Summary

In this chapter, the limited-feedback MISO system in our work is introduced. In particular, two codebooks of different sizes are considered in the system. Then, we introduce and analyze the first-order Markov channel model. Moreover, codebook construction in the IEEE 802.16e-2005 standard is also presented. Finally, since there are two kinds of sizes of codebooks in the standard, this gives us the motivation how to design a scheme to utilize them in time varying channels, and the proposed algorithm will be presented in Chapter 3.

Chapter 3

Dual-Mode Scheme and Mode Selection

In our work, we have two codebooks F and ₁ F corresponding to 6-bit ₂ quantization and 3-bit quantization at the receiver and the transmitter. For simplicity, we call the mode using the 6-bit codebook for limited feedback as Mode I and the mode using the 3-bit codebook as Mode II. The two modes are defined as follows: (1) Under Mode I scheme, the best precoder is chosen at the receiver, and then the index of the precoder, which is coded in 6 bits, are fed back to the transmitter per two frames; (2) For Mode II, this work is realized per one frame. Therefore, the two modes in our definition have the same feedback rate as 3 bits/time frame.

Instead of using Mode I or Mode II exclusively for the system architecture, we consider a dual-mode scheme in [11], in which two codebooks are available at both

3.1 Dual-Mode Scheme

The dual-mode limited feedback system has two codebooks of different sizes at the receiver and transmitter. In this thesis, we adopt the 3-bit and 6-bit codebooks in the IEEE 802.16e-2005 [10], which have been introduced in Chapter 2. In order to fairly compare between the two modes and select one for use, we consider a fixed-rate feedback scheme as described in [11]. Namely, 6 bits are fed back to the transmitter per two frames in Mode I; 3 bits are fed back to the transmitter per one frame in Mode II. As a result, an equal feedback rate of 3 bits/frame is maintained for each mode. In particular, we assume that the transmitter is able to distinguish between the two modes via feedback bits without extra ones for indicating which mode should be adopted. Also, we assume that the receiver has full knowledge of CSI at the start of each frame so that it can choose the best beamformer and better mode for use in time-varying channel. The dual-mode selection scheme in MISO system is shown in Fig. 3-1

frame1 frame2 frame3 frame4 frame5 frame6 frame7

f1 f3 f5 f7

Mode I

f2

f1 f3 f4 f5 f6 f7 f8

Mode II

6 bits 6 bits 6 bits 6 bits

3 bits 3 bits 3 bits 3 bits 3 bits 3 bits 3 bits 3 bits

frame8

h

1

h

₂

h

₃

h

₄

h

₅

h

₆

h

₇

h

₈

Fig. 3-1 Dual-mode selection scheme in MISO system

3.2 Throughput-Based Criterion

Before a further discussion, we should first define the throughput used in this work. The throughput is defined as (maximum reliable transmission rate) × (1–bit error rate), whose physical meaning is the number of correct bits per second that can be achieved without any error-control coding if the transmitter use the channel capacity as the transmission rate. The maximum reliable transmission rate is channel capacity

( )

( ) log 12

C γ = +γ (3.1)

where γ is given by (2.2).

The bit error rate (BER) generally has no closed-form solution, therefore we turn to its upper bound from the relation with symbol error rate (SER)

2

SER BER SER

log M ≤ ≤ . (3.2)

Considering the worst case, we choose SER as BER.

Finally the throughput is written as

( )

( ) ( ) 1 _M( )

T γ =C γ ⋅ −P γ (3.3)

where P γ is the SER for a given modulation scheme, which is M-QAM in our _M( ) case, and has the following formula [19]:

⎛ ⎛ ⎞⎞2

3.3 Modal Metric

In order to select a better mode for use at the transmitter, we should first define the modal metric used at the receiver. Modal metric, as implied by its name, is used as a gauge to compare between two modes, and a careful choice of it can increase the system performance. Our modal metric is defined as the average throughput between two consecutive frames:

{

, 1,

}

^{{ }}

1 ( ) [ ( )] , 1,2

m 2 Tk m γ Tk₊ m γ m

Γ = +E ∈ (3.6)

where T_{i j,} ( )γ is the throughput for the i th frame under the j th mode and [ ]⋅E denotes expectation over channel statistics. This type of modal metric is also defined in [11] in which the modal metric is defined as selection criterion in [11] is called SER-based criterion.

