3.3 MU-MISO Nonlinear Transmit Processing
3.3.1 MU-MISO Vector precoding (VP)
Fig. 3.10 shows the block diagram of a MU-MISO system with vector precoding. The transmitter has the freedom to add an arbitrary perturbation signal a[n] ∈ τZK + j τ ZK to the data signal prior to linear transformation with the filter F ∈ CN ×K, since the receivers apply the modulo operator M(•). Here, τ denotes a constant that depends on the modulation alphabet, so we setτ = 2√
2 for QPSK modulation (see Fig. 3.11) and τ = 8/√
10 for 16QAM modulation [83]. This constant is associated with the modulo operatorM(•). This nonlinear operation is defined as
M (x) = x− ℜ (x)
where⌊•⌋ denotes the floor operator which gives the largest integer smaller than or equal to the argument. The corresponding fundamental Voronoi region is
V=n
which means that the modulo operator constrains the real and imaginary part ofx to the interval [−τ/2, τ/2] by adding integer multiples of τ and j τ to the real and imaginary part, respectively. For example, forx = 3.4− 1.5 j and τ = 2, when the modulo operator is applied we getM(x) = −0.6 + 0.5 j. Note that if we apply the modulo operator to a multidimensional vector x= [x1, . . . , xK]T, it is satisfied that
M(x) = [M(x1), . . . , M(xK)]T ∈ VK whereM(xi), i = 1, . . . , K is defined as in Eq. (3.61).
As can be seen from Fig. 3.10, the data vector u[n] ∈ CK is first superimposed with the perturbation vector a[n], and the resulting vector is then processed by the linear filter F to form the transmit vector
x[n] = F d[n]∈ CN, n = 1, . . . , NB
where d[n] is the desired signal given by u[n] + a[n] and n is the symbol index in a block size ofNBdata symbols.
(a) (b)
Figure 3.11: Modulo Operator. (a) QPSK, (b) 16QAM.
Similar to linear precoding filters, we impose a transmit power constraint. Since the statistics of the transmit symbols are unknown, we average over the block instead of taking the expected value, i.e.
After passing through the channel and by superimposing the AWGN noise, the received signal is given by
y[n] = HF d[n] + η[n].
The weightg in Fig. 3.10 is assumed to be constant throughout the block of NBsymbols.
Note that we use a common weight for all the users. Thus, the weighted estimated signal is given by
d[n] = gHF d[n] + gη[n].ˆ (3.62)
The modulo operator at the receiver is used to compensate the effect of adding the perturbation a[n] at the transmitter.
The MSE can be expressed as [22, 23]
εVPWF(a[n], x[n], g) = 1
3.3 MU-MISO Nonlinear Transmit Processing 57
Note that the expectation is conditioned on the full knowledge of the symbols u[n] by the transmitter. But since the statistics of a[n] are unknown, we average the symbol MSE over the whole block.
Since a[n] is discrete, we cannot derive with respect to a. The optimization procedure is as follows. We start by fixing a, after which x andg are optimized taking into account the transmit power constraint. For these optimum x andg we choose the best a according to the MSE criterion. Although we optimize the continuous and discrete part separately, this procedure leads to the optimum minimization of Eq. (3.63) [23].
MU-MISO Wiener Spatial Vector Precoding (WF-VP)
We have to find the joint optimum of all the perturbation vectors a[n], all the transmit vectors x[n], and the gain factors g for n = 1, . . . , NB:
aVPWF[n], xVPWF[n], gVPWF
= argmin
{a[n],x[n],g}
εVPWF(a[n], x[n], g) s.t.: 1 NB
NB
X
n=1
kx[n]k22 ≤ Etx. (3.64) The MSEεVPWF(a[n], x[n], g) is given by Eq. (3.63) [22, 23] and can be rewritten as
εVPWF(a[n], x[n], g) = 1 NB
NB
X
n=1
dH[n]d[n]− g∗xH[n]HHd[n]− gdH[n]Hx[n]
+|g|2xH[n]HHHx[n] +|g|2tr (Cη)
(3.65)
where we useE[kd[n]k22|u[n]] = kd[n]k22andE[kx[n]k22|u[n]] = kx[n]k22, since the data signal u[n], the perturbation signal a[n] and consequently, the transmitted signal x[n] are known to the transmitter.
The Lagrangian function can be expressed as
L (a[n], x[n], g, λ) = εVPWF(a[n], x[n], g) + λ 1 NB
NB
X
n=1
xH[n]x[n]− Etx
!
