# Influence of Pulse Shaping Filters on PAPR Performance of Underwater 5G Communication System Technique: GFDM

Authors: Jinqiu Wu; Xuefei Ma; Xiaofei Qi; Zeeshan Babar; Wenting Zheng
Journal: Wireless Communications and Mobile Computing
 (2017)

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## Abstract

Generalized frequency division multiplexing (GFDM) is a new candidate technique for the fifth generation (5G) standard based on multibranch multicarrier filter bank. Unlike OFDM, it enables the frequency and time domain multiuser scheduling and can be implemented digitally. It is the generalization of traditional OFDM with several added advantages like the low PAPR (peak to average power ratio). In this paper, the influence of the pulse shaping filter on PAPR performance of the GFDM system is investigated and the comparison of PAPR in OFDM and GFDM is also demonstrated. The PAPR is restrained by selecting proper parameters and filters to make the underwater acoustic communication more efficient.


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## Body

## 1. Introduction

The bandwidth limitation of underwater acoustic (UWA) channel makes OFDM one of the most significant and widely used modulation techniques for UWA communication. Although OFDM has considerable advantages over single-carrier modulations in combating frequency selective fading, it does not suit well with the future requirements because of the need for precise synchronization and the large PAPR. The requirement for cyclic prefix (CP) in every OFDM symbol also limits its spectral efficiency. Consequently, new multicarrier transmission schemes are needed to address these problems. Among them, filter bank multicarrier (FBMC) [1] and generalized frequency division multiplexing (GFDM) [2] are new promising techniques for 5G applications. GFDM is a two-dimensional (time and frequency) block-based data structure with multicarrier transmission scheme that is also derived from the filter bank approach [3, 4]. It divides the transmitting data into subgroups and subcarriers and each subcarrier is pulse shaped with prototype filter. This process reduces the OOB emissions and PAPR, making fragmented spectrum and dynamic spectrum allocation feasible without severe interference in incumbent services or other users [5].It is a novel concept for flexible multicarrier transmission that introduces additional degrees of freedom when compared to traditional OFDM [6]. The CP insertion method of GFDM is different from OFDM, where CP is added in each block instead of each symbol, which increases the efficiency of the system.The flexibility of GFDM allows it to cover a single-carrier frequency domain equalization (SC-FDE) and CP-OFDM as special cases [5–7]. When M=1 and the filter bank A=FNH, where FN is a N×N Fourier matrix and FH is the Hermitian transpose of F, GFDM turns into OFDM. When K=1 and g→ is a Dirichlet pulse SC-FDE is obtained. Thus, GFDM retains all main benefits of OFDM at the cost of some additional implementation complexity. The block structure of GFDM can be designed according to requirements, especially for the bandwidth limited system. The flexibility of the selection in the number of subblocks and subcarriers makes GFDM utilize fragmented spectrum which greatly increases the spectrum efficiency. These aspects are relevant for the scheduling of users in a multiple access scenario [8]. One of the main advantages of GFDM over OFDM is the low PAPR [9]. The PAPR character is one of the most important parameters to measure the performance of GFDM communication system. References [7–9] that have been published focus on the BER performance of GFDM, while there has been no systematic research on the PAPR performance of GFDM. Based on this, the paper gives a systematical analysis and it is an important reference for further application of this technique about how to select the appropriate filter banks and parameters according to the linear dynamic range of the transmitter power amplifier especially for the band limited underwater communication field.The rest of this paper is organized as follows. In Section2, a typical GFDM system and its properties are demonstrated including the bock diagram, PAPR of GFDM, and pulse shaping filters used in GFDM system. Then, simulation results of the influence of the pulse shaping filter on PAPR are analyzed in Section 3. Finally, conclusions are drawn in Section 4.


