The result obtained for this case is therefore 28.
4.2 Limits of the methods and discussion
Other simulations that are not appropriate for the presentation suggest strongly that the number of used subcarriersK′ has to be really important face to the number of used sub-symbols M′ in order to be efficient, guarantee to maximize the out of band PSD, and furthermore, being more efficient that the last method. since the Method 3.2 is easy to understand and determine how it works. A few narrow cases, such as the Case 1, are how-ever working for this method, since we do not meet so many times different couples for a same result obtained for the objective function.
And that is precisely there that we finally expect to propose the last method (Method 3.3) which promises to be strong and efficient in calculation, and sure to find the results.
Knowing the already calculated values, the algorithm still compare them with the last cal-culated value, in order to see where a new potential solution should be encountered, in order to satisfy the resolution of the problem (2.9). We expect finally helping and propos-ing the literature another line of study in order to develop and advance in the search for solutions to this known problem.
Unfortunately, the limitation of the computer’s memory capacity does not allow us to offer more calculation results for much larger matrices, especially for Method 3.2.
How-ever, Method 3.3 will use the same process to solve the calculation for larger matrices, whose memory usage is different.
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Chapter 5 Conclusion
In this thesis, we used a matrix-based characterization of generalized frequency division multiplexing (GFDM), in order to design and improve prototype filters used here to max-imize the spectral efficiency. This characterization used properties derived from the con-ventional GFDM matrix transmitter, which were not easily exploitable from the conven-tional filter prototypes observed in the research. From here, we settled to manage the sets of used subcarriers K and used subsymbols M, in order to maximize the spectral effi-ciency in out of band.
In addition, the new optimization problem introducing, firstly solved by raw force, the product maximization of used subcarriers and used subsymbols for out-of-band radiation was addressed under two other main aspects: one resolution in one dimension, and the other in two dimensions. The results showed in the first case and for objective function values having the same result, that the higher the number of used subcarriersK′ is, the better the maximization result will be. Thus, we will choose to maximize this number of used subcarriersK′, within the physical limit possible, to keep an convenient compromise between the devices to maintain and maximize out-of-band spectral efficiency.
The third method, for its part, proposes a solution as convergent as the second, but currently requiring a much larger number of calculations than for the first method, in or-der to solve the proposed maximization problem (2.9).
Moreover, we have emphasized that a good management and design of pairing the couples given by used subcarriers and used subsymbols does permit the maximization of spectral efficiency, a demonstration of this is given is more detailed in the following 5. The contribution involved by this work thus show that an improvement can be studied and developed in order to brighten and perfect the utilization of various prototypes already encountered in the literature, whose primary choice of parameters studied in our main ob-jective function was not adequately defined.
For possible further work, a search for a new effective method for rapid calculation this type of problem certainly already encountered but not yet solved in a formal way, leading to a convergence known in advance. Also, other mathematical methods can be introduced in order to decrease, even drastically and effectively reduce the logarithmic complexity involved in this maximization calculation for a non-decreasing function in both directions (two-dimensional matrix). In view of the memory problem encountered for very large matrices, it could also be envisaged to propose other means of calculation in order to accommodate a larger number of users on a large scale.
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Bibliography
[1] G. Fettweis, M. Krondorf, and S. Bittner. GFDM - generalized frequency division multiplexing. Veh. Technol. Conf., 2009. VTC Spring 2009. IEEE 69th, pages 1–4, 04 2009.
[2] N. Michailow, M. Matthé, I. Gaspar, A. Caldevilla, L. Mendes, A. Festag, and G. Fet-tweis. Generalized frequency division multiplexing for 5th generation cellular net-works. IEEE Trans. Commun., 62(9):3045–3061, 09 2014.
[3] Po-Chih Chen, Borching Su, and Yenming Huang. Matrix characterization for GFDM:
Low complexity MMSE receivers and optimal filters. IEEE Transactions on Signal Processing, 65(18):4940–4955, 06 2017.
[4] Po-Chih Chen and Borching Su. Filter optimization of out-of-band radiation with performance constraints for GFDM systems. Signal Processing Advances in Wireless Communications (SPAWC), 12 2017.
[5] Ghaith Al-Juboori, Evgeny Tsimbalo, Angela Doufexi, and Andrew R. Nix. A com-parison of OFDM and GFDM-based MFSK modulation schemes for robust IoT ap-plications. 2017 IEEE 85th Vehicular Technology Conference (VTC Spring), 11 2017.