(3.66)
whereλ ∈ R0,+. Now, we set its derivative with respect to x[n], n = 1, . . . , NB andg to
zero, which leads to the necessary KKT conditions
since the optimization problem in Eq. (3.64) is not a convex programming problem.
Then, the transmit symbols are directly obtained from the first KKT condition and are given by
First of all, we have to show thatλ > 0, i.e. the power constraint as active. Multiplying the second KKT condition byg, we have
1
NB −gdH[n]Hx[n] +|g|2xH[n]HHHx[n] +|g|2tr (Cη)
= 0 (3.69)
and multiplying the Hermitian of the first KKT condition by x[n] from the right, we have 1
NB −gdH[n]Hx[n] +|g|2x[n]HHHHx[n] + λ
NB
xH[n]x[n] = 0. (3.70) With Eq. (3.69) and the transmit energy constraint, the Lagrangian multiplierλ is given by
Therefore, it becomes clear thatλ > 0 for the non–trivial case that∃n : x[n] 6= 0. Thus, the transmit energy constraint is active andλ =|g|2ξ with ξ = tr(Cη)/Etx.
Then, we reach the following solution for the WF-VP:
xVPWF[n] = 1
3.3 MU-MISO Nonlinear Transmit Processing 59
where gWFVP is directly obtained from the transmit energy constraint and ξ = tr(CE η)
tx . Remember thatg is chosen only once in each block.
We define a matrix Φ = (HHH + ξI)−1. Applying the matrix inversion lemma to Eq. (3.72) shows that xVPWF[n] = gVP1
WFHHΦd[n] and then, gVPWF = qPNB
n=1(dH[n]ΦHHHHΦd[n])/(EtxNB). Thus, when we plug these results into the MSE expression in Eq. (3.65) we obtain that
εVPWF(a[n], x[n], g) = ξ
Since Φ is positive definite, we can use the Cholesky factorization to obtain a lower triangular matrix L and a diagonal matrix D with the following relationship [22],
Φ= HHH+ ξI−1
= LHDL.
Thus, the perturbation signal can be found by the following search [22]
aVPWF[n] = argmin
This search can be solved by means of the Schnorr-Euchner sphere decoding [87, 88]
where the use of real-valued notation to represent vectors and matrices has been considered to run the final computer simulations (see Appendix B.5).
Note that due to the unit lower triangular structure of D1/2L, thei–th summand of the Euclidean norm||D1/2Lu[n] + D1/2La[n]||22 =PK
whereli,j corresponds to the element of thei-th row and j-th column of L and di,i is the i-th entry of the diagonal matrix D.
When the off-diagonal elements of L are approximately zero, i.e. li,j = 0, for j 6= i, we have
a[n] = argmin
a[n]∈τ ZK+j τ ZK||u[n] + a[n]||22 (3.76) which leads to a[n] = 0, i.e. we obtain the linear precoding approach described in Subsection 3.1.2.
u[n] P a[n]
d[n] v[n] x[n]
I − B
F
Figure 3.12: Linear Representation of Tomlinson Harashima Precoding.
When a1[n], . . . , aK[n] are computed successively, i.e. ai[n] is found for fixed a1[n], . . . , ai−1[n], the i–th element of a[n] is obtained as
ai[n] =− Qτ ZK+j τ ZK ui[n] + Xi−1
j=1
li,j(uj[n] + aj[n])
!
. (3.77)
This successive computation of a[n] enables us to obtain the scheme depicted in Fig. 3.12, which corresponds to the linear representation of THP. According to the definition of the modulo operator asM(x) = x− (⌊ℜ(x)τ +12⌋τ + j⌊ℑ(x)τ +12⌋τ), it is straightforward to see that the quantizerQτ ZK+j τ ZK(x) is equivalent to the term⌊ℜ(x)τ +12⌋τ + j⌊ℑ(x)τ +12⌋τ and then we can write Qτ ZK+j τ ZK(x) = x − M(x). Therefore, the perturbation signal a[n]
can easily be included inside the feedback loop (as can be seen in Fig. 3.13) by means of the modulo operator M(x). This leads to the well-known suboptimum approach of Tomlinson-Harashima precoding described in the following subsection.
The above result for a[n] can be transformed to aVPWF[n] = argmin
a[n]∈τ ZK+j τ ZK||D1/2Lu[n] + D1/2La[n]||22
= argmin
λ[n]∈ZK+j ZK||τD1/2Lλ[n]− (−D1/2Lu[n])||22
= argmin
λ[n]∈ZK+j ZK||Gλ[n] − z[n]||22 (3.78) where G = τ D1/2L and z[n] = −D1/2Lu[n]. This is the called a closest point search in the lattice generated by the matrix G [89].
