Capacity limit for faster-than-Nyquist non-orthogonal frequency-division multiplexing signaling

Faster-than-Nyquist (FTN) signal achieves higher spectral efficiency and capacity compared to Nyquist signal due to its smaller pulse interval or narrower subcarrier spacing. Shannon limit typically defines the upper-limit capacity of Nyquist signal. To the best of our knowledge, the mathematical expression for the capacity limit of FTN non-orthogonal frequency-division multiplexing (NOFDM) signal is first demonstrated in this paper. The mathematical expression shows that FTN NOFDM signal has the potential to achieve a higher capacity limit compared to Nyquist signal. In this paper, we demonstrate the principle of FTN NOFDM by taking fractional cosine transform-based NOFDM (FrCT-NOFDM) for instance. FrCT-NOFDM is first proposed and implemented by both simulation and experiment. When the bandwidth compression factor α is set to 0.8 in FrCT-NOFDM, the subcarrier spacing is equal to 40% of the symbol rate per subcarrier, thus the transmission rate is about 25% faster than Nyquist rate. FTN NOFDM with higher capacity would be promising in the future communication systems, especially in the bandwidth-limited applications.


Introduction
In 1940s, Shannon put forward the up limit of communication capacity in an additive white Gaussian noise (AWGN) channelthe Shannon limit 1,2 .With the exponential data traffic growth in bandwidth hungry applications such as high definition TV and mobile internet, the communication capacity gradually approaches Shannon limit nowadays, which makes the bandwidth hungry situation even more severe.Under this circumstance, increasing the spectral efficiency is a key challenge to meet the demand for high-capacity communications.As a highly spectral efficient scheme, faster-than-Nyquist (FTN) signal was firstly proposed by Mazo in 1970s 3 , but it gains great attraction in our bandwidth-strived world because it can transmit more data in the same bandwidth compared to conventional Nyquist schemes 4 .FTN signal has been widely investigated for the next-generation wireless and optical communications [5][6][7][8] , which reflect its great potential to be applied in high-capacity communications.
The existing FTN scheme can be mainly categorized into two general types, one is compressing the duration between two adjacent pulses in time domain 3 , and the other one is compressing the subcarrier spacing in frequency domain 8 .For the time-domain scheme, Mazo has firstly proposed a FTN binary sinc-pulse signal in 1975.To obtain the transmission rate faster than Nyquist rate, the pulses are accelerated with the time acceleration factor τ.However, it is no longer orthogonal between the accelerated pulses, so the FTN binary sinc-pulse signal can be considered as a kind of FTN non-orthogonal time-division multiplexing (NOTDM) signal.When τ is set to 0.8-the Mazo limit, twenty-five percent more bits can be carried in the same bandwidth and there is not obvious deterioration in bit error rate (BER) performance because the trellis decoding can effectively compensate the inter-symbol interference (ISI) 4 .While τ is lower than the Mazo limit, the BER performance will be degraded because ISI is so serious that the data cannot be accurately recovered.
Recently, non-orthogonal frequency division multiplexing (NOFDM) signal has been realized by further compressing the subcarrier spacing, which has the potential to increase the transmission capacity [9][10][11] .In conventional OFDM signal, the subcarrier spacing is equal to the symbol rate per subcarrer.Is there an increasing of transmission capacity whenever the subcarrier spacing is less than the symbol rate per subcarrier?In fact, we find that only while the subcarrier spacing is less than half of the symbol rate per subcarrier, the NOFDM signal can achieve a transmission rate faster than Nyquist rate.In our previous work, we proposed FTN fractional Hartley transform-based NOFDM (FrHT-NOFDM) signal for optical

