Chapter 3 Single-Mode Limited Feedback Precoding for MISO
3.3 Computer Simulations
In IEEE 802.16-2005 standard [25], all codebooks can be divided into two kinds:
3-bit codebooks and 6-bit codebooks. Under a fixed average feedback rate, two modes are defined as follows: (1) Mode I is defined in which the transmitter is precoded with the 6-bit beamformer and the receiver feeds back six bits per two frames; (2) Mode II is defined in which the transmitter is precoded with the 3-bit beamformer and the receiver feeds back three bits per frame. The schemes of Mode I and Mode II are illustrated in Figure 3.3.
Figure 3.3: Transmit beamforming schemes of Mode I and Mode II
In the following, we illustrate the performances of the single-mode limited feedback MISO system. Besides, we also consider a MISO-OFDM system in the simulations and illustrate the performances of the single-mode limited feedback MISO-OFDM system. Under a fixed average feedback rate at three bits per frame or per OFDM symbol period, the schemes of Mode I and Mode II for single-carrier systems and multi-carrier systems are commented in Table 3.4.
Table 3.4: Schemes of Mode I and Mode II for single-carrier and multi-carrier systems Fixed average feedback rate at 3 bits/frame (or OFDM symbol period)
Single-Carrier (MISO) Multi-Carrier (MISO-OFDM) Mode I feed back 6 bits per two frames feed back 6 bits per tone and per two
OFDM symbol periods
Mode II feed back 3 bits per frame feed back 3 bits per tone and per OFDM symbol period
(a) Single-mode limited feedback MISO system
Table 3.5 lists all parameters used in this simulation. This simulation uses 16-QAM modulation and one data substream on a 4× wireless system. The transmit 1 beamforming codebooks in IEEE 802.16-2005 are applied in the simulation. The average feedback rate is fixed at three bits per frame. In Mode I, the receiver feeds back six bits to the transmitter per two frames; in Mode II, the receiver feeds back three bits to the transmitter per frame. Each frame consists of 128 symbols and each transmission consists of 16 frames.
The channel is assumed to be quasi-static and Rayleigh fading. The channel gains of two consecutive frames follow the first-order Markov channel model (Equation (3.3)). The elements of the random perturbation Δh in Equation (3.3) are assumed to be independent identically distributed (i.i.d.) complex Gaussian random variables with zero-mean and variance (1−ρ). SNR is defined as the ratio of the average received signal power to the average received noise power. Perfect channel knowledge known at the receiver and the error-free feedback channel are also assumed in the simulation.
SER performances of the single-mode limited feedback MISO system (operating in Mode I or Mode II) with ρ =1, 0.9, 0.8, 0.7 are shown in Figures 3.4-3.7.
Table 3.5: Simulation environment of the single-mode limited feedback MISO system
Parameter Values
Channel Rayleigh fading channel
(First-order Markov channel model)
Modulation 16-QAM
Number of transmit antenna 4
Number of receive antenna 1
Fixed average feedback rate 3 bits per frame
Frame length 128 symbols
Number of frames 16 frames per transmission
Codebook Transmit beamforming codebooks in IEEE 802.16-2005
When ρ = , Figure 3.4 illustrates that the SER performance of Mode I (6-bit 1 beamformer) is better than that of Mode II (3-bit beamformer). The channel does not vary in the two consecutive frame periods. In Mode I, the optimal beamformer can be selected from the 6-bit codebook (sixty-four beamformers) at the start of the first frame of the two consecutive frames, and it is also optimal for the channel of the second frame. In Mode II, the optimal beamformer is selected from the 3-bit codebook (eight beamformers) at the start of the first frame, and it can be reselected at the start of the second frame. Because the optimal beamformer selected from the 6-bit codebook is closer to the unquantized optimal beamformer than that selected from the 3-bit codebook, the SER performance of Mode I is better than that of Mode II.
