Optimization Algorithms in Wireless and Quantum Communications

Multi-antenna system

Wireless broadcast system

Multi-antenna system

Mathematical model of multi-antenna system

Upper-triangular decomposition — the key component of the sphere decoder

Tree generated by the sphere decoder algorithm

Reduced tree of the branch and bound sphere decoding algorithm

Comparison of the number of points per level in the search tree visited by the

An H ∞ estimation analogy used in deriving a lower bound on integer least-

Computational complexity and the distribution of the points in the search

Flop count histograms for SD, SPHSD, GSPHSD, PLT, and SDsdp algorithms,

Single-input multiple-output (SIMO) system

Mathematical model of SIMO system

QR factorization

Tree search

Mathematical model of a wireless broadcast channel

Comparison of the sum rate of Method 2.1 to the sum rate of reg. pseudo-

A sketch of radial signal scaling

Comparison of BER, M=6 antennas/users, 8PSK-Method 2, 8PSK-Method

Comparison of rates, M=6 antennas/users, 8PSK-Method 2, 8PSK-Method

Comparison of BER, M=20 antennas/users, 8PSK-Method 3, 8PSK-regularized

Comparison of BER, M=6 antennas/users, 16PSK-Method 4, 16PSK-Method

Comparison of rates, M=6 antennas/users, 16PSK-Method 4, 16PSK-Method

VPT scheme

Thesis contributions

ML detection

In the first part of the thesis we consider multiple-input multiple-output (MIMO) systems. In Chapter 2 we consider the case of the so-called coherent ML detection in MIMO systems.

Broadcast channels

Quantum unambiguous detection

To allow portability of the model, it is usually assumed that the channel matrix coefficients are i.i.d. The case when the channel matrix is known at the receiver is usually referred to as the coherent case of the signal detection in multi-antenna systems.

Sphere decoder and its modification

In this chapter, we try to reduce the computational complexity of the sphere decoder while still finding the exact solution. We will show that the lower bound LBsdp significantly improves the expected complexity of the standard sphere decoder.

Figure 2.3: Upper-triangular decomposition — the key component of the sphere decoder algorithm

SDP-based lower bound

This essentially means that the SDsdp algorithm performs about four times more operations per each node of the tree than the standard sphere decoder algorithm. On subplot 3 of Figure 2.7, the comparison of the total number of points kept in the tree by SD and SDsdp algorithms is shown.

Figure 2.7 compares the expected complexity of the SDsdp algorithm to the expected complexity of the standard sphere decoder algorithm (SD-algorithm)

H ∞ -based lower bound

In particular, we show in Section 2.5 that the lower bound obtained by solving a related convex optimization problem, where the search space of integers is relaxed to a sphere, can be derived as a special case of the lower bound from Theorem 2.1. Then, in Section 2.6, we show that the lower bound is obtained by solving another convex optimization problem, where .

Figure 2.7: Computational complexity of the SD and SDsdp algorithms, m = 50, D = {− 1 2 , 12 } 50

Spherical relaxation

Generalized spherical relaxation

Since the generalized spherical bound is at least as tight as the spherical bound, we expect the GSPSD algorithm to prune more points from the search tree than the SPHSD algorithm. Clearly, finding the inverse of ˆαLL∗+I has a cubic complexity and is required at each node of the search tree (Lis is constant per level, but ˆα varies from node to node).

Polytope relaxation

The lower bound studied in this subsection is tighter than the spherical bound discussed previously. However, although the lower bound based on the polytope relaxation is tighter than that based on the spherical relaxation, the overall computational effort is not necessarily improved.

Performance comparison

Flop count

Flop count histogram

Eigen bound

Eigen bound performance comparison

In particular, Figure 2.11 compares the expected complexity and total number of points in the tree of the EIGSD algorithm with the expected complexity and total number of points of the standard spherical decoder algorithm. As you can see, the EIGSD algorithm has a significantly better shaped (shorter tail) flop count distribution than the SD algorithm.

Figure 2.11: Computational complexity of the SD and EIGSD algorithms, m = 12, D = {− 15 2 , − 132 ,

Summary and discussion

The results we present show potentially significant improvements in the speed of the sphere decoding algorithm. We have provided an explicit analytical upper bound on the expected complexity of an out-of-sphere decoder.

