Multipath First-Order Markov MISO Channel Model

Chapter 4 Dual-Mode Limited Feedback Precoding for MISO

4.3 SER Based Mode Selection Criterion for MISO-OFDM Systems

4.3.2 Multipath First-Order Markov MISO Channel Model

In this section, we will show that the multipath first-order Markov MISO channel model has the same property with the first-order Markov MISO channel model which is assumed and introduced in Chapter 3. Thus, the proposed SER based mode selection criterion can be applied to MISO-OFDM systems.

Consider a multipath MISO channel with channel order L_ch and temporal-correlation coefficient ρ . The time-varying property of the multipath MISO channel can be characterized by the first-order Markov channel model (Equation (3.3)).

The first-order Markov channel model is only characterized by the one-tap temporal-correlation coefficient ρ . From Equation (4.56), the channel matrix between the j th transmit antenna and the receive antenna for the (k +1)th OFDM symbol time can be written as

1 ,

,0 ,1 , coefficient of the n th tap between the j th transmit antenna and the receive antenna for the k th OFDM symbol time. We assume that the path loss vector is

loss,0,..., loss,_N_CP convenience. Δh_{j n}_, is a complex normally distributed random variable of the n th tap between the j th transmit antenna and the receive antenna with zero-mean and variance (1−ρ)⋅P_loss,_n. ρ is the temporal-correlation coefficient ( 0≤ ≤ ). A ρ 1 small ρ =( 0) implies the fast fading environment and a large (ρ =1) corresponds to the time-invariant channel. We also assume that the channel length is smaller than

NCP, that is L_ch + <1 N_CP.

Equation (4.63) can be rewritten as

1 , frequency response between the j th transmit antenna and the receive antenna for the k th OFDM symbol time. ΔΛ is the innovation matrix between the j th transmit _j

antenna and the receive antenna. Λ^k_j⁺¹, Λ , and ^k_j ΔΛ are all diagonal matrices. _j We assume that Δh_{j n}_, is a complex normally distributed random variable with zero-mean and variance (1−ρ)P_loss,_n and _loss, distributed random variable with zero-mean and variance (1−ρ). All elements are mutually dependent normal random variables with zero-mean and variance (1−ρ). Equation (4.64) can be rewritten as

, 1 , , , for 1,..., ,

k k

j nn⁺ ρ j nn j nn n NFFT

Λ = Λ + ΔΛ = (4.65)

where Λ^k_{j nn}_,⁺¹ is the channel frequency response of the n th subcarrier between the j th transmit antenna and the receive antenna for the (k +1)th OFDM symbol time, and Λ^k_{j nn}_, is the channel frequency response of the n th subcarrier between the j th transmit antenna and the receive antenna for the k th OFDM symbol time. ΔΛ_{j nn}_, is a complex normally distributed random variable with zero-mean and variance (1−ρ).

Consider a MISO-OFDM system with N transmit antenna and one receive _t antenna. For the n th subcarrier, we can collect Λ^k_{j nn}_,⁺¹, for j =1,...,N_t, as a row

vector Λ^k_{ch n}⁺_,¹ = Λ⎡⎢⎣ _1,^k⁺_nn¹,...,Λ^k_{N nn}⁺_t_,¹ ⎤⎥⎦. The MISO channel frequency response of the n th subcarrier for the (k +1)th OFDM symbol time can be written as

,1 , , , for 1,..., ,

k k

ch n⁺ = ρ ch n + Δ ch n n = NFFT

Λ Λ Λ (4.66)

where ΔΛ_{ch n}, ∼ CN

(

0, 1( −ρ)I_N_t

)

. All elements of ΔΛ_{ch n}_, are independent because of the independence between N transmit antennas. _t

Comparing Equation (4.66) with Equation (3.3), we see that the frequency response of the multipath first-order Markov MISO channel model has the same time-varying property with the first-order Markov MISO channel model defined in Equation (3.3). Thus the proposed SER based mode selection criterion can be used to select the better mode tone by tone in the MISO-OFDM system.