0 2 4 6 8 10 12 14 16 10-4
10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO System, R=4, ρ=1
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.4: SER performance of the single-mode 4× MISO system with 1 ρ = 1
When ρ =0.9, Figure 3.5 illustrates that the SER performance of Mode I becomes slightly poorer than that in Figure 3.4. The curve of Mode I crosses the curve of Mode II at SNR=14 dB. When ρ =0.8 , Figure 3.6 shows that the SER performance of Mode I becomes poorer than that in Figure 3.5. The curve of Mode I crosses the curve of Mode II at SNR=10 dB. When ρ =0.7, Figure 3.7 shows that the SER performance of Mode II is better than that of Mode I.
The reason that the SER performance of Mode I becomes poorer as ρ decreases is that the beamformer which is optimal for the channel of the k th frame is not always optimal for that of the (k +1)th frame. The nonoptimal beamformer decreases the average received power gain over the (k +1)th frame. The single-mode system operating in Mode I loses diversity gain as ρ decreases.
0 2 4 6 8 10 12 14 16 10-4
10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO System, R=4, ρ=0.9
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.5: SER performance of the single-mode 4× MISO system with 1 ρ =0.9
0 2 4 6 8 10 12 14 16
10-4 10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO System, R=4, ρ=0.8
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.6: SER performance of the single-mode 4× MISO system with 1 ρ =0.8
0 2 4 6 8 10 12 14 16 10-4
10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO System, R=4, ρ=0.7
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.7: SER performance of the single-mode 4× MISO system with 1 ρ =0.7
Figure 3.8 shows the SER performance comparison between Mode I and Mode II under a fixed SNR=12 dB. The two lines cross at about ρ = 0.85. Figure 3.8 shows that when the channel is subject to high temporal correlation (ρ ≥0.85), Mode I outperforms Mode II whereas with ρ <0.85 the latter yields better performance. It is obviously observed that Mode I is suitable to be used in the high temporal-correlation channel and Mode II is suitable to be used in the low temporal-correlation channel.
From the simulation results, we observed that the single-mode limited feedback system (either Mode I or Mode II) is not robust enough in mobile environments.
0 0.2 0.4 0.6 0.8 1 10-3
10-2 10-1
ρ
SER
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
ρ =0.85
Figure 3.8: SER performance comparison between Mode I and Mode II for 4× 1 MISO system
(b) Single-mode limited feedback MISO-OFDM system
We consider a MISO-OFDM system and Table 3.6 lists all parameters used in this simulation. This simulation uses 16-QAM modulation and 128 subcarriers in the MISO-OFDM system. The multipath Rayleigh fading channel is used in the simulation.
The channel gains of two consecutive OFDM symbol periods follow the first-order Markov channel model (Equation (3.3)), and the relative delays and the average power follow the parameters of the channel model defined in ITU-R M.1225 [37]. The elements of the random perturbation Δh in Equation (3.3) are assumed to be i.i.d.
zero-mean complex Gaussian with variance (1−ρ). SNR is defined as the ratio of the average received signal power to the average received noise power after DFT. Perfect channel knowledge known at the receiver and the error-free feedback channel are also assumed in the simulation.
The parameters of the simulation environment of the MISO-OFDM system follow IEEE 802.16-2005 [25] and are stated as follows: The FFT length is 128. The bandwidth is 2.5MHz and the sampling frequency is 2.8 MHz. The sampling time is 357.14 ns and the subcarrier spacing is 21.875 kHz. The ratio of CP time is 1/4. The OFDM symbol time is 57.143 us. Each frame consists of 16 OFDM symbols.
The transmit beamforming codebooks in IEEE 802.16-2005 are applied in the simulation. The average feedback rate is fixed at three bits per tone and per OFDM symbol period. In Mode I, the receiver feeds back six bits to the transmitter per tone and per two OFDM symbol periods; in Mode II, the receiver feeds back three bits to the transmitter per tone and per OFDM symbol period. The transmit symbols are precoded with the beamformers tone by tone and we assume that all subcarriers are independent. SER performances of the single-mode limited feedback MISO-OFDM systems with ρ =1, 0.9, 0.8, 0.7 are shown in Figures 3.9-3.12.
