Simulation Results for The Combined Fast BB and The Box-

We consider a high-loaded scenario with K = 28 and SF = 31 in an AWGN channel. The spreading sequences are randomly generated in every simulation trial. Fig.7.5 compares the performance of the combined BB-DCD detector with various complexity thresholds, with that of the box-constrained DCD detector, and the fast BB detector in terms of group detection error (GDE), which represents the probability that at least one user symbol is incorrect. As the DF detector is one of the simplest receivers, the detection performance of the DF detector is also shown under identical conditions for the purpose of compari- son. It can be seen that the DF detector provides a poor performance in the highly-loaded scenario. The box-constrained DCD detector gives a better performance but not as good as that of the fast BB detector. The BB-DCD detector with the threshold Tr = 104 _offers a better performance than the box-constrained DCD detector. With the increase of the complexity threshold by Tr = 2 × 104_{, Tr = 5 × 10}4 _{and Tr = 10 × 10}4_{, the GDE per-} formance of the BB-DCD detector is improved and approaches the optimal performance. For the complexity threshold of Tr = 5 × 104_{FLOPS, the difference compared to the fast} BB detector in the performance is 0.1 dB at GDE = 10−5_.

CHAPTER 7. COMBINED MULTI-USER DETECTION WITH IMPROVED “COMPLEXITY-DETECTION” PERFORMANCE 125 0 5 10 15 20 10−4 10−3 10−2 10−1 100 SNR,dB

Group Detection Error

DF Box−constrained DCD BB−DCD (Tr=104) BB−DCD (Tr=2×104) BB−DCD (Tr=5×104) BB−DCD (Tr=10×104) fast BB

Figure 7.5: Group detection error of the BBD-DCD vs. box-constrained DCD, fast BB and DF; K = 28, SF = 31.

Fig.7.6 shows the worst-case complexity of the combined detector for this scenario. Though the DF detector offers the lowest complexity compared with other detectors, it performs poorly for all SNRs. The box-constrained DCD detector shows a nearly con- stant complexity for all SNRs. The worst-case complexity of the fast BB detector in- creases when the SNR decreases. At SNR = 0 dB, its worst-case complexity is 1000 times of complexity of the box-constrained DCD detector. It can be seen that the worst- case complexity of the BB-DCD detector increases when the threshold increases by Tr = 104_{, Tr = 2 × 10}4_{, Tr = 5 × 10}4_{, Tr = 10 × 10}4 _{for SNR ≤ 13dB and is close to the} complexity of the fast BB detector for SNR ≥ 13 dB.

7.6 Conclusions

The fast BB detector provides the optimal detection performance. However, the worst- case computational complexity of the fast BB detector is prohibitive, which makes the fast BB algorithm difficult for real-time application. The box-constrained DCD algorithm

CHAPTER 7. COMBINED MULTI-USER DETECTION WITH IMPROVED “COMPLEXITY-DETECTION” PERFORMANCE 126 0 5 10 15 20 103 104 105 106 107 SNR,dB

Worst Computational Complexity (FLOPS)

DF Box−constrained DCD BB−DCD (Tr=104) BB−DCD (Tr=2×104) BB−DCD (Tr=5×104) BB−DCD (Tr=10×104) fast BB

Figure 7.6: Worst-case complexity of the BB-DCD detector vs. the box-constrained DCD, fast BB and DF detectors; K = 28, SF = 31.

shows a low computational complexity at any value of SNR. However, its detection per- formance is inferior to that of the fast BB detector. In order to give a good trade-off between the performance and the complexity, we proposed a combination of the fast BB detector and the box-constrained DCD detector. This approach has shown that in a highly loaded scenario, the complexity of the BB-DCD detector is reduced significantly with only a small loss in detection performance with respect to the ML detector, while still better than that of the box-constrained DCD detector.

Chapter 8

The DCD Based Simplified Matrix

Inversion for OFDM symbols

Contents

8.1 Introduction . . . 127 8.2 MIMO-OFDM Model . . . 128 8.3 MIMO OFDM Detection Scheme . . . 129 8.4 DCD-based Inversion for MIMO-OFDM Systems . . . 130 8.5 Simulation Results . . . 133 8.6 Discussion on application of the DCD based matrix inversion . . . . 139 8.7 Conclusions . . . 143

8.1 Introduction

Multiple Input Multiple Output (MIMO) systems have received much attention in recent years as a promising method for next-generation communication systems, because the use of multiple transmit and receive antennas significantly increases the system capacity and diversity [115]. The use of orthogonal frequency-division multiplexing (OFDM) drasti- cally simplifies receiver design in MIMO wireless systems when the channel is frequency selective [116]. In such systems, the linear MIMO receivers including zero-forcing (ZF) and minimum mean square error (MMSE) receivers need to perform channel correlation

CHAPTER 8. THE DCD BASED SIMPLIFIED MATRIX INVERSION FOR OFDM SYMBOLS 128

inversion. The MMSE detector, however, requires a high computational complexity. The main complexity of the MMSE-based detector is for the inversion calculation. In this chapter, we present an approach based on the DCD algorithm to simplify the inversion operations in MIMO-OFDM systems. The idea of the approach is that the DCD algo- rithm obtains separately the individual columns of the inverse of the matrix. Due to the low complexity of hardware implementation of the DCD algorithm, a block of DCD pro- cessors can be used to obtain the columns of the inverse of the channel correlation matrix in parallel.

In slow fading channels, the DCD-based inverse of the channel correlation matrix needs only to be performed in one OFDM symbol, and then the same solution can be used for other symbols over an OFDM frame. However, in fast fading channels, the DCD-based inverse needs to be performed separately for each symbol in the OFDM frame.

In the frequency-selective channels, the channel is decomposed into parallel flat channels in the frequency domain. The change of the channel frequency response can be slow between neighbouring subcarriers especially if the multipath delay spread is significantly smaller than the duration of an OFDM symbol. The DCD-based inverse of the channel correlation matrix obtained from the first-subcarrier can be used as an initialization for that of the second subcarrier, and so on, up to that of the last sub-carrier. This reduces the complexity compared to the case where the inverse of the channel correlation matrix is initialized to zero for each subcarrier.

This chapter is organised as follows. In Section 8.2, the MIMO-OFDM model is pre- sented. In Section 8.3, two detectors which require matrix inversion, are introduced. In Section 8.4, the DCD-based matrix inversion for MIMO-OFDM symbol is proposed. Var- ious situation results of this inversion approach are presented in Section 8.5. In Section 8.6, the DCD-based inversion is applied to the underwater acoustic communication. Sec- tion 8.7 concludes the chapter.