2.2 Definition of Prototype Filters for Maximizing Spectral Efficiency
2.2.2 Problem formulation
In this subsection, we bring the main problem that constitutes the main contribution of this master’s thesis. As mentioned in 5, some guard symbols and guard subcarriers are often used for GFDM [3].The design of the prototype filter based on the characteristic matrix together with the design of the number and position of guard subcarriers and subsymbols for maximizing spectral efficiency under OOB-radiation and performance constraints can be studied. The motivations presented above therefore invite us to pose the problem in this way.
Thus, we are led to introduce an optimization problem characterized by a maximiza-tion of parameters of subcarriers and subsymbols values, while the in-band filter part is efficient and the spectral efficiency is maximized. Let us introduce a GFDM system char-acterized by a GFDM transmitter matrix and a prototype filter g. Let K, M ,K, M and BO
be set according to the frequency sample used, whereBOdesignates the set of frequencies classified out of band of the filter waveform. It should again be noted thatK is the set in-cluding the subcarriers involved in a situation, called used subcarriers. Respectively,M is the set including the subsymbols involved in the same situation, called used subsymbols.
Let ξ be the noise enhancement factor, characterized as
ξ = 1 D
K∑−1 k=0
M∑−1 l=0
1
|[G]k,l|2
. Let ρ and η be some positive real numbers.
We formulate the optimization problem as following :
maximize
g,K, M |K||M| (2.9a)
subject to ∥g∥2F = 1, (2.9b)
fmax∈BO
Sa(f )≤ ρ, (2.9c)
ξ≤ η, (2.9d)
where the objective function (2.9a) induced about maximization, is written as the product of the absolute values of the numbers of used subcarriers and used subsymbols, according to the point of view adopted and willing to be satisfied. The constraint (2.9b) is the nor-malization of the prototype filter used, under the Frobenius norm. The constraint (2.9c) caracterized by maxf∈BOSa(f ), particularly specifies the maximization of the spectral ef-ficiency where Sa(f ) is the PSD detailed in 5, detailed calculations of which are proposed in 5, to ensure the veracity of the reformulations involved. Specifically, this is the PSD of the GFDM analog baseband transmitted signal, which needs to be maximized because this is one the main interests of this contribution work on designing prototype filters. Trivially, f designates the frequency used, and is taken here inBO. The physical representation of this constraint where the PSD maximization below the ρ parameter must be interpreted as the maximum power spectral efficiency desired by the designer. Finally, the constraint (2.9d) pertains to maintain a sufficiently good MSE or SER performance. This param-eter is set to 1, to have a minimized mean-square error (MSE) for the zero-forcing (ZF) receiver under AWGN channel.
12
Chapter 3
Proposed methods
With regard to the problem statement introduced at the end of the previous chapter, we propose to solve it here in this independent chapter. The objective function (2.9a) of this problem, that is to remind maximizeg,K,M|K||M|, is to maximize the product of the num-ber of used subcarriers in the setK and the number of used subsymbols in the set M involved. To deal with the under- statement g, in the objective function for the maximiza-tion problem, this will be settled as the prototype filter introduced in [4], so that it can be based on the characteristic matrix involved and used by the designer. Precisely, the trans-formation, is as the one proposed in 5. Since the constraint (2.9c) is the most important constraint to understand in terms of definition and calculation (see 5 and 5), it will there-fore be the heaviest mathematical part to exploit. By using an algorithm studied in [4]
and detailed in 5, which was recently designed to address the problem presented in the equation (2.8), we are now able to deal with the different constraints of the optimization problem (2.9) and to have an approach in order to solve the main problem of this thesis described in it.
Thus, we are suggesting to solve this problem under three different aspects. We intend to solve this optimization problem by first proposing a method by brute force, putting for-ward a matrix of size K× M, thus calculating all the elements of the matrix concerned to determine the couple solution of the problem. Secondly, another method is to transform the matrix of size K×M into a matrix of size 1×(K ∗M) knowing the possible
combina-tions of couples (K′;M′), whereK′andM′are respectively the number of used subcarri-ers in the setK and the number of used subsymbols in the set M, and the non-decreasing property of the PSD function involved. The last method is finally a two-dimensional cal-culation, of which we will present the details of calculations and the limits of the method.
3.1 Method 1: complete calculation by raw force
As mentioned earlier, we propose in this section to calculate the completeness of the K × M size matrix elements, and to determine the maximum PSD values in the out-of-band portion of the transmitted signal of the GFDM system. To do this, we use the algorithm introduced in [4], a detailed explanation of which is provided in 5, which is dealing with the constraint 2.9d introduced in our problem.
We make sure to choose the parameters of the sets of used subcarriersK and used sub-symbolsM involved one by one, and by incrementation, first of the subsymbols and then of the subcarriers, we complete the matrix of size K × M, by entering the value of PSD obtained in the out-of-band part after calculation and stop of the algorithm used, thanks to the fixed stopping criterion for the loop involved in 5.
Finally, in order to to find the optimal solution, and more precisely the optimal cou-ple (K′;M′) of the problem of maximization of these parameters that are the number of used subcarriersK and used subsymbols M involved to maximize the spectral density of power out of band, and thanks to the desired maximum solution of PSD involved by ρ, we can make a simple comparison about multiplied sets between them, under the constraints of the problem (2.9), so that we can point out the sets of used subcarriersK and used sub-symbolsM as the optimal solution.
14
The method is summarized in the following algorithm:
Algorithm 1 Raw force complete calculation Result: OptimalK′, OptimalM′
initialization: empty D = K× M matrix forK′ = 0 to K − 1 do Evolutionary loop
forM′ = 1 to M do Evolutionary loop D(K′,M′) = maxf∈BO Sa(f ) end
end
Find (OptimalK′; OptimalM′) such as max D(K′;M′)≤ ρ
Whatever other methods are introduced later, this method naturally holds its place here, since it guarantees the solution of this problem including a non-decreasing function, in all cases.
Note: Since the literature does not propose a good solution for this kind of known problem, the raw force is the universal method to solve the problem (2.9). That is why, in order to speed up the resolution of the problem, and given the function characterizing the power spectral density partly out of band, and knowing that the values of the subcarriers and subsymbols give this calculation a higher value plus their number increases, id est, the power spectral density function is a non decreasing function, it goes without saying that this method can be set a stop parameter during the calculation as soon as the desired PSD value for a couple solution (K′;M′) is exceeded in the solution search under this constraint (2.9c). To some extent, we expect to obtain a matrix rudely looking like a superior triangle shape.
Algorithm 2 Raw force complete calculation with stopping criterion Result: OptimalK′, OptimalM′
initialization: empty D = K× M matrix forK′ = 0 to K − 1 do Evolutionary loop
forM′ = 1 to M do Evolutionary loop D(K′,M′) = maxf∈BO Sa(f )
if maxf∈BO Sa(f )≥ ρ then K′ =K′+ 1
M′ = 1
Break current loop end
end end
Find (OptimalK′; OptimalM′) such as max D(K′;M′)≤ ρ