Table 3.6: Simulation environment of the single-mode limited feedback MISO-OFDM system
Parameter Values Channel Multipath Rayleigh fading channel
(First-order Markov channel model)
Tap (ITU-R M.1225) 1 2 3 4 5 6
Relative delay (ns) 0 310 710 1090 1730 2510
Average power (dB) 0 -1 -9 -10 -15 -20
Bandwidth 2.5 MHz
FFT length 128
Sampling frequency 2.8 MHz
Subcarrier spacing 21.875 kHz
Useful symbol time 45.714 us
Ratio of CP time 1/4
CP time 11.429 us
Symbol time 57.143 us
Sampling time 357.14 ns
Frame length 16 OFDM symbols
Modulation 16-QAM
Number of transmit antenna 4
Number of receive antenna 1
Fixed average feedback rate 3 bits per OFDM symbol period Codebook Transmit beamforming codebooks in IEEE
802.16-2005
0 2 4 6 8 10 12 14 16 10-4
10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO-OFDM System, R=4, ρ=1
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.9: SER performance of the single-mode 4× MISO-OFDM system 1 with ρ = 1
When ρ = , Figure 3.9 illustrates that the SER performance of Mode I is also 1 better than that of Mode II in the MISO-OFDM system. The channel does not vary in the two consecutive OFDM symbol periods. In Mode I, the optimal beamformers for all subcarriers can be selected from the 6-bit codebook at the start of the first OFDM symbol of the two consecutive OFDM symbols, and they are also optimal for the channels of the second OFDM symbol time. In Mode II, the optimal beamformers are selected from the 3-bit codebook at the start of the first OFDM symbol, and they will be reselected at the start of the second OFDM symbol. Because the optimal beamformers selected from the 6-bit codebook are closer to the unquantized optimal beamformers for all subcarriers than that selected from the 3-bit codebook, the SER performance of Mode I is always better than that of Mode II.
0 2 4 6 8 10 12 14 16 10-4
10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO-OFDM System, R=4, ρ=0.9
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.10: SER performance of the single-mode 4× MISO-OFDM system 1 with ρ =0.9
When ρ =0.9, Figure 3.10 illustrates that the SER performance of Mode I also becomes slightly poorer than that in Figure 3.9. Two lines cross at SNR=14 dB. When ρ =0.8, Figure 3.11 shows that the SER performance of Mode I becomes poorer than that in Figure 3.10. The curve of Mode I crosses the curve of Mode II at SNR=10 dB.
When ρ =0.7, Figure 3.12 shows that the SER performance of Mode II is better than that of Mode I.
The reason that the SER performance of Mode I becomes poorer as ρ decreases is that the beamformers which are optimal for the channel of the k th OFDM symbol time are not always optimal for the channel of the (k +1)th OFDM symbol time. The nonoptimal beamformers decrease the average received power gain and diversity gain over the (k +1)th OFDM symbol time as ρ decreases.
0 2 4 6 8 10 12 14 16 10-4
10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO-OFDM System, R=4, ρ=0.8
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.11: SER performance of the single-mode 4× MISO-OFDM system with 1 ρ =0.8
0 2 4 6 8 10 12 14 16
10-4 10-3 10-2 10-1 100
SNR (dB)
SER
4x1 MISO-OFDM System, R=4, ρ=0.7
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
Figure 3.12: SER performance of the single-mode 4× MISO-OFDM system with 1 ρ =0.7
0 0.2 0.4 0.6 0.8 1 10-3
10-2 10-1
ρ
SER
Mode II (3-bit Beamformer) Mode I (6-bit Beamformer)
ρ =0.85
Figure 3.13: SER performance comparison between Mode I and Mode II for 4 1× MISO-OFDM system
Figure 3.13 shows the SER performance comparison between Mode I and Mode II for 4× MISO-OFDM system under a fixed SNR=12 dB. Two lines cross at about 1 ρ =0.85. Figure 3.13 shows that when the channel is subject to high temporal correlation (ρ ≥0.85), Mode I outperforms Mode II whereas with ρ <0.85 the latter yields better performance.