[6] M. Matthé, N. Michailow, I. Gaspar, and G. Fettweis. Influence of pulse shaping on bit error rate performance and out of band radiation of generalized frequency division multiplexing. Proc. IEEE ICC Workshop, pages 43–48, 2014.
[7] M. Grant and S. Boyd. Cvx: Matlab software for disciplined convex programming, version 2.1.
[8] S. Boyd and L.Vandenberghe. Convex Optimization. Cambridge University Press, 2009.
[9] J. Dattorro. Convex optimization Euclidean distance geometry. Meboo Publishing, 2016.
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Power Spectral Density and OOB Leakage
In this section, which serves as an aid for simulation later, we define the OOB leakage O as a performance measure for the OOB radiation of transmit signals. To evaluate O for GFDM, we first address the power spectral density (PSD) of GFDM signals. We derive an analytical PSD expression encompassing an interpolation filter used in a D/A converter.
Actually, from the GFDM digital baseband transmit signal x[n], we obtain the analog baseband transmit signal xa(t) is obtained by passing x[n] through a D/A converter with a sampling interval Ts and an interpolation filter p(t), id est, xa(t) = ∑∞
gm[n]e−jwn be the discrete-time Fourier transform of gm[n], where gm[n] is defined in (2.7). Assuming the data symbols are zero-mean and i.i.d. with symbol energy ES, we can derive that With some derivations, we further obtain that
Gm(ejw) =
where wl′ = w− (2πl/D′) and
Since gf is the frequency-domain prototype transmit filter, defined as the D-point DFT of g, id est,
is the periodic sinc function for any positive integer p. Using (1), (2), (3) and (4), we can express the PSD with gf, which enables designing the PSD in terms of the frequency-domain prototype transmit filter. A special case that leads to a simple expression of Gm(ejw) is L = 0. When L = 0, (2) can be reduced to
To characterize the OOB radiation, we define the OOB leakage as
O = |BI|
In (7),BOandBIare the set of frequencies considered out of band and in band respec-tively, and|BO| and |BI| denote the lengths of the corresponding intervals. Recall that K is the set of subcarrier indices actually used. The nominal frequencies of the subcarriers inK lie in BI, several guard subcarriers are used betweenBOandBI, andBOis reserved
36
for the use of other users.
Finally, note that in (2.6), the setsK and M are not required to be K = {0, 1, · · · , K − 1} or M = {0, 1, · · · , M − 1}. This means some guard symbols or guard subcarriers can be used. GFDM is proposed to exhibit low OOB radiation. This advantage is particularly significant if some guard symbols and guard subcarriers are used [6].
Used algorithm
To solve the optimization problem (2.8),the techniques for solving convex optimization problems [8] have been used. Since the problem (2.8) is non convex, some transformations on the problem have been made. After introduction of some variable S∈ HD+, defined as S = vec(G)(vec(G))H, where G is defined in (2.3). By the definition of the energy ξH and Theorem 2, we have
ξH = 1
Using (8), we obtain an equivalent form of the problem
minimize
By a lemma used in [3], we further involve a derivation as
Therefore, the objective function (9a) is a supremum of affine functions of S, and thus convex in S. We can also show that the constraints (9b) and (9c) are convex. However, the problem (9) is still non convex because the rank constraint (9d) is non convex.
To approach the optimization problem with a rank constraint, an iterative algorithm [9]
is used. The problem (9) is tackled by iterating the optimal point ˜S of
minimize
with the optimal point ˜V of
minimize
until convergence, where w > 0. The algorithm in [9] is used only for real variables. For this part, the algorithm has been extended so that it can be used for complex variables.
Specifically, the domain is changed from the set of real symmetric positive semidefinite matrices to the set of complex Hermitian positive semidefinite matrices, and we intro-duce the operator| · | to assure that the objective functions are real-valued. According to the simulation results of this algorithm presented in the next section, the extension worked.
The problems (15) and (16) are convex, so the techniques for solving convex opti-mization problems can be applied. It is not difficult to show that all constraints of the two problems are convex. Beside this, one note that w > 0 and that maxf∈BO Sa(f ) is convex in S. Thus, to prove the convexity of the problems, it has been shown that tr(˜SV) is convex in V. Regarding the complex variables V as independent real variables, id est, their real parts and imaginary parts. Then, tr(˜SV) is a norm of a affine transformation of these real variables. Since any norm is convex, and composition with an affine transformation of these preserves and convexity, the problems are convex.
