A. Shannon capacity of wireless channels
Given a single channel corrupted by an additive white Gaussian noise (AWGN), at a level of SNR denoted by , the capacity (rate that can be achieved with no constraint on code or signaling complexity) can be written as
log (12 ) bps/Hz
power to obtain double capacity (To go from 1bps/Hz to 11bps/Hz the transmitter power must be increased by 1000 times). In practice wireless channels are time-varying and subject to random fading. In this case we denote h the unit-power complex Gaussian amplitude of the channel at the instant of observation.
The capacity, written as becomes a random quantity, whose distribution can be computed. The cumulative
distribution of this 1x1 case (one antenna on transmit and one on receive) is shown on the left in Fig.2-1 We notice that the capacity takes, at times, very small values, due to fading events.
B. Multiple antennas at one end
Given a set of M antennas at the receiver (SIMO system), the channel is now composed of M distinct coefficients h [ ;h ; h0 1 ; hM-1] where hi is the channel amplitude from the transmitter to the i-th receiver antenna. The expression for the random capacity can be generalized to
2 1 1
where [ ]Hrepresents Hermitian transposition, can be approximated as log (12 ) bps/Hz
CSIMO M
(4) M is the number of the receiver. Compared to the capacity of SISO system
lo g (12 ) b p s/H z
S ISO
C
It shows that slow logarithmic growth of the bandwidth efficiency limit.
In Fig. 2-1, we see the impact of multiple antennas on the capacity distribution with 4 and 10 antennas respectively. This is due to the spatial diversity, which reduces fading and the higher SNR of the combined antennas.
However going from 4 to 10 antennas does not give very significant improvement as spatial diversity benefits quickly level off. The increase in average capacity due to SNR improvement is also limited because the SNR is increasing inside the log function in (3). We also show that the results obtained in the case of multiple transmit antennas and one receive antenna, 4x1 and 10x1 when the transmitter does not know the channel in advance. In such circumstances the multiple transmit antennas cannot beamform blindly. Conventional multiple antenna systems are good at improving the outage capacity performance, attributable to the spatial diversity effect but this effect saturates with the number of antennas.
C. Capacity of MIMO links
We now consider a full MIMO link as Fig. 1 with N transmit antennas and M receive antennas respectively. The channel is represented by a matrix of size M x N with random elements denoted by HM xN , we have the now famous capacity boosting compared to SISO channels and fast growth compared to SIMO channels
Foschini [1] and Telatar [3] both demonstrated that the capacity in (5) grows linearly with k=min (M, N) rather than logarithmically. This result can be intuited as follows the determinant operator yields a product of min(M,N) nonzero eigenvalues of its (channel-dependent) matrix argument, each eigenvalue characterizing the SNR
using a pair of right and left singular vectors of the channel matrix as transmit and out along the range of this distribution. Hence, it is unlikely that most eigenvalues are very small and the linear growth is indeed achieved. In theory and in the case of idealized random channels, limitless capacities can be realized provide that we can afford the cost and space of many antennas and RF chains. In reality the performance will be dictated by the practical transmission algorithms selected and by the physical channel characteristics.
Applying MIMO techniques to wireless communication to meet the anticipated demand for high bit rate, real time services within limited bandwidths. MIMO propagation channel asymptotically gives M x N diversity and min (M, N) orthogonal communication channels for fully uncorrelated antennas. A large number of parallel channels is attractive since they are capable of carrying parallel information in the same bandwidth. The eigenvalue decomposition deduced from the propagation channel matrix (M x N) is an important parameter in this context because it determines the effective number of available parallel subchannels. Next we will introduce the concept of eigen-analysis on MIMO channel, a matrix solution leads itself to an analytical approach where the eigenvalues of the transmission system lead to a definition of the maximum gain as the largest eigenvalue.
Fig. 2-1 Shannon capacity as function of number of transmitter and receiver antennas
2.2 Spatial Correlation Coefficient
The complex correlation coefficient is a complex number that is less than unity in absolute value. Let a, b be two complex random variables, the complex correlation coefficient of a and b is defined as
* * elements in the two arrays have the same polarization and the same radiation pattern.
