信道状态信息(CSIT)已知的MIMO信道容量
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Project ID: P7 Author: Likai Ma z3326280 Yi Chen z3457081
Abstract— This project is inspired by the desire to study the underlying theory regarding capacity of MIMO channels with CSIT and investigate how channel matrices with different ranks will influence the resulting channel throughput. MATLAB simulations for various cases of channel matrices will be performed so as to verify that the rank of the channel matrix is in fact the decisive factor for channel capacity with CSIT. Moreover, the results obtained through both water-filling method and uniform power allocation will be compared in order to demonstrate that water-filling solution is indeed the optimal solution for power allocation when channel state information is known at transmitter.
A. MIMO CHANNEL MODEL [5] A MIMO system consisting of ������ transmit and ������ receive antennas is shown as in Figure 1.
1
Figure 1. Systematic diagram of a MIMO system
(1)
������ denotes the ������ × ������ channel matrix that takes the
following form
������ =
ℎ11 ⋯ ℎ1������
ℎ21 ⋯ ℎ2������ ⋮⋱ ⋮
ℎ������1 ⋯ ℎ������������
������ and ������ are the transmit and receive vectors with
Explanations
AWGN
Additive white Gaussian noise
SVD
Singular value decomposition
CSIT
Channel state information known at transmitter
Table 1. Table of abbreviations
(3)
where ������2 denotes the identical noise power at each of
the antennas on the receiver side.
B. CAPACITY FOR AWGN CHANNELS The information capacity of a Gaussian channel with power constraint ������ is defined as the maximum mutual information of random variables ������ and ������, which is shown by the following expression
������ = max ������(������; ������)
(4)
������ ������ :������ ������2 ≤������
The channel capacity of an AWGN channel with power constraint ������ and noise power ������2 is given by Equation (5)1
II. TABLE OF ABBREVIATIONS The following table (Table 1) gives a set of abbreviations used throughout the following sections of the report.
Abbreviations
However, the availability of spectral resources creates a limit upon the maximal achievable transmission rate. In recent years, MIMO has attracted considerable attention to researchers and engineers due to the significant advantage it demonstrates. The most compelling property that MIMO exhibits is that the theoretical channel capacity increases almost linearly with the number of transmit or receive antennas under ideal conditions.
complex AWGN vector with independent and equal
variance in both real and imaginary parts. Its
covariance matrix is thus given by
ຫໍສະໝຸດ Baidu
������ ������������������ = ������2������������
������ = max
������
1 2
log(1
+
������������ ������������2
)
(6)
with power constraint
������������ = ������
������
Figure 2. Parallel Gaussian channels
The optimal power allocation such that the above equation attains its maximum can be derived from applying so called Lagrange multiplier method and is given by the following
The following sections of this report are organized into six separate sections. In Section II, a number of
commonly defined and used abbreviations will be listed to the reader. In Section III, the system model of MIMO and the related mathematical theories will be presented. Section IV demonstrates the simulation results and provide them with detailed comments so as to explain what implications those results provide. A brief conclusion about the work of this project will be drawn in Section V. In Section VI and VII, namely the Reference and Appendix sections, the associated literature and MATLAB code will be presented respectively.
TELE 9754 Coding and Information Theory
Final Project Report
Project ID P7
Project Topic CAPACITY OF MIMO CHANNELS WITH KNOWN CHANNEL STATE INFORMATION AT TRANSMITTER (CSIT)
A MIMO system with ������ transmit and ������ receive
antennas is commonly represented by a linear
relationship that is given by the following equation
������ = ������������ + ������
III. BACKGROUND THEORY The analysis of MIMO system model and channel capacity requires knowledge across different areas, including linear algebra, probability theory and information theory. Some important and fundamental mathematical definitions and derivations associated with MIMO channel will be presented in this section.
Author Name Likai Ma Yi Chen
Student ID z3326280 z3457081
E-mail malikai0409@yeah.net 715436908@qq.com
CAPACITY OF MIMO CHANNELS WITH KNOWN CHANNEL STATE INFORMATION AT TRANSMITTER (CSIT)
their covariance matrices given by the following
expressions
������ ������������������ = ������������, ������ ������������������ = ������������
(2)
������ is a statistically independent zero-mean and
1
������
������ = 2 log(1 + σ2) nats
(5)
When there are multiple parallel AWGN channels,
shown in Figure 2, the overall channel capacity will be
given by maximising the total capacity, that is
I. INTRODUCTION Wireless communication has undergone developments over 2G, 3G and the newly adopted 4G with progressively increasing data transmission rates. A wide range of activities such as internet browsing, multimedia and etc which are otherwise unavailable in the past have been made possible on today’s mobile devices.