## 2. System Model and Properties

### 2.1. Sending Terminal

The block diagram of GFDM communication system is shown in Figure1. Vector b→c is the decoded data of the original binary data b→, while after modulation we obtain vector d→. The dimension of d→ is N×1 and it can be decomposed into K subcarriers with M subsymbols, which satisfies the equation N=K×M. Vector d→ can be expressed as d→=d→0T,…,d→M-1TT, where d→0=d→0,0T,…,d→K-1,0TT and d→m=d→0,mT,…,d→K-1,mTT. Therefore, dk,m is the data transmitting on the kth subcarrier and mth subsymbol of the block.(1)gk,mn=gn-mKmod⁡N·exp⁡-j2πkKn,where n denotes the sampling index. From (1), we can see that each gk,mn is the different time and frequency transformation of prototype filter gn.Figure 1
Block diagram of the transceiver.The transmitting datax→=xnT can be superimposed by all sending symbols.(2)xn=∑k=0K-1∑m=0M-1gk,mn·dk,m,n=0,…,N-1.Let g→k,m=gk,mnT; (2) can be [8](3)x→=Ad→,where the dimension of A is KM×KM and can be expressed as(4)A=g→0,0⋯g→K-1,0g→0,1⋯g→K-1,M-1,where g→k,m is the time and frequency shifted version of g→0,0. Finally, after adding CP, the sending signal can be expressed as x~→, which is the GFDM modulated data d→. g→1,0=An,2 and g→0,1=An,K+1 are circularly frequency and time shifted versions of g→0,0=An,1. The GFDM modulator is shown in Figure 2. Each dk,m is transmitted with the corresponding pulse shape gk,m.Figure 2
Modulator of the GFDM.


### 2.2. Receiving Terminal

Transmission though underwater channel can be modeled by(5)y→=Hx→+w→,where y→ is the receiving signal, x→ is the sending signal, the underwater channel matrix is H, and w→ denotes the additive white Gaussian noise (AWGN). At the receiver, time and frequency synchronization are performed and cyclic prefix is removed. Under the assumption of perfect synchronization, channel equalization used in OFDM can be adopted in GFDM.The dimension of channel matrixH is N×N. After channel estimation and equalization, the receiving signal is represented by z→ as(6)z→=H-1y→+w→=H-1HAd→+H-1w→=Ad→+w-→.After GFDM demodulation, the estimated data will be(7)d→=Bz→.The dimension of matrixB is KM×KM; the form of B will be different when using different equalization method such as (match filter) MF, (zero forcing) ZF, and (Minimum Mean Square Error) MMSE. The forms of B are given as follows:(8)BMF=AH,BZF=A-1,BMMSE=σn2σd2I+AHA-1AH,where σn2 and σd2 are noise and signal variance. After demodulation and decoding, we get the estimated binary data b^→.Figure1 shows the block diagram of GFDM in underwater acoustic communication system. In the first step, the source data vector b→ is encoded as b→c, followed by GFDM modulator, which modulates mapped data d→. The detailed structure of modulator is shown in Figure 2. The GFDM modulator plays a similar role as IFFT in OFDM. Vector d→ is the data block with dimension N×1, which is composed of K subcarriers and M subsymbols and N satisfies the equation N=K×M. After adding CP to the modulated data x→, the data x~→ is transmitted through the underwater acoustic channel. At the receiving terminal the channel estimation and equalization are performed after synchronization and CP removal. Finally, the estimated sending data b^→ is obtained after demapping and decoding.From the description of the sending and receiving terminal, it can be concluded that GFDM falls into the category of filtered multicarrier systems [10–12]. The name derives from the fact that the scheme offers more degrees of freedom than traditional OFDM or single carrier with SC-FDE.Figure2 shows the modulator of GFDM, in which dk,m represents data transmitted on the kth subcarrier and, in the mth subsymbol of the block, pulse shaped by the corresponding pulse shape filter gk,m. Time and frequency division of OFDM, SC-FDE, and GFDM are shown in Figure 3. From Figure 3, we can conclude that GFDM is the generalization of OFDM and SC-FDE. When M=1 and K=N GFDM turns into OFDM and whenever M=N and K=1 SC-FDE is obtained. Consequently, the PAPR of GFDM will range between OFDM and SC-FDE.Figure 3
Time and frequency division of different system.
(a)
OFDM (b)
SC-FDE (c)
GFDM


### 2.3. PAPR of GFDM

A discrete GFDM signal is given in(9)x~→n=∑k=0K-1∑m=0M-1gk,mndk,me-j2πnk/K,where K×M=N and gk,mn=gn-mkmod⁡N·exp⁡-j2πk/Kn is the time and frequency conversion of prototype filter gn. dk,m represents data on the kth subcarrier in the mth subsymbol of the block. In each block, M signals superimposed on M carriers to produce the GFDM waveform. M signals in one subsymbol block overlap with the same phase; GFDM signals will confront a peak power which is M times larger than the average power. Therefore, the PAPR of GFDM signal is expressed as(10)PAPRdB=10log10⁡max0≤n≤N-1⁡xn2Exn2,where E· is mathematical expectation and xn represents the discrete GFDM signal in time domain. From expression (9), the amplitude of the GFDM signal can be expressed as(11)An=Re2xn+Im2xn.The cumulative distribution function can be written as(12)Fz=1-e-z.The complementary cumulative distribution function (CCDF) can be defined as(13)PPAPR>z=1-PPAPR≤z=1-FzN=1-e-zN.CCDF is a curve to measure the distribution of system’s PAPR. The PAPR is influenced by many factors of pulse shaping filter used in GFDM communication system.