The spatial complex correlation coefficient at the BS between antenna m1 and m2 is
where < a, b> computes the correlation coefficient between a and b. The spatial complex correlation coefficient observed at the MS is similarly defined as
1 2 1
,
2MS
m m
h
mnh
mn(8) Given (7) and (8), we can define the following symmetrical complex correlation Matrices
2.3 Eigenanalysis on MIMO channel
A way to estimate the number of independent channels between two terminals in a rich scattering environment is to use the eigenvalue decomposition of the instantaneous correlation matrix R defined as
R=HHH
where H is the narrowband complex channel matrix and [ ]H represents Hermitian transposition. H is expressed as
1 1 1 2 1 receiver and the n-th antenna at the transmitter. Note that each element of the channel matrix H is a function of frequency. Then (5) becomes
( ) log (det(2 M ( ) ( ) ))H
C f I H f H f
N
For the frequency-selective case, the capacity needs to average over the frequencies:
, 2
where B is the bandwidth. Note that the channel matrix H is normalized such that
2 1
E hmn
From [7], in order to get the weight vector associated to the eigenvalue
i are real, nonnegative singular values.
m x n
where U and V are unitary matrices and u and v are the left and right singular vectors, respectively. is called the singular values of H. There is an important relationship between the SVD of H and the eigenvalue decomposition (EVD) of R such that where is the eigenvalue. A channel matrix H MxN offers k=min {M, N} parallel channels with different mean gains and correlated fast fading statistics. These k channels are accessible by applying the appropriate weight vectors u and v at both the transmitter and receiver antenna array. Then (11) is just a
compact way of writing the set of independent channels
1 1 1
The SVD is particularly useful for interpretation in the antenna context. If one estimates the response of each antenna element to a desired transmitted signal, one can optimally combine the elements with weights selected as a function of each
element response. For instance, choosing one particular eigenvalue, it is noted that
Vi is the transmit weight factor for excitation of the singular value i . A receive weight factor of U , a conjugate match, gives the receive voltage i* Sr and the square of that the received power
* diagonalized leading to a number of independent orthogonal modes of excitation, where the power gains of the i-th mode or channel is i. The weights applied to the arrays are given directly from the columns of the U and V matrices. Thus, the eigenvalues and their distributions are important properties of the arrays and the medium, and the maximum gain is given by the maximum eigenvalue. The number of nonzero eigenvalues may be shown to be the minimum value of M and N. The
Fig. 2-2 Transmission from three (N) transmit antennas to two (M) receive antennas. There are two independent channels (the minimum of M and N), which are excited by the V vectors on the transmit side and weighted by the U vectors on the
receive side. The power is divided between the two channels according to the water filling principle. This is a maximum capacity excitation of the medium. In
case only maximum array gain is wanted, only the maximum eigenvalue is chosen (one channel)
Once the channel matrix H is diagonalized by SVD and obtain the power gain in the k-th channel is given by the k-th eigenvalue i.e., the signal to noise ratio (SNR) for the k-th channel equals
2 k
k k
n
p
where p is the power assigned to the k-th channel, k kis the k-th eigenvalue and
2
n is the noise power. The number of independent eigenmode channels k depends on the number resolvable paths L and the number of antenna elements at the transmitter and the receiver. According to Shannon the maximum capacity of k parallel channels equals
where the mean SNR is defined as constraint on the powers is that
k tan
2.4 MIMO LOS Channels
In this section, we provide condition guaranteeing a high rank MIMO channel in real environment. We suggest that rank properties are governed by simple geometrical propagation parameters.
Considering the N transmitter, M receiver setup described in Fig. 2-3, only uniform linear arrays (ULA) are considered in this paper, but the analysis can be extended to other array topologies, like uniform circular arrays. We assume bore-sight propagation from the transmit array to the receive array. In addition, we assume the signal radiated by the k-th transmit antenna to impinge as a plane wave on the receive array at an angle of k. This assumption is justified when the antenna aperture is much smaller than the range R and the receive antenna array is within the far-field of the transmit antenna array. The propagation of a plane wave representing path k impinging on the antenna array causes a time delay Rxr k, at different antenna elements. This small time delay of the arrival of the wavefronts between different antennas results in a phase-shift r kRx, at these receive antennas. The delay r kRx, at receive antenna r compared to the first antenna is defined as:
,
The array propagation vector hkRx contains these phase shifts with respect to the first antenna for a certain path k. Denoting the signature vector induced by the k-th
where d r and dt are the receive and transmit antenna spacing, respectively, we have
Rx Rx Rx
1 2 N
H [h h h ] . The array propagation vector defines the spatial response of an antenna array. The common phase shift due to the distance R between transmitter
and receiver has no impact on capacity and is therefore ignored. Clearly, when the
k (k=1,2 N) (all other parameters being fixed) approach zero we find that H approaches the all ones matrices and therefore has rank 1. In practice, this happens for large R. As the range decrease, linear independence between the signature starts to build up. Since the capacity of a MIMO channel depends on the actual channel coefficients, which are random variables, the capacity is a random variable as well.
