Characterization of GFDM Signal with Timing Offset, CFO, Non-Linearity, and PN

In this paper, we study the joint eﬀects of timing oﬀset (TO), carrier frequency oﬀset (CFO), nonlinear power ampliﬁer distortion, and phase noise (PN) on generalized frequency division multiplexing (GFDM) system. Closed form expressions for signal-to-interference ratio (SIR) at GFDM receiver with synchronization errors and PN using a nonlinear power ampliﬁer is derived. Then, we have been conducted simulation studies to compare the performance of GFDM systems with orthogonal frequency division multiplexing (OFDM) systems using matched ﬁlter (MF) and zero forcing (ZF), in presence of these impairments. The results show that GFDM systems are more robust against TO and PN while they are more sensitive to CFO and nonlinear distortion compared to OFDM systems.

power amplifier for OFDMA is derived in [13], considering a non-uniform power distribution in subcarriers. In recent years, the impacts of some RF impairments on the GFDM waveforms have been studied. In [14], the SIR performance of GFDM and OFDM systems are compared with frequency and timing errors. Also, the SIR for OFDM and GFDM signals with PN, CFO, and TO is analyzed in [15]. In [16] SINR of GFDM Systems under I/Q imbalance is provided while the analysis of SIR with PN, CFO, and I/Q imbalance is investigated in [17]. The author in [18] derives the optimal filter for a GFDM system with CFO. For a third-order nonlinear power amplifier, the signal-to interference-plus noise ratio (SINR) for GFDM signal is studied in [19]. Despite all these efforts, however, the joint impact of TO, CFO, nonlinear power amplifier distortion, and PN on the performance of GFDM systems has not been investigated. Therefore, in this paper, joint impact of TO, CFO, nonlinear power amplifier distortion, and PN on the performance of GFDM waveform is analyzed. Assuming a polynomial model for the nonlinear behavior of power amplifier we derive closed-form expressions for SIR of GFDM signal under these impairments.
The rest of this paper is organized as follows. The GFDM system model is given in Section II. The SIR of GFDM signal with nonlinear power amplifier with synchronization errors and PN is obtained in Section III. Simulation results are presented in Section IV. Finally, this paper is concluded in Section V.
Notations: Vectors and matrices are represented by lower and upper case fonts (e.g. ⃗ x and X ), respectively. E [·], (·) * , (·) H , (·) T , and (·) −1 are the expectation, conjugate, Hermitian, transpose, and inverse operator, respectively. diag (X) represents a column vector of the main diagonal elements of the matrix X. ⊗ and • are convolution and Hadamard operators. The N × N identity matrix, N × N zero matrix, m × n zero matrix, and all one column vector of size N are denoted by I N , 0 N , 0 m×n , and 1 N , respectively.   . Therefore, d k,m is the transmitted data on the k-th subcarrier and m-th subsymbol, which is transmitted with a pulse shaping filter as follows

SIGNAL AND SYSTEM MODEL
where g k,m [n] represents the circularly shifted version of prototype pulse shaping filter g [n]. The transmitted samples are The matrix form of the equation above is where G is an N × N matrix, which can be expressed as Finally, after adding a cyclic prefix to the GFDM signal to prevent ISI, the transmitting signal can be expressed as ⃗ x. With the assumption of linear power amplifier, perfect synchronization, and removing the cyclic prefix the received signal can be written as where ⃗ x is the transmitted signal, ⃗ y is the received signal, H is the N × N channel matrix, and ⃗ w is additive white Gaussian noise (AWGN) with zero mean and variance σ 2 w . Assuming zero forcing equalization, the detected received signal is The estimated data, after GFDM demodulation, can be represented as where G r is the N × N GFDM demodulation matrix. In this paper, we consider both matched filter (MF) receiver G r = G H and zero-forcing (ZF) receiver G r = G −1 . Finally, the estimated transmitted signal after demapping and decoding is expressed as ⃗ b.

