Table of contents
Table of contents
The theory behind MIMO
1.1 Introduction
1.2 Channel models and diversity
1.2.1 Space diversity in the channel
1.2.2 Polarization diversity
1.2.3 Fading and fading models
1.2.4 The Nakagami m fading channel
1.2.5 Estimation of the m-parameter
1.3 Channel capacity
1.3.1 Maximum capacity, if there is no CSI at the transmitter
1.3.2 STBC, a method to achieve close to maximum capacity
1.3.3 Maximum capacity, if the channel is known to the transmitter
1.3.4 Beamforming
1.4 Mutual coupling
1.5 A novel method to increase the capacity…
MIMO measurement setup
2.1 Hardware
2.2 Calibration
2.3 Measurement environment
2.3.1 Description of the measurement locations
2.4 Antenna array configurations
2.5 The retrodirective antenna measurements
Results of the measurements in a static channel
3.1 Line of sight measurement with a linear array with all vertical polarization
3.2 Line of sight linear array with two elements horizontally polarized
3.3 LOS circular array with all elements vertically polarized
3.4 NLOS linear array with all elements in vertical polarization.
3.5 LOS frequency sweep with linear array and vertical polarization.
3.6 NLOS frequency sweep with linear array and vertical polarization.
3.7 Conclusions of measurements in static channel
Results of the measurements in a non-static channel
3.8 LOS linear array and vertical polarization
3.9 NLOS linear array and vertical polarization
3.10 Conclusions of measurements in a non-static channel
3.11 Measurement with retrodirective antenna in LOS non-static channel
3.12 Measurement with retrodirective antenna in NLOS case in a static channel
3.13 Conclusion of measurements with retrodirective in a non-static channel
Conclusions of MIMO measurements
4.1 When is CSI most usefulness at the transmitter?
4.2 What are the effects of mutual coupling at the antenna arrays?
4.3 How useful are the Nakagami m distribution?
4.4 When can a retrodirective antenna be useful in a MIMO channel?
Appendix A
Acknowledgement
References......
The theory behind MIMO
1.1 Introduction
In the never-ending search for increased capacity in a wireless communication channel it has been shown that by using an MIMO (Multiple Input Multiple Output) system architecture it is possible to increase that capacity substantially. Usually fading is considered as a problem in wireless communication but MIMO channels uses the fading to increase the capacity.
MIMO systems transmits different signals from each transmit element so that the receiving antenna array receives a superposition of all the transmitted signals. All signals are transmitted from all elements once and the receiver solves a linear equation system to demodulate the message. The idea is that since the receiver detects the same signal several times at different positions in space at least one position should not be in a fading dip.
If the transmitter have CSI (Channel State Information) then the transmitter can use the “Waterfilling technique” (see section 1.3.4) to optimize the power allocation between the antenna elements so that an optimal capacity is achieved.
When the CSI is supplied to the transmitter a decrease in spectral efficiency is unavoidable so therefore it is interesting to know in what cases it is important to have CSI and when the benefits are negligible. This will be answered after a series of measurements.
Consider two transmitting antennas where the first antenna is transmitting and the second does not. The electro-magnetic wave from the first antenna will induce a voltage in the other antenna and then the other antenna will also transmit a signal and so on, this is called “Mutual coupling”. The effects of Mutual coupling on the capacity will be investigated in this thesis.
The Rayleigh-distribution is a well-known estimation of the PDF (Probability Density Function) of the fading statistics in a radio channel. In this thesis another distribution will be used that is called the “Nakagami m distribution”. The Nakagami m distribution has different shapes depending on the m-value and for m=1 it equals the Rayleigh distribution. Two different ways to estimate the m-parameter is presented (see section 1.2.5) and by measurements it will be shown if the Nakagami m distribution is a good way to estimate the PDF of the fading statistics or not.
Since the MIMO system architecture uses the independent fading between different antenna-elements perhaps it could be possible to increase the independent fading by using some sort of mixer in the channel so that the channel doesn’t get stuck in a stat of low diversity gain. There will be some experiments made with a retrodirective antenna that should work as a mixer.
This thesis is divided into four chapters with the contents listed below:
- Explaining the theories that has been used in the thesis
- Explaining the measurement setup and the hardware that has been used.
- Presenting the results from the different measurements and making some conclusions.
- Answering the four questions: In which cases are CSI at the transmitter most useful? Can the effects of mutual coupling increase the capacity? How useful is the Nakagami m distribution and is it difficult to estimate the m parameter? Can a retrodirective antenna be used to increase the independency of the fading channel?
1.2 Channel models and diversity
In this section some assumptions will be made about the channel that makes it possible to calculate the capacity for the channel. The H matrix will be introduced, the H matrix represent the complex channel gain, which incorporates the fading influence of the channel. The H matrix will be normalized in a way that makes it possible to compare the capacity for the MIMO system with a SISO system.
