Performance Analysis of Parallel Concatenation of LDPC Coded SISO-GFDM System for Distinctive Pulse Shaping Filters using USRP 2901 Device and its Application to WiMAX

Abstract

The increasing demand for high data rates requires channel error control codes for the upcoming fifth generation. This article presents an investigation of the parallel concatenation of low-density parity-check codes (PC-LDPC) in the fifth generation proposed waveform candidate called generalized frequency division multiplexing (GFDM). PC-LDPC codes are obtained by dividing the long and high complexity single LDPC codes into small two lower complexity codes, and these designed codes are applied to the 5G-GFDM waveform. Since the GFDM signal transmits data in both the time and frequency domain, these PC-LDPC codes can deal with two-dimensional errors. This channel coded GFDM system is integrated into Universal software radio peripheral (USRP) device for real-time implementation. The Attainment of the proposed transceiver is verified by computation of BER under distinctive channel coding techniques like convolutional, Golay, Bose-Chaudhuri-Hochquenghem (BCH), extended length single LDPC code. The different pulse shaping filters such as Raised Cosine (RC), Root Raised Cosine (RRC), Gaussian, and Xia 4th order filter are applied to the GFDM under the Gaussian noise and Rayleigh fading channel to compute Out of band (OOB) power. The PC-LDPC coded GFDM outperforms LDPC by 6.5 dB in the RRC filter for roll-off factor rate 0.5 under the Rayleigh fading channel. PC-LDPC code outperforms LDPC code with a coding gain of 2 dB was observed in IEEE 802.16 Transceiver.

1 Introduction

The GFDM system accomplishment the white spaces in the spectrum. The brunt of non-orthogonal subcarriers of GFDM can be controlled by the properties of transmitter and digital receiver filters. The transmission is block-based on FFT [1]. The data arrangement in GFDM is made in blocks where every block has different subcarriers, and every subcarrier has various sub symbols. The pulse shaping filters are applied to every symbol separately [2]. This pulse shaping reduces the out of band radiation of transmitted signals [3]. This GFDM scheme can fulfil the requirements of 5G. The synchronization technique that reduces the spectral emission is proposed [4]. In order to improve the system performance and capacity of the channel, channel coding is required for future wireless communications. LDPC codes are used in the early 1990s and 2000 and used in digital video broadcasting, satellite communications applications. This article uses the LDPC code of block length 10⁷, approached the Shannon limit within 0.0045 dB [5]. The difference between hamming and LDPC codes are in the matrix size. The LDPC codes have a sparse matrix code with a few ones in the matrix. Whereas Hamming code has a length code of 2 ^m-1 columns. the LDPC codes parity check matrix depends on the transmitted data. The hamming code is related to cyclic coding theory [6, 7].

1.1 Associated Work

The forward error correction codes use redundancy bits to transmit the digital signal to detect and correct the received bits in the receiver side. The entire data block is employed to one code work in convolutional codes [8]. Other than convolutional codes, Golay codes, BCH codes, and RS codes [9] are also used as error correction and shows significant improvement in error-correcting capability [10]. The concatenation of codes is proposed in the channel encoder in the transmitter section. The convolutional codes are concatenated with reed Solomon codes to form turbo codes, which reaches performance near to Shannon limit. This turbo code uses iterative decoding algorithms in the receiver section. Another code proposed by Gallagher in 1961 is the low-density parity-check matrix; these codes show performance near the Shannon limit [11]. The turbo codes suffer from error floor in BER plots in high Signal-to-noise ratio areas because of low weight codeword [12]. The turbo receiver was designed in GFDM based cognitive radio to utilise feedback information for channel estimation [13]. The long block LDPC codes show better performance than turbo codes. However, the latency problem is identified in LDPC codes. In order to overcome that, concatenation of LDPC codes are proposed with the same iterative decoding algorithm, which is used in turbo codes as are the right choice for better performance [14]. The Dirichlet based pulse shaping filter reduces the OOB emission for the GFDM system; the BER analysis for guard symbols in the GFDM system was explained [15]. The novel proposed ramp filter is used as a pulse-shaping filter and reduces the OOB in GFDM corresponding to other filters [16]. The proposed pulse shaping filter has intense sharpness compare to RRC pulse in GFDM, and the Symbol error probability is calculated [17]. The SISO-GFDM transceiver was designed in a virtual and remote lab, and its characteristics are compared with the 4G-OFDM system [18]. A Quasi-cyclic LDPC decoder was designed using the NI USRP device and provides 2.4 Gb/s [19]. The audio signal transmission was tested in hard decoding in LDPC decoder for 32-bit codeword using USRP devices [20]. A 71% improvement in the complexity of receiver design was observed in computer-based simulation for LDPC code concatenated with staircase codes [21]. The parallel concatenation of LDPC codes with different compositions is elucidated in the survey article [22]. A significant reduction in BER was observed in 5G-GFDM when the concatenation of LDPC code with Turbo codes are used as channel coding schemes [23].

1.2 Scope of the Paper

The investigation of PC-LDPC codes with a 5G GFDM system over Gaussian noise and flat fading Rayleigh channel for distinctive prototype filters is analyzed in this paper. The BER performance is compared with other channel coding schemes. To identify the best pulse shaping filter, which provides less out of band power and efficient BER values in the GFDM system. To propose the novel GFDM transceiver design with high coding gain and less complexity decoder in receiver. They are furthermore exploring the PC-LDPC-GFDM transceiver by incorporating the WiMAX model for improvement in BER.

1.3 Organization of the Paper

This paper is categorized into Passage 1 accord the establishment about the GFDM, and its related work in the area of the field, the scope of the proposed system and organization of the article. Passage 2 explains the implementation and simulation parameters of LabVIEW based GFDM transceiver. Passage 3 explains the mathematical analysis of GFDM and parallel concatenation of LDPC codes using diagrams. Passage 4 discusses the USRP 2901 device, incorporation of PC-LDPC-GFDM system with IEEE 802.16 standard, and simulation parameters used by the user for GFDM transceiver. Passage 5 gives information about VI Programming for GFDM Using USRP Device, Sect. 6 and 7 explains the results achieved and Conclusions.