A communication system suffers only if the capacity is lower than needed for a transmission therefore usually the outage capacity is given. The 1% outage capacity defines the minimum capacity that is ensured over 99% of the transmission time.
Hence we choose to use the full orthogonality between the signatures of adjacent pairs of transmit antennas as a criterion for the
receiver to be able to separate the transmit signatures well hence implying high
number of receive antenna it will tend to be sufficient. In practice, for larger values of antenna spacing, the transmit antennas can fall into the grating lobes of the receive array in which case orthgonality is not realized. (13) can be written into
t r
d
Md R
which can be interpretated in terms of basic antenna theory as follows: The angular resolution of the receive array (inversely proportional to the aperture in wavelengths) should be less than the angular separation between two neighboring transmitter. Of course a similar condition in terms of transmit resolution can be by enforcing orthogonality between the rows of H. In a pure LOS situation orthogonality can only be achieved for very small values of range R. For example at a frequency of 2.44GHz with M=4, a maximum of R=15m is acceptable for 1m transmit antenna spacing (i.e.10 ).
Fig. 2-3 N-input M-output configuration
Chapter 3 Measurement campaign and set-up
Chapter 3
Measurement campaign and set-up
As described in the introduction, a vast measurement campaign was planned to investigate the MIMO channel in real environments. Published literatures [9] so far has most relied on assumptions about the statistical behavior of this wireless channel.
Although this has proven the concept and allowed further investigations to be undertaken, rigorous MIMO channel measurement campaign conducted is therefore necessary in order to characterize the performance of these systems in real environments. The objective of this chapter is therefore to describe the measurement set-up, the measurement campaign and extract the parameters from measurement raw data.
3.1 Measurement Set-up
For measurement of the time-varying and directional mobile radio channels, the RUSK vector channel sounder was employed [9]. The measurement equipment and its system diagram and are shown in Fig 3-1 and Fig 3-2. The sounder system consists of a mobile transmitter (Tx) that is omni-directional, and a fixed receiver (Rx) with an 8-element array antenna, each having a beamwidth of 120 . A fast 0 multiplexing system switches between each of these elements in turn in order to take a complete vector snapshot of the channel in 12t . Periodic multi-frequency p
Doppler bandwidth of up to 20kHz allows complete statistical analysis of the time varying radio channel with respect to different azimuthally directions of the impinging waves. In the case of a remote link measurement, Tx/Rx synchronization is maintained by two rubidium references. This calibration process of rubidium reference removes the tracking error of the measurement system and as a result of phase and delay normalization. Allowing the system a warm-up time about 60 minutes to stabilize oscillator and amplifier minimizes temporal drift of the measurement system. The telemetry allows remote control of the digital receiving unit (DRU) from portable transmitting station (PTS) location.
The channel impulse responses of the antenna array are recorded as vector snapshots in rapid succession. After receiving by Rx, signals are gathered to DRU and sent to a personal computer (PC) to analyze where AOA is estimated by using Unitary ESPRIT with sub-array smoothing technology. An overview about array signal processing including estimation of the AOA and a comparison of ESPRIT with other algorithm can be found in [10]. The receiving antenna was mounted on a rooftop at 2.44 GHz with the transmission power of 1w. The transmitter antenna was carried in a trolley and was 1.8 meter above the road. In order to get multipath components, we sampled data by moving measurements along selected routes with walking speed. We performed the measurements at the time of 10:00~20:00 with many pedestrians and vehicles, which may result in random scattering effects.
RFT
Fig. 3-1 System diagram of the RUSK vector channel sounder
(a) (b)
Fig. 3-2 RUSK vector channel sounder. (a) Transmitting unit. (b) Receiving unit.
3.2 Measurement Campaign
There are four measurement sites illustrated in Fig. 3-3. Detailed experimental setup or arrangement at each site is given as follows:
Measurement sites National Chiao Tung University Guang Fu campus Site 1 along route no.1 with total route length: 50m (12700 snapshots)
Site 2 along route no.2 with total route length: 170m (36700 snapshots)
Site 3 along route no.3 with total route length: 200m (4200 snapshots,)
Site4 along route no.4 with total route length: 250m (4800 snapshots)
Moving speed
route no.1 and route no.2 : Speed=2~3 km/hr route no.3 and route no.4 : Speed=10 km/hr
Tx-Rx distance route no.1 15~50m Propagation delay time 1.6 s for route no.1.
