On the Achievable Spectral Efficiency of Non-Orthogonal Frequency Division Multiplexing

We investigate the spectral efficiency of non-orthogonal frequency division multiplexing (NOFDM). NOFDM is regarded as a frequency-domain faster-than-Nyquist (FTN) signaling with block transmission, where each subcarrier is compressed in the frequency domain by utilizing the discrete fractional Fourier transform (DFRFT). Since pulse shaping filters are not employed, the effective bandwidth of the NOFDM signal should take into account the resulting out-of-band (OOB) radiation. Therefore, we derive a closed-form expression for the power spectral density (PSD) of the NOFDM signal with windowing for OOB reduction. Through numerical comparison, it is shown that NOFDM outperforms OFDM in view of the effective signal bandwidth if they are compared under the same number of subcarriers. On the other hand, by the capacity analysis based on the asymptotic spectral efficiency with a large number of subcarriers, we elucidate that NOFDM may not outperform OFDM from an information-theoretic perspective.

On the Achievable Spectral Efficiency of Non-Orthogonal Frequency Division Multiplexing Yuto Hama , Member, IEEE, and Hideki Ochiai , Fellow, IEEE Abstract-We investigate the spectral efficiency of nonorthogonal frequency division multiplexing (NOFDM).NOFDM is regarded as a frequency-domain faster-than-Nyquist (FTN) signaling with block transmission, where each subcarrier is compressed in the frequency domain by utilizing the discrete fractional Fourier transform (DFRFT).Since pulse shaping filters are not employed, the effective bandwidth of the NOFDM signal should take into account the resulting out-of-band (OOB) radiation.Therefore, we derive a closed-form expression for the power spectral density (PSD) of the NOFDM signal with windowing for OOB reduction.Through numerical comparison, it is shown that NOFDM outperforms OFDM in view of the effective signal bandwidth if they are compared under the same number of subcarriers.On the other hand, by the capacity analysis based on the asymptotic spectral efficiency with a large number of subcarriers, we elucidate that NOFDM may not outperform OFDM from an information-theoretic perspective.

