and the simulation result are shown in Figure 4.6.

If we regard MSE as noise, then the equivalent SNR would be SNR/(1 + MSE). So we can get theoretical SER under AWGN channel for QPSK as

SER = 2Q

Ãr E_s 1.4321N₀

!

. (4.13)

The result is as shown in Figure 4.7. Since STC using two data subcarriers to decode. The influence of MSE on two neighbor subcarriers would be average.

0 5 10 15 20 25 30

Linear Interpolation MSE in AWGN QPSK

MSE1

Figure 4.6: MSE performance for uncoded QPSK resulting with linear interpolation, antenna 0.

Figure 4.7: SER performance for uncoded QPSK resulting from linear interpolation.

Now we derive the theoretical MSE for Wiener filtering in AWGN channel. Suppose we know the autocorrelation Φ and the cross-correlation θ. Since the channel response h = 1 in AWGN and the noise power is σ²₀, we get

The received pilot vector p, containing noise, is given by

p^T =£

1 + n₀ 1 + n₁ 1 + n₂ 1 + n₃ ¤

. (4.16)

Hence the estimation is given by

ˆh =£

The theoretical MSE in AWGN channel is thus

MSE = E

But actually we do not know the autocorrelation and the cross-correlation. In our cal-culation, we sum the six tiles (one subchannel) instead to eatimate the autocorrelation and, further, use linear interpolation to approximate the channel response at the data subcarrier

locations to estimate the cross-correlation over the six tiles. These methods cause errors.

The estimated cross-correlation is then equal to, for one tile, ˆhˆp^∗_i =

If we ignore the second-order noise terms, then ˆhˆp^∗_i ≈1 + n^∗_i +

We add up all the estimates for the six tiles to estimate θ, resulting in

θ⁰ = θ +£

δ0 δ1 δ2 δ3

¤ (4.24)

where

θ =£

Similiarly, if we ignore the second-order noise terms in the estimation of the autocorrelation matrix Φ, then the estimated quantity containing noise is given by

Φ⁰ = Φ + ∆ (4.26)

The channel estimate by Wiener filtering using estimated autocorrelation and cross-correlation is given by

ˆh = θ⁰Φ⁰⁻¹

And the MSE would be MSE =Eh

The theoretical MSE and the simulation result are shown in Figure 4.8. In the simulation, we use the average over guard band subcarriers (subcarriers 0 through 89 and 933 through

0 5 10 15 20 25 30

Figure 4.8: MSE performance of Wiener filtering channel estimation for uncoded QPSK, antenna 0. Autocorrelation and cross-correlation are obtained by averaging over one sub-channel.

1023) to estimate the noise power. In the following simulations, each data point in an average over simulation of 420000 tiles and each symbol containing ten subcannel (average over three subchannels use 378000 tiles and nine subchannel instead).

From the simulation result, we can see that the performance of Wiener filtering is worse than linear interpolation if only six tiles are used to estimate the autocorrelation and the cross-correlation. The reason should be due to noise-induced model mismatch as the au-tocorrelation and the cross-correlation are both calculated from noisy signal. If we use ten subchannels to estimate the autocorrelation and the cross-correlation, then we can get better performance. And the performance is much closer to the theory under known autocorrelation and cross-correlation. Figure 4.9 shows the MSE simulation result where ten subchannels to estimate the autocorrelation and the cross-correlation.

Figer 4.10 and 4.11 shows the SER and MSE performance under channel estimation by linear interpolation and that by Wiener filtering with averages taken over one, three, five

0 5 10 15 20 25 30

Figure 4.9: MSE performance of Wiener filtering channel estimation for uncoded QPSK, antenna 0. Autocorrelation and cross-correlation are obtained by averaging over ten sub-channels.

SER in uplink STTD mode under AWGN

Linear MMSE One MMSE Three MMSE Five MMSE Ten

Figure 4.10: Comparrision of SER performance with using Wiener filtering and linear inter-polation channel estimation in STTD under QPSK modulation in AWGN.

and ten subchannels, separately, in AWGN channel. We can see that if we use only one subchannel to average, the performance of Wiener filter is worse than linear interpolation

0 5 10 15 20 25 30

Figure 4.11: Comparrision of MSE performance with using Wiener filtering and linear inter-polation channel estimation in STTD under QPSK modulation in AWGN.

and if we choose more subchannel to average, the performance is better. The performance of using five subchannel to average is close to using ten subchannels.

In SUI channels, the antenna correlation ρenv is defined as follows: The baseband signals are modeled as two complex random processes X(t) and Y (t) with an envelope correlation coefficient of

In our simulation, we consider to two different cases, one with correlation equal to zero and the other with nonzero antenna correlation. We can see that in 2-Tx transmission with zero correlation, the slope of SER is nearly equal to −2, meaning a diversity order of 2. The presence of antenna correlation will decrease the performance.

Fig. 4.12 shows the STTD transmission performance with channel esimation by linear interpolation and that by Wiener filtering under single-path Rayleigh fading at several

dif-ferent velocities, where the antenna correlation is equal to zero. Figs. 4.13 and 4.14 are under SUI2 and SUI3 respectively. In OFDMA, the tile allocation in frequency domain is not contiguous, if choose different subchannel to average to get correlation, the performance of Wiener filtering might be different. In Fig. 4.15 we simulate two different subchannel sets, each set containing ten subchannels, and using Wiener filtering with correlation average over one subchannel. In the simulation, we see no difference at SER and MSE in two different sets of subchannel. Thus we can ignore the influence of different subchanel.

