The NP-hard nature of the nonconvex quadratically constrained quadratic problems
P4.1 and P4.2 motivates the development of potentially suboptimal reformulations [210–
212]. In particular, the optimization problems P4.1 and P4.2 are recast as semidefinite
programs by exploiting their resemblance to the sensor network location problem [210,
213], where the aim is to maximize the MED between adjacent sensor nodes.
4.3.1 MED Maximization
This section concentrates on reformulating the non-convex optimization constraints
of P4.1 via semidefinite relaxation [211]. First, the number of meaningful constraints in
the second line of P4.1 and P4.2is determined. In particular, the number of non-identical
constraints is given by Nc= N X i=1 (N − i) =N (N − 1) 2 , (4.8)
where the fact that for any xi, xj, i 6= j, the operation kxi− xjk is equivalent to kxj− xik
has been considered, which makes unnecessary to analyze the symmetrical terms. More-
over, the left-hand side of the quadratic constraints can be decomposed as
khkwk− hmwmk2 = K X i=1 h(k,i)wk− h(m,i)wm 2 , (4.9)
where h(k,i) refers to the i-th entry of hk. The i-th term of the summation in (4.9) can
be re-formulated as [213] h(k,i)wk− h(m,i)wm 2 = Tr e(i)(k,m) e(i)(k,m) H wwH = Tr AE(i)(k,m) , (4.10)
where A , wwH, E(i)(k,m) , e(i)(k,m)
e(i)(k,m) H
, and e(i)(k,m) ∈ CN ×1 is a vector with two non-zero entries in the positions specified by k and m
e(i)(k,m)=0, . . . , h(k,i), . . . , −h(m,i), . . . , 0T
Let E(k,m)be defined as E(k,m), K X i=1 E(i)(k,m). (4.12)
Note that E(k,m) ∈ HN, ∀ k 6= m, where HN represents the set of (N × N )-element complex-valued Hermitian matrices. Then, by substituting (4.9), (4.10) and (4.12) into
(4.6), P4.1 can be recast as P4.1 : maximize A d subject to Tr E(k,m)A ≥ d, ∀ k 6= m Tr (A) ≤ (PtN ) A 0, rank (A) = 1. (4.13)
Here, A 0 indicates that A is positive semidefinite, and is has been considered that
kwk2 = Tr wwH = Tr (A). The above optimization problem is equivalent to that
formulated in (4.6) and still remains NP-hard. However, a relaxed convex version of
P4.1 can be obtained by dropping the non-convex constraint rank (A) = 1, which results
in
P4.10 : maximize
A d
subject to Tr E(k,m)A ≥ d, ∀ k 6= m
Tr (A) ≤ (PtN ) , A 0. (4.14)
The optimization problem (4.14) is convex in the optimization variables A and d, which
facilitates the employment of efficient convex solvers [211]. At this point, it should be
emphasized that the pre-scaling problem (4.14) differs from those conventionally em-
ployed for beamforming [214]. Specifically, there exists a larger number of optimization
constraints and these are distinct, since the information conveyed by SSK depends on
the distance between the symbols received when different antennas are activated. In-
stead, conventional beamforming problems aim at determining the beamforming vector
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Re Im
Received constellation points without pre-scaling
Re
Im
Received constellation points with pre-scaling
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (a) (b)
Figure 4.2: Impact of pre-scaling in the received constellations for a (4 × 1)- element MISO system. (a) Conventional SSK and (b) Pre-scaled SSK.
the information is solely carried out via the traditional amplitude-phase constellations.
4.3.2 Power Minimization
Following a procedure akin to that employed for deriving P4.10 from P4.1, the semidef-
inite relaxation of P4.2 yields
P4.20 : minimize
A Tr (A)
subject to Tr E(k,m)A ≥ d, ∀ k 6= m,
A 0. (4.15)
4.3.3 Effect of the Optimization on the Received Constellation
Prior to characterizing the performance of the scheme considered, the effect of solv-
ing the optimization problem P4.20 in the received constellation is illustrated using an
intuitive example. Specifically, Fig. 4.2(a) shows the received constellation, when con-
ventional SSK transmission is employed, whereas Fig. 4.2(b) represents that under the
same channel conditions, but applying the pre-scaling coefficients designed following
P4.20 using d = 0.3. In both figures, the distinct constellation symbols are illustrated by
tion symbols of Fig. 4.2(b) demonstrates that the approach considered is indeed capable
of enhancing the MED, hence improving the overall performance of conventional SSK.
Moreover, the average transmission power of E {Pt} = 1 W for SSK is reduced in this particular example to E {Pt} = 0.56 W thanks to the proposed pre-scaling. However, the solutions obtained by solving the relaxed problems might become suboptimal in
terms of the attainable MED or the transmission power in P4.10 and P4.20 , respectively,
as a consequence of removing the non-convex rank constraint. The characterization of
this aspect in the pre-scaling scheme considered motivates the following analysis.