I. INTRODUCTION
A DRASTIC enhancement of spectral efficiency is urgently required in future wireless communications such as the sixth generation mobile communications standard (6G) [1].In contrast, the spectral efficiency close to the Shannon limit has been achieved by state-of-the-art physical layer techniques.In modern high speed wireless communication systems such as cellular networks and wireless LAN, the block transmission scheme based on orthogonal frequency division multiplexing (OFDM) has been a de-facto standard, partly due to its efficient implementation based on the fast Fourier transform (FFT).
In order to enhance the utilization of limited frequency resources, several non-orthogonal multi-carrier-based techniques have been proposed, such as non-orthogonal frequency division multiplexing (NOFDM) (i.e., non-orthogonal counterpart of OFDM) [2] and spectrally efficient frequency division multiplexing (SEFDM) [3].These schemes are based on frequency division multiplexing (FDM) with nonorthogonal subcarriers, where the adjacent subcarriers are allocated with a smaller spacing compared to OFDM [4].
Nearly five decades ago, faster-than-Nyquist (FTN) signaling was proposed for single-carrier transmission with pulse shaping filters, where each symbol is transmitted at faster than the Nyquist rate by intentionally violating the Nyquist criterion [5].FTN signaling compresses adjacent orthogonal resources in the time domain, whereas NOFDM performs that in the frequency domain.Therefore, NOFDM has also been regarded as a variant of FTN signaling [6].Furthermore, FTN can be simultaneously implemented in the time and frequency domain, and this approach is called multiple-carrier FTN (MC-FTN) [7].In this paper, these non-orthogonal resource allocation approaches performed in the time-frequency domain are collectively referred to as FTN.
The major challenge of FTN signaling is how to deal with the interference among non-orthogonal resources, i.e., intersymbol interference (ISI) in the time-domain FTN (TD-FTN) and inter-carrier interference (ICI) in the frequency-domain FTN (FD-FTN).Furthermore, the amount of ISI and ICI depends on the type of filters employed [8].Therefore, in the previous studies, the achievable performance of FTN signaling with a variety of pulse shaping filters has been investigated from an information-theoretic perspective [9], [10], [11], [12].In practice, the root raised cosine (RRC) filter is employed for single-carrier transmission with pulse shaping.In this context, FTN signaling achieves higher capacity compared to the classical Nyquist transmission with the same RRC filter except for the case without roll-off, i.e., the ideal rectangular filter [13].This result also holds for FD-FTN employing RRC filtering [14], [15] based on filter bank multiple carrier (FBMC).In addition, MC-FTN with RRC and Gaussian filters achieves higher capacity compared to the Nyquist transmission with the same pulse shaping filters [16].
The potential advantage of FTN signaling over the Nyquist schemes has been analyzed in terms of capacity [13].The case for FD-FTN signaling with RRC filters has been investigated in [15].It is important to note that the conventional analysis for the achievable spectral efficiency of FTN signaling is based on the excess bandwidth introduced by the pulse shaping filter.In practice, however, pulse shaping filters are not employed in most existing broadband wireless communication standards, such as cellular networks (4G and 5G) and wireless LAN, since they adopt block transmission based on OFDM signaling [6].Therefore, the above-mentioned analysis is not applicable to these block transmission systems since the spectrum is controlled by windowing rather than filtering [17].Nevertheless, to the best of the authors' knowledge, the information-theoretic analysis of the FTN signaling with block transmission over the conventional OFDM signal has not been well investigated.
Motivated by the above fact, in this work, we focus on FTN signaling with block transmission, which corresponds to FD-FTN of OFDM, and we refer to this simply as NOFDM 1 .NOFDM can be efficiently implemented by discrete fractional Fourier transform (DFRFT) instead of discrete Fourier transform (DFT) in OFDM systems [3].Since pulse shaping filters are not employed due to the block transmission, outof-band (OOB) radiation of the NOFDM signal degrades its spectral efficiency similar to OFDM, and thus windowing should be employed at the transmitter [17].Therefore, we evaluate the spectral efficiency of the NOFDM signal in terms of the effective bandwidth determined based on its power spectral density (PSD) by taking the effect of windowing into account.To this end, we derive the closed-form expression of the PSD for the NOFDM signal and analyze the effective signal bandwidth.Furthermore, we develop a capacity of the NOFDM signal as an asymptotic spectral efficiency with a large number of subcarriers, based on which we conclude that OFDM (or orthogonal subcarrier allocation) is in fact the optimal form of NOFDM.The primary contributions of this paper are summarized as follows: • We analyze the spectral efficiency of FTN signaling with practical block transmission in terms of the effective signal bandwidth achieved by windowing.• The closed-form PSD expressions of the NOFDM signal are derived for rectangular and raised-cosine windows, based on which the spectral efficiency of NOFDM system is derived by utilizing the eigenvalue decomposition (EVD) of the equivalent channel matrix of NOFDM.
• We demonstrate that NOFDM may outperform OFDM in terms of spectral efficiency in view of the effective bandwidth if they are compared under the same number of subcarriers, but we also show that its benefit is limited to the case with a small number of subcarriers.
• We analyze the capacity of the NOFDM signal derived from the asymptotic behavior of the spectral efficiency with a large number of subcarriers, and show that OFDM is the optimal form of NOFDM.This paper is organized as follows.In Section II, we describe the system model of the NOFDM signal.Section III focuses on the effective bandwidth of the NOFDM signal based on the PSD derived in closed-form.Theoretical analysis of NOFDM is given in Section IV, where the spectral efficiency and capacity are investigated and compared with OFDM.Finally, Section V concludes this work.

II. SYSTEM MODEL
In this section, we describe the system model of FTN signaling with block transmission using DFRFT.For the purpose of the mathematical analysis presented later, we introduce the system in a framework of the precoded NOFDM signal [18], and its block diagram is shown in Fig. 1.We note that the classical OFDM corresponds to the special case of NOFDM.
Throughout this work, we focus on the fundamental performance of the NOFDM signal over an additive white Gaussian noise (AWGN) channel.Thus, the insertion of the cyclic prefix (CP) that compensates for the ISI caused by multi-path fading is not considered.