The reason for the expectation is based on the fact that we assume the receiver has full knowledge of channel statistics at k th frame but nothing about it for the coming frame. Taking average over two time frames can help the receiver exploit the information of channel correlation and further enhance the accuracy of our modal metric. If the channel correlation ρ is equal to 1, then there is no need to take the expectation over the (k +1)th frame, so the modal metric in this case becomes

, ( ) { }1,2

m Tk m γ m

Γ = ∈ (3.8)

3.3.1 Computation of Modal Metric in Mode I

The expected value of the throughput corresponding to Mode I conditioned on h can be formulated as k

1,1 0 1,1

[T_k+ ( )]γ =

∫

^∞T_k+ ( )γ ⋅f_Γ(γ ^h_k) dγ

E (3.9)

where γ = h^H_k+1f_k^s₊₁²/N₀ and f γ_Γ( h is the probability density function (PDF) _k) of the random variable Γ conditioned on h . In order to compute this integral, we _k need to find out the distribution of Γ .

In this mode, the selected precoder for the (k +1)th frame is the same to that for the previous one, i.e., f_k^s₊₁ =f , therefore, we can rewrite γ as _k^s h^H_k+1f_k^{s 2}/N₀. Furthermore, we transform (3.9) into another representation

1,1 0

( ) ( )

then we find that Z is a noncentral Chi-square distributed random variable with degrees of freedom being 2 and the noncentrality parameter being λ [20]

( )

I i is the modified Bessel function of the first kind of zero order [21]. Finally, The expected value of throughput corresponding to Mode I conditioned on h can be _k written as

The derivation given above is based on the assumption that the channel correlation ρ is not equal to 1, or Equation (3.12) does not make any sense. In this case, when ρ is equal to 1, the channel condition of the k th frame is the same to that of the (k +1)th frame, so the expectation is meaningless and the throughput of the (k + th frame is the same to that of the k th frame. 1)

3.3.2 Computation of Modal Metric in Mode II

The expected value of throughput corresponding to Mode II is much involved than that in Mode I since the best precoder is reselected in the (k +1)th frame.

Therefore, we have two uncertainties: h^H_{k +1} and f_{k +1}. The formula for the expected value of throughput is formulated in the same way to that under Mode I

1,2 0 1,2

[T_k+ ( )]γ =

∫

^∞T_k+ ( )γ ⋅f_Γ(γ ^h_k) dγ

E (3.17)

The receive SNR can then be derived as

Γ =SNR⋅ h^H_k+1f_k^s₊₁²

multiplication of two random variables U and V under the condition that h is _k given.

bound, thus the resulted expected value is a lower bound, which is questionable to be used as a modal metric in Mode selection. Therefore, in our work, we turn to use V , which is a random variable generated from V with h being averaged out, as our _k alternative. With the help of numerical experiments, the plot of f v is depicted in _V( ) Fig. 3-2. Then we approximate the PDF of random variable V with Gamma distribution [22, 23]

( ) an approximation of the distribution of V , thus

( ) _{[ ]}

To examine the tightness of this approximation, we compare the root mean square error (RMSE) of using the function ( )f v as well as that of using different order of polynomials, and the result is plotted in Fig. 3-3. As we can see, the approximation with ( )f v is tight enough.

Moreover, two random variables U and V are independent. Instead of giving a proof in detail, we provide an alternative way to demonstrate it. The chief reason for the independence lies in the fact that V has nothing do with h , thus making them _k an independent pair. To better illustrate this, we give the graphs of the jointly distribution of UV and the product of their distributions.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.5 1 1.5 2 2.5 3 3.5

Data

Density

V Gamma

Fig. 3-2 Histogram of the random variable V and its approximation with Gamma distribution

0.2 0.4 0.6 0.8 1 1.2 1.4

RMSE

Polynomial f(v)

Fig. 3-4 Joint PDF of random variables V and U

Fig. 3-5 Product of PDFs of random variables V and U

3.4 Selection of Good Mode and Beamformer

The rule for selecting a suitable mode and a specific precoder is described as follows: the receiver first determines the mode m for use based on the value of each modal metric, i.e., Mode I is chosen if Γ > Γ and Mode II on the other hand. _{1 2}

After choosing the mode type, the receiver chooses the precoder f^s maximizing the throughput

monotonically increasing function of γ , (3.23) becomes

2 { } The rule for selecting a good mode and a beamformer can be better described in the

following flow chart.

3.5 Numerical Results

In this section, we shall give some simulations to demonstrate the advantages of the dual-mode scheme and the proposed selection criterion. Table 3-1 lists all parameters in our simulation.