2 Channel Coded GFDM model

Figure 1 presents the block design of the GFDM system. The 16-bit PN sequence is used, which generates 65,535 bits. The generation of 16 bits can be done using a linear feedback shift register (LFSR), with tap positions at [1, 11, 13, 14, 16], as shown in Fig. 2. The PC-LDPC codes are formed by dividing the long length LDPC codes into two short block regular LDPC codes with H matrix as parity check matrix. The corresponding bits are converted to complex data symbols with QAM or BPSK modulation schemes. These complex data symbols are assigned to the resource mapper, which converts the data into the two-dimensional format using the GFDM modulation block structure, as shown in Fig. 3. One cyclic prefix is added for the entire GFDM block, whereas in OFDM, one cyclic prefix is added for each symbol. Since the USRP device is used, it converts the transmitted bits into IQ samples data. the verto antenna transfers the data into free space. The received data is a combination of AWGN channel or Rayleigh fading channel along with environmental noise. These data are processed through the FPGA receiver, and these IQ data samples are transferred through the GFDM receiver section. In the receiver section, removing of cyclic prefix symbol, if zero padding is included in the transmitter, the zero-padding is also removed in the receiver. The channel estimation is used to predict the estimation of the received symbol. The decoding of the PC-LDPC codes are also done with the help of two single LDPC decodes followed by deinterleaver.VI. Once the PC-LDPC decoder reaches the maximum number of iterations, the decoder stops performing, and the output is received.

Fig. 1

General Block diagram of Proposed GFDM transceiver using USRP 2901

Fig. 2

Galois Linear Feedback shift register (LFSR) generation of PN Sequence

Fig. 3

GFDM modulator

The VI hierarchy of the GFDM LabVIEW program is viewed in Fig. 4.

Fig. 4

Virtual Instrumentation Hierarchy

Figure 4 shows the list of Virtual instrumentation programs used in the proposed design. The sequence of steps followed by the transmitter, receiver its sub-VI's are mentioned in the above Figure. The sequence of Virtual instrumentation programming is explained.

3 List of VI’s in GFDM Transmitter

3.1 Galois Pseudo Random Noise Generator VI (Message Source)

The Galois PN sequence generation is shown in Fig. 2. With the help of LFSR, D flip flop has m input bits, the length of the PN sequence is N = 2 ^m−1. The tap positions are [1, 11, 13, 14, 16], the 16-bit input is used. Equation (1) is shown mathematical form of message bits generation.

$$ M^{{\left( {16} \right)}} \left( x \right) \, = \, 1 \, + x^{1} + \, x^{13} + \, x^{14} + \, x^{16} $$

(1)

3.2 Symbol Mapper and Prototype Filter VI’s

The source encoder instance VI maps the input bits to complex symbols. The modulation scheme used here is QAM and BPSK schemes. With ${2}^{\Delta }$ Constellation size $\Delta$ is the order of the modulation. The symbol map is an array which maps every symbol in the complex baseband modulated waveform. The VI calculates the impulse response of the filter using Eqs. (2) and (3). Mainly four pulse shaping filters are chosen for GFDM system design. These prototype filters and their impulse response are shown in Eqs. (2), and (3), (4), and (8). The filters are RC, RRC, Gaussian pulse, and Xia 4th order filter. The Eqs. (5), (6) and (7) are the functions of Xia 4th order filter. In every impulse response of the filter, $\alpha$ is the roll-off factor. For every pulse shaping filter, the roll-off factor plays an essential role in controlling the signals out of the band emission spectrum. It helps in reducing the latency of the system. However, we observed that the BER of the system also depends on roll-off factor values. Hence, we identified two unique values of roll factor values to compute BER and plot the response.

$$ P_{RC} \left( t \right) = \sin c\left( \frac{t}{T} \right)\frac{{\cos \left( {2\Pi \alpha t} \right)}}{{1 - (\frac{2\alpha t}{T})^{2} }} $$

(2)

$$ P_{RRC} \left( t \right) = \frac{{\sin \left( {\frac{\Pi t}{{T\left( {1 - \alpha } \right)}}} \right) + \frac{4\alpha t}{{T\cos \left[ {\frac{\Pi t}{T}\left( {1 + \alpha } \right)} \right]}}}}{{\Pi t/T[1 - \left( {4\alpha t/T )^{2} } \right]}} $$

(3)

$$ P_{gaussian} \left( t \right) = \frac{\sqrt \pi }{\alpha }e^{{\left( {\frac{{ - \pi^{2} t}}{{\alpha^{2} N}}} \right)}} $$

(4)

$$ Xia\left( t \right) = 1, \quad if \left| t \right| \le \frac{{\left( {1 - \alpha } \right)KT_{s} }}{2} $$

(5)

$$ Xia\left( t \right) = \frac{1}{2}(1 + e^{{j\pi P_{Xia} }} \left( t \right), \quad if \frac{{\left( {1 - \alpha } \right)KT_{s} }}{ 2} < \left| t \right| \le \frac{{\left( {1 + \alpha } \right)KT_{s} }}{2} $$

(6)

$$ Xia\left( t \right) = 0, \quad if \frac{{\left( {1 + \alpha } \right)KT_{s} }}{2} < \left| t \right| $$

(7)

$$ P_{Xia} \left( t \right) = \left( {\frac{{\left| t \right| - \left( {\frac{{\left( {1 - \alpha } \right)KT_{s} }}{2}} \right)}}{\alpha T}} \right) $$

(8)

These four prototype filters p [.] can be used for GFDM, which has a maximum impact on Out of band (OOB) power and BER plot response.