A. Discrete Fractional Fourier Transform
DFRFT is a linear transform similar to DFT, and the kth output sample of N -point DFRFT with the input b = where a ∈ R denotes the order of DFRFT.It is obvious from (1) that DFRFT is a generalization of DFT as they coincide when a = 1.Therefore, DFRFT can also be implemented efficiently with its complexity order O (N log N ) [20].
We define the DFRFT matrix with the order a and size N denoted by F N,a ∈ C N ×N as where ω ≜ e −j2π/N .The inverse DFRFT (IDFRFT) is defined by changing the sign of the DFRFT order [21], and the IDFRFT matrix is given by X H representing Hermitian transpose of a matrix X.Since the DFT and IDFT matrices correspond to the special cases of the DFRFT and IDFRFT matrices with a = 1, we define their shorthand notations as H , respectively.

B. Transmitter
In this subsection, we describe the NOFDM transmitter.Its signal processing is regarded as a generalization of the conventional OFDM, and they can be extended by generalizing DFT in OFDM to DFRFT as illustrated in Fig. 1.
We first consider the conventional OFDM signal with the symbol period T and N subcarriers.The frequency interval between the adjacent subcarriers, commonly referred to as the subcarrier spacing (SCS), is given by and the orthogonality among subcarriers is guaranteed in OFDM.As a result, N data symbols are transmitted by the OFDM signal with N subcarriers during the symbol period T .Hence, the rate per each data symbol corresponds to the Nyquist interval, which is given by The sampling frequency is then expressed as whereas the corresponding Nyquist frequency is defined as On the other hand, the SCS of NOFDM is adjusted by introducing the compression factor denoted by α = (0, 1] ∈ R as T ∈ C N ×1 denote the data symbol vector with x n ∈ C representing the nth complex data symbol.For information-theoretic analysis, it is assumed to be an independent and identically distributed (i.i.d.) complex Gaussian random variable with the average symbol power E s , i.e., x n ∼ CN (0, E s ).Thus, its covariance matrix is given by E xx H = E s I N with I N representing the identity matrix of size N .The transmitted signal of NOFDM with N subcarriers is expressed as x n e j2πntα/T , where the symbol period of NOFDM remains the same as that of OFDM, i.e., T .
From ( 8), the average transmission power of the NOFDM signal is calculated by where by assumption E {x n x * m } takes E s when n = m and 0 otherwise.Hence, the transmission power of the NOFDM signal in (8) does not depend on the compression factor α.
We now consider the baseband symbol representation of ( 8).In the NOFDM signal, data symbols with N subcarriers are transmitted with the symbol interval of T regardless of the compression factor α. Hence, the rate per each data symbol corresponds to the Nyquist interval of (4).As a result, the sampling at the receiver is performed in the same principle as OFDM.Thus, the kth element of the discrete-time baseband signal discretized with the Nyquist interval of ∆t = T /N is expressed as From ( 1) and ( 10), the baseband symbol of the NOFDM signal with the compression factor α is generated by IDFRFT operation with the order of a = α [22].The transmitted with linear precoding is expressed as where P ∈ C N ×N denotes the precoding matrix and represents the precoded data symbol vector.

C. Receiver
The received signal over an AWGN channel is expressed as where n (t) is the AWGN term.The receiver first samples the received signal with the Nyquist interval ∆t = T /N , which is matched with the sampling rate of the baseband symbol at the transmitter.The kth sample of the received baseband signal is given by where n k ≜ n (k∆t) represents the kth noise sample with zero mean and complex variance of N 0 , i.e., n k ∼ CN (0, N 0 ). where is the AWGN vector.Therefore, from ( 9), the received signal-to-noise power ratio (SNR) denoted by γ s is defined as which does not depend on the compression factor α either.Similar to the conventional precoded OFDM signal, such as DFT-precoded OFDM [23], the receiver employs the postcoding after NOFDM demodulation performed by DFRFT operation.Let W ∈ C N ×N denote the postcoding matrix.The received symbol vector of the NOFDM signal after DFRFT operation and postcoding is expressed as where the equivalent channel matrix of NOFDM with the compression factor α and N subcarriers, referred to as the NOFDM matrix, is defined as The We observe that setting α = 1 (for OFDM), the NOFDM matrix reduces to the identity matrix with size N as
Let λ k denote the kth eigenvalue of the NOFDM matrix, and all the eigenvalues are assumed to be sorted in descending order with respect to the index k.Since the NOFDM matrix H N,α is Hermitian, i.e., (H N,α ) , it can be rewritten by EVD as where is the unitary matrix whose kth column u k is the eigenvector corresponding to λ k .From (20), the precoding and postcoding matrices based on EVD are given by Therefore, from ( 16), the resulting received symbol vector of EVD-precoded NOFDM signal is expressed as where denotes the equivalent noise vector.Its covariance matrix is given by Since the precoding matrix of EVD-precoded NOFDM signal is unitary from (21), the covariance matrix of the precoded data symbol vector x = Px in ( 11) is given by Therefore, EVD-precoding does not affect the resulting transmission power in (9).