Table 3-1 Simulation parameters

Parameter Value

Channel Rayleigh fading channel

(First-order Markov channel model)

Modulation 16 QAM

Number of transmit antennas 4

Number of receive antennas 1

Fixed average feedback rate 3 bits per frame

Frame length 128 symbols

Number of frames 10000

Codebook

Transmit beamforming codebooks in IEEE 802.16-2005

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3.4

3.6 3.8 4 4.2 4.4 4.6 4.8 5

Channel Correlation ρ

Throughput (bits/s/Hz)

4x1 MISO System, 16 QAM , SNR = 10dB

16QAM (Mode Selection) 16QAM (3-bit Precoding) 16QAM (6-bit Precoding)

Fig. 3-7 Dual-mode scheme and single-mode scheme with SNR = 10dB and 16 QAM

In this figure, we can observe that the dual-mode scheme with the proposed mode selection criterion adequately switch between the other two single-mode schemes, which in our case are the 3-bit precoding scheme and the 6-bit precoding scheme. In low-correlation regime, the receiver prefers Mode II. Due to the rapid fluctuation of channel statistics, the system is able to achieve better performance if it reselects the precoder at the start of each frame. On the other hand, in high-correlation regime, the receiver turns to use Mode I. Because the channel statistics varies slowly, the high resolution of the 6-bit codebook is able to compensate for the performance degradation due to the use of same precoder over two frames.

3.6 Summary

In this chapter, we first introduce the dual-mode scheme in which two different sizes of codebooks are adopted. The goal of using dual-mode scheme is to achieve a better system performance on the premise that adequate selection criterion can be used to switch between the two. Afterwards, we propose our mode selection criterion by first defining the throughput in this work and then the modal metric.

In Section 3.2, we give the definition of the throughput whose physical meaning is the number of correct bits per second that can be achieved without any error-control coding if the transmitter use the channel capacity as the transmission rate. In Section 3.3, the definition of the modal metric is given. The modal metric is able to exploit the channel correlation between the consecutive two frames by taking average of the throughputs corresponding to each mode under the assumption that the receiver has full knowledge CSI. In particular, the modal metric involves the expectation of the throughput pertaining to the next frame since the receiver has no idea about the exact channel condition of it. The detailed analysis of modal metric corresponding to Mode I and Mode II is also derived. Then in Section 3.4, we describe the selection rule for the good mode and the precoder. Finally, a simulation is given to justify the proposed algorithm for the dual-mode scheme.

Chapter 4

Modal Metric Approximation

In Chapter 3, the definition and derivation of the modal metric corresponding to Mode I and Mode II are given; moreover, simulations justify the utility of the modal metric by showing that the dual-mode scheme outperforms the single-mode scheme over full range of channel correlations. The superior performance is ascribed to the consideration of channel correlation in the modal metric. Namely, the expected value of throughput of the coming frame is considered when performing the mode selection.

While the expected value of the throughput with respect to Mode I and Mode II are derived in Chapter 3, the integral are a little bit troublesome when performing this algorithm in practical, and to simply the computation of this integral is one of the future works in [11]. Therefore, we wonder if there are closed-form solutions for them.

Unfortunately, to our best knowledge, due to the combination of the various difficult-analyzing functions in the integrals, the answer is no. To seek what is less

4.1 Approximation of Modal Metric in Mode I

We start the approximation with the equation (3.10) and assume that ρ ≠ . 1 Since (3.10) can be rewritten as

0 0

Therefore (4.1) turns to

(

²

)

Since Q-function has no closed form representation, we turn to its approximation [24]:

2 2

With this approximation,

( )

fitting [25]:

Therefore, (4.9) becomes

( ) ¹

(

²

)

At this point, Let us postpone the computation of this integral for a moment and consider the following lemmas in [21]:

Lemma 1

With Lemma 1 and Lemma 2, (4.13) becomes

( )

( )

Therefore, with Lemma 2 to Lemma 6, we obtain the following results:

4.2 Approximation of Modal Metric in Mode II

Since the random variable W is noncentral chi-square distributed with 2N _t degrees of freedom and the non-centrality parameter λ being 2ρ h_k ²/(1−ρ),

Again, we use the approximation in (4.7) for Q-function, so

With numerical experiment, we find that H u decays very fast and is negligible for ^{( )} 20

x > , so we approximate D u α β with a polynomial ( , , )( , , ) f u α β using least squares fitting [25]:

7

( )

− . The closed form representation for Φ ⋅ can be obtained with Lemma 2 to Lemma 6. ( )