3.3 Mathematical Model of GFDM Modulator

The different operations are performed on the d_k,m (data vector). The serial to parallel converter converts this data vector into parallel data streams. The mapped data is divided into K subcarriers having M subsymbols, i.e., shown in Eq. (9). The mathematical modelling of GFDM was carrying out in the article [4].

$$ \overrightarrow {{d_{k,m} }} = (d_{0,m} ,......d_{K - 1,m} )^{T} $$

(9)

After converting the serial data to parallel, the pulse shaping operation is performed on each data symbol separately, as represented by Eq. (10).

$$ P_{k,m} \left[ n \right] = P\left[ {\left( {n - mk} \right)\bmod N} \right]e^{{ - j2\pi \frac{kn}{k}}} $$

(10)

The transmit samples x(n), as shown in Eq. (11), is attained by superpositioning the transmitted symbols, n = 0, 1,……..N-1

$$ x\left( n \right) = \mathop \sum \limits_{k = 0}^{K - 1} \cdot \mathop \sum \limits_{m = 0}^{M - 1} \cdot p_{k,m} \left[ n \right] * d_{k,m} $$

(11)

$$ y\left( n \right) = GF \cdot \overrightarrow {x\left( n \right)} + \overrightarrow {w\left( n \right)} $$

(12)

in which GF is a channel matrix

$$ GF = \left( {NS + Ns_{cp} + Ns_{ch} - 1} \right)x\left( {Ns + Ns_{cp} } \right),where \, Ns = KM $$

(13)

Ns is the number of symbols, which is equal to the multiplication of subcarriers and sub symbols. After modulation of the transmitted signal, the cyclic prefix is added at the end of the GFDM block with length K/4.

$$ y_{cp} \left( n \right) = x_{cp} \left( n \right) + w\left( n \right) $$

(14)

Due to the Intersymbolic interference problem in the channel, the zero-forcing equalizer is performed to get the desired signal.

$$ ZF = GF^{ - 1} GFx\left( n \right) + GF^{ - 1} w\left( n \right) $$

(15)

$$ ZF = x\left( n \right) + w\left( n \right) $$

(16)

$$ GF_{ZF} = (GF^{H} GF)^{ - 1} GF^{H} $$

(17)

$$ d_{r} = GF_{ZF} \cdot Z $$

(18)

The desired signal can be obtained from the noisy channel using zero forcing equalizer is explained mathematically using equations from (14), (15), (16), (17) and (18).

3.4 Encoding Sequence of Parallel Concatenation of LDPC codes (PC-LDPC)

the encoding process of PC-LDPC codes is explained with the parity check matrix H matrix of size 6 × 12. The number of ones is always greater than several zeros in the matrix. The code rate R is R = [(N-M)/N], N = 12 and M = 6, which are several columns and rows in the matrix. The Tanner graph, which helps in the decoding of LDPC, is shown in Fig. 5. The variable nodes and parity nodes are of lengths 12 and 6 as shown in Eq. (19). The variable nodes and parity nodes are represented by $N_{m} [m\varepsilon \{ 1....N\}$ and ${M}_{n}\left(n\varepsilon\left\{1....M\right\}\right)$

$$ H = \left[ {\begin{array}{*{20}c} 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 \\ \end{array} } \right] $$

(19)

Fig. 5

LDPC Constraint graph (Tanner)

The PC-LDPC codes are formed by using the same length regular LDPC encoders with R = ½ and MT Permute Interleaver.VI is used in between the two LDPC encoders. With the help of an old technique called the LU decomposition method, the decoding process of concatenation is explained below. The H network into two sections, i.e., H₁ and H₂ as shown in Eq. (20). Each partitioned H grid is again utilized in the LU decay framework. i.e., to discover the lower triangular network and upper triangular lattice. i.e., the lower triangular lattice comprises each of the zeros over the corner-to-corner components. Thus, the upper triangular network comprises each of the zeros underneath the diagonal components.

$$ H = \left[ {H_{1} } \right]\left[ {H_{2} } \right] $$

(20)

H₁ = [H_L1 H_U1], H₂ = [H_L2 H_U2], LXU = H, UXL # H,

The factorization of H₁ MXM.

H_L1= $\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 1& 1& 0& 0& 1& 0\\ 0& -1& 1& 0& 0& 1\end{array}\right]$ H_U1 =  $\left[\begin{array}{cccccc}1& 1& 0& 0& 1& 1\\ 0& -1& 1& 0& -1& -1\\ 0& 0& 1& 1& 0& 1\\ 0& 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 0& -1\\ 0& 0& 0& 0& 0& -2\end{array}\right]$

The factorization of ${H}_{2}$ MX(N−M).

H_L2= $\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 1& 1& 0& 0& 0\\ 1& 0& -1& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 1& 1& -1& 0.5& 0.5& 1\end{array}\right]$ H_U2 =  $\left[\begin{array}{cccccc}1& 0& 1& 0& 0& 0\\ 0& 1& 0& 0& 0& 1\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 2& 1& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& -1.5\end{array}\right]$

The codeword C_W is the concatenation of information block ${C}_{IB}$ and redundancy block ${C}_{RB}$ is shown in Eq. (21).

$$ C_{w} = \left[ {\begin{array}{*{20}c} {C_{RB} } & {C_{IB} } \\ \end{array} } \right] $$

(21)

From the properties of the H matrix, substitute Eq. (21) and Eq. (20) in Eq. (22). C_W = codeword, information block ${C}_{IB}$, redundancy block ${C}_{RB}$, parity check matrix is H, H_L1, H_L2 is the lower triangular matrixes, H_U1, H_U2 is the upper triangular matrixes. Transpose of H matrix multiplied by codeword is equal to zero. The redundancy bits allow the user to add extra bits to the information bits for ensuring no data is lost in the total block of data. this redundancy bits also helps the receiver to detect or correct the errors. To avoid the computational complexity, the parity check matrix is divided into LU matrix as shown in below equations.

