Performance Analysis of Generalized Frequency Division Multiplexing in Various Pulse-Shaping Filter with Raised Cosine and Root Raised Cosine Filter

Generalized frequency division multiplexing (GFDM) is non orthogonal multicarrier modulation scheme which is suitable for the fifth generation (5G) of wireless network. Pulse shaping filter design in GFDM system have effects on symbol error rate performance due to inter symbol interference. In this paper contribute to symbol error rate performance in GFDM system with additive white Gaussian noise channel, zero forcing channel Rayleigh fading has been analyzed for pulse shaping filter namely raised cosine and root raised cosine filter and also simulation is done and results are reported in terms of symbol error rate, signal to noise ratio, different value of roll off factor and different modulation technique. Comparison of simulation results of this method with existing methods is done and improvements in result are obtained as compared to existing.


Introduction
GFDM is the multiplexing technique proposed for the 5G networks. It is the generalization of digital multi-carrier concept of trans receiver [1]. In GFDM, the blocks of data are independently modulated. Every block consists of various sub-carriers and every sub-carrier has various sub symbols. The pulse shaping of sub-carriers are done using circularly shifted filter which is shifted in both domains time as well as frequency. Out of band emissions (OOB) get reduced by this process. The inter symbol interference (ISI) and inter carrier interference (ICI) may arise due to subcarrier filtering [2][3][4]. Overhead is small in GFDM block as there is a single cyclic prefix (CP) for a block having various sub symbols. GFDM trans receiver encoded symbols belong to line of 1 3 complicated constellation factors in which denotes the amount of bits in keeping with symbol, additionally referred to order of modulation [5][6][7][8].
GFDM trans receiver is given in Fig. 1. This is K = MN manufactured from two integers. The k factors are frequently visualized as disintegrated into N subcarrier and M sub symbol with the GFDM system overlapping factor. The NM X I column vector d = [d(0)………d(M − 1) denotes the transmitted data on the subcarrier. The baseband transmit signal in digital communication is achieved through the sum of all subcarrier and sub symbol signals according to where g(n) denotes the impulse response of prototype filter with N samples while k, m, n are subcarrier, sub symbol and time sample. This sequence can also be represented as a column vector g k,m and d k, m being dk, m being the data symbol transmitted in the mth sub symbol of the kth subcarrier. Collecting the filter samples in a vector allows formulating as � ⃗ x = A � ⃗ d whereas d represent data matrix and A represent transmit matrix [9][10][11][12][13][14].

Pulse Shaping Filter
In GFDM modulator various operation is performed serial-to-parallel conversion, pulse shaping operation is performed on each data symbol separately as represented by the Eq. (2) where n is the sampling index, p[n] is the pulse shaping used and p k,m [n] is the pulse shaping filter, p[n] after shifting in both time and frequency domain (2). In this shifted version of pulse shaping filter, complex exponential performs shifting in frequency domain and modulo operation performs shifting in time domain [15][16][17].

Raised Cosine Filter
Raised cosine filter (RC) is the pulse shaping filter and it is used to minimize the inter symbol interference (ISI). The impulse response of this filter as given (3) (1) In this pulse shaping filter, α is the roll off factor and T is the transmission symbol period.

Root Raised Cosine Filter
In digital communication, RRC is mostly used as transmit and receive filtering. The equivalent response of these two filters is equal to that of RC filter.
The impulse response of RRC is given as where α is the roll off factor and T is the transmission symbol period [18][19][20].

Symbol Error Rate Analysis
Symbol error rate (SER) is defined as the number of symbols in error when the symbols are transmitted through the channel. In the generalized frequency division multiplexing model data symbols are transmitted through the additive white Gaussian noise (AWGN) and Rayleigh channel using different pulse shaping filters at various values of roll off factor. The analysis of SER versus Es/N0 is done in case of zero forcing (ZF) receiver [21][22][23][24]. The analytical expressions of SER in AWGN and Rayleigh channel are also calculated in this section [25,26]. The ZF receiver eliminates self-interference but introduces noise enhancement which depends on the choice of the pulse shaping [27][28][29][30][31][32]. The noise enhancement factor (NEF) denoted by ξ which causes reduction in signal-to-noise ratio (SNR) while using ZF receiver is given by the Eq. (6)

