First, we derive an upper bound for the average PEP, assuming that the decoding consists of a linear equalization followed by an symbol-wise quantization into the signal constellation A . See also [11] and [17]. To see that, let us formulate the detection problem as follows. Based on (3.6), we have Y=Ωc+W with the matrix Ω
(
K)
2KNMr KNΛ I ⊗F ∈ × being responsible for yielding the frequency-domain output of the time-domain input signal blocks c cT1 cT2 cTK ∈T KN . Hence we can address the ML-detection problem as follows:
values of K and N, where A denotes size of the employed constellation. So we seek a suboptimal linear equalization approach. Note that W is a white Gaussian vector since DFT serves as a unitary operation only. Let us define Z ΛHY∈KN ,
ΛH ∈KN
W W and ΘΛHΩ∈KN KN× . Then we can rewrite (3.6) as
Z =Θc+W. (4.2)
We assume rank( )Λ =KN . Since IK ⊗F is of full rank, rank( )Θ =KN by definition of Θ . Hence the inverse Θ−1 exists and can be used as a linear equalizer.
Let Z =Θ−1Z denote the output of the linear equalizer Θ−1, then from (4.2) we readily arrive at
Z = +c Θ−1W. (4.3)
Notice that the diversity gain weighting has been normalized at the output of Θ−1 and hence the detection problem can be formulated with respect to the employed
where c is an estimate of c after equalization. Note that the so obtained estimate minimizing the metric in (4.4) is suboptimal since the underlying noise is no longer white and its covariance depends on the matrix Θ−1ΛH. We see that minimizing the metric in (4.4) amounts to a symbol-wise hard-decision into A since
− c2
Z is the sum of KN nonnegative terms and each of which can be minimized with respect to an entry of c , i.e., minimization on symbol-level. Therefore, using Θ−1 as an equalizer and (4.4) as the detector amounts to having a linear equalization scheme followed by a hard decision on each entry of Z into the constellation A . It is noteworthy that the computational cost of (4.4) is linear in composite block length KN, which is computationally cheaper than that of (4.1).
The PEP analysis considers the probability that a symbol block c∈AKN is transmitted while another c is detected in the minimum-distance perspective. Given the channel realization h , and hence the matrix Λ , the conditional PEP is defined as all
[ ]
2 2Pr c→ c hall =Prc Z- < c-Z Λ. (4.5)
Note that proving that the average PEP has maximum diversity is equivalent to proving that the error rate performance exhibits maximum diversity, by virtue of the union
the Euclidean distance between c and c , as well as the corresponding normalized
As the variable ξ , when conditioned on channel realizations, is a zero-mean Gaussian random variable, the error probability in (4.6) is completely determined by the variance
2
Notice that since w is white, the entries of w are i.i.d. circular symmetric Gaussian, with real part and imaginary part of each of the entries being N(0, 0.5σw2) distributed.
. Since multiplication by unitary matrix dose not change Frobenius norm, we arrive at
Let us define the following terms: the encoder structure, and by (4.8) it can be easily verified that 2 2
F = F
With the definition of h jm
Before we proceed further, let us make one more assumption deduced from the rank premise: condition number of Dps, denoted as K
( )
Dps , is upper bounded by a finite real number Ku ∈ for all possible channel realizations, i.e.,( )
Dps ≤ u for all Dps.K K (4.14)
( )
i) For all possible system configurations (N,Mr,Mt,L,K), the probability of singular occurrence rank( )Λ <KN is very low in practice. As one can perform as many simulations as possible to find the above highly plausible, our rank premise is reasonable. Hence K
( )
Dps < ∞.ii) Given known channel order L and desired transmission rate, selecting high values for Mt and Mr can effectively suppress K
( )
Dps .iii) It agrees with the intuition that as N increases, the probability of K
( )
Dps → ∞ drops. This is because each of the diagonal entries, which are correlated, of Dps is sum of nonnegative terms.From the above three observations we deduce that K
( )
Dps can be universally upper bounded by a finite real number in practice.On the other hand, suppose B∈n n× is a nonsingular diagonal matrix, then for any
Therefore, by assuming (4.14) and along with (4.15), there exist infinitely many full rank
{ }
Ψ ∞ ∈ N N L× −( ) satisfying Ψ 2 ≤ m such that D−1 2 ≤(
D Ψ)
+ 2 forall channel realizations. Now let us find such a matrix Ψ among these possible candidates so that any (N-L) of the rows of Ψ are linearly independent. Note that this additional constraint can be very easily satisfied and hence does not in any way composite frequency responses such that
( ) ( )
whose entries are given by( )
(
ΨβDβ)(
DpsΨ)
=IN L− . (4.19)Let ( ) MIMO multipaths, hjm, are statistically independent of each other. The distribution of the channel vector h can then be expressed as a multi-dimensional Gaussian pdf[18, all
pp. 502],
Now we can average the upper bound with respect to the channel pdf:
[ ]
all all all all all
m which is essentially a circular symmetric Gaussian random variable. Let qi =Re
{ }
αi , and qi =Im{ }
αi . Therefore by symmetry between the real part and imaginary part ofαi, we can further factorize the integration as
[ ]
( )2 ( 1) ( 1)( 2 ) 1
Hence, at high SNR region, we can expect the PEP be bounded by
{
Pr[ ] }
1( )2 ( 1)( SNR) ( 1).It is seen from (4.27) that at high SNR, full spatial (transmit-receive) diversity gain of order MrMt and full multipath diversity of order L+1 are achieved. Note that the above results counterbalance the deduction in [26] that the “CP-only” scheme with extended GOSTBC cannot achieve full multipath diversity.
It is noteworthy that the derivation from (4.7) to (4.25) really relies on the structure of GOSTBC. From the perspective in [17], the particular relation Λ+ F−2 ≥C hall 22 for some real constant C in (4.24) requires certain precoder design, whose purpose is to compensate the detrimental effect due to channel zeros/nulls. However, as we see from the above derivation, the precoding mechanism could be not necessary. Note that Ψ can be regarded as a virtual precoder introduced for the purpose of deriving PEP.
Remark 4.1 By noting the slope change at relatively higher SNR region within Fig. 5, we see that the possibility of delivering full multipath diversity under the proposed scheme is evidently noticeable even with small numbers of antennas, i.e., as the number of taps (L+1) increases by 1.5 times, so does the slope of error probability above 4 dB. This in turn justifies the deduction behind (4.16) upon which our PEP analysis is built. For simulations in Fig. 5, we set N=250 and QPSK for 1000 consecutive transmission during which the channel remains fixed. The channel taps of each of them assume power delay profiles whose sum are normalized to 2 for a fair comparison. Other details for simulations are stated in Chapter 6.
Chapter 5
Training-Based Channel Estimation
In this chapter, the training-based channel estimation by M. Biguesh[3] will be generalized to frequency-selective fading scenario by exploiting the structure in (3.3).
Optimality criteria for the frequency-domain channel estimation in LS sense and under a given power constraint will be derived. Also, conditions under which the training blocks are designed so as to apply a linear MMSE approach for time-domain channel estimation are derived. First, let us denote the training matrix as Π = t t1 t2 tK, where tk ∈ is the N kth training block. For simplicity, we reuse the notations adopted in chapter 3 for the subsequent discussion on channel estimation. Hence the encoder output becomes
Since the channel estimation is processed on a per receiver basis, we consider MISO in the subsequent sections of this chapter, i.e., the index j equals one and one only, and will hence be discarded.