Summary - MIMO-OFDM Technique Overview - 用於多輸入多輸出正交分頻多工系統之波束形成輔助多用戶適應性無線電資源管理技術

Chapter 2 MIMO-OFDM Technique Overview

2.7 Summary

Information theory shows that MIMO communication systems can significantly increase the capacity of band-limited wireless channels by a factor of the minimum number of transmit and receive antennas, provided that a rich multipath scattering environment is considered. In Sections 2.2 MIMO system model and MIMO channel capacity are introduced.

New degrees of freedom in spatial domain provided by multiple antennas are introduced in Section 2.3. The diversity makes us choose the better subchannel to transmit data which increasing the system performance in BER curve.

In order to achieve a high data rate in MIMO systems, spatial multiplexing technique is presented in Section 2.4. Spatial multiplexing allows significant data rate enhancement in a wireless radio link without additional power or bandwidth consumption. It is realized by transmitting independent data signals from the individual transmit antennas. Two typical spatial multiplexing schemes, D-BLAST and V-BLAST, are introduced in Sections 2.4.1 and 2.4.2.

Another technique called beamforming is described in Section 2.5. Beamforming is implemented by multiplying the symbol(s) with appropriate beamforming vactor(s) both at the transmitter and the receiver. While CSI is available both at the transmitter and the receiver, MIMO systems can benefit from significant diversity and codign gains by using beamforming.

OFDM has gained wide acceptance in wireless communications as an appropriate broadband modulation scheme. OFDM systems have the desirable immunity to intersymbol interference (ISI) caused by the delay spread of wireless channels.

Therefore, it is a promising technique for high data rate transmission over frequency-selective fading channels. In Section 2.6, OFDM systems are introduced.

Chapter 3 Joint Beamforming and Subcarrier Allocation for Multiuser

MIMO-OFDM Systems

Multiple-input multiple-output (MIMO) systems where multiple antennas are used at both the transmitter and receiver have been acknowledged as one of the most promising techniques to achieve dramatic improvement in physical-layer performance.

Moreover the use of multiple antennas enables space-division multiple access (SDMA), which allows intracell bandwidth reuse by multiplexing spatially separable users [36], [37]. Channel variation in the spatial domain also provides an inherent DOF (Degree of Freedom) for adaptive transmissions. Recently, multiuser adaptive transmission in multiple-antenna systems has been reported in [38] and [39] to exploit the space and multiuser diversity from an information theory point of view.

In Chapter 3, we will propose two algorithms: the first one is to design transmit and receive beamforming in order to cancel interference between users in downlink case and use SVD (Singular Value Decomposition) decomposition to simplify the design procedure and ensure good performance. The second method is to select N users transmitting data on the same subcarrier where N is the number of transmit and receive

This chapter is organized as follows. First, SVD decomposition is introduced whose mathematical form and physical meaning will be given. Second, we formulate the problem of beamforming design and show how to find the beamformer with SVD.

Finally, we propose a method of allocating subcarriers to maximize the system transmission rate and show the simulation results of above algorithms.

Subscriber 1 Subscriber 2

Subscriber 3

Subscriber 1 Subscriber 2

Subscriber 3

Figure 3.1: Scenario of transmit and receive beamformer to null interference

3.1 System Model and Problem Formulation

In multiuser systems, interference is the most important factor that dominates the system performance. Consequently, we try to design the transmit and receive beamformer in downlink case for interference cancellation. The algorithm is SVD based because it can simplify the design. The scenario is shown in Figure 3.1.

3.1.1 Singular Value Decomposition (SVD)

From basic linear algebra, every linear transformation can be represented as a composition of three operations: a rotation operation, a scaling operation, and another rotation operation. In the notation of matrices, the matrix H has a singular value decomposition (SVD):

,∀ ∈C^{M N}^×

H = UΛV* H , (3.1)

where U∈C^{M M}^× and V∈C^{N N}^× are unitary matrix, Λ∈R^{M N}^× is diagonal matrix whose diagonal elements are nonnegative real numbers and whose off-diagonal elements are zero. The diagonal elements

1 2 ... _Nmin

λ λ≥ ≥ ≥λ are the ordered singular

values of the matrix H, where N_min min(M N, ). We can rewrite the SVD as

min

* N

i i i i

∑

H u v (3.2)

i.e., the sum of rank-one matrices λ_iu v ’s. It can be seen that the rank of H is _i ^*_i precisely the number of non-zero singular values.