4.3 Numerical Results

In this section, we give some simulation results to demonstrate the effectiveness of the throughput-based criterion. For the sake of simplicity, we denote E[T_k+1,1( )]γ

as T ; approximated ₁ E[T_k+1,1( )]γ as T ; ₁^* E[T_k+1,2( )]γ as T ; approximated ₂

[T_k+1,2( )]γ

E as T . Moreover, the throughput performance with respect to the ₂^* dual-mode scheme, the single-mode scheme with 3-bit precodeing, and the single-mode scheme with 6-bit precoding are also given. Finally the proposed mode selection criterion is compared with other criteria. Table 4-1 lists all parameters in our simulations.

Table 4-1 Simulation parameters

Parameter Value

Channel Rayleigh fading channel

(First-order Markov channel model)

Modulation 16QAM, 64 QAM

Number of transmit antennas 4

Number of receive antennas 1

Fixed average feedback rate 3 bits per frame

Frame length 128 symbols

Number of frames 10000

4.3.1 Approximation vs. Exact Value (Mode I)

Fig. 4-1 shows the expected value of throughput over the (k +1)th frame under Mode I with 16-QAM modulation. Three different channel correlations: ρ =0.1, ρ =0.5, and ρ =0.9 are considered. First, the expected value of the throughput over the (k +1)th frame is an increasing function of SNR. Second, As the channel correlation gets larger, the performance gets better. The reason is that the best precoder for beamforming over the k th frame is near optimal for the (k +1)th frame. Moreover, the approximation of the expected value is shown to be very tight for different channel correlations and SNRs. The proposed approximation is also shown to be very tight for different modulations in Fig. 4-2

0 2 4 6 8 10 12 14 16

4x1 MISO System, 16 QAM 6-bits precoding

ρ = 0.9

ρ = 0.9 (Approximation) ρ = 0.5

ρ = 0.5 (Approximation) ρ = 0.1

ρ = 0.1 (Approximation)

Fig. 4-1 T vs. ₁ T with 16 QAM and different channel correlations ₁^*

0 2 4 6 8 10 12 14 16 0

1 2 3 4 5 6 7

SNR

E[Throughput] (bits/s/Hz)

4x1 MISO System, 64 QAM 6-bits precoding

ρ = 0.9

ρ = 0.9 (Approximation) ρ = 0.5

ρ = 0.5 (Approximation) ρ = 0.1

ρ = 0.1 (Approximation)

Fig. 4-2 T vs. ₁ T with 64 QAM and different channel correlations ₁^*

4.3.2 Approximation vs. Exact Value (Mode II)

Fig. 4-3 shows the expected value of throughput over the (k +1)th frame under Mode II with 16-QAM modulation. Three different channel correlations: ρ =0.1, ρ =0.5, and ρ =0.9 are considered. First, the expected value of the throughput over the (k +1)th frame is an increasing function of SNR. Second, we can observe that unlike Mode I, in which the performance gets better as the channel correlation gets larger, the performance corresponding to different channel correlations are almost the same. The reason is that the best precoder for beamforming over the k th frame is

0 2 4 6 8 10 12 14 16

4x1 MISO System, 16 QAM 3-bit precoding

ρ = 0.9

ρ = 0.9 (Approximation) ρ = 0.5

ρ = 0.5 (Approximation) ρ = 0.1

ρ = 0.1 (Approximation)

Fig. 4-3 T vs. ₂ T with 16 QAM and different channel correlations ₂^*

4x1 MISO System, 64 QAM 3-bit precoding

ρ = 0.9

ρ = 0.9 (Approximation) ρ = 0.5

ρ = 0.5 (Approximation) ρ = 0.1

ρ = 0.1 (Approximation)

Fig. 4-4 T vs. ₂ T with 64 QAM and different channel correlations ₂^*

4.3.3 Dual-Mode Scheme vs. Single-Mode Scheme

To better examine the improvement of the dual-mode scheme over the single-mode scheme, average throughput over different channel correlations at a fixed transmit SNR is compared. In Fig. 4-5 and Fig. 4-6, each figure has four lines corresponding to the single-mode schemes with 3-bit and 6-bit codebook respectively, the dual-mode scheme using exact modal metric, and the dual-mode scheme using the approximated modal metric.

In each figure, we can see that the system performance does not remain well in all channel correlations using Mode I or Mode II exclusively. Mode I is good at low channel correlations since the receiver selects the best precoder at each time frame.

