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**CHAPTERTWO ****CHANNEL MODELLING: AN OVERVIEW**

**MIMO COMMUNICATION SYSTEM MODEL**

A block diagram for a generalized MIMO wireless communication system can be represented as that shown in Figure 2.1, where the system is divided into the space-time coding and signal processing parts and then multiplexed onto the N transmitter elements and finally through the channel. It is assumed that a set of Q independent data streams represented by the symbol vector b(n)is encoded into NT discrete-time complex baseband streamsx(n) at the transmitter as described in [19], where n is the time index. These input symbols are then converted into discrete-time samples by coding in space across the NT outputs and over time through sampling. This is then converted into a continuous-time baseband waveform, X(ω), with ω being the frequency. This signal is then up-converted and fed into the NT antennas after any filtering and amplification. At the receiver the signal is combined from the NR channels to produce the continuous output vector waveform Y(ω), which is then matched-filtered and sampled to produce the discrete-time baseband sample stream y(n). This y (n) signal is finally space-time decoded to generate the estimated transmitted symbols, bˆ(n) for the Q independent data streams.

Hence the baseband MIMO input-output relationship can be expressed as y(t) = H(t) ∗ x(t) + n(t) (2.1) where n(t) is additive white Gaussian noise (AWGN) and ∗ denotes the convolution. H(t) is an N×M channel impulse response matrix. Equation (2.1) can also be written in the frequency Each element Hij (ω) represents the transfer function between the j th transmit and ith receive antenna. If the signal bandwidth is sufficiently narrow so that the channel response is constant over the system bandwidth (frequency flat channel), or when the wideband signals can be divided into narrowband frequency bins and processed independently, then equation (2.2) can be written as y = Hx + n (2.3) where H is the narrowband MIMO channel matrix. In order to study the MIMO channel capacity, the elements of the narrowband MIMO channel matrix is assumed to be independent, identically distributed (iid) and at best, the data streams Q, would be equal to the rank of H. However, in real MIMO systems factors such as antenna impedance matching, antenna element spacing, polarization properties and degree of scattering influence these and affect the system performance.

**MIMO SYSTEM CAPACITY**

In the single-input-single-output (SISO) system, the ergodic (mean) capacity of a random channel with NT = NR = 1 and an average transmit power constrained by PT can be written

as where P is the average power of a single channel codeword transmitted over the channel and EH denotes the expectation over all channel realizations. The channel is thus defined [20] as the maximum of the mutual information between the input and the output over all statistical distributions on the input, that satisfy the power constraint. If each symbol at the transmitter is denoted by b, the average power constraint can be written as Using (2.4), the ergodic (mean) capacity of a SISO system (NT = NR = 1) with a random complex channel gain h11 is given in ( [15]) as where ρ is the average signal-to-noise ratio (SNR) at the receiver branch. If |h11| is Rayleigh, |h11|2 follows the chi-squared distribution with two degrees of freedom [21]. Thus equation (2.6) can be written as [15] where χ2 is a chi-squared distribution random variable with two degrees of freedom. Thus, using equation (2.7) one can show as in Figure 2.2 the Shannon capacity for a Gaussian channel and the capacity of a Rayleigh fading channel for a SISO channel, where the Rayleigh fading plot is as in [22]. For the MIMO system we can write the capacity of a random channel with power constraint PT and transmit vector X(ω), whose elements are complex Gaussian-distributed random variables, as where Rx = E ©xxHª is the diagonal elements of the transmit covariance representing the transmit power from each antenna, tr(.) denotes the trace, PT is the total transmit power limited by the constraint tr(R) ≤ PT and E{·} is the expectation.

**CHAPTER ONE – INTRODUCTION **

**1.1 Background and Motivation**

**1.2 Author’s Contributions and Outputs **

1.2.1 Research Contribution

1.2.2 Journal Publications

1.2.3 Conference Proceedings

1.2.4 Invited Paper

1.2.5 Additional Contributions

**1.3 Outline of Thesis**

**CHAPTER TWO – CHANNEL MODELLING: AN OVERVIEW **

**2.1 MIMO Communication System Model**

**2.2 MIMO System Capacity**

2.2.1 Water-Filling Capacity

2.2.2 Uninformed Transmitter Capacity

2.2.3 Diversity and Spatial Multiplexing

**2.3 Multipath Characterization**

2.3.1 Beamforming

2.3.2 Bartlett Beamformer

2.3.3 Capon Beamformer

2.3.4 Double-Directional Channel Model

2.3.5 Ray Tracing

2.3.6 Geometric Models

**2.4 Conclusion **

**CHAPTER THREE – GEOMETRIC MODELLING**

**3.1 Introduction**

**3.2 Model Description **

**3.3 Model Analysis **

**3.4 Results**

**3.5 Conclusion**

**CHAPTER FOUR – WIDEBAND MIMO MEASUREMENT SYSTEM **

**4.1 Introduction **

**4.2 System Overview**

**4.3 System Components **

4.3.1 Transmitter Subsystem

4.3.2 Receiver Subsystem

4.3.3 Synchronization Module

4.3.4 Monopole Antennas

**4.4 System Deployment **

4.4.1 Wideband Probing

4.4.2 Calibration Procedure

4.4.3 Measurement Environment

4.4.4 Data Collection

**4.5 Conclusion**

**CHAPTER FIVE – DATA ANALYSIS AND MODEL ASSESSMENT**

**5.1 Capacity Modelling **

5.1.1 Introduction

5.1.2 Model Description

5.1.3 Results

**5.2 Modelling Spatial Correlation **

5.2.1 Introduction

5.2.2 Model Description

5.2.3 Results

**5.3 Double Directional Channel Modelling .**

5.3.1 Introduction

5.3.2 Model Description

5.3.3 Results

**5.4 Conclusion**

5.4.1 Capacity Modelling

5.4.2 Spatial Correlation

5.4.3 Double Directional Channel

**CHAPTER SIX – MAXIMUM ENTROPY MODELLING **

**6.1 Introduction**

**6.2 Model Description **

**6.3 Data Processing **

**6.4 Results**

**6.5 Conclusion**

**CHAPTER SEVEN – CONCLUSION**

**7.1 Summary **

**7.2 Future Recommendations**

**REFERENCES **

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MIMO CHANNEL MODELLING FOR INDOOR WIRELESS COMMUNICATIONS