6.4 s for route no.2, route no.3 and route no.4
We name site-ij as the particular propagation condition i along the measurement distance in route -j i.e., site12 means the LOS condition along route no.2, site23 means the OLOS condition along route no.3, site34 means the NLOS condition along route no.4
The propagation environment at each site is described as the following table.
Table 1- descriptions of the propagation environment at each route.
route no. Propagation situation Local environment
route no.1
Fig. 3-3 Measurement sites in the NCTU campus
MIMO channels can be modelled either as double directional channels or as vector (matrix) channels. The former method is more related to the physical propagation effects, while the latter is more emphasized on the effect of the channel on the system. Another distinction is whether to treat the channel deterministically or stochastically. In the following, we outline the relations between those description methods.
The deterministic double directional channel is characterized by its double directional impulse response. It consists of L propagation paths between the transmitter and receiver sites. Each path is delayed in accordance to its excess-delay
i, weighted with the proper complex amplitude a ei j i and each direction of departure (AOD) T i, associated with the corresponding direction of arrival (AOA)
,
R i. The channel impulse response matrix h is
, ,
absolute time t; also the set of multipath components (MPCs) contributing to the propagation will vary, N N(t). The variations with time can occur both because of movements of scatters, and movement of the transmitter. The number of paths L can become very large if all possible paths are taken into account. In our experiments, the total number of resolvable multipath components was between 193 and 769. We simulate the deterministic channel applying the site-specific method to describe the direct wave, specular reflection waves, and single and multiple-over-rooftop diffracted waves. Once the site-specific method, i.e. deterministic method, is finished, the field strength distribution, power delay profile and power azimuth profile are shown in Fig. 3-4, we survey the multipaths in propagation, only one path is single rooftop diffracted wave accompanied with other 31 corner diffracted multipaths and acquire different realizations of the channel and proceed this procedure 15 times to obtain the complete channel matrix H4 4. Based on the theory of reciprocity of antenna, we obtain AOD by interchanging the position the transmitter and receiver.
AOA T , AOD R in route no.1 case is approximately 0 . Repeating the 0 procedure above for 100 times gives an ensemble of channel realization and computes the capacity and plots a cumulative distribution function (CDF) for the MIMO channel capacity. Fig.3-4 gives the power delay profile and power azimuth profile of measurement for LOS of route no.1 and Fig 3-5 gives the time averaged Delay-Azimuth Spectrum of measurement of route no.1.
Fig 3-4. The field strength distribution of propagation, power delay profile and power azimuth profile for LOS of route no.1 in the left straight side, the ones which
transmitter and receiver are interchanged shown in the right straight.
Fig 3-5 Time Averaged Delay-Azimuth Spectrum of measurement of route no.1
Fig 3-6 (a)
Fig 3-6 (b)
Fig 3-6 (a) Field strength distribution of route no.2 and the Power Delay Profile and Power Azimuth Profile of route no.2 (b) Time Averaged Delay-Azimuth Spectrum of
measurement of route no.2
Fig 3-7 (a)
Fig 3-7 (b)
Fig 3-7 (a) Field strength distribution of route no.3 and the Power Delay Profile and Power Azimuth Profile of route no.3 (b) Time Averaged Delay-Azimuth Spectrum of
measurement of route no.3
Fig 3-8 (a)
Fig 3-8 (b)
Fig 3-8 (a) Field strength distribution of route no.4 and the Power Delay Profile and Power Azimuth Profile of route no.4 (b) Time Averaged Delay-Azimuth Spectrum of
measurement of route no.4
3.3 Measurement data extraction
By examining the measurement raw data, we extracted the angle-of-arrival (AOAs), angle-of-departure (AODs), delays and azimuths of the multipath components [10].
Using the commercial software Matsys to obtain
(1) Time-variant Impulse Response h t( , , )s , where t represents observation time, represents delay time, s represent channel, we take this to evaluate whether the environment is clean i.e. observing the Power Delay Profile had a trend of decaying along the propagation distance as time is going. In Fig. 3-9, we observe that (a)~(c) power level increases as time goes by and (c) appear apparent decaying situation at some measurement, the same bandwidth. In Fig. 3-9 (c), there is a time difference between the strongest receive signal power and next strong one around 0.25~0.3us, i.e.
the multipath propagate to arrival receive array more over 75~90m. From these Power Delay Profiles, we recognize (a)~(c) as OLOS, NLOS and LOS, respectively and take these snapshots for data processing.
Fig. 3-9 The impulse response of (a) top (b) bottom left (c) bottom right figure presents the measurement during different observation time
Fig. 3-9 The impulse response of (a) top (b) bottom left (c) bottom right figure presents the measurement during different observation time