SIR ANALYSIS IN GFDM
In this section we derive the SIR expression for GFDM signal by considering RF impairments, namely TO, CFO, nonlinear distortion, and PN. We use m as TO, L as normalized maximum channel delay spread, N cp as the length of CP, ε as normalized CFO, and ϕ n as PN. Also to model the PN, discrete Brownian motion is considered, i.e., [ϕ[n] − ϕ[n − 1]] ∼ N (0, 2πβT s ), where T s is the symbol period and β is 3-dB bandwidth. Also, the transmitted signal with the baseband polynomial model for a memoryless nonlinear power amplifier could be expressed as [19] where z [n] is the baseband power amplifier output signal, x [n] is the power amplifier input signal, a 2i+1 is the complex polynomial coefficient, and 2N p + 1 is the nonlinearity order. Since distortion due to even terms can be easily removed by filters, only odd terms are considered. Here we consider N p = 1 because the third-order polynomial is enough to model the power amplifier in most practical cases [19]. Then, the power amplifier output is as follows where w [n] is the nonlinear term.
As shown in Fig. 2, TO can be divided into four cases depending on symbol starting point estimation. Therefore, these cases are: symbol starting point estimation after the actual starting point (m > 0) and symbol starting point estimation before the actual starting point, which include L − N cp < m < 0, −N cp < m < L − N cp , and m < −N cp . Also, we assume that phase noise, TO, and CFO occur at the receiver. The received GFDM signal with perfect time and frequency synchronization and a linear power amplifier, can be written as (5). We first consider the case that the symbol starting point estimation is after the actual starting point with m > 0 as depicted in Fig. 2-(a). Then, by extending (5) to consider the synchronization errors, phase noise, and the power amplifier distortion model of (9), the received signal in vector form can be expressed as where C ∈ C N ×N is the CFO matrix, d next is the vector of the next GFDM symbol, P ∈ C N ×N is the PN matrix, R 1 , R 2 ∈ C N ×N are TO matrices, G 1 ∈ C (N −L)×N and G 2 ∈ C L×N are submatrices of G, and W is the vector of nonlinear distortion. These matrices can be expressed as Using (10), after zero forcing equalization and GFDM demodulation, the estimated datad becomes where H 1 ∈ C N ×(N −L) and H 2 ∈ C N ×L are the submatrices of H = H 1 H 2 , and B 1 , B 2 , B 3 , andn are given respectively as We can rewrite B 1 , B 2 , B 3 , andd as where Theorem 1: The SIR for GFDM by considering RF impairments, namely TO, CFO, nonlinear distortion, and PN for the case of m > 0 can be obtained as follows with Proof: The proof is provided in Appendix.
3.2 Symbol starting point estimation before the actual starting point ((L − N cp ) < m < 0): We first consider the case that the symbol starting point estimation is before the actual starting point with (L − N cp ) < m < 0. As depicted in Fig. 2-(b), TO does not occur in this case, however, ISI and ICI are created due to non-orthogonal waveform of GFDM. Then, by extending (5) to consider the synchronization errors, phase noise, and the power amplifier distortion model of (9), the received signal in vector form is obtained as follows where R l 1 and R l 2 ∈ C N ×N are TO matrices and M Ncp is the matrix that shifts the elements of Gd circularly, which are given as Using (16), after zero forcing equalization and GFDM demodulation, the estimated datad iŝ where B l1 = R l 1 H + R l 2 HM N cp G and B l2 = H. Theorem 2: The SIR for GFDM by considering RF impairments, namely TO, CFO, nonlinear distortion, and PN for the case of (L − N cp ) < m < 0 can be obtained as follows with Proof: The proof is similar to that of Theorem 1.
3.3 Symbol starting point estimation before the actual starting point (−N cp < m < (L − N cp )): We first consider the case that the symbol starting point estimation is before the actual starting point with −N cp < m < (L − N cp ). As depicted in Fig. 2-(c), ISI and ICI exist in this case due to the missing samples of the current GFDM symbol. Then, by extending (5) to consider the synchronization errors, phase noise, and the power amplifier distortion model of (9), the received signal in vector form in this case is obtained as follows where R l 3 and R l 4 ∈ C N ×N are TO matrices given as Using (20), the estimated datad after zero forcing equalization iŝ where B l3 = R l 1 HG + R l 3 HM N cp G + R l 4 H 1 G 1 , B l4 = R l 4 H 2 G 2 , and B l5 = H. Since the equation above is similar to (14), its SIR expression is obtained in a similar manner to (15) as: with

Symbol starting point estimation before the actual starting point (m < −N cp ):
We first consider the case that the symbol starting point estimation is before the actual starting point with m < −N cp . As depicted in Fig. 2-(d), ISI exist in this case due to TO choosing some samples of the previous GFDM symbol. Then, by extending (5) to consider the synchronization errors, phase noise, and the power amplifier distortion model of (9), the received signal in vector form is obtained as follows where R l 5 and R l 6 ∈ C N ×N are TO matrices given as Using (23), the estimated datadafter zero forcing equalization iŝ where B l6 = R l 1 HG + R l 3 HM Ncp G + R l 5 H 1 G 1 , B l7 = R l 5 H 2 G 2 + R l 6 HG, and B l8 = H. SIR in this case is similar to (15). with