### 2.4. Pulse Shaping Filter

The frequency response of pulse shaping filters can be used in GFDM system which are shown in Table1 [13, 14], in which linα⁡x is a truncated linear function.(14)linα⁡x=min1,max0,1+α2α+xα.Equation (14) is used systematically to describe the roll-off area defined by α in frequency domain. And p4x=x435-84x+70x2-20x3 is a polynomial that maps the range (0,1) onto itself.Table 1
Frequency response of different filters.
Name Frequency response RC G R C f = 1 2 1 - cos ⁡ π  lin α ⁡ f M  RRC G R R C f = G R C f  1st Xia G X i a 1 f = 1 2 1 - exp ⁡ - j  lin α ⁡ f M sign ⁡ f  4th Xia G X i a 4 f = 1 2 1 - exp ⁡ - j π p 4 lin α ⁡ f M sign ⁡ f


## 2.1. Sending Terminal

The block diagram of GFDM communication system is shown in Figure1. Vector b→c is the decoded data of the original binary data b→, while after modulation we obtain vector d→. The dimension of d→ is N×1 and it can be decomposed into K subcarriers with M subsymbols, which satisfies the equation N=K×M. Vector d→ can be expressed as d→=d→0T,…,d→M-1TT, where d→0=d→0,0T,…,d→K-1,0TT and d→m=d→0,mT,…,d→K-1,mTT. Therefore, dk,m is the data transmitting on the kth subcarrier and mth subsymbol of the block.(1)gk,mn=gn-mKmod⁡N·exp⁡-j2πkKn,where n denotes the sampling index. From (1), we can see that each gk,mn is the different time and frequency transformation of prototype filter gn.Figure 1
Block diagram of the transceiver.The transmitting datax→=xnT can be superimposed by all sending symbols.(2)xn=∑k=0K-1∑m=0M-1gk,mn·dk,m,n=0,…,N-1.Let g→k,m=gk,mnT; (2) can be [8](3)x→=Ad→,where the dimension of A is KM×KM and can be expressed as(4)A=g→0,0⋯g→K-1,0g→0,1⋯g→K-1,M-1,where g→k,m is the time and frequency shifted version of g→0,0. Finally, after adding CP, the sending signal can be expressed as x~→, which is the GFDM modulated data d→. g→1,0=An,2 and g→0,1=An,K+1 are circularly frequency and time shifted versions of g→0,0=An,1. The GFDM modulator is shown in Figure 2. Each dk,m is transmitted with the corresponding pulse shape gk,m.Figure 2
Modulator of the GFDM.


