Scientific Paper / Artículo Científico 



https://doi.org/10.17163/ings.n23.2020.04 


pISSN: 1390650X / eISSN: 1390860X 

NUMBER OF SUBCARRIER FILTER COEFFICIENTS IN GFDM SYSTEM: EFFECT ON PERFORMANCE 

NÚMERO DE COEFICIENTES DEL FILTRO DE LAS SUBPORTADORAS EN EL SISTEMA GFDM: 
Randy Verdecia Peña^{1,* }, Humberto Millán Vega^{2} 
Abstract 
Resumen 
Generalized Frequency
Division Multiplexing (GFDM) is a nonorthogonal multicarrier transmission
scheme proposed for fifth (5G) and future generation wireless networks. Due
to its attractive properties, it has been recently discussed as a candidate
waveform for the future wireless communication systems. GFDM is introduced as
a generalized form of the widely used Orthogonal Frequency Division
Multiplexing (OFDM) modulation scheme and it uses only one cyclic prefix (CP)
for a group of symbols. The main focus of this work is to present like impact
on the system performance the coefficient quantity of the subcarrier filter.
A simple method for the computation of the coefficients of the prototype
filter is employed. Besides, it is presented a structure for the GFDM by
taking advantage of the arrangement in the modulation matrix. We evaluated
the Bit Error Rate (BER) using the receiver models presented in this work.
The results showed that the BER is affected according to the coefficients
quantity of the prototype filter. Based on the obtained results, the
coefficients quantity has a relation with the number of time slots of the
GFDM system. 
El GFDM (Generalized Frequency Division Multiplexing) es un esquema de transmisión multiportadora no ortogonal propuesta para la quinta (5G) y futura generación de redes inalámbricas. Por sus atractivas propiedades, está siendo investigada como una forma de onda a ser considerada para los futuros sistemas de redes de comunicaciones. La GFDM es introducida como una generalización del ampliamente utilizado esquema de modulación OFDM (Orthogonal Frequency Division Multiplexing) y usa un único prefijo cíclico (Cyclic Prefix, CP) para un grupo de símbolos. El objetivo principal de este trabajo es presentar cómo impacta la cantidad de coeficientes del filtro de las subportadoras en el desempeño del sistema. Se emplea un método simple para el cálculo de los coeficientes del filtro prototipo. Además, se presenta una estructura para la GFDM aprovechando la estructura de modulación matricial. Se evaluó la tasa de error de bit (Bit Error Rate, BER) usando los modelos de receptores presentados en este trabajo. Los resultados muestran que el BER es afectado según la cantidad de coeficientes del filtro prototipo. Basado en los resultados obtenidos, la cantidad de coeficientes tiene relación con el número de intervalos de tiempo del sistema GFDM. 


Keywords: GFDM, number of coefficients,
prototype filter, BER. 
Palabras clave: GFDM, número de coeficientes, filtro prototipo, BER. 
^{ } ^{1,* }Signals, Systems and Radiocommunications Department, Escuela Técnica Superior de Ingenieros de Telecomunicaciones (ETSIT), Universidad Politécnica de Madrid, España. Corresponding author ): randy.verdecia@upm.es http://orcid.org/0000000347982681 ^{2}Physical Departament (Retired), Universidad de Granma – Cuba. http://orcid.org/0000000194217494^{ } 