$$ C_{W} H^{T} = 0 $$

(22)

$$ C_{RB} H_{1}^{T} + C_{IB} H_{1}^{T} = 0 $$

$$ C_{RB} \left[ {H_{L1}^{T} .H_{U1}^{T} } \right] + C_{IB} H_{U1}^{T} = 0 $$

$$ C_{RB} H_{L1}^{T} \cdot H_{U1}^{T} = C_{IB} H_{U1}^{T} $$

$$ H_{L1}^{T} \left( {C_{RB} \cdot H_{U1}^{T} } \right) = C_{IB} H_{U1}^{T} $$

$$ letY = C_{RB} \cdot H_{U1}^{T} \,and\,Z = C_{IB} H_{U1}^{T} $$

$$ H_{L1}^{T} Y = Z,Y = \frac{Z}{{H_{L1}^{T} }} $$

$$ C_{RB} = \frac{Y}{{H_{U1}^{T} }} $$

(23)

The redundancy block can be obtained by using above Eq. (23).

The first component, encoder1 encodes the information block.

$$ C^{1}_{RB} C^{1}_{IB} = \left[ {C^{1}_{RB1} C^{1}_{RB2} .....C^{1}_{RBM} C^{1}_{I} C^{1}_{2} .....C^{1}_{N - M} } \right] $$

(24)

${{C}^{1}}_{RB}$ is the parity matrix block of the first encoder output. The output of the first encoder is applied to interleaver.VI to generate the ${{C}^{1}}_{I}$.interleaved data of the systematic block. The encoder2 uses only the interleaved output data. The output of the interleaved circuit is given as ${{C}^{2}}_{IB}={C}_{IB}$ interleaved is applied to the second encoder. Figure 6 shows the encoding process.

$$ \left[ {C^{2}_{RB} C^{2}_{IB} } \right] = \left[ {C^{2}_{RB1} C^{2}_{RB2} \ldots ..C^{2}_{RBM} C^{2}_{1} C^{2}_{2} \ldots \ldots C^{2}_{N - M} } \right] $$

(25)

Fig. 6

The parallel concatenation of LDPC using two regular LDPC codes with R = ½

The second encodes the information block ${C}_{I}$

$$ C_{I } = \left[ {C_{1} C_{2} ......C_{N - M} } \right] $$

(26)

$$ \left[ {C^{1}_{RB} C^{2}_{RB} C^{2}_{I} } \right] = \left[ {C^{1}_{RB1} C^{2}_{RB2} ......C^{1}_{RBM} C^{2}_{RB1} C^{2}_{RB2} ....C^{2}_{RBM} C^{2}_{1} C^{2}_{2} .. \ldots C^{2}_{N - M} } \right] $$

(27)

Equation (27) is concatenation of information block, redundancy parity blocks.

3.5 PC-LDPC Decoding Process

Soft in soft out (SISO) message-passing decoding algorithm is used for decoding single LDPC code.

Each Part of the code is decoded by the soft in the soft out (SISO) algorithm. The decoder receives the soft outputs ${{C}^{2}}_{RB}$,${{C}^{1}}_{RB},{{C}^{2}}_{IB}$. In which ${{C}^{2}}_{IB}$, ${{C}^{1}}_{RB}$ and ${{C}^{2}}_{RB}$ Represents the received block corresponding to the interleaved block and received blocks and parity blocks of second and first decoders are shown in Eq. (28). The first iteration generates the soft information block${I}^{2}$, i.e., the LDPC decoder2 results ${in I}^{2}=[{{I}^{2}}_{1}{{I}^{2}}_{2}.....{{I}^{2}}_{N-M}]$ with the input Eq. (29) as shown in Fig. 7.

$$ \left[ {C^{2}_{RB} C^{2}_{IB} } \right] = \left[ {C^{2}_{RB1} C^{2}_{RB2} \ldots ..C^{2}_{RBM} C^{2}_{1} C^{2}_{2} \ldots .C^{2}_{N - M} } \right] $$

(28)

Fig. 7

Decoding process of PC-LDPC using SISO

The LDPC decoder1 results in the soft information ${I}^{1}$ using the received block ${{C}^{1}}_{RB}$ and the soft information ${I}^{2}$ generated by first LDPC decoder 2; the resultant equation is shown as (30)

$$ \left[ {C_{RB}^{1} I_{d} } \right] = \left[ {C_{RB1}^{1} C_{RB2}^{1} ...C_{RBM}^{1} I_{d}^{1} I_{d}^{2} ....I_{dN - M} } \right] $$

(29)

Considering the next iteration, the first LDPC decoder2 uses the soft information ${I}^{1}$ generated by the second LDPC decoder1 to do the decoding process, the input for the decoder is shown in Eq. (31)

$$ \left[ {Y^{2} I} \right] = \left[ {Y_{1}^{2} Y_{2}^{2} ....Y_{M}^{2} I_{1} I_{2} ...I_{N - M} } \right] $$

(30)

$$ I = x^{2} + I_{INTERLEAVER} $$

(31)

The interleaver and deinterleaver blocks in the output of LDPC decoder1 and decoder 2 are used to decorrelate the soft decision at the output of each decoder. After the maximum number of iterations (MaxIT) reach the decoder stops working, and LDPC decoder1 generates the output Y. the output along with concatenation of soft information I are shown in Eq. (30). The sequence of decoding steps followed in this design is the SUM-PRODUCT algorithm [24].

