y = <(˜y) =(˜y) M ×1 , st= <(˜st) =(˜st) N ×1 , v = <(˜v) =(˜v) M ×1 , H = <( ˜H) −=( ˜H) =( ˜H) <( ˜H) M ×N (3.5) and where M = 2m and N = 2n. In the transformed MIMO system model y, H, st are all real-valued quantities and Ω = D is a real-valued signal set.
3.3
MIMO capacity
The notion of channel capacity was introduced by Shannon in [53]. The capacity of a channel, denoted by C, is the maximum rate at which reliable communication can be performed, and is equal to the maximum mutual information between the channel input and output vectors. Shannon proved two fundamental theorems: (i) for any rate R < C and any desired non-zero probability of error Pe there exists a rate R code that achieves
Pe and (ii) the error probability of rates R > C higher then the channel capacity is
bounded away from zero. As a result, the channel capacity is a fundamental limit of communication systems.
Several channel capacity definitions are available in the literature depending on: (i) what is known about the state of the channel, referred to as channel state information (CSI), or the distribution of the channel, referred to as channel distribution information (CDI), and the time scale of the fading process. For a time-varying channel where CSI is available at both the transmitter and receiver, namely the channel matrix ˜H is known, the
transmitter can adapt its rate or power based on the CSI. In this case the ergodic capacity is defined as the maximum mutual information averaged over all the channel states. Ergodic capacity is a relevant metric for quickly varying channels, since the channel experiences all possible channel states over the duration of a codeword. In case of perfect CSI at both the transmitter and receiver the outage capacity is defined as the maximum rate of reliable communication at a certain outage probability. Outage capacity requires a fixed data rate in all non-outage channel states and no data is transmitted when the channel is in outage since the transmitter knows this information. Outage capacity is the appropriate capacity metric in slowly varying channels, where the channel coherence time exceeds the duration of a codeword, thus each codeword is affected by only one channel realization.
In the following the capacity of single-user MIMO channel is considered for the case DOI:10.15774/PPKE.ITK.2015.010
3.3. MIMO CAPACITY
when: (i) CSI is available at the transmitter (CSIT) and receiver (CSIR) and the channel is constant, and (ii) CDIT and CSIR is available and fading channel is assumed.
When CSIT is available the estimated distribution or channel state is sent to the transmitter through a feedback channel. If the transmitter adapts to these time-varying short-term channel statistics then capacity is increased relative to the transmission strat- egy associated with just the long-term channel statistics. The average transmit power is constrained across all transmit antennas as E[˜sHt ˜st] ≤ P . When the channel is constant,
and CSIT, CSIR is available, the capacity is defined as
C = max
˜
Q:tr( ˜Q)=P
log det(Im+ ˜H ˜Q ˜HH) (3.6)
where ˜Q is the input covariance matrix, which is n × n positive semi-definite complex
matrix.
With the help of the singular value decomposition (SVD) the channel matrix can be factorized as
˜
H = ˜UΣ ˜VH, (3.7) where ˜U is Cm×m unitary matrix, Σ is Rm×n diagonal matrix with real non-negative entries, and ˜V is Cn×nunitary matrix. The diagonal elements of matrix Σ, denoted by σ
i,
are the singular values of ˜H and a descending order is assumed σ1 ≥ σ2≥ · · · ≥ σmin(n,m). The matrix ˜H has exactly r positive singular values, where r is the rank of ˜H, and
r ≤ min(n, m). In [44], [8] it was shown that the SVD can convert channel ˜H into
min(n, m) parallel, noninterfering SISO channels after precoding the input, i.e. st= ˜V˜st, and the received vector is multiplied by matrix ˜UH, resulting in y = ˜UH˜y. The system
model is transformed as y = ˜UH( ˜Hst+ ˜n) = ˜UH( ˜UΣ ˜VH( ˜V˜st) + ˜n) = Σ˜st+ ˜UH˜n = Σ˜st+ n (3.8)
where n = ˜UHn. Since ˜˜ U and ˜V are unitary ˜n and n have the same distribution, and
the transformations are power preserving E[k˜stk2] = E[ks
tk2]. Since Σ is diagonal and
the singular values σi are strictly positive, the non-interfering channels are described as
yi = σi˜sti+ ni, for i = 1, . . . , min(n, m). (3.9)
3.3. MIMO CAPACITY
Note, that the above is possible only if the transmitter knows the channel, because the precoding requires the exact channel matrix state information. The water-filling algorithm [54] can be used to optimally allocate power over the different quality parallel channels, leading to the following allocation:
Pi= max µ −
1
σ2i, 0
!