III. POWER SPECTRAL DENSITY AND EFFECTIVE BANDWIDTH
The primary issue in block transmission of the NOFDM signal is that its power does not localize in the frequency domain since pulse shaping filters are not employed.It leads to high OOB radiation and degrades the spectral efficiency unless a suitable windowing is applied [17].Motivated by this fact, we define the effective bandwidth of the NOFDM signal for a given acceptable OOB radiation value.To this end, we derive closed-form expressions of the PSD for the NOFDM signal with and without windowing.Based on this PSD property, we evaluate the bandwidth of the NOFDM signal compared to the conventional OFDM signal.

A. Power Spectral Density
We first consider the power spectrum of the continuoustime signal so as to estimate the effective bandwidth of the NOFDM signal.
The transmitted NOFDM signal with windowing is expressed as where g(t) is a window function, T w (≥ T ) represents the total NOFDM symbol length with windowing, and s (ℓ) (t) denotes the ℓth NOFDM symbol.By writing the nth data symbol of the ℓth NOFDM symbol as x n , we have From ( 26), the autocorrelation function of the NOFDM signal is given by where R g (τ ) represents the autocorrelation function of the window function [17].It should be noted that EVD-precoding considered in this work does not affect the autocorrelation function since, from ( 25), E x(ℓ) n x(ℓ) * m in (28) takes the same value as that without precoding 2 , i.e., E x(ℓ) . Thus, the PSD of the NOFDM signal is calculated as where G (f ) is the Fourier transform (or spectrum) of g(t).
2 As in the conventional precoded FTN signaling, the power allocation may improve the spectral efficiency of the NOFDM signal since the eigenvalues of the NOFDM matrix are unbalanced except when N → ∞.However, it affects the resulting PSD property even under the constant transmission power condition [24] since the spectrum corresponding to the nth subcarrier depends on its allocated power, denoted by pn, due to E {xn x * n } = pnEs from the covariance matrix of the precoded data symbol vector given in (25).Moreover, its optimization should be performed for each combination of the number of subcarriers N and the compression factor α. Therefore, in this work, we do not pursue the power allocation optimization but leave it as our future work.Replacing the index k by k − (N − 1)/2 leads to the symmetric PSD expressed as In what follows, as a practical windowing approach, we consider the rectangular window and the raised-cosine window illustrated in Fig. 2 [25].
1) Rectangular Window: The block transmission without smooth windowing is equivalent to employing a rectangular pulse, and it is known as the rectangular window.As illustrated in Fig. 2(a), the symbol length with rectangular window T w is identical to that of the original NOFDM signal, i.e., T w = T .With the NOFDM symbol interval of T , it is given by Hence, its Fourier transform (or spectrum) is expressed as where From ( 30) and (32), the symmetric PSD of the NOFDM signal with rectangular window is given by 2) Raised-Cosine Window: In practical block transmission with the OFDM signal, smooth windowing is applied so as to mitigate the OOB radiation caused by the discontinuity of successive OFDM symbols.In this work, we consider the raised-cosine window (also known as Hann window) as an example [25], which is commonly employed in practical systems.
The NOFDM signal structure with raised-cosine window is illustrated in Fig. 2(b), where δT represents the transition period and δ denotes a fraction of the symbol length corresponding to the transition period.This transition period of δT is inserted at the head and tail of each NOFDM symbol.Hence, the resulting symbol length with raised-cosine window is given Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
by T w = T + 2δT .Its window function is expressed as [17] and its Fourier transform is given by Therefore, from (30) and (36), the symmetric PSD of the NOFDM signal with raised-cosine window is expressed as