We compare the above simulation results with that of MISO systems. It is obviously observed that the results of the MISO-OFDM system are similar to that of the MISO system because the multipath channel response can be reduced into a multiplicative constant on a tone-by-tone basis by DFT at the receiver and all subcarriers are independent. The properties of the OFDM system will be given in Chapter 4.
3.4 Summary
The single-mode limited feedback system has been introduced in this chapter. We consider the transmit beamforming scheme and MISO channel model in this thesis and use the first-order Markov channel to model the temporal-correlation channels. The transmit beamforming codebooks in IEEE 802.16-2005 standard are applied in our simulation. The results show that the performance of Mode I becomes poorer as ρ decreases in the MISO system and in the MISO-OFDM system. Obviously, the single-mode limited feedback system is not robust enough in mobile environments.
Mode I is suitable to be used in the high temporal-correlation channel and Mode II is suitable to be used in the low temporal-correlation channel.
Equation Section (Next)
Chapter 4
Dual-Mode Limited Feedback Precoding for MISO Systems
In Chapter 3, we introduced the single-mode limited feedback precoding for MISO systems. It is obviously observed that the performance of the single-mode limited feedback system is affected by the codebook size and the temporal-correlation coefficient of the channel. Under a fixed average feedback rate, the single-mode limited feedback system is not robust enough in mobile environments. The system with a large-size codebook is suitable to be used in the high temporal-correlation channel and that with a small-size codebook is suitable to be used in the low temporal-correlation channel. To solve this problem, we propose a dual-mode limited feedback system with an SER based mode selection criterion in this chapter. The dual-mode limited feedback system has two codebooks with different sizes. The proposed SER based mode selection criterion can select the better mode for current transmission.
In this chapter, the dual-mode limited feedback MISO systems are presented first.
Then the proposed SER based mode selection criterion will be introduced and derived.
We will show that the proposed SER based mode selection criterion also can be applied to the MISO-OFDM system. Finally, the simulation results are given.
4.1 Dual-Mode Limited Feedback Systems
The dual-mode limited feedback system is a system that has two codebooks with different sizes. Under a fixed average feedback rate, the receiver feeds back the index of the optimal beamformer from the codebook selected by the mode selection criterion.
The dual-mode limited feedback system can slelect the better mode according to the current CSI and the temporal-correlation coefficient of the channel. In the high temporal-correlation channel, the dual-mode limited feedback system will select Mode I for current transmission; in the low temporal-correlation channel, the dual-mode limited feedback system will select Mode II. The dual-mode limited feedback MISO system architecture is shown in Figure 4.1 and the mode selection scheme of the dual-mode limited feedback system is illustrated in Figure 4.2.
Figure 4.1: Dual-mode limited feedback system architecture
Figure 4.2: Mode selection scheme of the dual-mode limited feedback system
The receiver uses the mode selection criterion to select the better mode according to the current CSI and the temporal-correlation coefficient of the channel at the start of the k th frame. Under a fixed feedback rate at three bits per frame, if Mode I is selected, the receiver feeds back six bits at the start of the k th frame and the selected beamformer will be also used for the (k +1)th frame. It feeds back once every two frames; if Mode II is selected, the receiver feeds back three bits at the start of the k th frame and the (k +1)th frame. It feeds back once every frame. The dual-mode limited feedback system can switch between Mode I and Mode II by using the proposed mode selection criterion.
In the following, we will present and derive the proposed SER based mode selection criterion.
4.2 SER Based Mode Selection Criterion for MISO Systems
For selecting the better mode from Mode I and Mode II, we compare the average SER over two consecutive frames in Mode I with that in Mode II. We propose a mode selection criterion based on the average SER over two consecutive frames. The proposed SER based mode selection criterion is stated as follows:
SER Mode Selection Criterion (SER-MSC) Select Mode I if
The proposed SER based mode selection criterion will be effective if the average SER over two consecutive frames can be determined. In the following, we will derive the exact values of SERk,mode1 , SERk,mode2 , E
{
SERk+1,mode1}
, and{
SERk+1,mode2}
E .