Consider a communication link with nT transmitting antennas and nR receiving antennas. Some important assumptions are made:
There is only a single user transmitting at any given time, so the received signal is corrupted by AWGN (Additive White Gaussian Noise) only.
The communication is carried out in packets that are of shorter timespan then the coherence time of the channel. This means that the channel is constant during the transmission of a packet.
The channel fading is frequency-flat. This means that the channel gain can be represented by a complex number. This also means that the transmission is very narrowband and the complex number, which represents the fading, is constant over the bandwidth.
All of these assumptions will be proven to be adequate by measurements later.
With these assumptions we can use this mathematical model
(1)
Where rt=is the received signal at time instant t, stis the sent signal and vt is AWGN with unit variance and uncorrelated between the nr receiver antennas. Receiver antenna i receive a superposition of every sent messages from transmitter j, weighted by the channel response and some AWGN are added.
The nRnT transition matrix is made up of elements hi,j as follows
, (2)
where hi,jdenotes the complex channel coefficient between the j:th transmit antenna and the i:th receiver antenna.
When comparing systems of different sizes, we need to normalize the channel matrix. The channel matrix is normalized such that , where represent the Frobenius norm. This normalization removes the influence of the variation in time and frequency but keeps the spatial characteristics, which is of interest here. Also the increased antenna gain due to the use of multiple antennas is not included in (13).
1.2.1 Space diversity in the channel
When the receiving antenna is moved relative to the transmitter there will be temporary local minimum in the amplitude of the incoming electromagnetic wave. This is called fading. This will of course happened twice every wavelength in the standing wave scenario. This effect can be reduced by using two receiving antennas at a distance of quarter of a wavelength apart from each other. The receiver then checks from which antenna it gets the strongest signal and uses that one. This is called space diversity.
Suppose a receiver has one antenna at point A in the Figure 1.1 below and one antenna at point B, it is obvious that the amplitude in the received signal from antenna B is much greater then the received signal from antenna A. It is also evident that there is a great advantage to be able to choose between these two antennas.
Figure 1.1. This figure shows that although point A and B are close to each other the received signal strength is very different. This is because of fading in the channel.
1.2.2 Polarization diversity
Even if the signal is sent with linear vertical polarization and the receiver has linear vertical polarization it is not evident that the incident signal at the Rx is of linear vertical polarization due to reflections in the channel. If the received signal is low it is often possible to receive a stronger signal by changing the polarization direction. This can be obtained by having two antennas with 90-separated polarizations.
1.2.3 Fading and fading models
In this section the concept of fading will be explained and two different fading distributions, Rayleigh and Nakagami m will be introduced.
Consider a simple communication system with one transmitting and one receiving antenna, the fading in this system can be approximated by the so-called “Two ray model”. When the transmitting antenna transmit a signal the receiving antenna will receive a signal that comes directly from the transmitter and a short while later there will come another signal that has been reflected by something, perhaps the ground or a large building.
Figure 1.2This figure shows that in the “Two ray model” the receiver will first receive a direct beam from the transmitter and a moment later it will also receive a reflected beam.
This is called multipath propagation and when many reflections exists this makes the nature of the mobile radio channel to be more easily described statistically. The ability to predict this fading behavior is very important to the receiver. If this is not possible there will not be an optimal reception of the signals at the receiver, which will be shown later. The received signal strength affected by channel fading, due to multipath propagation and shadow fading, due to large obstacles. The received signal is also affected by noise, both internal and external interference. In Figure 1.3 it is shown that globally the signal-strength decreases proportionally to , and as in free space (not shown here). By closely examine the graph it can be seen that it varies more rapidly and this behavior comes from the quick fading and the multipath behavior of the channel. And in the receiver AWGN is added.
Figure 1.3This figure shows that globally the signal strength decreases proportionally to 1/distance3.5. By closely examine the graph it can be seen that it varies more rapidly because of fading and the multipath behavior of the channel.
The received signal can be expressed as eq. (1) where H is the fading matrix and v is noise. Making a histogram of the received signal gives us an approximation of the PDF of the fading coefficients in the channel if the noise is negligible. There are several distributions used to model the fading statistics. The most commonly used distribution functions for the fading envelopes are Rice, Rayleigh and Nakagami-m. Rayleigh is a special case of Nakagami-m, when m equals one. The fading models are related to some physical conditions that determine what distribution that best describe the channel.
The Rayleigh distribution assumes that there are a sufficiently large number of equal power multipath components with different and independent phase.
The Nakagami one distribution equals the Rayleigh distribution above. It is a general observation that an increased m value corresponds to a lesser amount of fading and a stronger direct path.