Intensity
Mod. communication 8 .When the subcarrier spacing is 40% less than the symbol rate per subcarrier, twenty percent baseband bandwidth saving can be obtained and the transmission rate is 25% faster than the Nyquist rate.For Nyquist signal, the capacity limit was shown in the pioneering work of Shannon.However, different from Nyquist signal, FTN signal has a smaller pulse interval or narrower subcarrier spacing.What is the capacity limit of FTN signal?In this paper, we first give the mathematical expression for the capacity limit of FTN NOFDM signal, which can be also applied to FTN NOTDM signal.The mathematical expression shows that FTN signal have higher capacity limit compared to Nyquist signal.However, the interference degrades the BER performance and decrease the capacity of the FTN signal.In FTN NOTDM signal, the ISI can be effectively compensated by trellis decoding when the τ is greater than the Mazo limit.In FTN NOFDM signal, how to eliminate the inter-carrier interference (ICI) is important.We demonstrate the principle of FTN NOFDM by taking fractional cosine transform-based NOFDM (FrCT-NOFDM) for instance.To the best of our knowledge, FrCT-NOFDM is first proposed in this paper.The simulations and experiments have been demonstrated to verify the feasibility of the FrCT-NOFDM.While the bandwidth compression factor α is set to 0.8, the subcarrier spacing is equal to 40% of the symbol rate per subcarrier and the ICI can be effectively compensated by iterative detection algorithm.The transmission rate is about 25% faster than Nyquist rate and the capacity limit is also 25% higher than Shannon limit.

Results
In this paper, we demonstrate the principle of FTN NOFDM by taking FrCT-NOFDM for instance.A block diagram of the FrCT-NOFDM system is depicted in Figure 1.Different from traditional OFDM system, the inverse FrCT (IFrCT)/FrCT algorithm is employed to realize the multiplexing/demultiplexing processing.The N-order IFrCT and FrCT are defined as where 0 and α is the bandwidth compression factor.The α less than 1 determines the level of the bandwidth compression.When α is equal to 1, equation 2 is the Type-II discrete cosine transform (DCT) in which the matrix is orthogonal.The Type-II DCT is probably the most commonly used form, and is often simply referred to as "the DCT" 12 .DCT-based OFDM (DCT-OFDM) signal can be generated by Type-II DCT 13,14 .In DCT-OFDM, the subcarrier spacing is equal to half of the symbol rate per subcarrier, which is half of the subcarrier spacing in discrete Fourier transform-based OFDM (DFT-OFDM).Therefore, the subcarrier spacing of DCT-OFDM is equal to 1/2T where T denotes the time duration of one DCT-OFDM symbol 15,16 .Due to the compression of subcarrier spacing, the subcarrier spacing of FrCT-NOFDM should be smaller than 1/2T .