B. Effective Bandwidth
In this subsection, we discuss the bandwidth of the NOFDM signal.From (34) and (37), we observe that the bandwidth corresponding to the sidelobe spreads without limitation.In other words, the energy of the NOFDM signal does not localize in the frequency domain.Therefore, in this work, we the bandwidth of the NOFDM signal with respect to the acceptable OOB radiation value, which we define as the effective bandwidth.
Let p target denote the target (or acceptable) PSD value for NOFDM systems.We define the maximum frequency f max as the maximum frequency that exceeds this target PSD p target derived from (34) and (37).It is expressed as where R + represents a set of real positive numbers.By the use of the maximum frequency f max , we define the effective bandwidth of the NOFDM signal for a given PSD value p target as

C. Spectrum of Baseband NOFDM Signal
We consider the spectrum of discrete-time NOFDM signal.Fig. 3 illustrates the spectrum of the baseband symbol of OFDM and NOFDM signals sampled at the Nyquist interval of ∆t = T /N in (10).By the sampling theorem, the spectrum of the baseband NOFDM signal should be periodic, and its period corresponds to the sampling frequency f s .In Fig. 3, W main denotes the bandwidth corresponding to the mainlobe in the NOFDM signal with the compression factor α. It is generally defined as where f c,k represents the central frequency of the kth subcarrier in the NOFDM signal, which is given by It is clear from (40) that the bandwidth of the mainlobe decreases as the compression factor α decreases.Hence, it is considered in the previous studies on FD-FTN signaling that its bandwidth can be reduced arbitrarily by compressing the subcarrier spacing [3], [15], [26].However, this often misleads the actual spectral efficiency of FD-FTN signaling since, in practice, the effect of sampling should be taken into account.In the NOFDM signal, the sampling frequency and the Nyquist frequency are given by f s = N/T in (5) and (6), respectively, regardless of the compression factor α, as its baseband symbol is sampled at the Nyquist interval of ∆t = T /N so as to detect the symbols corresponding to all subcarriers.Therefore, the sampling rate is always larger than the bandwidth corresponding to the mainlobe for the NOFDM signal, i.e., f s ≥ W main due to the fact that 0 < α ≤ 1.As a result, the spectrum of the baseband NOFDM signal with α < 1 contains its sidelobe below the Nyquist frequency as illustrated in Fig. 3, since the sidelobe cannot be removed in the NOFDM signal due to the block transmission structure.In other words, the bandwidth of the NOFDM signal should be determined not by the compression factor α but by the sampling frequency f s = N/T , and its bandwidth should never be smaller than the sampling frequency f s since the sidelobe is still a part of the baseband signal.This fact indicates that if there exists any interference in the range of |f | ≤ f n , it may not be negligible, even if it falls outside the mainlobe.
In summary, the main difference between the NOFDM and OFDM signals is that the mainlobe of the NOFDM signal occupies a bandwidth smaller than that of OFDM.Therefore, it is important to note that the effective bandwidth of the NOFDM signal, according to the definition of (39), is no smaller than the sampling frequency f s , even though its mainlobe can be made narrower by controlling α.