4.4 Computer Simulations

In the following, we will illustrate the performances of the single-mode and dual-mode limited feedback MISO systems. We also consider a MISO-OFDM system in the simulations and illustrate the performances of the single-mode and the dual-mode limited feedback MISO-OFDM systems. The schemes of Mode I and Mode II for single-carrier systems and multi-carrier systems are commented in Table 3.4

(a) Dual-mode limited feedback MISO system

We consider a dual-mode limited feedback system in this simulation. Table 4.1 lists all parameters used in our simulation. This simulation uses 16-QAM modulation and one data substream on a 4× wireless system. The transmit beamforming 1 codebooks in IEEE 802.16-2005 are applied in the simulation. The average feedback rate is fixed at three bits per frame. The dual-mode system uses the proposed SER based mode selection criterion to select the better mode for current transmission. 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 also follow the first-order Markov channel model (3.3). 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. 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 systems (operating in Mode I and Mode II) and the dual-mode system with the SER based mode selection criterion are shown in Figures 4.4-4.7 with ρ =1, 0.9, 0.8, 0.7.

Table 4.1: Simulation environment of the dual-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 4.4 illustrates the SER performances of the dual-mode 1 system and the single-mode system. The performance of the dual-mode system is close to that of Mode I (6-bit beamformer) and better than that of Mode II (3-bit beamformer).

It is observed that the dual-mode system mainly selects Mode I for current transmission but not Mode II.

When ρ =0.9 , Figure 4.5 illustrates that the SER performance of the dual-mode system still is better than that of the single-mode system. The curve of Mode II crosses the curve of Mode I at SNR=14 dB. The single-mode system operating in Mode I losses some diversity gain, but the dual-mode does not. At SNR=14 dB, the dual-mode system can switch between Mode I and Mode II, so it has better performance than the single-mode system operating either in Mode I or Mode II.

When ρ =0.8, Figure 4.6 shows that the SER performance of Mode I becomes poorer than that of Mode I in Figure 4.5. The curve Mode II crosses the curve of Mode I at SNR=10 dB. The dual-mode system with SER-MSC is more robust than the

single-mode system. At low SNR region, the dual-mode system mainly selects Mode I for transmission. At SNR=10 dB, the dual-mode system can switch between Mode I and Mode II, so it has better performance than the single-mode systems. At high SNR region, the performance of the dual-mode system is close to that of Mode II. It states that the dual-mode system mainly selects Mode II at high SNR region.

When ρ =0.7, Figure 4.7 shows that the performance of the dual-mode system is still better than that of the single-mode systems. At low SNR region, the dual-mode system mainly selects Mode I because the performance of Mode I is better than that of Mode II. At high SNR region, the dual-mode system mainly selects Mode II because the performance of Mode II is better than that of Mode I. It states that the dual-mode system is more robust than the single-mode system in the time-varying channel.

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) Dual-Mode (SER-MSC)

Figure 4.4: SER performance of the dual-mode 4× MISO system with 1 ρ = 1

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) Dual-Mode (SER-MSC)

Figure 4.5:SER performance of the dual-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) Dual-Mode (SER-MSC)

Figure 4.6: SER performance of the dual-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) Dual-Mode (SER-MSC)

Figure 4.7: SER performance of the dual-mode 4× MISO system with 1 ρ =0.7

Figure 4.8 shows the SER performance comparison between the single-mode and the dual-mode systems under a fixed SNR=12 dB. Figure 4.9 shows the SER performance comparison between the single-mode and the dual-mode systems under a fixed SNR=12 dB for 0.7≤ ≤ρ 0.95. As we can see, 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. The dual-mode system with SER-MSC is more robust than Mode I and Mode II for any ρ . The curve of Mode I crosses the curve of Mode II at about ρ =0.85. In the region (0.7≤ ≤ρ 0.95), the performance of the dual-mode system is better than that of Mode I and Mode II because the dual-mode system with SER-MSC can select the better mode for current transmission.