Spectral efficiency and capacity limit
Figure 2 shows the sketched spectra of DCT-OFDM (i.e., α = 1) and FrCT-NOFDM when the subcarrier number is set to 4.
The subcarrier spacing of FrCT-NOFDM is equal to α/2T where T denotes the time duration of one FrCT-NOFDM symbol.
All the subcarriers locate in the positive frequency domain.Meanwhile, their images fall into the negative frequency domain.
The baseband bandwidth of FrCT-NOFDM can be calculated by where N denotes the subcarrier number.Nyquist frequency is equal to half of the sample rate.Therefore, the Nyquist frequency is equal to N/2T .When N is large enough, the baseband bandwidth of DCT-OFDM (i.e., α = 1) is almost the same with Nyquist frequency and the baseband bandwidth of FrCT-NOFDM (i.e., α < 1) is lower than Nyquist frequency.It is worth noting that Nyquist frequency should not be confused with Nyquist rate.Nyquist rate is twice of the baseband bandwidth.In general, the symbol rate is equal to the sample rate when all the subcarriers are valid.Therefore, the symbol rate of FrCT-NOFDM is faster than Nyquist rate.In other word, FrCT-NOFDM can be considered as a kind of FTN signal.When α is set to 0.8, the subcarrier spacing is equal to 40% of symbol rate per subcarrier.Twenty percent baseband bandwidth saving can be obtained and the transmission rate is about 25% faster than the Nyquist rate.
Afterwards, we discuss the capacity limit for FrCT-NOFDM signal.The channel is in the presence of AWGN.The well-known Shannon limit can be calculated by where W is the signal bandwidth, P s is the signal power and P n is the noise power.This equation depicts the capacity limit for Nyquist signal 2 .
To obtain the capacity limit of FrCT-NOFDM signal, we will apply the analysis method for single-carrier signal in Shannon's papers 1,2 to the multi-carrier signal.However, there is great difference between single-carrier and multi-carrier signal.The capacity limit of the single-carrier signal can be carried out in the time domain, but the capacity limit of multi-carrier signal should be derived out in the frequency domain.The derivation of capacity limit needs to employ the geometric methods 17 .In the following, we will briefly give the derivation of the capacity limit for FrCT-NOFDM.This derivation is similar to that in Shannon's papers.
In the frequency domain, there are W /(subcarrier spacing) independent amplitudes where W is the baseband bandwidth and the subcarrier spacing is equal to α/2T .Therefore, the number of independent amplitudes is equal to 2TW /α, which determines the dimensions of signals.For large W , the perturbation caused by noise can be considered as some points near the surface of a sphere with radius 2TW P n /α centered at the signal point.The power of received signal is P s + P n .Similar to the perturbation, the received signal can be considered as some points whose positions are on the surface of a sphere with radius 2TW (P s + P n )/α.The number of the distinguishable signals is no more than the volume of the sphere with radius 2TW (P s + P n )/α divided by the volume of the sphere with radius 2TW P n /α.The volume of an n-dimensional sphere of radius r can be calculated by 17 where Γ n/2 + 1 = ∞ 0 e −t × t n/2 dt.Therefore, the upper limit for the number M of distinguishable signals is given by Consequently, the capacity limit of FrCT-NOFDM signal can be bounded by where SNR is the signal-to-noise power ratio.
The capacity limit of FrCT-NOFDM is higher than Shannon limit of Nyquist signal because the α is less than 1.When α is set to 0.8, the capacity limit is 25% higher than Shannon limit.Without loss of generality, equation ( 8) can be applied to the other kinds of FTN NOFDM signal.In FTN NOTDM, more samples can be transmitted in the same period 3,4 , thus more information can be transmitted in the same period.In our opinion, equation ( 8) not only is applicable for the FTN NOFDM signal but FTN NOTDM as well.The α in equation ( 8) should be replaced by the time acceleration factor τ for FTN NOTDM signal.Furthermore, we think this equation can be also applied to the other types of FTN signal.It is worth noting that, in FrCT-NOFDM signal, ICI becomes a serious problem that should not be neglected because the subcarriers are no longer orthogonal.ICI will degrade the BER performance, thus decrease the capacity of FrCT-NOFDM.Therefore, the capacity of FrCT-NOFDM signal is likely to approach the limit only when ICI has been effectively eliminated.How to eliminate the ICI is crucial for FrCT-NOFDM signal.