D. Numerical Results
Without loss of generality, we introduce the normalized frequency, denoted by f , defined as and the corresponding normalized Nyquist frequency is introduced as Fig. 4 shows the PSD of the NOFDM signal with rectangular window as a function of f , where the compression factor is set as α = 0.25, 0.5, 0.75, and 1 (OFDM), and the number of subcarriers is set as N = 256.From Fig. 4, the bandwidth corresponding to the mainlobe of the NOFDM signal becomes narrower by reducing the compression factor α according to (40).As a result, OOB radiation is also improved.The maximum frequency f max in (38) and corresponding effective bandwidth W eff in (39) are also illustrated in Fig. 4 for the target PSD of p target = −30 dB, where we observe that f max of the NOFDM signal becomes smaller than that of the OFDM signal evaluated at the same value of p target .It follows that the effective bandwidth can be improved by NOFDM.On the other hand, in the case of α = 0.25, the maximum frequency f max is smaller than the Nyquist frequency f n for the target PSD value of p target = −30 dB.Thus, further reduction of the compression factor α does not provide any gain in terms of the effective bandwidth W eff for p target = −30 dB.
We next evaluate the PSD of OFDM and NOFDM (α = 0.5) signals achieved by the different windows in Fig. 5, where rectangular and raised-cosine windows are compared for N = 256.Here, we set the transition period of raised-cosine window as δ = 1/64.As a result, raised-cosine window significantly improves OOB radiation compared to rectangular window for both OFDM and NOFDM signals.
Based on the above observation on the PSD, we evaluate the bandwidth with respect to the compression factor α. The effective bandwidth W eff in (39) and maximum frequency f max in (38) are plotted for the normalized frequency f in Fig. 6, where the cases with N = 128, 256, and 512 are evaluated.For reference, the normalized Nyquist frequency fn in (43) and the normalized sampling frequency fs = f s /f s = 1 are also shown in the same figure.In all cases compared in Fig. 6, the effective bandwidth achieved by raised-cosine window is smaller than that with rectangular window, since OOB radiation is improved as shown in Fig. 5. From (39), when f max > f n , the effective bandwidth of the NOFDM signal is twice as large as the maximum frequency, i.e., W eff = 2f max .Since this condition is satisfied for all α in the case of rectangular window with N = 128, the resulting effective bandwidth can be improved as the compression factor decreases.In contrast, the maximum frequency becomes lower than the Nyquist frequency with the small compression factor for the other cases.As a result, the effective bandwidth is determined by the sampling frequency as the compression factor decreases.Therefore, the optimal compression factor may exist for each number of subcarriers N and each window.

IV. SPECTRAL EFFICIENCY AND CAPACITY
Based on the discussion on the bandwidth of the NOFDM signal in the previous section, we derive the achievable spectral efficiency and capacity based on EVD-precoding.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 6.The effective bandwidth and maximum frequency of the NOFDM signal with respect to the normalized frequency f , which are plotted by solid and dashed lines, respectively.The number of subcarriers is set as N = 128, 256, and 512.

A. Mutual Information
From (23), the kth received symbol of EVD-precoded NOFDM signal is expressed as where z k is an i.i.d.Gaussian random variable with zero mean and complex variance of λ k N 0 from (24), i.e., z k ∼ CN (0, λ k N 0 ).Therefore, the NOFDM signal with N subcarriers can be divided into N parallel complex Gaussian channels, and the equivalent SNR corresponding to the kth NOFDM channel is given by The mutual information between the data symbol vector x ∈ C N ×1 and the received symbol vector y ∈ C N ×1 of the NOFDM signal in ( 23) is expressed as where h (•) represents the differential entropy.For mathematical analysis, we assume that each data symbol x k is an i.i.d.complex Gaussian random variable, i.e., x k ∼ CN (0, E s ).
Hence, the mutual information in (46) can be rewritten as Since the diagonal element of the NOFDM matrix is given by H N,α [k, k] = 1 from ( 18), the sum of the eigenvalues is expressed as where tr (X) represents the trace of a matrix X.Therefore, from Jensen's inequality and (48), the mutual information of the NOFDM signal in (47) can be bounded as where the equality holds if and only if the compression factor is chosen as α = 1 from (19).Therefore, NOFDM with α < 1 is inferior to OFDM (α = 1) in terms of mutual information.

B. Spectral Efficiency
We develop the achievable spectral efficiency of the NOFDM signal considering its PSD.The spectral efficiency of the NOFDM signal is expressed from (47) as where W eff is the effective bandwidth of the NOFDM signal for a given target PSD value p target defined in (39).In the case of raised-cosine window, the time duration T in (50) is replaced by (1 + δ)T since the support of g(t) in ( 35) is (0, T + 2δT ) and the tail part of δT overlaps with the head part of the next NOFDM symbol [17].Hence, the rate loss by raised-cosine window is expressed as T / (T + 2δT − δT ) = 1/ (1 + δ).
When α = 1, since λ k = 1 holds for all k ∈ {0, 1, • • • , N − 1} from (19), the spectral efficiency of the OFDM signal is derived as As demonstrated in Section III, NOFDM can be made superior to OFDM in terms of the effective bandwidth compared at the same target PSD value.Hence, there may be the case that NOFDM outperforms OFDM in terms of spectral efficiency, whereas it is not advantageous in terms of mutual information as described in the previous subsection.