The dual-mode limited feedback system is more robust in mobile environments.

0 0.2 0.4 0.6 0.8 1 10^-3

10^-2 10^-1

SER

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.8: SER performance comparison between the single-mode and the dual-mode 4 1× MISO systems

0.7 0.75 0.8 0.85 0.9 0.95

10^-2

SER

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.9: SER performance comparison between the single-mode and the dual-mode 4 1× MISO systems with 0.7≤ ≤ρ 0.95

(b) Dual-mode limited feedback MISO-OFDM system

We consider a MISO-OFDM system and Table 4.2 lists all parameters used in this simulation. This simulation uses 16-QAM modulation and 128 subcarriers in the MISO-OFDM system. 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 dual-mode limited feedback system selects the better mode between Mode I and Mode II by using SER-MSC. The transmit symbols are precoded tone by tone and we assume that all subcarriers are independent. SER performances of the dual-mode limited feedback MISO-OFDM system with ρ =1, 0.9, 0.8, 0.7 are shown in Figures 4.10-4.13.

Table 4.2: Simulation environment of the dual-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

When ρ = , Figure 4.10 illustrates the SER performances of the dual-mode 1 system and the single-mode system. The performance of the dual-mode system is close to that of Mode I and better than that of Mode II. It is observed that the dual-mode system mainly selects Mode I for transmission but not Mode II.

When ρ =0.9 , Figure 4.11 illustrates that the SER performance of the dual-mode system still is better than that of the single-mode system. The curve of Mode II crosses the curve of Mode I at SNR=14 dB. The single-mode system operating in Mode I losses a little diversity gain, but the dual-mode does not. The performance of the dual-mode system is still better than that of the single-mode systems.

When ρ =0.8, Figure 4.12 shows that the SER performance of Mode I becomes poorer than that of Mode I in Figure 4.11. The dual-mode system with SER-MSC is more robust than the single-mode system. At low SNR region, the dual-mode system mainly selects Mode I for transmission. At SNR=10 dB, the dual-mode system can switch between Mode I and Mode II by using SER-MSC, so it has better performance than the single-mode system. At high SNR region, the performance of the dual-mode system is close to that of Mode II. It states that the dual-mode system mainly selects Mode II at high SNR region.

When ρ =0.7, Figure 4.13 shows that the performance of the dual-mode system is still better than that of the single-mode system. At low SNR region, the dual-mode system mainly selects Mode I; at high SNR region, the dual-mode system mainly selects Mode II. It states that the dual-mode system is more robust than the single-mode system in the time-varying channel.

0 2 4 6 8 10 12 14 16

4x1 MISO-OFDM System, R=4, ρ=1

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.10: SER performance of the dual-mode 4× MISO-OFDM system with 1 ρ =1

4x1 MISO-OFDM System, R=4, ρ=0.9

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.11: SER performance of the dual-mode 4× MISO-OFDM system with 1 ρ =0.9

0 2 4 6 8 10 12 14 16

4x1 MISO-OFDM System, R=4, ρ=0.8

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.12: SER performance of the dual-mode 4× MISO-OFDM system with 1 ρ =0.8

4x1 MISO-OFDM System, R=4, ρ=0.7

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.13: SER performance of the dual-mode 4× MISO-OFDM system with 1 ρ =0.7

Figure 4.14 shows the SER performance comparison between the single-mode and the dual-mode MISO-OFDM systems under a fixed SNR=12 dB. Figure 4.15 shows the SER performance comparison between the single-mode and the dual-mode MISO-OFDM systems under a fixed SNR=12 dB for 0.7≤ ≤ρ 0.95. As we can see, 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. The dual-mode system with SER-MSC is more robust than Mode I and Mode II for any ρ , especially in the region (0.7≤ ≤ρ 0.95). The dual-mode limited feedback system with SER-MSC can select the better mode for current transmission in mobile environments.