Received:
25062019, accepted after review: 25112019 Suggested
citation: Verdecia Peña, R. and Millán Vega, H. (2020). «Number of subcarrier
filter coefficients in GFDM system: effect on
performance». Ingenius. N._ 23, (januaryjune). pp. 5361. doi: https://doi.org/10.17163/ings.n23.2020.05. 
Wireless and Mobile
communication have become essential tools for the life and modern society.
The future wireless networks of telecommunication need higher throughput
based on very high spectral and energy efficiencies, very low latency and
very high data rate. That requires a more effective physical layer (PHY)
[1–3]. The core of the physical layer of fourth generation (4G) is the
Orthogonal Frequency Division Multiplexing (OFDM). These systems allow high
data throughput. OFDM modulation is widely adopted due to its favorable
features like a simple implementation built on the Fast Fourier Transform
(FFT) and robustness against fading channels [2,4]. However, the application
scenario previewed for fifth generation (5G) networks have challenges where
OFDM could have limitations. The low latency needed for Vehicle to Vehicle
Communications and Tactile Internet applications require a data cutoff where
OFDM packet with one cyclic prefix (CP) per symbol have a low spectral
efficient [1, 4–6]. The requirement of OFDM to preserve the orthogonality
between individual subcarriers is essential for the machineto machine (M2M)
communication. Due to the need of low power consumption which influence the
negative form on the synchronization process, this procedure is not possible
by OFDM modulation [4,7]. Other disadvantage of the OFDM system is the high
outofband (OOB) radiation resulting from rectangular pulse shaping [8].
OFDM can fulfill the requirements of 5G in a limited way, due to these
shortcomings.
In
recent years, several waveforms have been proposed to overcome the above
limitations of OFDM, this is the case of FBMC, UFMC, GFDM in references
[9–14] are suggested many waveforms. Filter Bank Multicarrier (FBMC) the
subcarriers are pulse shaped individually to reduce the OOB emissions, this
is caused because the subcarriers have narrow bandwidth and the length of the
transmit filter impulse response is long. The applications that to need a
number of transmit of large symbols are benefit with this modulation. But it
is clear, this modulation scheme is not suitable for low latency scenarios,
where high efficiency must be achieved with short burst transmissions [1,
5–7]. Universal Filtered Multicarrier (UFMC) a group of subcarriers is
filtered to reduce the OOB emission. A principal characteristic of this
modulation is the impulse response can be short obtaining high spectral
efficiency in short transmissions [1]. The disadvantage of UFMC does not
require a CP, then is more sensitive to small time misalignment than CPOFDM
and might not be suitable for applications that need loose time synchronization
to save energy [1, 5, 6]. In this context, the Generalized Frequency Division
Multiplexing (GFDM) is one alternative multicarrier scheme that is currently
under evaluation as a candidate of the PHY layer for the next generation of
mobile 
communication systems.
It is interesting that one of the main relevance of the GFDM is that its
generalized form of OFDM preserves most of the valuables properties of OFDM
while addressing its limitations. The GFDM can provide a very low OOB
radiation. It is more bandwidth efficient than OFDM as it uses only CP for
group of symbols in its block rather than a CP per symbol as for the case of
OFDM [9, 15]. The
GFDM modulation is foreseen for the modulation of independent blocks where
each block consists of a number of subcarriers and symbols. The data symbols
belonging to the subcarriers are filters with a prototype that is circularly
shifted in time and frequency domains. The subcarrier filtering results in
nonorthogonal subcarriers, then intersymbol (ISI) and intercarrier (ICI)
might arise. Filter Impulse Response (FIR) can be employed for filtering the
subcarriers and this choice has a negative impact on the Bit Error Rate (BER)
performance and the OOB emissions as shown in [1]. In this work, we present BER
curves to compare the influence that to have the selection of the total
number of coefficient’s filter and is shown to exist a relationship between
the number of time slots and the
coefficients of the filter in the GFDM systems. It is necessary to present
this aspect because performance degrades when the total coefficient is not
chosen correctly. A
GFDM symbol consists of a block structure of MN samples, where each N
subcarrier carries M timeslots. In a GFDM block, the overhead is kept small
by adding a single CP for an entire block that contains multiple subcarriers.
Thus period that benefit can be used to improve the spectral efficiency of
the system. The remaining sections are organized as follows. The systems
model and properties of the GFDM transmitter are presented in Section 2.
Section 3 presents different receiver structures. Section 4 shows the
expression of the prototype filter as obtained from the subcarrier filter
coefficients. Section 5 analyzes the BER performance of GFDM including the
theoretical equations assuming ZeroForcing (ZF), Matched Filter (MF) and
Matched Filter–Parallel Interference Cancellation (MFPIC) receivers. We used
the coefficients obtained in Section 4. Finally, Section 6 presents some
conclusions. The main objective of this work is to present a structure for
the GFDM by taking advantage of the arrangement in the modulation matrix. Notation: Bold
lower case is used for column vectors and bold upper case for matrices. All
vectors are in column form. The vector and matrix transpose and Hermitian are
indicated by the superscripts ‘T’ and ‘H’, respectively. We use W_{MN}
to denote the discrete Fourier transform (DFT) matrix of size MN. We also
assume that W_{MN} is normalized, such thatW_{MN
}_{ }= I_{MN}, where I_{MN}
denotes the identity matrix of size MN. Hence, = . The terms FFT e iFFT refer to
the fast implementation of DFT and inverse DFT (iDFT), respectively. 
2.
Materials and Methods 2.1. System model
and properties of GFDM The Generalized
Frequency Division Multiplexing is a multicarrier system. The data packet in
GFDM is such that only one CP per block of transmitted symbols is required
[10]. Figure 1 presents the structure of a GFDM data packet. In the system
GFDM the data symbols over each subcarrier are filtered through a
welllocalized bandpass filter with the aim of limiting the InterCarrier
Interference (ICI) [16]. The GFDM data packet is organized in M timeslots
and N subcarriers.
Figure
1. GFDM data
packet. The
OFDM system can provide a high OutOfBand (OOB) radiation and a least
bandwidth efficiency in comparison with GFDM [1, 8] due to the fact that OFDM
system uses a CP per symbol as is presents in Figure 2.
Figure
2. OFDM data packet. Consider
the block diagram of the transceiver depicted in Figure 3. A mapper, e.g QAM
[7], maps the encoded bits to symbols from a 2_ valued complex
constellation, where _ is the modulation order. The s vector denotes a
data block that contains MN symbols, which can be decomposed into M
timeslots and N subcarrier each according to and ,
Figure
3. Block diagram
of the transceiver for GFDM. 
The
data symbols are taken from a zero mean independent and identically
distributed (i.i.d) process with unit variance. The expression that relates
the input data symbols s[m] and GFDM transmitter output x[m],
may be expressed as [16].
where
is the iDFT matrix of size (MN ×
MN), Cf is circular matrix of the size (MN ×MN),
with the first column composed by the vector . The coefficients cf are
the components of the discrete spectrum of the formatter pulse, with y [17, 18]. It
will be shown in this work that the coefficients quantity influences the GFDM
system performance. is the expanded vector of the data
symbols s[m] that can be organized as , where epresents the column vector of the size
(M − 1 × 1) [17, 18]. The expression (1) is performed in two steps. First of all it is performed as the circular convolution of c and s_{ex}[m] for obtaining (MN ×MN) to the result of the first step for obtaining the vector x[m] of size (MN × 1). It is useful to comment that the computational complexity represented in (1) is dominantly determined by an iFFT of dimension (MN × MN). The C_{f }_{s}_{ex}[m] procedure can be calculated by:
where
ʘ is an operator denoting the pointwise multiplication, and the circular
convolution of the vectors c and sex[m] are
performed through pointwise multiplication of their respective iDFTs and
later it is applied applying a DFT to the result. If one considers the
expressions (1) and (2) the vector x[m] reduces to:
where
is the vector of the prototype filter
coefficients that to influencing on the GFDM performance. The computational
complexity in (3) can be reduced significantly by taken into consideration
that the vector sex[m] is the expanded version of the
vector symbol s[m]. However, product can be obtained by M
repetitions of in a column. Then the computational
complexity in (3) can be calculated through an iFFT of dimension (N × N).