3.6 AWGN and Rayleigh Channel VI

The two channels are employed to GFDM complex waveform, and the two channels are time-varying channels. The first channel is Gaussian noise, and the second channel is Rayleigh fading channel. These channel VI’s are already inbuilt in the LabVIEW programming models; the environmental noise is also added to the GFDM waveform along with two channels. The AWGN VI is utilized in which the user can determine E_b/N₀. The qualities of the channel can be varied with channel frequencies from 100 to 900 Hz. Figure 8 shows MT add AWGN.VI with IQ impairments. The data type of signal is pink color, a complex IQ GFDM signal applied to MT Apply IQ impairments.VI program.

Fig. 8

MT add AWGN.VI and MT Apply IQ Impairments.VI

The GFDM complex signal is applied to selective fading.VI, the fading profile is selected, such as flat fading or fast fading or slow fading, frequency selective fading. The MT generates a fading profile.VI is applied to the selective fading profile.VI block. The selective Rayleigh fading (jakes model) has been selected for this transceiver. The Jake model consists of frequency oscillators, and they generate sinusoidal oscillations based on components present in the feedback circuit as shown in Fig. 9.

Fig. 9

MT fading profile.VI (Rayleigh Fading channel) and Apply fading profile.VI

4 USRP 2901 Radio Transceiver Working Sequence for PC-LDPC Codes with GFDM

This section explains the two USRP 2901 devices used, which transmits the IQ data from the verto antenna and receives the combination of IQ data with AWGN channel noise along with environmental noise. This USRP radio transceiver is tested for different channel coding techniques for determining the performance analysis of the GFDM system under the AWGN channel. The exact process is again repeated for the Rayleigh fading channel.

5 Transceiver Blocks in USRP Device

The user PC propagates the IQ signals and send them to a radio device through the universal serial cable. The upconverter changes the IQ signals into 64 Mega samples per sec by using a mixer circuit. The BPF is used to pass the bandlimited signal to the frequencies of the USRP device ranging from 75 kHz to 2 GHz. The user can use any frequency for his application design. The verto antennas transmit the IQ samples to the free space environment. The receiver captures the IQ samples mixed with free space noise and sends them to receiver blocks. The Low noise amplifier amplifies the signal, mixer down-converts the IQ samples and Analog to digital converter to convert the signal into user-specified rate [25]. Figure 10 is the schematic diagram of the hardware setup.

Fig. 10

Hardware setup of two USRP devices with GFDM modulator and demodulator

5.1 PC-LDPC/LDPC Codes based GFDM/OFDM-WIMAX System Design and Specifications

This system model is based on a 5G candidate waveform called GFDM, but the IEEE standard 802.16e supports only the OFDM model. Hence we proposed to convert the GFDM system into the OFDM model and then apply PC-LDPC codes are forward error-correcting codes before the interleaving section in the scenario. To convert the GFDM system into the OFDM model, the number of subsymbols should be equal to 1. i.e.(M = 1), then K increases to N, Number of subcarriers(K) goes to symbols(N), (without altering bandwidth), then the sufficient spacing between the subcarriers is present, then GFDM is converted to OFDM.

The term WiMAX is termed as Worldwide Interoperability for Microwave Access. It is formed in 2001 to increase the interoperability of wireless MAN, also called the IEEE 802.16 standard. WiMAX is an alternative to digital subscriber lines and cable. Wireless fidelity (Wi-Fi) uses carrier sense multiple access collision avoidance schemes in the MAC layer. But in WiMAX, the MAC layer uses a scheduling algorithm, which helps overcome the drawbacks of Wi-Fi. WiMAX operates in the frequency range of 2–11 GHz. OFDM technique is used to implement a WiMAX technique that overcomes all the drawbacks of the existing system. In this scenario, the parallel concatenation of LDPC codes are used as FEC codes; the MT permute interleaver.VI is used as interleaver, as shown in Fig. 11. The performance of this system is compared with single long length high complexity LDPC codes. The simulation parameters used to generate results are taken from Table 1.

Fig. 11

PC-LDPC/LDPC as forward Error Correcting code in physical layer model of WiMAX

Table 1 Simulation parameters used in WiMAX model using PC-LDPC codes

5.1.1 VI Programming for GFDM System using USRP Device

Figure 12 shows the USRP 2901 device starts transmission session; the VI required are niUSRP open Tx session.VI opens the session, followed by the configuration of the signal.VI, the IQ rate, carrier frequency, device names, a gain of the antenna in decibels, and active antennas numbers are user-defined.

Fig. 12

Opening a USRP 2901 transmitter session

Figure 13 shows the sequence of steps followed in the receiver section, the Ni USRP open Rx session.VI is used to open the receiver session, Ni USRP Fetch RX Data.VI Fetches complex, double-precision floating-point data in a waveform data type from the specified channel.

Fig. 13

Opening a USRP 2901 receiver session

Figure 14 shows the GFDM system design with the parameters such as IQ gain imbalance, quadrature skew, roll-off factor value and a frequency offset value. This VI programming generates results for Tables 2 and 3, where the BER values are generated without employing channel coding schemes.

Fig. 14

GFDM system model design using LabVIEW programming

Table 2 BER computation for E_b/N₀ at 5 dB for prototype filter in GFDM with roll-off factor values

Table 3 BER computation for E_b/N₀ at 20 dB for prototype filter in GFDM with roll-off factor values

Figure 14 is the transmitter section for the WiMAX model. The number of subsymbols is equal to one, making the GFDM system model converted to the OFDM model. The simulation parameters used are shown in Table 1. Figure 15 shows the receiver model for the WiMAX model. Here, the LabVIEW programming based VI is used to generate the received signal and compute the BER value. The list of VI’s is used are CP removal.VI, which works to remove the cyclic prefix, CFO correction.VI works to correct the carrier frequency offset rate, BER.VI calculates the average BER for a given Galois PN sequence.