Result and Discussion
Simulation results of the impulse response of the pulse shaping filters used in the GFDM model at various roll off factor values, PSD and SER of the GFDM using different pulse shaping filters. The simulation results symbol error rate (SER) are calculated using AWGN channel and Rayleigh channel and at various values of roll off factor. The simulation results are obtained using software MATLAB 2017a. The parameters used for the computation of results are given in table (Table 1).
The Fig. 2 shows a spectrum analysis of the GFDM system. Number of symbol is 4, number of subcarrier is 4 roll of factor is 0.1, quadrature amplitude modulation are used. In this case signal passes through additive white Gaussian noise channel and zero forcing cannel. Raised cosine filter are used to improve the performance ( Table 2). Fig. 3 represent spectrum analyzer of GFDM system. Number of symbol is 2, length is 50, number of subcarrier is 4 and roll off factor is 0.1. Quadrature phase shift keying are used this signal passes through additive white Gaussian noise channel. Raised cosine filter are used to improve the performance and result shows performance improvement ( Table 3).
The Fig. 4 indicates the signal to noise ratio at x-axis and y-axis represent the symbol error rate. Quadrature Amplitude modulation is used at the transmitter with different     Table 4).
The Fig. 6 shows x axis represent signal to noise ratio (SNR) and y axis represent the symbol error rate (SER) with different roll off factor 0.05, 0.      The Fig. 9 shows x axis represent roll off factor and y axis represent mean symbol error rate. 0.1 roll off factor get symbol error rate 0.22 and 0.15 roll off factor get symbol error rate 0.16, 0.2 roll off factor get symbol error rate 0.5. Here obtained improvement result (Table 6).
In this table shows that the result this is comparison table between earlier method and present method. x axis represent signal to noise ratio and y axis represent symbol error rate. In case of signal to noise is 8 get the symbol error rate 0.7 in earlier method and present method get symbol error rate 0.5.Similarly signal to noise is 5 get the symbol error rate 0.3 in earlier method and present method get symbol error rate is 0.2. Signal to noise is 24 get the symbol error rate is 0.001 in earlier method and present method get symbol error rate is 0.0003. It conclude that in present method get improvement result (Table 7). In this table represent the error minimization of spectrum signal. x axis shows signal to noise ratio and y axis represent symbol error rate. When signal to the noise ratio is 8 and get error rate 0.6 in earlier method and signal to noise ratio is 8 and get error rate 0.5 in present method. In earlier method when signal to noise ratio is 8 and error rate 0.5 in present method. When signal to noise is 5 and error rate is 0.3 in earlier method and signal to noise ratio is 5 and error rate is 0.2 in present method. When signal to noise ratio is 24 and error rate is 0.001 in earlier method and present method signal to noise ratio is 24 and error rate is 0.001 and present method signal to noise ratio is 24 and error rate is 0.0003. It conclude that spectrum analysis get improvement result.

Conclusion
In this paper a detailed study of the spectrum analysis, roll off factor and symbol error rate, signal to noise ratio done using pulse shaping raised cosine and root raised cosine filters. Considering the number of subsymbol, subcarrier, roll off factor, modulation technique mapping, channel type like Rayleigh channel, AWGN channel, zero forcing receiver channel. Reported results are comparing present and existing methods. The standard pulse shaping filter raised cosine filter are compared with spectrum analysis and different roll of factor. Simulation results for spectum analysis and different roll of factor are reported. The overall research work carried out has given a significant contribution for raised cosine filter in spectrum analysis. Paper result will be useful in quality improvement and error minimization of spectrum analysis and different roll off factor testing with symbol error rate. Raised cosine and root raised cosine filter are used to improve the performance symbol error rate. Proposed technique reported this paper are very much useful in quality improvement and error minimization of GFDM.

Author Contributions
All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by MG. The first draft of the manuscript was written by MG and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Funding No funding was received from an organization for conducting the study of the submitted work and preparation of this manuscript.

Data Availability
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code Availability
The code of the algorithm has been run in MATLAB software.