To understand the concept of the algorithm, it would have been beneficial to know the solution of the problem (16). In fact, this problem can be solved analytically [9]. Specif-ically, with the ordered (in the order of non-increasing eigenvalues) eigendecomposition
˜S = QΛQH, the optimal point is ˜V = UUH, where U is the submatrix of Q obtained by removing the first column of Q. In other words, the first D− 1 eigenvectors of ˜V are the same as the last D− 1 eigenvectors of ˜S, and all correspond to eigenvalue 1. The last eigenvalue of ˜V is 0. Therefore, the term w|tr(S˜V)| introduced in the objective function (15a) can be considered as favoring the direction uuH, in the vector spaceHD+, where u is the first column of Q. In this way, we expect that the algorithm can converge to a point corresponding to a rank-1 S.
This iterative algorithm starts with the problem (15), so the initial value of ˜V is a parameter that can be designed. The choice of the weight w, which is also a parameter to
be designed, can affect the rate of convergence and the result S at convergence. A small report of the influence of w will be introduced.
Influence of the weight w The obtained prototype filter g at convergence is the same for all w in the range. The rank constraint is not met if w is too small, and the obtained objective gets greater if w is too large. As w increases above 0.003, NI increases nearly proportionally. Thus, to minimize the obtained objective and maximize the rate of con-vergence, w has been set to 0.003. In fact, w in [4] has been selected in this way for each case.
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Proof of non-convexity for (B.2d)
We remind below the main involved algorithm lacking of explanation about non-convexity about one element.
minimize S∈ HD+
fmax∈BO
Sa(f ) (17a)
subject to tr(S) = D, (17b)
tr(S◦−1)≤ Dη, (17c)
rank(S) = 1. (17d)
whom rank-constraint is not convex.
Proof (as an example) : Let
J11 =
1 0 0 0
J22=
0 0 0 1
(18)
Both matrices are rank-1, but the average of those two matrices is rank-2. The rank con-straint is known to be a matrix analogue to theL0-norm on vectors, that is to say :
• L0-norm counts the number of zero-entries ;
• rank constraint counts the number of non-zero singular values.
Remind : L0-norm of x is :
∥x∥0 =√∑0
i
x0i (19)
whom optimization problem should be :
min ∥x∥0
subject to Ax = b
(20)
where x∈ Rn, A∈ Rm×n, m ≤ n and b ∈ Rm.
However, doing so is not an easy task. Because the lack ofL0-norm’s mathematical rep-resentation, L0-minimisation is regarded by computer scientist as an NP-hard problem, simply says that it is too complex and almost impossible to solve.
In many cases, L0-minimisation problem is relaxed to be higher-order norm problem such asL1-minimisation andL2-minimisation.
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Add-ons about about g settlement and PSD calculation
.1 Proof of discrete-time Fourier transform of g
m[n]gm[n] is defined as :
The discrete Fourier transform of gm[n] is :
Gm(ejω) =
.1.1 No cyclic prefix (L=0)
To simplify the problem, let first L = 0, so that D′ = D. Thus :
=
DWHDgf. Knowing that WD is the normalized D-point discrete-Fourier-transform (DFT) matrix with [WD]m,n = e−j2πmn/D/√
D, and interpreting the discrete-time-Fourier-transform (DTFT) of the transmitted signal, we could rewrite every DTFT Gm-element as following :
Gm(ejω) = 1 The Dirichlet function is defined as :
Dx = sincp(x) =
Following that expression, we can write Gmas :
Gm(ejω) =
Gm(ejω) =
D∑−1 l=0
[gf]
le−j2πlmMsincD(ωl)e−jωlD−12
.1.2 With cyclic prefix (L=0)
Now, we keep the cyclic prefix, so that D′ = D + L. Thus :
DWHDgf. Knowing that WD is the normalized D-point discrete-Fourier-transform (DFT) matrix with [WD]m,n = e−j2πmn/D/√
D, and interpreting the discrete-time-Fourier-transform (DTFT) of the transmitted signal, we could rewrite every DTFT Gm-element.
Given that the CP has been introduced, one length is added in the transmitted part of the system. To keep this length, the operation is iterated once again (so that we can eliminate inter-symbol interference (ISI)). That is why we repeat the end of the symbol as :
We can use one again the Dirichlet function to write Gm as : that can be rewritten as :
Sa(f ) = ES|P (f)|2
Using the proposed Lemma in [3], saying that : The frequency-domain prototype filter gf =√
DWDg can be expressed as
48
gf = vec (
GTWK
)
, we can obtain a further derivation :
Sa(f ) = ES|P (f)|2 Then, we propose to rewrite vec
(
= ES|P (f)|2