We compare above simulation results with that of MISO systems. It is clearly observed that the results of MISO-OFDM systems are similar to that of MISO systems because the multipath channel response can be reduced into a multiplicative constant on a tone-by-tone basis by discrete Fourier transform (DFT) at the receiver and all subcarriers are independent. The transmitter is precoded with the optimal beamformer tone by tone and the proposed SER-MSC can be equally used in MISO-OFDM systems.

0 0.2 0.4 0.6 0.8 1 10^-3

10^-2 10^-1

SER

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.14: SER performance comparison between the single-mode and the dual-mode 4× MISO-OFDM systems 1

0.7 0.75 0.8 0.85 0.9 0.95

10^-2

SER

Mode II (3-bit Beamformer) Mode I (6-bit Beamformer) Dual-Mode (SER-MSC)

Figure 4.15: SER performance comparison between the single-mode and the dual-mode 4× MISO-OFDM systems with 0.71 ≤ ≤ρ 0.95

4.5 Summary

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. Under this condition, we propose a dual-mode limited feedback system which has two codebooks with different sizes and an SER based mode selection criterion.

From the derivation of SER-MSC, we know that the received signal power gain is a noncentral Chi-square distributed random variable. The simulation results show that the proposed dual-mode limited feedback system is more robust than the single-mode limited feedback system for any temporal-correlation coefficient ρ in mobile environments. Finally, we show the proposed SER-MSC can be equally used in MISO-OFDM systems.

Equation Section (Next)

Chapter 5 Conclusion

In this thesis, we introduce two popular approaches for transmission in the MIMO channel: diversity and spatial multiplexing. Precoding is one of the popular schemes to provide diversity gain and additional array gain for improving system performances, but it needs current CSI at the transmitter. Limited feedback systems are proposed to solve the problem. We survey the limited feedback system and explore the precoder selection criteria for selecting the optimal precoder in the limited feedback system. In Chapter 2, we also introduce the codebooks for MIMO precoding schemes in IEEE 802.16-2005, which provides codebooks with different dimensions and sizes depending on the number of transmit antennas and number of data substreams.

In Chapter 3, we introduce the single-mode limited feedback system which has only one codebook for precoding. We consider the MISO channel model and the transmit beamforming scheme because the practical cellular communication system has multiple transmit antennas and one receive antenna in general (e.g., GSM, DCS1800).

The first-order Markov channel is used to model the temporal-correlation channels because the model is only characterized by the one-tap temporal-correlation coefficient

ρ . The channel can be modeled as a time-invariant channel with a large ρ or a

time-varying channel with a small ρ . All codebooks defined in IEEE 802.16-2005 can be divided into two kinds: 3-bit codebooks and 6-bit codebooks. Under a fixed

feedback rate at three bits per frame, 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;

Mode II is defined in which the transmitter is precoded with the 3-bit beamformer and the receiver feeds back three bits per frame. Under this condition, the simulation results show that the single-mode limited feedback systems are not robust 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.

In Chapter 4, we propose a dual-mode limited feedback system which has two codebooks with different sizes and an SER based mode selection criterion. Based on the average SER over two consecutive frames, the dual-mode limited feedback system can switch between Mode I and Mode II by using SER-MSC. The simulation results show that the dual-mode limited feedback system is more robust and has better performances in mobile environments. We also show that the proposed SER-MSC can be applied to MISO systems and MISO-OFDM systems.

In this thesis, we proposed a dual-mode limited feedback system and derived a SER-MSC used to select the better mode. The dual-mode system can improve system performances based on current CSI in MISO systems or MISO-OFDM systems. It is found that SER-MSC has a high computation cost because of the integration of the PDF of the received power gain. It is not easy to implement this algorithm in hardware.

Besides, the assumptions that perfect CSI is available at the receiver, and the feedback channel has zero delay and is error free, may not always hold true. Moreover, it is not practical that each subcarrier uses the individual mode or beamformer in MISO-OFDM systems. Thus, one possible extension of this work is to design a simpler mode

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