Figure 2.13: Computational complexity of the SD and EIGSD algorithms, m varies, D = {− M 2 − 1 , − M 2 − 3 ,

Non-coherent ML detection

In section 3.5, we summarize the results obtained and suggest several possible directions for future work. Since ∗s= 1, exactly the same optimization problem is achieved if the optimization criterion used is joint channel estimation and signal detection [87]).

Exact non-coherent ML detection

Out-sphere decoder

Although (3.7) resembles the standard sphere decoder used for the minimization problem, it is fundamentally different. Thus, the out-of-sphere decoder algorithm generates a tree (see Figure 3.5), where the branches at level (m−k+ 1) of the tree correspond to all points of the (m−k + 1) dimensional lattice that satisfy (3.9). ).

Expected complexity of the out-sphere decoder

The real case
The complex case

Its expected complexity ECosd (averaged over channel and noise statistics) can be upper bounded as follows. After the derivation from the previous subsection, we can easily see that (3.17) has the following dual in the complex case. 3.22) Since both Rands are complex, we have that the kRsk2 chi-square is distributed with 2m degrees of freedom and the kRk:m,k:msk:mk2 chi-square is distributed with 2(m−k+ 1) degrees of freedom.

Approximate non-coherent ML detection

A simple rounding algorithm

Of course, since the AR algorithm is only an approximation, the exact ML algorithm still has a significant advantage in obtaining the encoding. This explains why the performance (symbol error rate) of the AR algorithm is slightly worse than that of exact ML.

Figure 3.6: Comparison of symbol error rate, AR, ML, MRC, and SDP q=4, T=N=10

SDP relaxation

Recall that the AR algorithm described in the previous section was based on the relaxation (3.26) of the same original ML detection problem (3.3). Therefore, the error rate of the AR algorithm cannot be lower than the error rate of the SDP algorithm.

Computing the PEP

Since the only case where the two differ is that Genie is sure to make an error, it cannot have a lower error probability than SDPLS. Then the probability that an error occurred if the SDPLS algorithm was used to solve (3.34) can be upper bounded as follows.

Asymptotic analysis, T → ∞

q = 2
General q

Let Bpep(ρ) defined in (3.57) be the PEP type bound on the probability that an error occurred if SDPLS algorithm was applied to solve (3.34). LetBpep(ρ) defined in (3.57) be the PEP-type upper bound on the probability that an error occurred if SDPLS algorithm was applied to solve (3.34).

Computational complexity

Therefore, in communications where the SIMO system dimension is less than 60 and 2-PSK signaling is used, or where the SIMO system dimension is less than 24 and 4-PSK signaling is used, the required number of arithmetic operations per the algorithm is similar to the corresponding number of operations required for the basic SDP.

Discussion and conclusion

As can be seen immediately, the main idea behind the concept is the broadcast channel. A significant progress in the analysis of the capacity region and the sum rate capacity of the broadcast channel has recently been achieved.

Finding optimal preprocessing matrix G

Maximizing the sum rate over G

In Figure 4.3, the comparison of the sum rate achieved by Method 2.1 and the sum rate achieved by the adjusted pseudo-inverse are compared with the sum capacity of the transmission channel. As can be seen, although we have no formal proof of this, Method 2.1 significantly narrows the gap between the regularized pseudo-inverse and the sum capacity.

Maximizing the minimum rate over G

The technique described in section 4.2.1 maximizes the sum rate of the multi-antenna broadcast system under the constraint of linear data processing. For example, in section 4.2.1 the total amount is maximized at the expense of the weakest users, who are ignored.

Figure 4.4: Comparison of the max-min rate of Method 2.2 to the max-min rate of reg.

Finding the optimal scaling coefficient k

Now we want to optimize the scaling factor, while keeping the magnitudes of u greater than or equal to 1. This will result in magnitudes of the components of the received vector r that will be at least as large as if there was no signal scaling at all.

Combined method

The above problem is convex and can therefore be solved exactly by efficient convex optimization techniques.

Simulation results

Conclusion

Moreover, we obtain similar results in the case where the number of users is much larger than the number of transmitting antennas. As we mentioned in the previous chapter, the channel status information (CSI) is complete.