While at high channel correlations, Mode II works better since the high resolution of the codebook compensates for the loss caused by choosing precoder seldom. Finally, the proposed algorithm of model selection works well for the dual-mode scheme which outperforms each single-mode scheme in full range of channel correlations because it always selects a better mode for use. Moreover, the proposed approximation for the modal metric is very tight over different channel correlations and modulations.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Channel Correlation ρ

Throughput (bits/s/Hz)

4x1 MISO System, 16 QAM , SNR = 10dB

16QAM (Mode Selection) 16QAM (Mode Selection App) 16QAM (3-bit Precoding) 16QAM (6-bit Precoding)

Fig. 4-5 Dual-mode scheme, Dual-mode scheme with approximated modal metric and single-mode scheme with SNR = 10dB and 16 QAM

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Channel Correlation ρ

Throughput (bits/s/Hz)

4x1 MISO System, 64 QAM , SNR = 10dB

64QAM (Mode Selection) 64QAM (Mode Selection App) 64QAM (3-bit Precoding) 64QAM (6-bit Precoding)

Fig. 4-6 Dual-mode scheme, Dual-mode scheme with approximated modal metric and single-mode scheme with SNR = 10dB and 64 QAM

4.3.4 Throughput-Based Criterion vs. Other Criteria

To compare the proposed criterion with other selection criteria, we give the following simulations. First, we consider a randomly-selecting criterion which randomly selects one mode for use per two frames and the simulation result is given in Fig. 4-7. As expected, the system performance with randomly-selecting criterion is between 3-bit precoding scheme and 6-bit precoding scheme since two modes are used with the same probability. Also, the proposed throughput-based criterion outperforms the randomly-selecting criterion.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3.4 3.6 3.8 4 4.2 4.4 4.6 4.8

Channel Correlation

Throughput (bits/s/Hz)

4x1 MISO System, 16 QAM , SNR = 10dB

16QAM (Mode Selection THR) 16QAM (6-bit precoding)

16QAM (Mode Selection Random) 16QAM (3-bit precoding)

Fig. 4-7 Throughput-based criterion and random selection

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 16QAM (Mode Selection SER) 16QAM (Mode Selection THR)

Fig. 4-8 Throughput-based criterion and SER-based criterion in the dual-mode scheme with SNR = 10dB and 16 QAM

16QAM (Mode Selection THR) 16QAM (Mode Selection SER) 16QAM (3-bit precoding) 16QAM (6-bit precoding)

Fig. 4-9 Throughput-based criterion and SER-based criterion in the dual-mode scheme with SNR = 10dB and 16 QAM

4.4 Summary

In this chapter, we try to analyze the formulas of the modal metric with respect to Mode I and Mode II given in Chapter 3. Our goal is to find the closed form solutions for them, but unfortunately, to our best knowledge this cannot be obtained. As an alternative, we derive the approximations of the expected value of throughput corresponding to Mode I and Mode II respectively, giving a much practical way to implement the algorithm of selecting the mode at the receiver. The simulation results show that the approximations are very tight throughout all channel correlations, SNRs, and modulations in consideration. Moreover, the proposed criterion performs comparably to other existing mode selection criteria.

Chapter 5

Conclusion

Limited feedback communication is a prevailing technique to enhance the system performance through relaying back the CSI over a band-limited channel to the transmitter. In MIMO systems, feedback can be used to assign a pre-designed matrix, i.e., precoder to the transmitter, hence activating the strongest channel mode. One of the approaches to implement this is the codebook-based limited feedback scheme. In this scheme, a pre-designed codebook containing finite numbers of quantized channel matrices is provided at the transmitter and the receiver, so the receiver can pick one precoder from the codebook by some criterions and inform the transmitter of the index of the precoder.

In this thesis, we consider a limited feedback system in which two different sizes of codebooks are used to enhance the system performance. The large codebook contains more precoders, hence a near-optimal precoder can be selected with a higher probability. On the other hand, while the small coderbook contains fewer precoders, it takes fewer bits for the system using Mode II to relay back the corresponding index.

In this thesis, we consider a limited feedback system in which two different sizes of codebooks are used to enhance the system performance. The large codebook contains more precoders, hence a near-optimal precoder can be selected with a higher probability. On the other hand, while the small coderbook contains fewer precoders, it takes fewer bits for the system using Mode II to relay back the corresponding index.

在文檔中 用於時變通道下有限回饋傳送波束形成之基於吞吐量模式選擇準則 (頁 26-0)