GFDM is a block-based communication scheme as shown in Figure 2.1 [2], which are proposed to satisfy the requirements of 5G. In this section, we summarize some important results of a previous work that we will use in this thesis to build up further investigations.
In a GFDM block, M complex-valued subsymbols are transmitted on each of the K sub-carriers, so a total of D = KM data symbols are transmitted. The data symbol vector d[l]
is decomposed as
d[l] = [d0,0[l]· · · dK−1,0[l] d0,1[l]· · · dK−1,1[l]· · · dK−1,M−1[l]]T ,
where dk,m[l] is the data symbol on the k-th subcarrier and m-th subsymbol in the l-th block, taken from a complex settlement. Therefore, it is possible to engineer the spectrum regarding given requirements and enabling pulse shaping on a subcarrier basis. By con-sidering that the data symbols are zero-mean and independent and identically distributed (i.i.d.) with symbol energy ES, id est, E{d[l]dH[n]} = ESIDδln. Each data symbol dk,m[l]
is thus transmitted via a pulse-shaped filter by the vector gk,mwhose n-th entry is a circu-larly shifted version of gk,0, and the complex exponential designates the shifting operation in the frequency domain :
[gk,m]
n = [g]⟨n−mK⟩
Dej2πkn/K, n = 0, 1,· · · , D − 1 (2.1)
where g is a D× 1 vector, referred to as the prototype transmit filter [2]. Let x[l] = [x0[l] x1[l]· · · xD−1[l]]T be the vector containing the transmit samples. They are under-stood as the superposition of all transmit symbols. Then, the GFDM modulator can be formulated as the transmitter matrix [2]
A =[
g0,0· · · gK−1,0g0,1· · · gK−1,1· · · gK−1,M−1
] (2.2)
such that x[l] = Ad[l]. The matrix A as defined in (2.2) is called hereafter a GFDM matrix with a prototype filter g. The vector x[l] is further added a cyclic prefix (CP) before sending to the receiver through a linerar time-invariant (LTI) channel. The complexity of
Figure 2.1: Block diagram of the transceiver.
this form is in O(KM log KM ). Yet, it has been shown in [3] that this implementation is advantageous for receiver implementation.
2.1.1 Characterization of GFDM Matrices: Basic Definition
In the literature, GFDM transmitter matrices are often characterized by a prototype trans-mit filter g. We will mainly use this notation in order to continue our study.
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In this thesis, we use another means for characterizing a GFDM transmitter matrix, namely, the characteristic matrix G of size K× M. A formal definition of this character-ization of a GFDM transmitter matrix necessary for the understanding and continuation of the study, is given as follows.
Definition (Characteristic matrix) Consider a KM × KM GFDM matrix A in (2.2) with a prototype filter g. We define the characteristic matrix G of the GFDM matrix A as
G =√
D reshape(g, K, M )WM (2.3)
where reshape(g,K,M) is a K× M matrix whose (k,m)-entry is [g]k+mK, ∀ 0 ≤ k < K, 0 ≤ m < M. The reshape process needs to be used many times, especially for the de-velopment of our future methods. Since we will limit our study to a 2 dimensional matrix, we simply explain that the processus of reshaping proposes to rearrange the order of the elements of an initial matrix, under new conditions of size, so that the number of elements is still kept.
2.1.2 Unitary and Invertible GFDM Matrices
With the characteristic-matrix-domain implementation, we can also easily identify the class of unitary GFDM matrices as follows.
Theorem 1 (Unitary GFDM matrices) Let A be a GFDM matrix with a K× M char-acteristic matrix G. Then, A is unitary if and only if G contains unit-magnitude entries:
|[G]k,l| = 1 ∀ 0 ≤ k < K, 0 ≤ l < M.
The following theorem introduced in [3] is needed to express the conditions for the non-singularity of a GFDM matrix in terms of its characteristic matrix and related prop-erties.
Theorem 2 (Properties of A−1) Let A be a GFDM with a K× M characteristic matrix G. Then,
a A is invertible if and only if G has no zero entries ;
b if A is invertible, then A−H is also a GFDM matrix whose characteristic matrix H satisfies [H]k,l = 1/[G]∗k,l,∀ k, l, id est,
H = (G∗)◦−1 ; (2.4)
c if A is invertible, the squared norm of each row of A−1 equals the energy of A−H, ξH =∥H∥2F/D.
Note: a proof of this theorem is proposed in [3].
2.1.3 GFDM Transmitter Implementations
The transmitter simply modulates the data symbol vector by
x[l] = Ad[l]. (2.5)
Then, x[l] is passed through a parallel-to-serial (P/S) converter, and a CP of length L is added, as shown in 2.1. DenoteK ⊆ {0, 1, · · · , K − 1} and M ⊆ {0, 1, · · · , M − 1}
as the set of used subcarriers and set of used subsymbols respectively, that are actually enrolled.
The digital baseband transmit signal of GFDM is expressed as
x[n] =
∑∞ l=−∞
∑
k∈K
∑
m∈M
dk,m[l]gm[n− lD′]ej2πk(n−lD′)K , (2.6)
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where D′ = D + L and
gm[n] =
[g]⟨n−mK−L⟩
D, n = 0, 1 . . . , D′− 1 0, otherwise
(2.7)
which is finally the transmitted part as shown in Figure 2.1.
Because GFDM is confined in a block structure of D samples, with K subcarriers car-rying M subsymbols each, it is possible to design the time-frequency structure to match the time constraints of low latency applications. Different filter impulse responses can be used to filter the subcarriers and this choice affects the OOB emissions and the SER performance. However, a simple example of M-ary Frequency Shift Keying (MFSK) has shown [5] that increasing the number of symbols leads to a poor spectral efficiency. Spec-tral efficiency means to utilize the available spectrum as efficient as possible. One could say that maximizing the spectral efficiency should base on the channel gains of the users.
It is also known that using by convention guard subcarriers or subsymbols for engineer-ing is still affordable and understandable, since it generally stems from a inter-symbol interference. Instead of just dividing the spectrum into subcarrier and separating them by introducing guard bands these carriers overlap but are orthogonal due to the nature of the pulse shaping.
As a reminder, we denoteK ⊆ {0, 1, · · · , K − 1} and M ⊆ {0, 1, · · · , M − 1} as the set of used subcarriers and set of used subsymbols. In [4], K has been selected as an odd number for each case because RC filters are essentially not applicable to cases where both K and M are even, as GFDM transmitter matrices under such cases are singular. By computing the method in [4], we can also notice that setting different couples (|K|; |M|) give a better spectral efficiency for a same result of a product of these parameters. Since it is still unclear to obtain a true determination of a design for the utilization of involved subcarriers for the setK and involved subsymbols for the set M in a GFDM system, this raises a main consequent problem, that is to say how to settle and designK and M so that
we can maximize the number of users through maximizing the spectral efficiency.