C. Capacity
Finally, we investigate the asymptotic behavior of the spectral efficiency of the NOFDM signal with a large number of subcarriers.In this work, we define the capacity of block transmission systems as the asymptotic spectral efficiency achieved by N → ∞.Since the normalized spectrum of the NOFDM signal is strictly limited to [−α/2, α/2] (see Appendix A), the NOFDM matrix H N,α with N → ∞ has only ⌈αN ⌉ eigenvalues of 1/α, while the remaining eigenvalues approach zero [27].From (50), the capacity of the NOFDM signal is expressed in terms of α as where Winf ≜ lim N →∞ (W eff /f n ) = 1 denotes the normalized bandwidth with rectangular window when N → ∞ (see Appendix A) 3 .By substituting α = 1 into (52), the capacity of the OFDM signal agrees with the Shannon capacity: As a result, the capacity of the NOFDM signal in (52) takes the maximum value of (53) when the compression factor is α = 1.In other words, the capacity of the OFDM signal is identical to the Shannon capacity, which cannot be achieved by the NOFDM signal with α < 1.

D. Numerical Results
We evaluate the spectral efficiency of the NOFDM signal derived in (50) compared to that of OFDM in (51).The results Fig. 7.The spectral efficiency of NOFDM signal with respect to the compression factor α, which is evaluated for the target PSD value ptarget = −30 dB and the SNR of γs = 0, 10, and 20 dB.The number of subcarriers is set as N = 128, 256, and 512.The Shannon capacity and NOFDM capacity are also plotted by chain and solid lines, respectively.with rectangular and raised-cosine (δ = 1/64) window are shown in Fig. 7 evaluated for the target PSD of p target = −30 dB at the different SNR of γ s = 0, 10, and 20 dB, and the number of subcarriers is set to be N = 128, 256, and 512.As a reference, NOFDM capacity in (52) and the Shannon capacity in (53) are also plotted in the same figure.The smaller compression factor α improves the effective bandwidth given by (39), but it degrades the mutual information derived in (47).As a result, as observed from Fig. 7, the optimal compression factor exists for each parameter, i.e., the number of subcarriers N and the received SNR γ s .Furthermore, the spectral efficiency achieved with raised-cosine window significantly outperforms that with rectangular window compared at the same number of subcarriers.It should be noted that the optimal compression factor also depends on the type of window employed at the transmitter.In both windowing approaches, the spectral efficiency η NOFDM approaches the capacity C NOFDM (α) as N increases according to (52).We finally evaluate the spectral efficiency of the NOFDM signal with respect to γ s .Fig. 8 shows the spectral efficiency and capacity of OFDM and NOFDM signals with rectangular window, where we set the number of subcarriers as N = 64, 256, and 1024.From Fig. 7, the compression factor takes different optimal values depending on the parameters.Thus, we evaluate the case with α = 0.92 in Fig. 8 for the NOFDM signal as an example.The corresponding NOFDM capacity in (52) with α = 0.92 is also plotted along with the Shannon capacity in (53).As the number of subcarriers N increases, the spectral efficiency of both OFDM and NOFDM improves since the OOB radiation can be decreased with the number of subcarriers.In contrast, the superiority of NOFDM over OFDM decreases as N increases.For example, the spectral efficiency of NOFDM is 1.24 times higher than that of OFDM compared with N = 64, but it becomes 1.07 times when N = 1024.Furthermore, the spectral efficiency of the NOFDM signal almost approaches its capacity when N = 1024, and its further improvement may not be expected.The spectral efficiency in the case of raised-cosine window with transition period δ = 1/64 is compared in Fig. 9.For all cases compared here, the spectral efficiency improves by raisedcosine window due to lower OOB radiation shown in Fig. 5.However, the advantage of NOFDM over OFDM is decreased even with N = 64 and 256.Furthermore, when N = 1024, the spectral efficiency of NOFDM with α = 0.92 is inferior to OFDM, since OFDM achieves spectral efficiency higher than the NOFDM capacity with α = 0.92.As a result, the optimal compression factor with raised-cosine window should be larger in this case.
In conclusion, NOFDM may have the potential advantage over OFDM in terms of spectral efficiency when they are compared with the same small number of subcarriers since non-orthogonal subcarrier allocation may reduce the effective bandwidth defined by (39) at the expense of the mutual information in (46).However, the gain achieved by non-orthogonal subcarrier allocation decreases as the number of subcarriers The spectral efficiency of NOFDM signal (α = 0.92) with raised-cosine window (δ = 1/64) for the target PSD value ptarget = −30 dB compared to that of OFDM signal, where the number of subcarriers is set as N = 64, 1024.The Shannon capacity and NOFDM capacity are also plotted.
increases.Furthermore, OFDM (α = 1) is superior to NOFDM with α < 1 in terms of mutual information and capacity.In other words, OFDM is the optimal form of NOFDM from the viewpoint of information-theoretic analysis 4 .
Remark: The gain of NOFDM in terms of spectral efficiency can be achieved when we consider the degradation of spectral efficiency caused by OOB radiation associated with block transmission [17].However, the performance of NOFDM is bounded by the Shannon capacity in terms of both spectral efficiency in (50) and capacity in (52), as demonstrated in Figs. 8 and 9. Therefore, our derived results do not contradict the existing analysis for FTN signaling with pulse shaping filters [13].As a reference, the connection between our analysis of the NOFDM signal and that of conventional FTN signaling is described in Appendix B.