The SVD decomposition can be interpreted as two coordinate transformations: it says that if the input is expressed in terms of a coordinate system defined by the columns of V and the output is expressed in terms of a coordinate system defined by the columns of U, then the input-output relationship is very simple. The equivalence is summarized in Figure 3.2

V* U

Figure 3.2: Converting H into a parallel channel through the SVD

Signal for

Figure 3.3: Block diagram of proposed MIMO-OFDM system with adaptive beamforming

3.1.2 System Model

The system under consideration is given in Figure 3.3. We consider the downlink case. For each user k we want to design the transmit beamformer

w and receive

beamformer

w to null the interference between users occupying the same subcarrier.

We assume that each user and base station are equipped with N antennas.

We consider the downlink case. There are N subscribers in one cell. The signal transmitted for user k is S_k.

w and

w represent the transmit and receive beamforming vector of user k. The total transmit signal after beamforming can be written as

If the length of the cyclic prefix is longer than the maximum time dispersion of the channel response, then the channel appears to be flat on each subcarrier. Denote the channel matrix of user k by

where h_{i j}^k_, is the channel gain from the jth transmit antenna to the lth receive antenna.

The signal received at user k is

After receive beamforming, the signal received at user k becomes

3.2 Proposed Beamforming Scheme with Interference Nulling

In ZF algorithm, we hope that the gain of desired signal is normalized to 1 and the gain of the interference is zero. This concept can be written as matrix form as follows:

1 1 1 2 1

where the (m,n)th element of matrix is the gain of signal n received by user m. The above matrix equation can be decomposed and the right identity matrix can be viewed as a diagonal matrix in general form. In this reason, we rewrite the equation as

Let transmit beamformer be right singular vector of H (ex:

Ti = i

w v ). Therefore the

equation (3.9) becomes

1 1 1

w to satisfy the above

equation. One possible solution can be find by satisfying the equation

1 2

Consequently

The SNR_k after beamforming is given by

2 2 2

SNR = /( )

k αk wTk σn (3.14)

3.3 Computer Simulations of ZF Algorithm

Before giving the simulation results, we define the relation between SNR and

SNR noise power 1

s b t bandwidth and M is the modulation order. Throughout the following simulations, the system transmit power is normalized to 1, and hence the noise power corresponding to a specific E_b N₀ is generated by

This result is utilized in the following of this thesis.

The parameter of simulation is shown in Table 3.1. Here we assume the number of users in one cell is equal to the number of antennas they are equipped with which is not a reasonable assumption. In the next section, we propose a method to choose N users from the K users in the cell. With such a method, the above assumption becomes acceptable.

Table 3.1: Simulation parameters of ZF algorithm

Number of antennas 2/4

Number of users in one cell 2/4

Modulation order 2,4,8,16,32,64

Family of modulation QAM

Comparing Figure 3.4 and Figure 3.5, we can observe that while the number of transmit antennas increases, the rate of the system increases proportionally. However, the performance of the proposed algorithm gets worse. More users are allowed to transmit data at the same time, such that the level of influence of interference is higher.

Although the number of degrees of freedom (number of transmit antennas) also increases proportionally, the performance still degrades slightly.

0 5 10 15 20 25 30 10^-4

10^-3 10^-2 10^-1 10⁰

Eb/No

BER

2 users 2x2 ZF

4QAM 8QAM 16QAM 32QAM 64QAM

Figure 3.4: BER performances of 2x2 ZF algorithm

0 5 10 15 20 25 30

10^-4 10^-3 10^-2 10^-1 10⁰

Eb/No

BER

4 users 4x4 QAM

4 QAM 8 QAM 16 QAM 32 QAM 64 QAM

Figure 3.5: BER performances of 4x4 ZF algorithm

3.4 Algorithm of Subcarrier Selection with Beamforming

In the above section we introduce an algorithm of designing beamforming vector to null interference. With this approach applied, we can find a good method to choose subcarriers for each user to transmit data streams. In this section, we propose a method to maximize total transmission rate with each user’s BER constraint satisfied. The block diagram is shown in Figure 3.6. We also extend the single carrier problem to multicarrier problem.