The data symbols s of the GFDM packet to transmit in Figure 2 can be obtained by superposition of the M vectors x[m]. We can describe this operation mathematically as:
where
circshift(·) means downward circular shift. The packet to transmit is
completed by adding the CP samples to obtain the vector . An interesting model in GFDM system is a
matrix model with the aim to have likeness with the OFDM system. The model
presented in [17–22] and the vector x_{G} can be
expressed in matrix form as:
where
s is the column vector that contains all the data symbols of the GFDM
packet of M time slots and N subcarrier as is illustrated in
Figure 2. A is the matrix of the GFDM system that is composed by the
coefficient of the prototype filter c_{f} that affect the
performance of the system. The c_{f} coefficient will be
calculated in other section. 2.2. Receiver Implementation The vector x_{G}
is the output of the GFDM modulator (see Figure 3), x_{G}
contains the transmitted samples that correspond to the GFDM data block s
of size (MN ×1). Finally, we added on the transmitter side a
cyclic prefix of LCP samples to produce . After that, the signal is
affected by the Additive Gaussian White Noise (AWGN),, where is the noise variance. The receiver signal
after CP samples removal can be expressed as:
where H represents the circular matrix of the channel of size (MN ×MN). The first column is shaped by the vector that corresponds to the impulse response of the discrete lowpass filter equivalent to the channel of size ch (completed with zeros). The circular matrix, H, can be expressed as:

From
the matrix as represented by equation (6), we can use two standard GFDM
receiver types, i.e. Zero Forcing (ZF) and Matched Filter (MF) receiver
[4,22,23]. We has defined the B matrix as the product of the H and A.
Then equation (6) can be rewritten as:
The
equalization scheme employed in this work is presented in Figure 4. In the
block scheme Q(·) is a function that maps each component of the transmitted
signal vector to the symbol nearest to the signal constellation of the
modulation employed and D(·) determines the minimum distance of the
estimative first that is employed like a metric in the PIC detector. The
switch in the figure defines the receiver employed in each state to obtain
the final estimate. Here ZF and MF are linear detectors and PIC is the
Parallel Interference Cancellation detector, respectively. Hence, the PIC is
employed as a first estimation of the output signal of the MF block.
Figure
4. Block diagram
of the receiver for GFDM. The
ZF receiver is characterized by the BZF matrix that represents
the B inverse matrix. After obtaining the ZF equalization, the linear
demodulation of the received signal can be expressed as:
where
nZF = BZFn is the noise after the ZF
equalization that affects the received signal of size (MN×1). The
second type receiver, i.e. the MF, is described by the BMF = BH
matrix.When it is applied the received vector in (8), the received signal
can be expressed as:

where
n_{MF} = B_{MF}n is the
noise after the MF equalization of size (MN × 1). The PIC detector implementation presents the least computational complexity as compared with other cancellation detectors as SIC [17–20]. The first estimation of the data symbols to the PIC detector is obtained as the output signal of the MF detector. This receiver can be implemented by the equations:
where
(B^{H}B)z corresponds to the matrix B^{H}B
with zeros in the main diagonal. Symbols estimation using equations (11) and (12) are sequentially generated up to a maximum number, J, of iterations. In the present work it was considered that the process can be interrupted after jth iterations (1 ≤ j ≤ J) depending on the quality of the generated estimates. The Maximum Likelihood (ML) metric employed here corresponds to the Minimum Distance (MD) metric. It can be computed as:
If
one detects a reduction in the quality of a given estimate, that is, , the estimate is adopted as the final one. 2.3. Calculating
Filter Coefficients of the Subcarriers The filtered of the
subcarriers in the GFDM modulator block presented in Figure 3 is essential to
the performance of the system. In this section presented like determine its
coefficients. It is presented in [16, 24–26] the prototype filter
corresponding to a class of real lowpass filters whose impulse response can
be express as:
where
P = FK, and cl(0 ≤ l < F) are real
coefficients, the overlap factor F is a positive integer and K is the number
of channels in the TMUX system. The
requirements for the coefficients cl(0 ≤ l < F), after
Mirabbasi and Martin [25] should meet the following conditions:

If
coefficients cl are chosen such that expression in (15) hold, then the
−3 dB frequency of the prototype filter would be approximately , when F is even. The minimum stopband
attenuation (MSA) and the approximate rate of falloff (ARF) of the sidelobes
depend of the overlap factor F and independent of the filter order [25]. It is required to find the F coefficients cl, and to solve a system for determining F coefficients cl. It was obtained in [24] the auxiliary equation:
If equation (16) is satisfied, then the side lobes of the discrete Fourier transform in equation (14) have the approximate falloff rate of ω^{−3}, with ω defining the uniformly–spaced frequencies around the unit circle. It can be written as:
By
using equations (15) and (16) it is possible to construct a system of
equations with the same number of unknowns. Furthermore, equation (18) can be
used to construct the remaining equations necessary to have a system of F equations.
with
the above equations it is possible to obtain the values of the prototype
filter coefficients for F = 2, F = 3 and F = 15. These
are shown in the Table 1. Table
1. Coefficients
of the prototype Filter F