A. SER Analysis for kth Frame in Mode I
In this part, we will analyze the exact value of SERk,mode1. We take the M-QAM constellation as the example in the thesis. From [32], the symbol error rate of M-QAM constellation is
and Q ⋅ is the Q-function. The symbol error rate of M-QAM constellation in ( ) Equation (4.4) can be expressed as a function of the SNR γ :
From Equation (3.1), the overall power gain for the received signal is h fk k2. The
beamformer selection criterion is used to maximize this power gain h fk k2 to minimize the SER. Thus, the beamformer selection criterion is stated as follows:
Maximum Signal-to-Noise Ratio Selection Criterion (MSNR-SC):
Pick f such that k,1 the beamformer set used in Mode I. By Equation (4.6), the average SER over the kth
frame in Mode I is
The exact value of SERk,mode1 can be calculated by Equations (4.9) and (4.10).
B. SER Analysis for kth Frame in Mode II
By the same derivation of the exact value of SERk,mode1, the average SER over the kth frame in Mode II can be written as beamformer set used in Mode II. Thus, the first terms of Equation (4.2) and (4.3) can be determined by Equations (4.9) and (4.11). In the following, we will derive the expected value of SERk+1,mode1 and SERk+1,mode2.
C. SER Analysis for (k+1)th Frame in Mode I
From Equation (4.9), we know that the exact value of the average SER over the (k +1)th frame in Mode I, SERk+1,mode1, can be written as start of the k th frame in Equation (4.15). Thus, the exact value of the average SER over the (k +1)th frame cannot be determined at the start of the k th frame.
SERk+ will be determined. SERk+1,mode1 will be substituted by
{
SERk+1,mode1}
0 1 2 3 4 5 6 7
analyze the probability distribution of the received power gain Y . First, we define a random variable X as follows:
1 1,1 , and Δ are normally distributed random variables with zero-mean and variance hi (1−ρ)/2. Thus, X and r X are normally distributed random variables with mean i
where
Equation (4.25) shows that the received power gain Y is the sum of two squared normally distributed random variables, X and r2 X . i2
Noncentral Chi-square distributed random variable is a linear combination of several normally distributed random variables [33]-[35]. The noncentral Chi-square distributed random variable Z with n degrees-of-freedom can be defined as
2 2 where λ is the non-centrality parameter. X is an independent, normally distributed k
random variable with mean μ and variance k σ . The normally distributed random k2 variables X are substituted by k X and r X and set i n = in our case. The 2 noncentral Chi-square distributed random variable Z can be expressed as
( )
( )
The probability density function of the random variable Y is
where f z is the PDF of the noncentral Chi-square distributed random variable with Z( )
2
where f z is the PDF of the noncentral Chi-square distribution with Z( )
2
E by the above derivation. Thus the average SER over two consecutive frames for Mode I can be determined.
D. SER Analysis for (k+1)th Frame in Mode II
know which beamformer will be selected. In this part, we will follow the analysis methods in [12] and calculate the expected value of SERk+1,mode2.Define a normalized channel vector as
1 1
Equation (4.37) can be rewritten as obtain. To solve this problem, we use a lower bound on the outage probability derived in [12] to substitute the exact FU( )u and then derive the exact fγh( )γ in the h function F u is derived under the geometrical framework presented in [36]. The U( )
proof of F uU( )≤F uU( ) is given in [12]. We use F u derived in [12] to substitute U( )
h . By the same results derived from Equations (4.19) to
(4.26), it is observed that the random variable γ is a linear combination of h N × t 2
γ is a noncentral Chi-square distributed random variable with h N × t 2 degrees-of-freedom, 1 2
where Z is a noncentral Chi-square distributed random variable with N × t 2 degrees-of-freedom, 1 2
a k
λ= ρh , and 1
a = −2ρ . The probability density function of the random variable γ can be written as h
( ) ( | , )
where f z is the PDF of the noncentral Chi-square distributed random variable with Z( )
t 2
N × degrees-of-freedom and 1 2 a k
λ= ρh , and it is defined in Equation (4.33).
λ= ρh , and it is defined in Equation (4.33).