Figure 3.6: Block diagram of beamforming aided subcarrier selection in single carrier case

From the gap approximation [40], the relation between the modulation order and the SNR under BER constraint can be expressed as

The above approximation is applied for M-QAM modulation. We find that this

approximation is tight to within 1 dB at the BER less than 10^-3. If the BER constraints are the same for all users, there exists a one-to-one mapping between the SNR and the modulation order. Therefore, the total number of transmit bits in one subcarrier can be written as

Observing the above equation, we find that the rate is proportional to the product of SNRs. For this reason, we choose the product of SNRs as the metric of the system performance. With ZF beamforming, the SNR of user k is ²/( ² ²)

k Tk n

α w σ which

makes the product of SNRs become ² ² ²

1( / )

the metric for choosing which users could occupy this subcarrier. The detail procedure is described in the following steps:

ZF Beamformer Aided Subcarrier Selection Algorithm

Step 1) Choose any N users from K total users. Calculate ² ²

1( / )

Step 2) Choose the maximal one. Let the selected users transmit data on this subcarrier.

Step 3) Do the above two steps for each subcarrier. Finally, the result of subcarrier allocation indicates which users can occupy each subcarrier. On the other hand, it also shows which subcarriers can be used by one user. In addition, the result makes each subcarrier occupied by N users.

3.5 Computer Simulations

Figure 3.7 shows the simulation results of the beamforming aided subcarrier selection algorithm. Here each user is equipped with two antennas such that each subcarrier only can be occupied by two users. The number of candidates is larger such that more multiuser diversity can be obtained. Channel model is assumed as independent Rayleigh fading channel. As shown in Figure 3.7, we find that when the number of candidates is larger, the BER curve behaves more like that under the AWGN channel. Because we always choose the two best users to occupy the subcarrier, the shape of distribution of selected path gain becomes narrow. As the number of candidates approaches infinity, the shape of distribution of selected channel fading gain becomes like an impulse. Hence its BER curve approaches that under the AWGN channel.

0 5 10 15 20 25 30

10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Eb/No

BER

2x2 16QAM ZF

2 users 4 users 6 users

Figure 3.7: BER performances of beamforming aided subcarrier selection with 2x2 antennas and 16QAM modulation

3.6 Summary

In this chapter, we propose a ZF beamforming design algorithm to null the interference between users in multiuser MIMO systems. SVD decomposition is utilized to simplify our design procedure. Finally, with the ZF beamforming design method, a subcarrier selection algorithm is proposed in order to maximize the total transmission rate under the BER constraint. The simulation result shows that the performance improves significantly due to multiuser diversity.

Chapter 4 Multiuser Adaptive Power and Bit Allocation for OFDM Systems

In multicarrier systems, a loading algorithm is used to allocate bits and power to subchannels. This task is a constrained optimization problem and two cases are of interest, namely, margin maximization and rate maximization.

Margin maximization is equivalent to power minimization subject to a fixed target rate; and rate maximization is equivalent to rate maximization subject to a fixed target power. In both cases, system specifications impose additional constraints including total power budget, power spectral density mask, maximum bit error rate, integer bit assignments, and maximum size of the embedded QAM constellations. Although there are several single-user bit-loading algorithms, not all of them are optimal under the integer bit constraint, and not all of the above constrains are encountered in the problems.

In [41]-[43], the bit-loading solution is generally noninteger; therefore, a suboptimal integer rounded bit allocation is provided. In [44]-[45], the discrete bit-loading problem is addressed using an iterative bi-filling or bit-removal procedure based on a subchannel-cost criterion for a total power budget constraint. We will review these methods in Section 4.2.1.

In this chapter, we propose a computationally efficient discrete bit-loading algorithm for single-user margin maximization problem. The new algorithm exploits the differences between the subchannel gain-to-noise ratios measured during channel training in order to calculate an initial bit allocation and then performs a bit-removal algorithm with fewer candidates to reach the optimal target-rate solution.

This chapter is organized as follows. Section 4.1 formulates the bit and power loading problem in mathematical form and describes the system model. In Section 4.2, we review the conventional method for solving bit and power loading problem. After that, we present the new bit loading algorithm in detail. Finally Section 4.3 gives the simulation result of the proposed algorithm.

4.1 System Model and Problem Formulation

The choice of constellations (and equivalently, of the allocated power) is considered for transmission over a set of parallel subchannels:

, 1

i i i i i

y = Pα x +n ≤ ≤i N (4.1) where E[x x_i ^*_j]=δ_ijand E[n n_i ^*_j]=σ δ . The CNR (channel gain to noise ratio) of the ² _ij

ith subchannel is given by CNR_i =α_i²/σ².