3.
Results and Discussions The simulation results
along with the derived theoretical obtained expressions are presented in this
section. In order to study the effect of the filter coefficients quantity of
the subcarrier on the BER in the system GFDM, we have considered the case of
the ZF, MF and MFPIC receivers. 3.1. BIT Error Rate
Analysis In this subsection we
analyze the performance of the GFDM system in terms of BER versus E_{b}/N_{0}
assuming that ZF, MF and MFPIC are employed. The ZF is able to remove
selfgeneration interference at the cost of introducing noise enhancement
[1]. The MFPIC receiver was the most flexible and adaptable to different
configurations of the data package GFDM [17, 18] as described in Figure 1.
The system parameters used for the simulations are presented in Table 2,
while Table 3 shows the channel impulse response used in the BER performance
evaluation. The impulse response of the multipath channel is normalized to
unitary energy and the length of the CP guard band is G = ch.
Table 2. System Parameters
Table 3. Channel Model
Figure
5 compares the BER performance of the classical ZF in the system GFDM with
different quantity of filter coefficients of the subcarriers considering the
system parameters from Table 2 and the multipath channel from Table 3. The
results presented in Figure 5 suggest that the system GFDM achieved the best 
performance when F =
3. In this case the Bit Error Rate was in the order of 2 × 10^{−2},
when compared to the results presented for F = 2 while F = 15
had more than 3 dB of advantage.
Figure
5. BER simulation
result for ZF receiver in GFDM (I), channel I.
Figure
6. BER simulation
result for MF receiver in GFDM (I), channel I. Figures 6 and 7 illustrate the BER performance for the MF and MFPIC receivers considering the three F cases. The figures showed that performance of the system GFDM depends strongly on the quantity of coefficients of the prototype filter of the subcarriers. The case F = M − 1 rendered the best choice of the quantity of filter coefficients in the system GFDM. Here, M represents the time slots of the system as depicted in Figure 2.
Figure 7. BER simulation result for MFPIC receiver in GFDM (I), channel I. 
The results shown in Figure 8 suggest that employing F = 3 for the different receivers the MFPIC detector had the best performance of the GFDM system. We found that the MF receiver is 4 dB more efficient than ZF with less computational complexity. The MFPIC is more complex than ZF and MF due to the number of iterations [18]. Furthermore, in Figure 8 we present (as a comparison) two curves of the performance, a 4QAM theoretical and other ZF CPOFDM with 64 FFT. It is found that BER performance of the MFPIC scheme is approximately the same with ZF CPOFDM FFT 64, where the difference in the performance is 0,5 dB in favor of the CPOFDM system. The cause behind this small difference is that the GFDM system is affected by the transmission matrix that depends on the coefficients quantity. However, both systems have the same computational complexity in the signal generation as they need FFT 64 but GFDM is more efficient than OFDM in terms of spectrum because the need of just only one CP to transmit a data packet of 256 symbols. On the other hand, the great difference in the BER performance of the 4QAM is produced because it is considered as a system with AWGN. In the simulations both systems have the same computational complexity in the signal generation as they need FFT 64 but GFDM is more efficient than OFDM in terms of spectrum because the need of just only one CP to transmit a data packet of 256 symbols.
Figure 8. Comparison of the simulation result for ZF, MF and MF receiver in GFDM (I) with F=3, 4QAM Theoretical and ZF CPOFDM FFT 64, channel I.
Other simulations are presented in Figure 9 considering that the GFDM system has dimension matrices (512 × 512) and the properties described in Table 2 for GFDM (II). The impulse responses of the channel have 8 taps like exhibited in Table 3. Figure 9 is shown that performance of the MFPIC scheme detector is approximately the same with ZF CPOFDM FFT 32, where the difference 
in the performance is 0,5 dB in favor of the CPOFDM system. Here, it is possible to verify again that there is an intrinsic relationship with the total coefficients of the subcarrier filter, because the best performance that can reach the GFDM system is equal to the OFDM system. The degradation of the performance in Figure 9 in comparison with Figure 8 is given by channel effect by increase the number of taps.
Figure 9. Comparison of the simulation result for ZF, MF and MFPIC receiver in GFDM (I) with F=15 and ZF CPOFDM FFT 32, channel II.
4. Conclusions
The expected implementation scenarios for the 5G wireless networks have challenges as the available physical layer technologies show a limited performance due to their shortcoming. The GFDM system seems a useful candidate by its rendering with the OFDM system. The key property of the GFDM system is the flexibility such that different applications can have a simple solution. This way, it is important to guarantee the coexistence with other technologies, as the current 4G. We produced modulation and demodulation schemes for GFDM system. The presented schemes have a matrix structure that reduces the computational complexity without incurring in any performance loss penalty. By employing the matrix structure of the transmitter and receiver GFDM systems, we analyzed and compared the BER performance for the different calculated coefficients. It was shown that the BER performance of the GFDM system depends on the coefficients quantity of the filter and prototype filter. In the GFDM system, to increase the total number of the coefficient’s filter not improve the performance in the GFDM system. The coefficient total depends on the number of subcarriers because it might filter symbols of others packets and generates interference. The performance of the system is conditional on accurate the coefficient total. 
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