Abstract— This project is inspired by the desire to study the underlying theory regarding capacity of MIMO channels with CSIT and investigate how channel matrices with different ranks will influence the resulting channel throughput. MATLAB simulations for various cases of channel matrices will be performed so as to verify that the rank of the channel matrix is in fact the decisive factor for channel capacity with CSIT. Moreover, the results obtained through both water-filling method and uniform power allocation will be compared in order to demonstrate that water-filling solution is indeed the optimal solution for power allocation when channel state information is known at transmitter.
A. MIMO CHANNEL MODEL [5] A MIMO system consisting of ������ transmit and ������ receive antennas is shown as in Figure 1.
1
Figure 1. Systematic diagram of a MIMO system
(1)
������ denotes the ������ × ������ channel matrix that takes the
following form
������ =
ℎ11 ⋯ ℎ1������
ℎ21 ⋯ ℎ2������ ⋮⋱ ⋮
ℎ������1 ⋯ ℎ������������
������ and ������ are the transmit and receive vectors with
Explanations
AWGN
Additive white Gaussian noise
SVD
Singular value decomposition
CSIT
Channel state information known at transmitter
Table 1. Table of abbreviations
(3)
where ������2 denotes the identical noise power at each of
the antennas on the receiver side.
B. CAPACITY FOR AWGN CHANNELS The information capacity of a Gaussian channel with power constraint ������ is defined as the maximum mutual information of random variables ������ and ������, which is shown by the following expression
������ = max ������(������; ������)
(4)
������ ������ :������ ������2 ≤������
The channel capacity of an AWGN channel with power constraint ������ and noise power ������2 is given by Equation (5)1
II. TABLE OF ABBREVIATIONS The following table (Table 1) gives a set of abbreviations used throughout the following sections of the report.
Abbreviations
However, the availability of spectral resources creates a limit upon the maximal achievable transmission rate. In recent years, MIMO has attracted considerable attention to researchers and engineers due to the significant advantage it demonstrates. The most compelling property that MIMO exhibits is that the theoretical channel capacity increases almost linearly with the number of transmit or receive antennas under ideal conditions.
complex AWGN vector with independent and equal
variance in both real and imaginary parts. Its
covariance matrix is thus given by
ຫໍສະໝຸດ Baidu
������ ������������������ = ������2������������
������ = max
������
1 2
log(1
+
������������ ������������2
)
(6)
with power constraint
������������ = ������
������
Figure 2. Parallel Gaussian channels
The optimal power allocation such that the above equation attains its maximum can be derived from applying so called Lagrange multiplier method and is given by the following
The following sections of this report are organized into six separate sections. In Section II, a number of
commonly defined and used abbreviations will be listed to the reader. In Section III, the system model of MIMO and the related mathematical theories will be presented. Section IV demonstrates the simulation results and provide them with detailed comments so as to explain what implications those results provide. A brief conclusion about the work of this project will be drawn in Section V. In Section VI and VII, namely the Reference and Appendix sections, the associated literature and MATLAB code will be presented respectively.
TELE 9754 Coding and Information Theory
Final Project Report
Project ID P7
Project Topic CAPACITY OF MIMO CHANNELS WITH KNOWN CHANNEL STATE INFORMATION AT TRANSMITTER (CSIT)
A MIMO system with ������ transmit and ������ receive
antennas is commonly represented by a linear
relationship that is given by the following equation
������ = ������������ + ������
III. BACKGROUND THEORY The analysis of MIMO system model and channel capacity requires knowledge across different areas, including linear algebra, probability theory and information theory. Some important and fundamental mathematical definitions and derivations associated with MIMO channel will be presented in this section.
Author Name Likai Ma Yi Chen
Student ID z3326280 z3457081
E-mail malikai0409@yeah.net 715436908@qq.com
CAPACITY OF MIMO CHANNELS WITH KNOWN CHANNEL STATE INFORMATION AT TRANSMITTER (CSIT)
their covariance matrices given by the following
expressions
������ ������������������ = ������������, ������ ������������������ = ������������
(2)
������ is a statistically independent zero-mean and
1
������
������ = 2 log(1 + σ2) nats
(5)
When there are multiple parallel AWGN channels,
shown in Figure 2, the overall channel capacity will be
given by maximising the total capacity, that is
I. INTRODUCTION Wireless communication has undergone developments over 2G, 3G and the newly adopted 4G with progressively increasing data transmission rates. A wide range of activities such as internet browsing, multimedia and etc which are otherwise unavailable in the past have been made possible on today’s mobile devices.