According to the Lenstra-Lenstra-Lov´asz (LLL) algorithm [90], this matrix G can be decomposed into a matrix Γ and a unimodular integer matrix T , i.e. the absolute value of its determinant is equal to one, as follows
G= Γ T−1. Note that the inverse of T is also unimodular integer.
Thus, Gλ[n] = Γ T−1λ[n] = Γ λ′[n] with integer λ′[n] ∈ ZK + j ZK. Based on the above factorization of the generator matrix G, the lattice search of Eq. (3.78) can be
3.3 MU-MISO Nonlinear Transmit Processing 61
rewritten as follows
λopt[n] = T argmin
λ′[n]∈ZK+j ZK||Γ λ′[n]− x[n]||22 = T λ′opt[n].
Since the columns of Γ are closer to orthogonal than those of the original G, this search can be solved more efficiently than in Eq. (3.78).
In order to find λ′opt[n] we employ the Schnorr-Euchner algorithm [87, 88], where a sphere decoder performs this lattice search. Then, the vector aVPWF[n] is simply given by
aVPWF[n] = τ λopt[n]∈ τZK+ j τ ZK.
Note that the complexity of the sphere decoder grows exponentially with the number of users [89] which implies that the implementation of VP in real systems may be questionable.
MU-MISO Zero-Forcing Spatial Vector Precoding (ZF-VP)
By considering the ZF constraintE[ ˆd[n]| d[n]] = gHF d[n], for n = 1, . . . , NB, with [cf. Eq. (3.63)]
d[n] = gHF d[n] + gη[n]ˆ the MSE in Eq. (3.63) reduces to
εVPZF(a[n], x[n], g) =|g|2tr (Cη) . (3.79) Thus, the optimization problem is expressed as
aVPZF[n], xVPZF[n], gVPZF
= argmin
{a[n],x[n],g}|g|2tr (Cη) s.t.: 1
NB NB
X
n=1
kx[n]k22 ≤ Etx and E[ ˆd[n]| d[n]] = d[n].
(3.80) We can form the Lagrangian function as follows,
L (x[n], µ[n], g, λ) = εVPZF(a[n], x[n], g) + 2ℜ tr 1 NB
NB
X
n=1
µT[n] (gHx[n]− d[n])
!!
+ λ 1 NB
NB
X
n=1
xH[n]x[n]− Etx
!
(3.81)
whereλ∈ R0,+and µ[n]∈ CK,n = 1, . . . , NB.
From the Lagrangian function, we can obtain the following KKT conditions:
that are only necessary to find the solution including the zero-forcing constraint.
It is easy to show that the transmit energy constraint is active. Indeed, multiplying the first KKT condition by xH[n] from the left, summing over n = 1, . . . , NB, and taking into account again the transmit energy constraint, we get
λ 1
where the last equality is obtained from the ZF constraint [third KKT condition in Eq. (3.81)]. With this result and the second KKT condition, we get
λ = |g|2tr (Cη)
1 NB
PNB
n=1xH[n]x[n]
and thereforeλ > 0 as long as |g|2 6= 0. So, the transmit energy constraint is always active.
Combining the first KKT condition with the third and with the transmit energy constraint, we can obtain, respectively, the transmit symbols and the receive weights as follows of the zero–forcing solution, i.e. HHH is regular or the zero–forcing condition can be
3.3 MU-MISO Nonlinear Transmit Processing 63
u[n] P v[n] x[n]
I − B η[n]
F H gI
M(•)
M(•) d[n]ˆ u[n]ˆ
Q(•) u[n]˜
Figure 3.13: MU–MISO System with Tomlinson Harashima Precoding.
fulfilled, it is necessary that N ≥ K. We assume that g is real and positive to ensure a unique solution. Note that by applying the matrix inversion lemma to Eq. (3.72), we get that (HHH + ξI)−1HH = HH(HHH + ξI)−1 and it is easy to see that for ξ = tr(Cη)/Etx → 0 the Wiener VP solution converges to the ZF approach.
Pluggingg into the MSE in Eq. (3.79) yields
εVPZF(a[n], x[n], g) = ξ NB
NB
X
n=1
dH[n] HHH−1 d[n].
Due to d[n] = u[n] + a[n], the optimum perturbation vectors are found by the following closest point search in a lattice [23, 91]
aVPZF[n] = argmin
a[n]∈τ ZK+jτ ZK
HH HHH−1
(u[n] + a[n])
2
2 (3.83)
since(HHH)−1 = (HHH)−1HHH(HHH)−1 and zHAHAz=kAzk22.