|Year : 2010 | Volume
| Issue : 3 | Page : 175-178
Adaptive Modulation and Beam-forming for OFDM MIMO Systems with Rate-limited Feedback
Electronics and Radar Development Establishment, CV Raman Nagar, Bangalore, India
|Date of Web Publication||4-Aug-2010|
Electronics and Radar Development Establishment, CV Raman Nagar, Bangalore
| Abstract|| |
In this paper, an adaptive modulation and beam-forming strategy for orthogonal frequency division multiplexing (OFDM) multiple-input multiple-output (MIMO) systems with rate-limited feedback is presented. It utilizes the correlation between OFDM subcarriers to reduce the amount of feedback of channel states from a receiver to a transmitter. An expression for the correlation coefficient between the squared gains of channel taps of the OFDM subcarrier for a Rayleigh fading channel with uniform and exponential multipath profile is derived. The performance of the proposed feedback strategy is compared with the one proposed.
Keywords: Beam-forming, OFDM, MIMO
|How to cite this article:|
Deb D. Adaptive Modulation and Beam-forming for OFDM MIMO Systems with Rate-limited Feedback. IETE J Res 2010;56:175-8
| 1. Introduction|| |
It is known that the performance of multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems can be drastically improved by adapting transmission parameters, e.g., constellation size, power, and transmit beam-forming vector to the channel condition  . For incorporating these, the feedback of channel states (CS) to the transmitter becomes a necessity. In general, the information rate on the feedback link is limited. Designing a strategy for rate-limited feedback systems boils down to the design of a codebook. Results have been reported in the literature about different types of codebook design. Some rely on the codebook design based on the Grassmannian algorithm for beam-forming vector quantization , and others on the Lloyd algorithm for vector quantization of all the transmission parameters together  . The first method was studied for MIMO-OFDM systems but only the beam-forming vector was considered for adaptation. In order to reduce the amount of feedback, the information about the beam-forming vectors of all the subcarriers was not sent to the transmitter; rather highly correlated subcarriers were grouped together and representative beam-forming vectors were fed back to the transmitter. In the Llyod algorithm-based design, the beam-forming vector, constellation size, and power allocation were optimized and quantized jointly and were fed back thereon  . But this method was studied only for flat fading channel. The design of the feedback strategy for OFDM systems based on joint optimization of transmission parameters is not studied in the literature.
In this paper, the codebook design methodology for OFDM-MIMO systems is studied, wherein the work done on joint optimization of transmission parameters for a flat fading rate-limited MIMO system is extended to a rate-limited OFDM-MIMO system. As suggested in  the correlation between OFDM subcarriers is used to reduce feedback requirement. An expression for the correlation co-efficient between the squared gains of the channel taps of OFDM subcarriers for the Rayleigh fading channel with uniform and exponential multipath intensity profile is also derived.
The remainder of the paper is organized as follows: Section 2 presents the system model used. Section 3 narrates the strategy for the design of the codebook for OFDM systems. Section 4 presents the expression of the correlation coefficient between the squared gains of the channel taps of OFDM subcarriers. Section 5 gives the simulation results and a comparison between the proposed technique and the one already existing in the literature. Finally, a conclusion is drawn in Section 6. 
| 2. System Model|| |
Let us consider an L-tap time and a frequency selective AWGN channel; then
where L is the length of the channel impulse response specified by h 0 , ..., h L-1, x(m) is the input signal, y(m) is the received signal, and v(m) is white noise with CN (0,N0 ). The channel taps in the frequency domain are obtained by passing this L-tap long channel impulse response through a N point DFT, where N is the number of subcarriers of the OFDM system and H(n) denotes the channel tap of the nth OFDM subcarrier. Incorporating adaptive transmission parameters, the input output relation for the nth OFDM subcarrier can be written using 1 as
where s(n) is an information symbol from an appropriate signal constellation of size M, w is the beam-forming vector with unit norm ║ w ║=1, P denotes transmit power, Y(n) is the signal received by the nth OFDM subcarrier, [.]H denotes the Hermitian operator, and v(n) is white noise with CN (0,N0 ).
| 3. Codebook Design Methodology|| |
As mentioned earlier, in this paper, the codebook design methodology developed by Giannakis is adapted for OFDM systems  .
For designing a codebook, the training vectors with the same distribution as that of H(n)s are generated. The distribution of H(n) is derived in  . With these training vectors, a codebook for a flat fading system is designed with a specified BER following the steps in  .
In this paper, this codebook is adapted for OFDM systems. To exploit the correlation between the neighboring OFDM subcarriers, the subcarriers are grouped into blocks to reduce the feedback requirement. The channel is considered to have a binary correlation model, i.e., if the value of the correlation coefficient between two subcarriers is found to be greater than or equal to a threshold γ, then the subcarriers are considered to have the same channel state and are put in the same group. Moreover, the beam-forming vectors of subcarriers within the block will also be correlated as the channel states are correlated. Within the block, the channel states are represented by the channel state of a representative OFDM subcarrier of that block. Based on channel states of the representative OFDM subcarrier, the constellation size, power allocated, and beam-forming vector are changed at the transmitter for the subcarriers.
The assumed binary correlation model induces a loss in system performance. To compensate for this, the specified BER is scaled by a factor α and the codebook design is iterated with this scaled BER requirement and the process continues until the system meets the BER performance. The value of α is estimated through Monte Carlo simulation for a given value of threshold γ. The values of α, γ and the group size are interdependent.
The algorithm is described below.