## 2.2. Receiving Terminal

Transmission though underwater channel can be modeled by(5)y→=Hx→+w→,where y→ is the receiving signal, x→ is the sending signal, the underwater channel matrix is H, and w→ denotes the additive white Gaussian noise (AWGN). At the receiver, time and frequency synchronization are performed and cyclic prefix is removed. Under the assumption of perfect synchronization, channel equalization used in OFDM can be adopted in GFDM.The dimension of channel matrixH is N×N. After channel estimation and equalization, the receiving signal is represented by z→ as(6)z→=H-1y→+w→=H-1HAd→+H-1w→=Ad→+w-→.After GFDM demodulation, the estimated data will be(7)d→=Bz→.The dimension of matrixB is KM×KM; the form of B will be different when using different equalization method such as (match filter) MF, (zero forcing) ZF, and (Minimum Mean Square Error) MMSE. The forms of B are given as follows:(8)BMF=AH,BZF=A-1,BMMSE=σn2σd2I+AHA-1AH,where σn2 and σd2 are noise and signal variance. After demodulation and decoding, we get the estimated binary data b^→.Figure1 shows the block diagram of GFDM in underwater acoustic communication system. In the first step, the source data vector b→ is encoded as b→c, followed by GFDM modulator, which modulates mapped data d→. The detailed structure of modulator is shown in Figure 2. The GFDM modulator plays a similar role as IFFT in OFDM. Vector d→ is the data block with dimension N×1, which is composed of K subcarriers and M subsymbols and N satisfies the equation N=K×M. After adding CP to the modulated data x→, the data x~→ is transmitted through the underwater acoustic channel. At the receiving terminal the channel estimation and equalization are performed after synchronization and CP removal. Finally, the estimated sending data b^→ is obtained after demapping and decoding.From the description of the sending and receiving terminal, it can be concluded that GFDM falls into the category of filtered multicarrier systems [10–12]. The name derives from the fact that the scheme offers more degrees of freedom than traditional OFDM or single carrier with SC-FDE.Figure2 shows the modulator of GFDM, in which dk,m represents data transmitted on the kth subcarrier and, in the mth subsymbol of the block, pulse shaped by the corresponding pulse shape filter gk,m. Time and frequency division of OFDM, SC-FDE, and GFDM are shown in Figure 3. From Figure 3, we can conclude that GFDM is the generalization of OFDM and SC-FDE. When M=1 and K=N GFDM turns into OFDM and whenever M=N and K=1 SC-FDE is obtained. Consequently, the PAPR of GFDM will range between OFDM and SC-FDE.Figure 3
Time and frequency division of different system.
(a)
OFDM (b)
SC-FDE (c)
GFDM


## 2.3. PAPR of GFDM

A discrete GFDM signal is given in(9)x~→n=∑k=0K-1∑m=0M-1gk,mndk,me-j2πnk/K,where K×M=N and gk,mn=gn-mkmod⁡N·exp⁡-j2πk/Kn is the time and frequency conversion of prototype filter gn. dk,m represents data on the kth subcarrier in the mth subsymbol of the block. In each block, M signals superimposed on M carriers to produce the GFDM waveform. M signals in one subsymbol block overlap with the same phase; GFDM signals will confront a peak power which is M times larger than the average power. Therefore, the PAPR of GFDM signal is expressed as(10)PAPRdB=10log10⁡max0≤n≤N-1⁡xn2Exn2,where E· is mathematical expectation and xn represents the discrete GFDM signal in time domain. From expression (9), the amplitude of the GFDM signal can be expressed as(11)An=Re2xn+Im2xn.The cumulative distribution function can be written as(12)Fz=1-e-z.The complementary cumulative distribution function (CCDF) can be defined as(13)PPAPR>z=1-PPAPR≤z=1-FzN=1-e-zN.CCDF is a curve to measure the distribution of system’s PAPR. The PAPR is influenced by many factors of pulse shaping filter used in GFDM communication system.


## 2.4. Pulse Shaping Filter

The frequency response of pulse shaping filters can be used in GFDM system which are shown in Table1 [13, 14], in which linα⁡x is a truncated linear function.(14)linα⁡x=min1,max0,1+α2α+xα.Equation (14) is used systematically to describe the roll-off area defined by α in frequency domain. And p4x=x435-84x+70x2-20x3 is a polynomial that maps the range (0,1) onto itself.Table 1
Frequency response of different filters.
Name Frequency response RC G R C f = 1 2 1 - cos ⁡ π  lin α ⁡ f M  RRC G R R C f = G R C f  1st Xia G X i a 1 f = 1 2 1 - exp ⁡ - j  lin α ⁡ f M sign ⁡ f  4th Xia G X i a 4 f = 1 2 1 - exp ⁡ - j π p 4 lin α ⁡ f M sign ⁡ f