SIMULATION RESULTS
We compare the effects of synchronization errors, PN, and nonlinear distortion on SIR of GFDM and OFDM systems. In our simulations, we consider K = 32, M = 5, N cp = 24, and for GFDM, a pulse shaping filter of root raised cosine with roll-off factor of 0.1. We use also 32-point fast fourier transform (FFT) and N cp = 24 for OFDM to ensure a fair comparison with GFDM. The Rayleigh fading channel is assumed with 10 channel taps and exponential power delay profile with βe −l/L for 0 ≤ l ≤ L where β is L l=0 (βe −l/L ) 2 = 1. Also, to place the transmitter in the nonlinear region, we considered the maximum input power of the PA to be 2.5 dB less than the saturation point of the nonlinear PA model.
We use ε = 0.1, βT s = 0.01, m = 1, α 1 = 1.0108 + 0.0858j, α 3 = 0.0879 − 0.1583j, also we consider both matched filter (MF) receiver G r = G H and zero-forcing (ZF) receiver G r = G −1 for our simulations. As mentioned in [15] and can be seen in our simulation results, GFDM is most sensitive to CFO, while it is robust to TO and PN. Fig. 3 shows SIR versus TO. As can be seen, OFDM performs better than GFDM for low TOs, since TO in addition to ICI and ISI creates inter-subsymbol interference in GFDM. However, for high TOs, the OFDM performance decreases since its symbol length is less than that of GFDM, which in turn increases the interference with the next symbol.
In Fig. 4, the effects of CFO on SIR is analyzed. In Equation (11), CFO matrix in GFDM shows, as the symbol length increases, the phase shift of data caused by CFO increases which leads to inferior performance of GFDM compared to OFDM.
The SIR versus PN variance is presented in Fig. 5. The GFDM performance is better with PN compared to OFDM since the symbol period, T s of OFDM is longer than that of GFDM. It can be observed from the PN model introduced in Section III that the phase noise impact on the OFDM signal is enhanced by the relatively long symbol period of T s compared with GFDM signal. Fig. 6 represents the SIR versus nonlinear distortion. High PAPR degrades the performance of communication systems. As mentioned in [3], a lower PAPR is achieved employing GFDM compared to OFDM, assuming linear power amplifiers. However, as shown in [20,21], GFDM systems have high PAPR with nonlinear amplifiers, which leads to an inferior performance compared to the OFDM system, which is also observed in Fig. 6.This increase in PAPR (hence decrease in performance) can be due to larger symbol length of GFDM compared to OFDM.
Note that here we have just analyzed the GFDM performance for two common receiver filters, namely MF and ZF. However, by designing a SIR maximizing filter (obtaining the filter by solving an optimization with the objective of SIR), GFDM can outperform OFDM, though with higher computational complexity.

CONCLUSION
In this paper, we have derived close-form expressions of the SIR for GFDM waveforms by considering RF impairments, namely TO, CFO, PN, and nonlinear distortion and analyzed the effects of synchronization errors, PN, and nonlinear distortion for GFDM and OFDM systems. Based on the simulations, GFDM based systems are more sensitive to CFO and nonlinear distortion than OFDM based systems, while the GFDM based systems are more robust to TO and PN. Therefore, the results presented in this paper can be a good guideline for waveform design in the next generation of communication systems. An extension of this work can include the comparison of combined effects of TO, CFO, PN and non-linearity with the common additive aggregate linear models for hardware impairments.

APPENDIX
The estimated data for l th symbol in (12) iŝ where d K is the zero-mean i.i.d. data symbol with unit variance on the K-th symbol and S l is the desired signal term, I l , and P l are unwanted signals, which represent the ISI and ICI introduced by synchronization errors and PN, and N l is the nonlinear distortion noise due to the nonlinearity of the power amplifier. where Then, B 1 can be written as where b L,i−1 is the i th column vector of B L and b T r1,i−1 is the i th row vector of B r1 . Using (28), the power of S l in (27) can be written as where a r1,i = diag b L,i b T r1,i . The power of I l by using (28) can be written as where A r1,i = b L,i b T r1,i . The power of P l by using (28) can be calculated as where A r2,i = b L,i b T r2,i . The power of N l can be calculated as where A r3,i = b L,i b T r3,i . Also, the power of the nonlinear distortion is Lemma: Suppose z n for n = 1, 2, · · · , N are zero-mean complex Gaussian RVs. a) If n ̸ = m, then E [z 1 z 2 , ... , z n z * 1 z * 2 , . . . , z * m ] = 0, n, m = 1, 2, · · · , N (36) b) If n = m, then E [z 1 z 2 , ... , z n z * 1 z * 2 , . . . , z * m ] = π E z π(1) z * 1 E z π(2) z * 2 , . . . , E z π(n) z * m , where π is a permutation of {1, 2, 3, · · · , m} (a set of integers).