Step 1. Evaluate ρ(d) for d =1, 2, ..., N, where ρ(d) denotes the correlation between kth and mth carrier with
d =│k-m│. Find the group size from ρ(d) for a given γ
Step 2. Set α =1.
Step 3. Generate training vectors for a single OFDM carrier.
Step 4. Set BERm = αBER.
Step 5. Design the codebook based on the Giannakis method with target BER as BERm .
Step 6. Generate channel taps for N OFDM subcarriers.
Step 7. Evaluate the BER for these channel taps with the designed codebook.
Step 8. Is the BER obtained in step 7 meets the specified BER? If yes, go to step 10, else continue.
Step 9. α = α + δα; go to step 4.
Step 10. Designed codebook is taken as the final codebook.
In the next section the correlation coefficient between the squared gains of OFDM subcarriers is derived.
| 4. Correlation Between OFDM Subcarriers|| |
For the purpose of evaluating the correlation coefficient between OFDM subcarriers, the exponential multipath intensity profile (MIP) is considered and the amplitude of the channel taps is considered to have Rayleigh distribution. For other types of MIP and intensity distribution, the correlation coefficient can be found numerically.
Let us derive the correlation coefficient for the exponential MIP, i.e., E(/h(k)/2 )= E(/h(0)/2 )e-kδ where δ is a constant  . It can be noted that the correlation coefficient for the uniform multipath intensity profile can be derived by setting are considered to have zero mean and σ2 variance.
Let H(n) be the complex channel tap of the nth OFDM subcarrier and ρ denote the correlation co-efficient between the squared gain of the kth and mth OFDM subcarriers, then
Let us denote the numerator by N u, the denominator by D(k) D(m) and introduce the intermediate variable φik =2πik/N, where i can take any integer value between 0 and L-1. Now rewrite this equation as
where D (k) and D (m) are obtained by replacing n by k and m, respectively, in the above equation. These analytical expressions are used to evaluate the correlation between OFDM subcarriers in Section 5.
| 5. Simulation Results|| |
For the purpose of simulation, the channel is considered to have Rayleigh fading, the number of OFDM subcarriers is taken to be 64, and the number of taps is taken to be 8. The correlation coefficient between the squared channel gains of SISO OFDM subcarriers is evaluated from the analytical expression derived in Section 4 and is compared with the simulation results. [Figure 1] shows the correlation coefficient between the squared gain of OFDM SISO subcarriers from analytical expression as well as simulation.
The performances of both the proposed and existing feedback schemes are compared in terms of spectral efficiency for a given average power, target ABER, and number of feedback bits. The target BER is taken to be 1e-3. For each representative subcarrier, 2 bits are used for feedback and γ is taken to be 0.8. The simulation is done for a 2Χ1 system. The channels are spatially correlated with the correlation coefficient value to be 0.7. [Figure 2] shows the comparison of the existing technique with the proposed one. The proposed technique shows improvement over the existing technique. There are two points in the graph where both the existing and proposed methods perform the same; this is because of the reason that at those points the constellation sizes for all the partitions came out to be the same from the codebook design, so there was no adaptive modulation for those cases.
Further, the performance of the proposed algorithm can be improved if extra-restriction is put on transmission based on the effective gain of the channel  . Thus for example, no transmission is carried out if the gain in the direction of the beam-forming vector of the channel is less than some predefined value called the outage amplitude. It is worth to suspend transmission if the channel is in deep fade.
| 6. Conclusion|| |
The proposed technique achieves a reduction in the amount of feedback requirement by exploiting the correlation between the subcarriers. The performance of the proposed technique is also compared with the existing technique given in  where the correlation between the subcarriers was exploited but there was no adaptive modulation. The comparison shows that the proposed technique offers considerable improvement in terms of spectral efficiency.
The performance of the proposed technique can be further improved by using a more sophisticated correlation model instead of a binary correlation model. In that model, the constellation size and the beam-forming vector of OFDM subcarriers within the block can be interpolated using the information of the representative carrier. This will reduce the loss in performance because of the assumption of the binary correlation model. This remains as an open research problem.
| References|| |
|1.||J Choi, and R W Heath Jr, Interpolation Based Transmit Beamforming for MIMO-OFDM With Limited Feedback, IEEE Trans. on Signal Processing, Vol. 53, No. 11, Nov 2005. |
|2.||D J Love, R W Heath Jr, and T Strohmer, Grassmannian beam-forming for multiple-input multiple-output wireless systems, IEEE Trans. Inf. Theory, Vol. 49, No. 10, pp. 27352747, Oct. 2003. |
|3.|| P Xia, S Zhou, and G B Giannakis, Multiantenna Adaptive Modulation With beam-forming based on bandwidth-bonstrained feedback, IEEE Trans. Commun, Vol. 53, No. 3, Mar. 2005. |
|4.||K Zhang, Z Song, and Y L Guan, Simulation of Nakagami Fading Channels With Arbitrary Cross-Correlation and Fading Parameters, IEEE Trans. on Wireless Comm., Vol. 3, No. 5, Sep 2004. |
|5.||Z Kang, K Yao, F Lorenzelli, Nakagami-m Fading Modeling in the Frequency Domain for OFDM System Analysis, IEEE Comm. Letters, Vol. 7, No. 10, Oct. 2003. |
| Authors|| |
Debasish Deb received his B.E degree from R.E.C, Silchar and M.Tech from IIT Kanpur. He is working with Electronics and Radar Development Establishment since 1999. His field of interest includes radar system design, algorithm design for radar and communication systems. He is currently working in the field of MIMO systems.
[Figure 1], [Figure 2]