4.3.4 Measuring the Impact of the Problem Relaxation
The pre-scaling vectors obtained by the SDR of the optimization problems P4.1 and
P4.2 only coincide with those of P4.10 and P4.20 when rank(A) = 1 [210]. This implies that
the pre-scaling vector w employed for transmission can be straightforwardly derived as
w = wopt = UΣ1/2, (4.16)
where woptdenotes the optimal pre-scaling vector solution to P4.1and P4.2, while U and
Σ correspond to the eigenvectors and eigenvalues of A respectively, i.e., A = UΣUH.
However, the above-mentioned ideal condition rank(A) = 1 is not always satisfied,
and therefore randomization strategies have be employed for finding close-to-optimal
solutions [210, 212]. Specifically, the pre-scaling vectors are obtained as [212]
w = cUΣ1/2v, (4.17)
where v is a vector comprised of the exponential random variables characterized by,
vi = ej bθi, bθi ∼ U (0, 2π], which are uniformly distributed on the unit circle of the complex plane satisfying EvvH
= IN. Here, the constant c guarantees that the problem
constraints are satisfied. It should be remarked that the solutions w obtained as a result
of (4.16) and (4.17) are sub-optimal when rank(A) 6= 1, i.e., fW , wwH 6= woptwHopt and A 6= wwH [210, 212].
Table 4.1: Mean (µ) and standard deviation (σ) of F4.1 with Pt= 1. N/K SSK-SDR SSK-CR SSK-MMD SSK µ σ µ σ µ σ µ σ 2/1 1 0 1.27 0.39 1 0 6.35 21.1 4/2 1.4 0.37 1.96 0.55 1.08 0.09 4.66 9.8 4/3 1.31 0.32 1.8 0.43 1.07 0.08 3.03 2.61
An accurate characterization of the impact of the above degradation should rely on
contrasting the resultant value of the objective function, namely the MED attained or
the optimal transmission power, obtained by the optimization problems P4.1, P4.2 and
their relaxed versions P4.10 , P4.20 . This characterization is, however, impractical due to
the computational hardness of deriving the optimal solution to the original problems
P4.1 and P4.2 [210, 212]. Alternatively, the impact of relaxation can be characterized
by exploiting that the value, f0?, of the objective function delivered by SDR provides a useful bound to the optimal problem [212]. Therefore, a relevant figure of merit F can
be defined as [212] F{4.1,4.2}= f 0? {4.1,4.2} f{4.1,4.2} , (4.18)
where f{4.1,4.2}0? denotes the specific value of the objective function in P{4.1,4.2}0 , when the solution directly retrieved by the solver A is employed, while f{4.1,4.2} corresponds to
the particular value of the objective function obtained after applying (4.16) or (4.17),
i.e. by employing fW , wwH. In the above expressions, the subscripts refer to the optimization problem considered, i.e., P4.1 or P4.2. At this point it should be clarified
that the solutions retrieved by the solver A are different from those obtained after
randomization fW when rank(A) 6= 1. The figure of merit F in (4.18) can also be
generalized both to SSK and to CR-aided SSK (SSK-CR) [183], as well as to SSK
maximum minimum distance (SSK-MMD) [209] to determine the solution’s proximity
to the optimal one.
Table 4.1 characterizes both the expectation and the standard deviation of F4.1, ex-
plicitly quantifying the degradation of the solutions provided by the schemes considered
in this work. In this particular case, f4.10? and f4.1 correspond to the MED obtained by employing A and fW respectively. Note that F4.1 ≥ 1, since the MED bound f
0?
obtained by the solver is always larger than or equal to the MED f4.1 attained after
randomization. Ds= 20 candidate scaling vectors are considered for SSK-CR [183, 209].
Remarkably, the results of Table 4.1 show that both the proposed semidefinite relaxation
(SSK-SDR) and SSK-MMD always achieve the optimal solution for N = 2, which is a
consequence of the Shapiro-Barvinok-Pataki bound [184]. It can also be observed that
the proposed SSK-SDR pre-scaling is capable of reducing both the expectation and the
standard deviation of the figure of merit F , when compared to conventional SSK and
SSK-CR. In other words, the solutions retrieved by the proposed SSK-SDR are closer to
the solution of the optimal problem P4.1. Indeed, the results of Table 4.1 indicate that
the benefits offered by the proposed SDR-based technique become more pronounced for
reduced system dimensions. Simultaneously, it can be seen that the SSK-MMD algo-
rithm, which was published throughout the development of this work [209], is capable
of providing better solutions than the proposed SSK-SDR scheme. Nonetheless, in the
following it is shown that the closer proximity of the SSK-MMD solutions to the optimal
ones is achieved at the expense of a substantial increase in their computational com-
plexity, and that the performance differences remain modest for the small scale antenna
systems considered in this chapter.