Fig. 15

GFDM/OFDM receiver model for WiMAX simulation parameters

As clarified in the GFDM and VI Hierarchy system model in Fig. 4, the prototype filters such as RC, RRC, Gaussian, and Xia are chosen, the list of VI's utilized in the channel coded GFDM handset is shown in Fig. 16. The PC-LDPC encoders with block interleaver are connected to the GFDM modulator. PC-LDPC is constructed by breaking a long and high complexity of conventional single LDPC code into two smaller and lower LDPC codes. The regular parity check matrix is used as input for LDPC code and has three inputs n, j and k. where each variable represents iterations size, rows, and columns size. The generated GFDM symbol is again added to the AWGN/ Rayleigh fading channel. Table 4 shows all the simulation parameters used in GFDM.

Fig. 16

VI for Proposed Transceiver for PC-LDPC coded GFDM system

Table 4 Transceiver simulation table

The receiver of GFDM virtual programming as shown in Fig. 17 consists of PC-LDPC decoder, deinterleaver, the receiver blocks are already briefed in passage 2 and the list of VI hierarchy used. The primary algorithm used here is MIN SUM, the message passing with soft input and soft output iterations are explained in the passage 3.4 and 3.5. the same regular two LDPC decoders are used with deinterleaver present between them. The maximum iterations for the decoder are set as 30, 100, and 200. For different pulse shaping filters RC, RRC, and Xia 4^th order filter with different roll-off factor values, corresponding BER is calculated, the expressions are explained in Sect. 6. The mathematical equation of the Symbol error probability (SEP) performance of GFDM having 16-QAM data transmission in AWGN channel and Rayleigh fading channel is given in Eqs. (32), (33) and (34).

$$ SEP = 2\left( {\frac{K - 1}{K}} \right)erfc\left( {\sqrt a } \right) - \left( {\frac{K - 1}{K}} \right)erfc^{2} \left( {\sqrt a } \right) $$

(32)

$$ \Upsilon = \frac{{3\frac{NS}{{NS + N_{cp} + N_{cs} }}}}{{2\left( {2^{\nabla } - 1} \right)}}\frac{{E_{s} }}{{\varphi N_{0} }} $$

(33)

$$ SEP_{.} = \frac{2}{{\left( {\left( {\frac{K - 1}{K}} \right)\left( {1 - \sqrt {\frac{ \Upsilon }{1 + \Upsilon }} } \right) - \left( {\frac{K - 1}{K}} \right)^{2} \left[ {1 - \frac{4}{\pi }\sqrt {\frac{ \Upsilon }{1 + \Upsilon }} a\tan \left( {\sqrt {\frac{1 + \Upsilon }{ \Upsilon }} } \right)} \right]} \right)}} $$

(34)

Fig. 17

GFDM Receiver VI

6 Result Analysis

Figure 18 shows the front panel output screenshot, in which k, j are rows and columns in the LDPC decoder. The maximum iterations of the decoder can be changed with the help of indicators, and the rayleigh envelope is plotted; the QAM constellation is plotted using the XY graph. The specifications of LDPC codes are n,j,k, MaxIT. n gives no of columns used, j gives no of ones in the column, j is always an odd number, k gives no of ones in a row, MaxIT specifies maximum no of iterations in a matrix that is not rank efficient, the PC-LDPC decoder will stops operation when the number of iterations exceeds the MaxIT.

Fig. 18

GFDM-PC-LDPC decoder Receiver front panel

Figure 19 and 20 shows the BER analysis, in which the GFDM system model is tested for different prototype filters such as RC, RRC, Gaussian, Xia 4^th order filter. With the higher alpha values, the system's performance degrades, for low values of alpha, performance is not efficient. For two values of alpha 0.2 and 0.5, BER improvement is observed in all prototype filters. Hence, the two values are chosen for BER computation of GFDM system and channel coding techniques under both Guassian Noise and Rayleigh channels and USRP radio transceiver environment noise.

Fig. 19

BER analysis for roll-off factor values in prototype filters

Fig. 20

BER analysis for the roll-off factor values in prototype filters

Figure 21 shows the GFDM out of band power is − 40 dB, and the OFDM signal’s power is − 20 dB. Since less emission of band power is observed in GFDM, it is an essential candidate waveform for the fifth generation. Figure 21a is the OOB of Raised cosine filter characteristics with several symbols are 1920, and its OOB power is − 36 dB. Figure 21b shows RRC filter characteristics, and its OOB power is − 40 dB, hence in digital communication RRC filter is much prefer than RC filter. Figure 21c shows Xia 4^th order characteristics, the OOB value is − 37 dB, it is not preferable for spectrum, Fig. 21d shows Gaussian filter prototype filter characteristics, and its OOB is -28 dB.

Fig. 21

Out of band emission performance of GFDM signal with RC, RRC, Guaasian, and Xia 4th order prototype filters are plotted

Table 5 shows the USRP device results of convolutional coded GFDM system, the specifications are constraint length K is 15, and code rate r is ½.

Table 5 BER computation of different prototype filters under the AWGN channel for Convolutional encoder and hard decision-based decoder of GFDM system (AWGN channel), with four types of prototype filters such as RC, RRC, Gaussian and Xia 4th order filter

The Parity check matrix of LDPC code is set as 256 × 512, the code rate of ½., maximum iterations of LDPC decoder is set to 30, 100, and 200 for the simulation results. All the BER values are computed from the NI RF hardware device called USRP 2901. Figures 22, 23, 24, 25, 26, 27, 28 and 29 shows the BER plots of values present in Tables 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Figure 30 and 31 shows the BER plots of values present in Tables 16, 17, 18, 19.