In Figs. 4.16, 4.17, and 4.18, we compare the SER with zero and nonzero antenna corre-lations. From the simulation, we can see that, since the power delay profile does not exceed the CP length, the MSEs for different power delay profiles have little difference. We also notice that at high SNR, the MSE saturates because of channel fading. Comparing linear interpolation and Wiener filtering, at low SNR, the Wiener filter has better performance if the samples averaged are enough. But in high SNR, the performance is almost the same. If there is nonzero antenna correlation, the performance would degrade.

0 5 10 15 20 25 30

MSE in UplinkPUSC QPSK channel 0 at SinglePathChan

Linear V60

SER in Uplink QPSK at Single Path Rayleigh

Linear V60 Linear V90 Linear V120

Wiener Average One subchan. V60 Wiener Average One subchan. V90 Wiener Average One subchan. V120 Wiener Average Ten subchan. V60 Wiener Average Ten subchan. V90 Wiener Average Ten subchan. V120

(b)

Figure 4.12: MSE and SER performance for uncoded QPSK under Wiener filtering and linear interpolation channel estimations at different velocities in single-path Rayleigh fading channel with ρ = 0. (a) MSE. (b) SER.

0 5 10 15 20 25 30

MSE in UplinkPUSC QPSK channel 0 at SUI−2

Linear V60

SER in Uplink QPSK at SUI−2

Linear V60 Linear V90 Linear V120

Wiener Average One subchan. V60 Wiener Average One subchan. V90 Wiener Average One subchan. V120 Wiener Average Ten subchan. V60 Wiener Average Ten subchan. V90 Wiener Average Ten subchan. V120

(b)

Figure 4.13: MSE and SER performance for uncoded QPSK under Wiener filtering and linear interpolation channel estimation at different velocities in SUI-2 channel with channel

0 5 10 15 20 25 30

MSE in UplinkPUSC QPSK channel 0 at SUI−3

Linear V60

SER in Uplink QPSK at SUI−3

Linear V60 Linear V90 Linear V120

Wiener Average One subchan. V60 Wiener Average One subchan. V90 Wiener Average One subchan. 120 Wiener Average Ten subchan. V60 Wiener Average Ten subchan. V90 Wiener Average Ten subchan. V120

(b)

Figure 4.14: MSE and SER performance for uncoded QPSK under Wiener filtering and linear interpolation channel estimation at different velocities in SUI-3 channel with channel

0 5 10 15 20 25 30

MSE in UpinkPUSC QPSK channel 0 at SUI−3

MMSEOneV60

SER in uplink STTD mode under SUI−2

V60 setA

Figure 4.15: Two different subchannel sets of MSE and SER performance for uncoded QPSK under Wiener filtering averaging over one subchannel at different velocities in SUI-2 channel with channel correlation ρ_env= 0. (a) MSE. (b) SER.

0 2 4 6 8 10 12 14 16 18 20

SER in Uplink QPSK at SinglePath with Cor=0.7

Linear V60 Linear V90 Linear V120

Wiener Average One subchan. V60 Wiener Average One subchan. V90 Wiener Average One subchan. 120 Wiener Average Ten subchan. V60 Wiener Average Ten subchan. V90 Wiener Average Ten subchan. V120 Corr Linear V60 Corr

Linear V90 Corr Linear V120 Corr

Wiener One subchan. V60 Corr Wiener One subchan. V90 Corr Wiener One subchan. 120 Corr Wiener Ten subchan. V60 Corr Wiener Ten subchan. V90 Corr Wiener Ten subchan. V120 Corr

Figure 4.16: SER comparison between zero and nonzero antenna correlation (ρ_env = 0.7) in single-path Rayleigh fading.

0 2 4 6 8 10 12 14 16 18 20

SER in Uplink QPSK at SUI−2 with Cor=0.5

Linear V60 Linear V90 Linear V120

Wiener Average One subchan. V60 Wiener Average One subchan. V90 Wiener Average One subchan. 120 Wiener Average Ten subchan. V60 Wiener Average Ten subchan. V90 Wiener Average Ten subchan. V120 Corr Linear V60 Corr

Linear V90 Corr Linear V120 Corr

Wiener One subchan. V60 Corr Wiener One subchan. V90 Corr Wiener One subchan. 120 Corr Wiener Ten subchan. V60 Corr Wiener Ten subchan. V90 Corr Wiener Ten subchan. V120 Corr

Figure 4.17: SER comparison between zero and nonzero antenna correlation (ρ_env = 0.5) in SUI-2 channel.

0 5 10 15 20 25 30

SER in Uplink QPSK at SUI−3 with Cor=0.4

Linear V60 Linear V90 Linear V120

Wiener Average One subchan. V60 Wiener Average One subchan. V90 Wiener Average One subchan. 120 Wiener Average Ten subchan. V60 Wiener Average Ten subchan. V90 Wiener Average Ten subchan. V120 Corr Linear V60 Corr

Linear V90 Corr Linear V120 Corr

Wiener One subchan. V60 Corr Wiener One subchan. V90 Corr Wiener One subchan. 120 Corr Wiener Ten subchan. V60 Corr Wiener Ten subchan. V90 Corr Wiener Ten subchan. V120 Corr

Figure 4.18: SER comparison between zero and nonzero antenna correlation (ρ_env = 0.4) in SUI-3 channel.

Chapter 5

Simulation of STC Downlink PUSC