Simulation and theoretical analysis
Afterwards, we will introduce an ICI cancellation algorithm based on iterative operation for FrCT-NOFDM.The simulations are demonstrated to verify the feasibility of FrCT-NOFDM.In the simulation system, the AWGN channel is employed.The Figure 3 shows the |C l,128 | versus l for FrCT-NOFDM with different α.When α is set to 1, the auto-correlation values for the 128 th subcarrier is 1.Meanwhile, the cross-correlation values are almost equal to 0. Therefore, when α is set to 1, the C is an identity matrix and the signal generated by DCT is a kind of OFDM signal.To achieve the smaller subcarrier spacing, α can be set to smaller than 1.However, when α is less than 1, the orthogonality is destroyed.As shown in Fig. 3, the interference caused by the adjacent subcarriers is larger than that caused by farside subcarriers.The ICI will increase with the decreasing of α.Thus, BER performance of FrCT-NOFDM will degrade with the decreasing of α.When the subcarrier spacing is equal to the half of symbol rate per subcarrier, the subcarriers is orthogonal in DCT-OFDM (i.e., α = 1 in FrCT-NOFDM), but the subcarriers is no longer orthogonal in FrHT-NOFDM 8 or fractional Fourier transform-based NOFDM (FrFT-NOFDM) 9 .Therefore, under the same subcarrier spacing, the ICI in FrCT-NOFDM should be smaller than that in FrHT-NOFDM or FrFT-NOFDM.
Recently, iterative detection (ID) algorithm is used to reduce the ICI for FrFT-NOFDM [9][10][11] .Two-dimensional constellation (e.g., M-QAM) is modulated in FrFT-NOFDM.However, as the system in Figure 1 depicts, one-dimensional constellation (e.g., M-PAM) is modulated in FrCT-NOFDM.The ID algorithm for FrCT-NOFDM with one-dimensional constellation can been given by where S i is an N-dimensional vector of recovered symbols after i th iteration, R is an N-dimensional vector of received symbols demodulated by FrCT, and e is an N × N identity matrix.The mapping strategy of 2-PAM is demonstrated in Fig. 4. The uncertainty interval is defined by d = 1 − i/I where i is the i th iteration and I is the total iterative number.Only points that fall in the blue area can be mapped to the corresponding constellation points.The points that fall in the pink area are unchanged.The iterative operation is shown in equation (10).After each iteration, ICI should be reduced, thereby the pink area can be reduced.The algorithm is completed while d is equal to zero.Because the decision for one-dimensional constellation is more simple than that for two-dimensional constellation, ID algorithm of FrCT-NOFDM should have lower complexity than that of FrFT-NOFDM.
Figure 5 depicts the BER performance of DCT-OFDM (i.e., α = 1) and FrCT-NOFDM.The modulated constellation employs 2-PAM.I depicts the iterative number of ID algorithm.When α is less than 1, the BER performance of FrCT-NOFDM without ID algorithm (i.e., I is set to 0) is seriously influenced by ICI.When I is set to 20, FrCT-NOFDM with α of 0.9 has the same BER performance compared to the DCT-OFDM.This is because that ID algorithm can effectively eliminate the ICI in FrCT-NOFDM with α of 0.9.When α is set to 0.8 or 0.7, the BER performance is still influenced by the residual ICI although the ID algorithm is employed.At the 7% forward error correction (FEC) limit, the required E b /N 0 for FrCT-NOFDM with α of 0.8 is about 2 dB higher than that for DCT-OFDM.When the FEC coding technique is employed, FrCT-NOFDM with α of 0.8 should have almost the same performance with DCT-OFDM.When α is set to 0.7, the BER of FrCT-NOFDM only achieves the FEC limit and there are a clear gap compared to DCT-OFDM.The BER performance degrades with the decreasing of α due to the increasing of residual ICI.Fig. 6 shows the BER against iterative number of ID algorithm for FrCT-NOFDM when E b /N 0 is set to 20 dB.When α is set to 0.8, BER performance can be improved by the increasing of iterative number, so the ID algorithm can effectively  eliminate the ICI.However, when α is set to 0.7, the increasing of iterative number cannot significantly improve the BER performance.This is because the ICI is too large to be completely eliminated by ID algorithm.Therefore, it needs a more effective algorithm to compensate the interference.The ID-FSD algorithm has better compensation performance than ID algorithm, which combines the ID algorithm and fixed sphere decoder (FSD) 9 .Furthermore, the channel coding have a good performance in resisting the ICI and have been investigated for improving the BER performance of NOFDM 4,18 .
The simulation results verify the feasibility of the FrCT-NOFDM.When α is larger than a certain value (this value is no more than 0.8), ICI can be effectively compensated and FrCT-NOFDM can achieve almost the same BER performance compared to DCT-OFDM.This certain value is similar to the Mazo limit in FTN-NOTDM 3 .Therefore, when the α is greater than the certain value, the capacity of FrCT-NOFDM can approach to the capacity limit shown in equation (8).