1.2.4 The Nakagami m fading channel
Extensive empirical measurement has confirmed the usefulness of Nakagami m distribution for modeling radio links [1], [2]. The probability density function of a Nakagami m fading channel is given by
, (3)
Where is the second moment, i.e. =, this means that if and and then . The m parameter, also known as the fading figure is defined as
, (4)
The Nakagami m distribution covers a wide range of fading conditions; when m=1/2, it is a one-sided Gaussian distribution and when m=1 it is a Rayleigh distribution. In the limit when m approaches infinity, the channel becomes static hence, an AWGN channel. And its corresponding PDF becomes an impulsive function located at . The kth moment of the Nakagami m distribution is given by
=
Finally it can be said that there is a relationship between the m-parameter in Nakagami-m distribution and the K parameter in the Rice distribution if m>1. The Rice distribution is not used and will not be explained in this thesis.
, iff (5)
1.2.5 Estimation of the m-parameter
In this section two different methods to calculate the m-parameter in the Nakagami m distribution will be introduced. The methods are the moment method and the maximum likelihood method.
It is proposed in [3] that it is possible to use the moment method to estimate the m parameter. The estimator is
(6)
Where
, k=2 or 4 (7)
In [4] it is a suggested to use a maximum-likelihood (ML) estimation to obtain the ML-optimal-m value of the measured channel amplitude. Let be random i.i.d. variables that corresponds to a Nakagami m fading channel. The log-likelihood function (LLF) of the independent multivariate Nakagami-m distribution based on is given by
=
Where {ri, i = 1,2,…,N} are samples of {Ri, i=1,2,…,N}. Taking the derivative of the LLF with respect to m, and setting it equal to zero, we obtain
Where is the psi function, also called the digamma function, defined in [3, p.258, eq 6.3.1] as , where . The statistic for m in (4) requires knowledge of which is usually not known. Substitution of the unbiased maximum-likelihood estimators of , , in (4) yields
(8)
Where the approximation in (6) becomes exact as N approaches infinity, and where
(9)
The parameter is determined by the observed samples only and it is independent of m. The ML estimation of the m parameter requires solving the nonlinear equation (8), which does not lead to a closed-form solution for the estimator. An asymptotic expansion of the psi function is given as [3, p.259, eq 6.3.18]
Using the second order approximation in (8) and solving for m, we obtain
(10)
as the ML-estimator for m. In eq(10) all negative solutions have been discarded since only positive m values are of interest.
It is stated in [4] that the works badly when the m-value is low but gets better with higher m-values but never gets as good as . It should also be considered that leads to a more complicated implementation. In section 3.8 & 3.9 the two methods have been used to approximate the m-parameter for an experimental dataset.
1.3 Channel capacity
In this section a definition of the channel capacity is made and there is also a discussion about how the capacity varies with the number of transmitting and receiving antennas.
An interesting discussion on how many transmitting and receiving antennas one should use can be found in [5]. Using a singular value decomposition (SVD), the channel matrix H can be decomposed into a product of three matrices as follows
(11)
Where U are a unitary matrix of dimension , V are also a unitary matrix of dimension and S is a matrix whose elements are all zero except for the diagonal where there will be min(,) of the H matrix eigenvalues. The V* represents the complex conjugate transpose of the matrix V. The three matrices corresponds to three different steps
1, Projection into Tx eigenmodes
Each of the first min(,) columns in V is a unit-norm vector corresponding to each of the transmit eigenmodes. The relative phases and amplitudes between transmit elements required to excite each eigenmode are described by each of these column vectors. The first min(, ) components of the vector V1 are therefore the projection of the transmit vector x into the transmit eigenvector subspace. When , the remaining - components belong to the subspace orthogonal to the transmit eigenvector subspace and cannot influence the received vector r.
2, Weighting by singular values
Each of the min(,) components corresponding to the transmit eigenmodes are weighted by it’s associated singular value contained in the main diagonal of the matrix S.
3, Mapping into Rx eigenmodes
Each of the first min(,) columns in U is a unit-norm vector corresponding to the mapping of each eigenmode in the receiving space. The remaining N-min(,) vectors are not Rx eigenmodes because they cannot be excited by the transmitter.
The discussion above leads to some interesting conclusions:
If , some power is wasted on exciting a subspace orthogonal to the receiver. The receiver cannot interpret this subspace and it’s totally unnecessary. If the power is allocated uniformly over the transmitter there will be an average power loss of
If , there is no power loss. There will be dimensions in the receiver space, which are not excited by the receiver eigenvectors.
1.3.1 Maximum capacity, if there is no CSI at the transmitter
In this section the capacity for a MIMO channel will be defined. There will also be shown that by supplying the transmitter with channel state information (CSI) it is a possible to greatly increase the capacity
The channel is estimated in the receiver through to use of a short known transmitted training sequence.
For a one-antenna system one can apply the Shannon formula:
(12)
Where C is the channel and is measured in [Bits/(sec*Hz)], B is the bandwidth and S/N is the signal-to-noise ratio. This is the maximum rate the channel can give with arbitrary low probability of bit errors (allowing infinite coding delay). Hence, it is an upper limit on the practical achievable bit-rate. When one uses a MIMO system one have to use a generalized version of Shannon’s formula:
= (13)