## 3. Simulation Results Analysis

OFDM and GFDM UWA communication system’s parameters are shown in Table2.Table 2
Simulation parameter.
Parameter Value Signal modulation type BPSK The cyclic prefix length 20.8 ms The sample frequency 48 kHz Bandwidth 6 kHz Starting frequency 6 kHzThe sample frequency of the system is 48 kHz and the bandwidth of OFDM signal isB=6 kHz. The transmitted signal occupies the frequency band between 6 kHz and 12 kHz. We use cyclic-prefixed OFDM and GFDM with a cycle-prefix Tg=20.8 ms per OFDM block. The setting of cycle-prefix is according to the maximum multipath delay of the underwater channel which generally ranges from 10 ms to 20 ms. BPSK modulation is used. The signal parameters are summarized in Table 2.Figure4 is the frequency domain of 1st Xia, 4th Xia, RC (Raised Cosine), and RRC (Root Raised Cosine) shaping filters with the roll-off factor α equal to 0.1 and 0.9. From the picture, we can draw the conclusion that when α is 0.1, the frequencies of the four filters are almost the same. When α approach 0.9, their performances become different. Figures 5 and 6 are the value of the matrix A with the roll-off factor α=0.1 and 0.9. From Figures 4, 5, and 6, we can draw the conclusion that with the increase of α the side lobe of the filter bank becomes lower. Figure 6 shows the detailed influence of pulse shaping filters on PAPR performance.Figure 4
The frequency domain of 4 pulse shaping filters withα 0.1 and 0.9.Figure 5
Value of matrixA with α=0.1, K=8, and M=9.Figure 6
Value of matrixA with α=0.9, K=8, and M=9.Figure7 shows the comparison of the CCDF of OFDM and GFDM communication systems for the same transmitted data. The complementary cumulative distribution function (CCDF) is one of the most frequently used parameters for analyzing PAPR reduction by measuring its distribution.Figure 7
CCDF of OFDM and GFDM communication system with different number of subcarriers.GFDM is a block structure with the dimensionsM×K, where M is the subblock number and K is the subcarrier number. Simulations are performed for different number of subcarriers of GFDM including 516, 258, 129, and 8. Figure 7 proves that GFDM has less PAPR than OFDM for the same parameters; for example, for CCDF less than 0.2, the PAPR of OFDM is larger than GFDM by 17.5 dB when the subcarrier number is 8 and the subblock number is 129. In order to keep the same transmitting data bits, there are 8 subblocks for 129 subcarriers and the PAPR of GFDM is about 9 dB lower than OFDM. When the number of subcarriers is 258 and the corresponding number of subblocks is 4, the PAPR of GFDM is about 7 dB lower than that of OFDM and about 2 dB higher than the scenario when the number of subcarriers is 129. When the number of subcarriers rises to 516, the PAPR of GFDM rises to 30 dB which is still 3 dB lower than OFDM. The influence of different filters on PAPR is depicted in Table 1, while Figure 8 shows the influence of the filters on PAPR. The comparison is based on the condition that the number of subcarriers in different systems is the same. Figure 8 shows that the PAPR of GFDM using 1st Xia filter and 4th Xia filter are almost the same, with the PAPR of 15.4 dB, lower than the other filters. However, RRC filter has the highest PAPR of 15.8 dB followed by RC filter with a PAPR of 15.5 dB, which concludes that when the CCDF is lower than 0.2, the GFDM system using RRC filter is higher than RC filter by 0.3 dB.Figure 8
CCDF of GFDM communication system with different filters.Figure9 shows the PAPR of GFDM communication system of RC filter with different roll-off factor α. The increase step of roll-off factor α is 0.2. When the roll-off factor α equals 0.9, the PAPR of GFDM system is 0.4 dB higher than the case when α equals 0.1. Obviously, the PAPR of GFDM increases with the increase in roll-off factor α. Whenever α rises by one step, the PAPR will increase by 0.1 dB. The results prove that the selection of the pulse shaping filter has a great influence on the performance of GFDM communication system. Thus, PAPR is influenced by either the same filter with different roll-off factors α or different filters with the same roll-off factor.Figure 9
CCDF of GFDM communication system with differentα.


## 4. Conclusion

In this paper, we investigated the influence of pulse shaping filters on the PAPR of the GFDM system. GFDM is an emerging candidate for future 5G networks. Taking advantage of the high degree of flexibility in frequency band selection, it apparently seems more suitable for the bandwidth limited UWA systems. The multicarrier scheme has both the advantage of making full use of the fragment spectrum and the merit of low PAPR, which will reduce the power consumption and save the hardware cost. The detailed PAPR of GFDM is analyzed. Different block structures, different filters, and different roll-off factors are factors that influence the PAPR of GFDM.However, foreseen scenarios for future 5G networks have requirements that undoubtedly go beyond higher data rates. Because of the flexibility of GFDM, it becomes a promising solution for 5G communication and can satisfy diverse requirements. This paper gives a systematical analysis and it is an important reference for further application of this technique about how to select the appropriate filter banks and parameters according to the linear dynamic range of the transmitter power amplifier. The research of the application of algorithms developed for OFDM in underwater acoustic GFDM, as well as other filters which can restrain the PAPR, will be the further research.



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