Proof-of-concept experiment
To verify the feasibility of FrCT-NOFDM, we set up an experiment as shown in Fig. 1.At transmitter, the size of FrCT was 256 and 2-PAM was modulated.Sixteen cyclic prefix samples were employed.One frame included 128 FrCT-NOFDM symbols, 10 training symbols and 1 synchronization symbol.The digital signal was uploaded into an arbitrary waveform generator (Tektronix AWG7122C) operating at 10 GS/s to realize digital-to-analog conversion.The resolution of arbitrary waveform generator was set to 8 bits.The overall link rate was approximately 10 Gbit/s and the net bit rate was approximately 8.7 Gbit/s (1 bit/sample × 10 GS/s × 256/(256 + 16) × 128/(128 + 10 + 1) ≈ 8.7 Gbit/s).An external cavity laser (ECL) with a linewidth of 100 kHz was used to generate the optical carrier.A Mach-Zehnder modulator (MZM) was used to modulate the optical carrier with the generated analog signal.The V π of the MZM is about 1.5 V and the bias voltage is set to about 1.5 V.
The launch optical power was set to 3 dBm.The length of standard signal mode fiber (SSMF) was 50 km.Its total loss is approximately 10 dB.A variable optical attenuator (VOA) was employed to change the received optical power.
At receiver, the received optical signal can be converted into an electrical signal by the photodiode (PD) (Discovery DSC-R401HG).The electrical signal was then filtered by a low-pass filter (LPF) with a 3-dB bandwidth of 10 GHz.The filtered electrical signal was captured by a real-time digital phosphor oscilloscope (Tektronix DPO72004C) operating at 50 GS/s to implement analog-to-digital conversion.The generated digital signal was decoded by off-line processing in MATLAB.
The training symbols were used to estimate the channel characteristics for intra-symbol frequency-domain averaging (ISFA) algorithm and frequency-domain equalization 19 .After equalization, ID algorithm was employed to reduce the ICI.
Figure 7 shows the electrical spectra of DCT-OFDM and FrCT-NOFDM signal with different α.The baseband bandwidth of DCT-OFDM is equal to 5 GHz which is the Nyquist frequency of the signal with a 10-GS/s sample rate.In FrCT-NOFDM, the baseband bandwidth is compressed to 4.5, 4, and 3.5 GHz when α is set to 0.9, 0.8, and 0.7, respectively.The corresponding Nyquist rate is 9, 8, and 7 Gbit/s, respectively.In the experiment, the link rate of FrCT-NOFDM is 10 Gbit/s, which is faster than the corresponding Nyquist rate.Therefore, the FrCT-NOFDM signal is a kind of FTN signal.When α is set to 0.8, 20% baseband bandwidth saving can be obtained and the transmission rate is about 25% faster than the Nyquist rate.These results verify the above theory analysis.
Figure 8 depicts the BER curves for DCT-OFDM and FrCT-NOFDM after back-to-back (BTB) and 50-km SSMF transmission.The iterative number of the ID algorithm is set to 20.FrCT-NOFDM with α of 0.9 has the same BER performance with DCT-OFDM.After 50-km SSMF transmission, the required received power at the 7% FEC limit was measured to be approximately −12 dBm for FrCT-NOFDM with α of 0.9.Compared to BTB transmission, the power penalty for 50-km SSMF transmission is about 3 dB.When α is set to 0.8, the ICI degrades the BER performance.After 50-km SSMF transmission, the required received power at the 7% FEC limit was measured to be approximately −10 dBm for FrCT-NOFDM with α of 0.8.Compared to DCO-OFDM, the power penalty for FrCT-NOFDM with α of 0.8 is about 2 dB at the 7% FEC limit.When α is set to 0.7, the ICI severely degrades the BER performance, the BER can only achieve the 20% FEC limit.
Figure 9 depicts the BER against iterative number of the ID algorithm after 50-km SSMF transmission.The received power is set to −8 dBm.The α is set to 0.8 and 0.7, respectively.In theory, ID algorithm with more iterative number can eliminate more ICI.Therefore, the BER decreases with the increasing of iterative number.However, when iterative number is larger than 20, the increasing of iterative number cannot significantly improve the BER performance.This may be because the other distortions such as chromatic dispersion (CD) mainly influence the BER performance while the ICI is small.The experiment results agree well with the simulation results.The feasibility of FrCT-NOFDM has been verified by both the experiment and simulation.