We want to minimize the total transmit power subject to satisfy user’s rate and BER requirements. The problem can be formulated as

1 req

req

minimize

subject to , {0,1,..., }, 1 BER BER

N i i

i i

b B b b i N

= = ∈ ∀ ≤ ≤

∑

^(4.2)

where b is the maximum number of bits that can be assigned in the system. In order

1.6 CNR

BER 0.2 exp( )

2^b 1

− × ×P

= ×

− (4.3)

where b means modulation order. This approximation is applied for M-QAM and the error is less than 1 dB at the BER less than 10^-3.

4.2 Power and Bit Allocation Algorithm

In this section the waterfilling solution from the view point of the capacity is first introduced. The modified form, gap approximation, is also recommended. After that, we review the conventional power and bit allocation algorithms and point out the drawbacks and advantages of the conventional algorithms. Finally, the proposed algorithm is described which combines the advantage of the conventional methods.

4.2.1 Conventional Power and Bit Allocation Algorithm

In this part, two kinds of famous conventional power and bit allocation algorithms are introduced. The first one is a capacity-like approach whose solution is continuous.

So the final bit distribution is rounded from the continuous waterfilling solution. The second one is a greedy algorithm. It tries to find the power change after adding or removing one bit on subcarriers. Find the minimal cost bit and to add or remove iteratively it until rate requirement is achieved.

4.2.1.1 Gap-Approximation with Water-Filling Solution

We start this part with well-known capacity achieving solution since the following methods are strongly based on it. From the landmark work by Shannon in 1948, the achievable information rate through a channel with a given SNR is given by

log₂(1+SNR) bits/transmission. For the set of parallel subchannels in Equation (4.1), the achievable information rate is given by the sum ₂

1log (1 )

i i

i₌ + pλ

∑

^{, and the}

capacity is given by the maximum achievable rate over all possible power allocation strategies { }p_i . The optimum power distribution is the well-known waterfilling solution:

max(0, 1 ) 1

P µ i N

= −α ≤ ≤ , (4.4)

where µ is the waterlevel chosen to satisfy the power constraint (the waterlevel can be alternatively chosen to satisfy rate requirement with minimum power)

To achieve the channel capacity, however, it is necessary to use ideal Gaussian codes, which are not practical for real systems, hence the need to employ simpler and more practical constellations such as QAM or pulse amplitude modulation (PAM).

So we use the gap approximation. The basic idea of the gap approximation is to avoid the need for ideal Gaussian codes inherent in the capacity-achieving solution.

Instead, a family of practical constellations, such as QAM or PAM, is employed. The number of bits that can be transmitted for a given family of constellations and a given probability of detection error Pe is approximately given by log (1 SNR / )₂ + Γ , where Γ ≥ is the gap which depends only on the family of constellations and on P1 e.. Interestingly, there is a constant gap between the Shannon capacity and the spectral efficiency of realistic constellations, which can be interpreted as a penalty for not using ideal Gaussian codes. For QAM constellations, for example, the number of bits at each subchannel is given by

log (12 ⁱ ⁱ)

P CNR

b = + ⋅

Γ . (4.5)

This expression is like the capacity expression. So the first conventional power and bit allocation algorithm solution is the waterfilling solution. The solution can be written as

max(0, 2) 1

P µ i N

= − Γ ≤ ≤ . (4.6)

This solution is continuous; hence, rounded distribution is the solution of the first conventional algorithm. This procedure is simple however the solution is not optimal.

The second conventional method is the greedy algorithm. From gap approximation we can find power increment with one bit increment per subchannel and power decrement with one bit remove per subchannel. The bit-filling algorithm starts from an initial all-zeros bit allocation (b_i =0 for 1≤ ≤i N) and adds one bit at a time to the subchannel that requires the minimum additional power until the target rate is achieved. On the other hand, the bit-removal algorithm starts from an initial maximum bit allocation and removes one bit at a time from the subchannel that saves the maximum power until the target rate is achieved.

The logarithmic rate expression in Equation (4.5) is a strictly increasing and concave function of Pi and vanishes at Pi=0. These conditions guarantee the optimality of the above greedy bit-filling and bit-removal methods. Moreover, it can be proved that both methods result to the same bit allocation. However, the computational load associated with each algorithm depends mainly on the requested target rate. If B_req is closer to the rate achieved by the b allocation, then bit-removal converges faster.

These methods are optimal but complex.

4.2.2 Proposed Power and Bit Allocation Algorithm with Low Complexity

In this section we will propose a new power and bit allocation algorithm which combines the advantages of the above two conventional algorithms which are optimal and computationally efficient.

0 5 10 15 20 25 0

1 2 3 4 5 6

Subcarriers sorted by CNR