Fig. 22

BER analysis of different channel coded GFDM system under AWGN channel, and PC-LDPC-GFDM system with frequency-selective Rayleigh channel for RC prototype filter only with roll-off factor value 0.2

Fig. 23

BER analysis of RC pulse shaped filter with roll-off factor value 0.5 for different channel coded GFDM system under AWGN channel, and PC-LDPC-GFDM system with frequency-selective Rayleigh channel

Fig. 24

BER analysis of RRC prototype filter based different channel coded GFDM system under AWGN channel, and PC-LDPC-GFDM system with frequency-selective Rayleigh channel with roll-off factor value 0.2

Fig. 25

BER analysis of channel coded GFDM system under AWGN channel, and PC-LDPC-GFDM system with frequency-selective Rayleigh channel for RRC prototype filter only with roll-off factor value 0.5

Fig. 26

BER analysis of Gaussian prototype filter-based channel coded GFDM system under AWGN channel, and PC-LDPC-GFDM system with frequency-selective Rayleigh channel with roll-off factor value 0.2

Fig. 27

BER analysis of channel coded GFDM system under AWGN channel, and PC-LDPC-GFDM system with frequency-selective Rayleigh channel for Gaussian prototype filter only with roll-off factor value 0.5

Fig. 28

BER analysis of Xia 4^th order filter for roll-off factor value 0.2 based channel coded GFDM system under AWGN channel, and PC-LDPC coded GFDM system with frequency-selective Rayleigh channel

Fig. 29

BER analysis of channel coded GFDM system under AWGN channel, and PC-LDPC-GFDM system with frequency-selective Rayleigh channel for Xia 4th order prototype filter only with roll-off factor value 0.5

Table 6 BER computation of different prototype filters under AWGN channel for Golay coded GFDM system

Table 7 BER computation of different prototype filters under the AWGN channel for BCH coded GFDM system, BCH code specifications are (63,36,5), (n,k,t)

Table 8 BER computation of Raised cosine prototype filters under AWGN channel for single long length high complexity LDPC encoder and decoder

Table 9 BER computation of Root Raised Cosine prototype filter under AWGN channel for single long length high complexity LDPC encoder and decoder

Table 10 BER computation of Gaussian prototype filter under AWGN channel for single long length high complexity LDPC encoder and decoder

Table 11 BER computation of Xia 4th order prototype filter under AWGN channel for single long length high complexity LDPC encoder and decoder

Table 12 BER computation of Raised Cosine prototype filter under AWGN channel based Parallel concatenation of LDPC decoder

Table 13 BER computation of Root Raised Cosine prototype filter under AWGN channel based Parallel concatenation of LDPC decoder

Table 14 BER computation of Guassian prototype filter under AWGN channel based Parallel concatenation of LDPC decoder

Table 15 BER computation of Xia 4th order prototype filter under AWGN channel based Parallel concatenation of LDPC decoder

Fig. 30

BER analysis of under AWGN channel, and frequency-selective Rayleigh channel for RC, RRC, Gaussian and Xia 4th order prototype filters with roll-off factor value 0.5., and the MaxIT represents the maximum number of iterations in the decoder

Fig. 31

BER analysis of PC-LDPC coded GFDM system with frequency-selective Rayleigh fading channel for RC, RRC, Gaussian, and Xia 4th order prototype filters with roll-off factor value 0.5., the MaxIT represents a maximum number of iterations in the decoder

Table 16 BER computation of Parallel concatenation of LDPC decoder, serial scheduling/message-passing algorithm is used in decoder section, with Raised Cosine prototype filter under Rayleigh Fading channel (Jakes model)

Table 17 BER computation of Parallel concatenation of LDPC decoder, with serial scheduling/message-passing algorithm, is used as a decoder section, with Root Raised Cosine prototype filter under Rayleigh Fading channel (Jakes model) is used as configuration

Table 18 BER computation of different prototype filters of Parallel concatenation of LDPC decoder, with serial scheduling/message-passing algorithm, is used as a decoder section, with Gaussian prototype filter under Rayleigh Fading channel (Jakes model) is used as configuration

Table 19 BER computation of Parallel concatenation of LDPC decoder, serial scheduling/message-passing algorithm is used as decoder section, with Xia 4th order prototype filter under Rayleigh Fading channel (Jakes model) is used as configuration

Figure 22 shows BER analysis with SNR for AWGN channel, Raised cosine filter configuration with roll-off factor value 0.2, the PC-LDPC results at BER = 10^–3 were 16 dB and 22 dB, respectively. The best value of SNR is 15 dB when two LDPC codes are connected in the parallel concatenation. The worst performance is seen in BCH coded GFDM system, i.e. BER = 10^–2 were 21 dB respectively and convolutional coded GFDM has BER = 10^–2 were 18 dB, Golay coded GFDM system has BER = 10^–2 were 19 dB. The result is shown in Fig. 22 that PC-LDPC code outperforms LDPC code up to about 0.5 dB with RC pulse shaping filter.

Figure 23 shows BER analysis with SNR for the AWGN channel. The PC-LDPC results at BER = 10^–4 were 24 dB and 21 dB, respectively. The best value of SNR is 16 dB for less BER value in PC-LDPC. The performance in BCH coded GFDM system is not efficient, i.e. BER = 10^–2 were 18 dB respectively and convolutional coded GFDM has BER = 10^–3 were 21 dB, Golay coded GFDM system has BER = 10^–2 were 15 dB. The single LDPC codes have BER = 10^–3 were 21 dB, and the PC LDPC code has BER = 10^–3 were 20 dB. The result is shown in Fig. 23 that PC-LDPC code outperforms LDPC code with a coding gain of 1 dB.

The PC-LDPC code under the Rayleigh fading channel results at BER = 10^–4 for 12 dB and 15 dB, respectively. The best value of SNR is 12 dB for less BER value in PC-LDPC. The BCH coded GFDM, Convolutional, Golay, single LDPC code system is not efficient, i.e., BER = 10^–3 were 21 dB respectively. The result is shown in Fig. 24 that the PC-LDPC code of Rayleigh fading channel outperforms LDPC code up to about 6 dB with RRC pulse shaping filter with 0.2 as roll-off factor value.