V. CONCLUSION
In this work, we have investigated the achievable spectral efficiency of the NOFDM signal, which is defined as the frequency-domain FTN signal with block transmission generated by DFRFT.To this end, we have derived the closed-form expressions of the PSD for the NOFDM signal with rectangular and raised-cosine windows, and the achievable spectral efficiency has been developed by utilizing EVD precoding.As a result, NOFDM can achieve higher spectral efficiency compared to OFDM with the same number of subcarriers.On the other hand, by the capacity analysis of the NOFDM signal, we have elucidated the fact that OFDM turns out to be optimal when the number of subcarriers is large.
The optimal power allocation should be investigated for the NOFDM signal with the limited number of subcarriers and left as our future work.(i.e., pulse shaping filter), reducing the symbol spacing controlled by τ will also improve the overall efficiency as the guard time between the successive frames (i.e., the effective frame duration T eff ) can be reduced.In conclusion, by switching the domains, our analysis and results are equally applicable to conventional FTN signaling with pulse shaping filters.

Fig. 1 .
Fig. 1.A block diagram of the NOFDM transmitter and receiver with the compression factor α.

Fig. 3 .
Fig. 3.The spectrum of baseband OFDM and NOFDM signals sampled at the sampling frequency of fs = N/T .

Fig. 4 .
Fig. 4. The PSDs of OFDM (α = 1) and NOFDM (α = 0.25, 0.5, 0.75) signals with respect to the normalized frequency f , where the number of subcarriers is set as N = 256.Note that rectangular window is employed for all the signals evaluated here.

Fig. 5 .
Fig. 5.The PSDs of OFDM (α = 1) and NOFDM (α = 0.5) signals with respect to the normalized frequency f , where the number of subcarriers is as N = 256.The results with rectangular and raised-cosine (δ = 1/64) windows are plotted by dashed and solid lines, respectively.

Fig. 8 .
Fig.8.The spectral efficiency of NOFDM signal (α = 0.92) with rectangular window for the target PSD value ptarget = −30 dB compared to that of OFDM signal, where the number of subcarriers is set as N = 64, 256, and 1024.The Shannon capacity and NOFDM capacity are also plotted.

Fig. 9 .
Fig.9.The spectral efficiency of NOFDM signal (α = 0.92) with raised-cosine window (δ = 1/64) for the target PSD value ptarget = −30 dB compared to that of OFDM signal, where the number of subcarriers is set as N = 64, 1024.The Shannon capacity and NOFDM capacity are also plotted.

Fig. 10 .
Fig. 10.NOFDM signal with N subcarriers compressed by a factor of α for the orthogonal frequency spacing of 1/T : (a) time domain window and (b) power spectrum, and conventional time-domain FTN signaling with N symbols compressed by a factor of τ for the orthogonal time spacing of T : (c) power spectrum and (d) time-domain frame structure.