, 1 ≤ i ≤ r, (3.10)
where Pi is the power of ˜sti, and the waterfill level µ is chosen such that Pr
i=1Pi = P .
Thus, the covariance that achieves the maximum capacity defined in Eq. 3.6 is
˜
Q = ˜VP ˜VH, where P ∈ Rn×nand P = diag(P1, . . . , Pr, 0, . . . , 0). The resulting capacity
is given by C = r X i=1 max(log(µσi2), 0). (3.11)
The constant channel model is easy to analyze from a mathematical point of view, however, wireless channels are time-varying. In this case a more common assumption is that the receiver is able to correctly estimate the channel state, but only the CDI is fed back to the transmitter. The perfect CSIR and CDIT model is motivated by the scenario where the channel state can be accurately tracked at the receiver and the statistical channel model at the transmitter is based on channel distribution information fed back from the receiver. The channel coefficients are typically assumed to be jointly Gaussian, so the channel distribution is specified by the channel mean and covariance matrices. When the transmitter has knowledge only on the channel distribution the precoding is not possible, thus an optimal strategy is to maintain a fixed-rate transmission that is optimized with the respect to the CDI. In [8] and [55] it was shown that the optimal transmit strategy is to allocate equal power in every spatial direction, thus the optimum input covariance matrix that maximizes the ergodic capacity is Q = PnIn. As a result,
the ergodic capacity is defined as
C = EH log det Im+P n ˜ H ˜HH . (3.12)
A useful approach to get insight of the multiplexing gain of MIMO systems is to analyze the asymptotic behavior of the ergodic capacity as either the SNR or the number of antennas are increasing. If n and m are fixed and SNR is taken to infinity, the capacity grows as C ≈ min(n, m) log2P + O(1). Consequently, a 3 dB increase in the SNR leads
to an increase of min(n, m) bps/Hz in spectral efficiency, thus the multiplexing gain of a MIMO system is min(n, m) times better compared to the SISO case.
Chapter 4
MIMO detection methods and
algorithms
4.1
Introduction
In this chapter the focus is on detection methods applied on MIMO systems when spatial multiplexing is used. The complexity of detection algorithms depends on many factors, such as antenna configuration, modulation order, channel, coding, etc. Several detection techniques are investigated such as: (i) the linear detectors, (ii) non-linear de- tectors based on the successive interference cancellation, (iii) tree-search based detectors and (iv) the ML detector offering the best BER performance.
The ML detector offers the best BER performance, however, its exponential com- plexity is not suitable for real-time applications. The SD algorithm, discussed in Sec. 4.6.1, was proposed to significantly reduce the search space without degrading the BER performance of the ML detector. The main disadvantage of the ML detector is the ex- ponentially growing complexity with both the number of elements in the signal set and the number of antennas.
In non-optimal detectors the complexity of the SD algorithm is reduced by intro- ducing some approximations such as (i) early termination of the search, (ii) introducing constraints on the maximum number of nodes that the detector algorithm is allowed to visit or (iii) on the run-time of the detector. Each of these strategies introduces some approximations which prevents these detectors from achieving ML performance. For non-optimal detectors a trade-off has to be made between computational complexity and the quality of detection. Papers [56], [20], [57], [58], [59] and [60] focus on finding a near-ML solution with a significant decrease in the computational complexity. These