The PC-LDPC code under Rayleigh fading channel results at BER = 10^–4 were 15 dB, respectively. The best value of SNR is 15 dB for low BER value in PC-LDPC. The BCH coded GFDM, Convolutional, Golay, single LDPC code system is not efficient, i.e., BER = 10^–2 were 18 dB respectively. The result is shown in Fig. 25 that the PC-LDPC code of Rayleigh fading channel outperforms LDPC code up to about 3 dB with RRC pulse shaping filter with 0.5 as roll-off factor value.

The PC-LDPC code under Rayleigh fading channel results at BER = 10^–3 were 15 dB, respectively. The best value of SNR is 18 dB for less BER value in PC-LDPC. The performance in BCH coded GFDM, Convolutional, Golay, single LDPC code system is not efficient, i.e., BER = 10^–2 were 15 dB, respectively. The performance of all the channel coding schemes is the same for high SNR values. The result is shown in Fig. 26 that the PC-LDPC code of the Rayleigh fading channel outperforms LDPC code up to about 0.7 dB with a Gaussian filter with 0.2 as a roll-off factor value.

The LDPC code under AWGN channel results at BER = 10^–4 was 15 dB respectively. The best value of SNR is 15 dB for less BER value in LDPC. BCH coded GFDM, Convolutional, Golay, single LDPC code system is not efficient, i.e., BER = 10^–3 were 21 dB respectively. The result is shown in Fig. 27 that the LDPC code of the Gaussian channel outperforms the PC-LDPC code up to about 2.5 dB.

The PC-LDPC code under Rayleigh fading channel results at BER = 10^–4 were 12 dB respectively. The best value of SNR is 12 dB for less BER value in PC-LDPC. The BCH coded GFDM, Convolutional, Golay, single LDPC code system is not efficient, i.e., BER = 10^–3 were 21 dB, respectively, the BCH coded GFDM has worst BER performance for SNR below 18 dB values. The result is shown in Fig. 28 that the PC-LDPC code of Xia 4^th order filter under Rayleigh fading channel outperforms PC-LDPC code under the AWGN channel up to about 3 dB.

The LDPC code under AWGN channel results at BER = 10^–4 was 15 dB respectively. The best value of SNR is 15 dB for less BER value in LDPC. The performance in BCH coded GFDM, Convolutional, Golay, single LDPC code system is not efficient, i.e., BER = 10^–2 were 21 dB respectively The result is shown in Fig. 29 that LDPC code under AWGN channel outperforms PC-LDPC code up to about 2 dB.

The PC-LDPC decoder maximum iterations are kept as 100 and 200. For different pulse shaping filters such as RC, RRC, and Gaussian, Xia 4^th order filters are tested with maximum iterations. BER = 10^–5 was SNR = 10 dB is observed for Xia filter, with Maximum iterations kept as 200. The remaining prototype filters with roll-off factor value 0.5 provides BER = 10^–4 were SNR = 18 dB, the coding gain of 6.5 dB is observed in PC-LDPC codes with Xia 4th order corresponding to PC-LDPC codes for other prototype filters.

The PC-LDPC decoder maximum iterations are kept as 100 and 200. For different prototype filters such as RC, RRC, and Gaussian, Xia 4th order filters are tested with maximum iterations. BER = 10^–5 was SNR of 9 dB is observed for Xia filter, with Maximum iterations kept as 200. PC-LDPC-GFDM system in the combination of Xia prototype filter with roll-off factor value 0.5 outperforms other PC-LDPC-GFDM of RC, RRC, Gaussian filters with a coding gain of 3 dB.

Figure 32 shows the BER analysis of PC-LDPC codes in an OFDM system compared with a single LDPC code. This Fig. shows BER analysis with SNR for AWGN channel, RC pulse filter configuration with α = 0.52 the LDPC code under AWGN channel results at BER = 10^–3 was 5 dB respectively. The best value of SNR is 8 dB for less BER value in PC-LDPC. The result is shown in Fig. 31 that the PC-LDPC code outperforms the LDPC code with a coding gain of 2 dB.

Fig. 32

BER analysis of PC-LDPC codes of OFDM/GFDM system under AWGN channel for WiMAX based simulation using RT Math script module LabVIEW programming

7 Conclusions

GFDM waveform is one of the waveforms as a candidate for 5G. the spectral characteristics of waveform depend on pulse shaping filters. The latency of the system depends on the power spectral density characteristics of the signal. The BER performance of the fifth-generation system depends on the channel coding schemes; the BER computation is required for SNR values in the communication system. In this article, the different prototype filters such as RC, RRC, Gaussian and Xia 4^th order filters are applied to the GFDM system under the AWGN channel. The RRC provides out of band power as  − 37 dB, and Xia's filter generates out of band power as − 40 dB, the improvement of 7 dB OOB is observed in RRC, Xia prototype filters corresponding to RC filter. The improvement of 20 dB is observed in GFDM corresponding to the 4G-OFDM signal. Hence, GFDM has less latency than OFDM, is suitable for industry 4.0 and latency applications, and is suitable for 5G candidate waveform. The coding gain of 6.5 dB was observed in the RRC prototype filter based GFDM system under Rayleigh fading channel in PC-LDPC codes corresponding to LDPC codes here BER = 10^–4 were 12 dB was observed respectively. PC-LDPC-GFDM system in the combination of Xia prototype filter with roll-off factor value 0.5 outperforms other PC-LDPC-GFDM of RC, RRC, Gaussian filters with a coding gain of 3 dB maximum iterations kept as 100. The BER = 10^–5 for SNR = 9 dB is observed for the Xia filter, with Maximum iterations kept as 200. Hence the coding gain of 12 dB was observed in PC-LDPC codes with Xia 4th order filter corresponding to other prototype filters. PC-LDPC code outperforms the LDPC code up to about 2 dB in the WiMAX scenario with OFDM Transceiver.