Discussion
The history of FTN signal began with the paper of James Mazo in 1975 3 , who investigated the time-domain binary sinc-pulse case.The FTN binary sinc-pulse scheme compresses the time period to obtain the transmission rate faster than Nyquist rate.The pulses are accelerated with the time acceleration factor τ and become no longer non-orthogonal, so the FTN sinc-pulse signal can be considered as a kind of FTN NOTDM signal.Twenty-five percent more bits can be carried in the same bandwidth while τ is set to 0.8-the Mazo limit.In this paper, we demonstrate the principle of FTN NOFDM by taking FrCT-NOFDM for instance.The subcarrier spacing is less than 50% of the symbol rate per subcarrier.When the bandwidth compression factor α is set to 0.8, the subcarrier spacing is equal to 40% of the symbol rate per subcarrier, a 20% baseband bandwidth saving can be obtained, and the transmission rate is about 25% faster than the Nyquist rate.
In 1940s, Shannon put forward the up limit of communication capacity for the Nyquist signal 1, 2 .As we know, the FTN signal should have a higher capacity than Nyquist signal.To the best of our knowledge, we first give the mathematical expression for capacity limit of FrCT-NOFDM signal, which also can be applied to the other FTN signals.The capacity limit of FrCT-NOFDM signal is higher than the Shannon limit at 1/α times.The capacity of FrCT-NOFDM signal is likely to approach the limit only when ICI has been effectively eliminated.How to eliminate the ICI is crucial for FrCT-NOFDM.We report the simulations and experiments of FrCT-NOFDM system.The results of simulations and experiments demonstrate the feasibility of the FrCT-NOFDM.When α is larger than a certain value (this value is no more than 0.8), ICI can be effectively compensated and FrCT-NOFDM can achieve almost the same BER performance compared to DCT-OFDM.This certain value is similar to the Mazo limit in FTN-NOTDM 3 .Therefore, when the α is greater than the certain value, the capacity of FrCT-NOFDM has the potential to approach to the capacity limit shown in equation (8).In the further work, more effective algorithm can be employed to eliminate the ICI and higher-order M-PAM constellation can be investigated to obtain higher spectral efficiency.FrCT-NOFDM can be potentially used in next-generation wireless and optical communications which require higher spectral efficiency and capacity.

Methods
The simulations and off-line processing in the experiment were both realized by MATLAB.The encoding and decoding were shown in Fig. 1.In the simulation, the AWGN channel was employed.The channel equalization is not required for the AWGN channel.In the experiment, the training symbols were used to estimate the channel characteristics.Meanwhile, the ISFA algorithm can improve the efficiency of channel estimation 19 .The frequency-domain equalization can compensate the channel distortion by using the estimated channel characteristics.
In NOFDM system, ID algorithm is employed to eliminate the ICI, which is critical to improve the BER performance.The ID algorithm for 2-PAM constellation is shown in Algorithm 1.Compared to two-dimensional constellation (i.e., M-QAM), the constellation mapping for one dimensional constellation (i.e., M-PAM) is more simple and accurate.

Figure 6 .
Figure 6.BER against iterative number of the ID algorithm.

Figure 9 .
Figure 9. BER against iterative number of the ID algorithm after 50-km SSMF transmission.