Figure 4.6: Comparison of BER, M=6 antennas/users, 8PSK-Method 2, 8PSK-Method 2.1, 4-PSK-Method 2.2, 4PSK-regularized pseudo-inverse

The vector-perturbation technique

The main idea behind (5.3) is to eliminate (or reduce) the power penalty that occurs in the case where the so-called zero-forcing (ZF) scheme (obtained for ˆl = 0 in (5.2)) is used. Decoding these signals is simple and the only processing required by receivers is scaling.

Case K = M

Low SNR regime (ρ → 0)

The results given in and (5.13) indicate that the sum rate of the diagonal vector perturbation technique varies linearly with the number of users. Consider communication in the low-SNR regime (ρ→0) over a square Gaussian broadcast channel using the diagonal vector interference technique with parameters β ≥1 and τ ≥w, where w is the width of the QAM constellation.

General SNR

Consider communication over quadratic Gaussian broadcast channel using diagonal vector perturbation technique with parameters β = 0 and τ > w2, where w is the width of a QAM constellation. Furthermore, the scaling law in the high-SNR regime is not only linear in the number of users, but also optimal, i.e. equal to the capacity-achieving DPC technique.

Case K M

The previous corollary states that the sum rate of VPT asymptotically achieves the same sum rate as DPC. Note: We emphasize that using the same selection of users as proposed in this section, it is easy to show that under the assumptions of the previous corollary, even the ZF scheme asymptotically achieves the same degree of summation as DPC.

Conclusion

The problem of detecting information stored in the state of a quantum system is a fundamental problem of quantum information theory. They also develop a closed-form solution for the optimal measurement in the case where both core states have dimension 1.

Problem formulation

Given that the state of the system is ρj, the probability of obtaining outcome i is Tr(ρjΠi). A worst-case analysis of interior point methods shows that the effort required to solve a semi-final program with a given accuracy grows no faster than a polynomial of the size of the problem.

Conditions for optimality

Dual problem formulation

Optimality conditions

Once we have found the optimal Zb that minimizes the dual problem (6.16), the constraints (6.22) and (6.23) are necessary and sufficient conditions for the optimal gauge operatorsΠbi. Necessary and sufficient conditions on the optimal measurement operators Πbi are that there exists a Z∈Γ such that.

Special cases

Orthogonal null spaces S i

In Appendix 6.9 we show that in this case, the optimal solution of the dual problem can be expressed as

Null spaces of dimension 1

The optimal measurement operators for this case were developed in [82] and can be written as.

Optimal detection of symmetric states

GU state sets

Therefore, to find the optimal gauge operators, all we need is to find the optimal generator Π. Since the optimal gauge operators satisfy Πi = UiΠUi∗,1 ≤ i ≤ m and ρi = UiρUi∗, Tr(ρiΠi) = Tr(ρΠ), so problem (6.9) reduces to a maximization problem.

CGU state sets

Since this problem is a (convex) semi-definite programming problem, the optimal generators Πk can be computed very efficiently in polynomial time within any desired accuracy. Note that the problem of (6.50) and (6.51) has rn2 real unknowns and lr+ 1 constraints, unlike the original maximization, which has lrn2 real unknowns and (lr)2+ 1 constraints.

Conclusion

Proof of (6.30)

Proof of (6.31)

Moreover, in [82] a bound for the maximum probability of correct detection in the case of clear discrimination of two general mixed states is derived. In [7] the authors derive an analytical solution for the clear discrimination of a special class of two mixed states.

Problem formulation

While in the previous chapter we considered the general problem of unambiguous quantum detection without limiting the number of quantum states at hand, in this chapter we limit ourselves to a specific case of unambiguous discrimination of two mixed quantum states. However, in the most general case, the analytical solution of the unambiguous discrimination of two mixed states still remains a very difficult task.

The dual problem

To solve (7.3), we will first solve the dual problem and then find the solution of the primal from the conclusions about the optimality conditions given in [31]. From now on, to avoid tedious discussion of degenerative low-rank cases, we will assume that E is invertible.

Optimality conditions

Solving the primal and dual problems

Rank-2 ∆s

One of ∆s is zero

One of ∆s has rank 2, the other rank 1

This concludes the case where one of the ∆s is of rank 1 and the other is of rank 2.

Both ∆s are rank 1

In the first part of the thesis (chapters 2 and 3) we discussed the problems of ML detection in MIMO (multiple-input multiple-output) systems in wireless communications. In this thesis, we developed an improved branch-and-bound version of the standard sphere decoding algorithm and demonstrated its performance through simulations.