FTN-NOFDM: Faster-Than-Nyquist Multicarrier

• FTN-NOFDM is a non-orthogonal multicarrier signaling technique that compresses subcarrier spacing below the Nyquist rate, increasing spectral efficiency while introducing controlled inter-carrier interference.
• The FrCT-NOFDM model employs fractional cosine transforms to achieve the faster-than-Nyquist operation, enabling accelerated symbol rates and necessitating advanced receiver designs for interference cancellation.
• Practical implementations in visible light and coherent optical communications highlight the tradeoff between spectral compactness and interference, driving ongoing research in detection and channel estimation techniques.

Searching arXiv for FTN-NOFDM papers to ground the article in primary sources. Faster-than-Nyquist non-orthogonal frequency-division multiplexing (FTN-NOFDM) is a class of non-orthogonal multicarrier signaling schemes in which adjacent subcarriers are packed more densely than the orthogonal or Nyquist spacing, so that the signal occupies less bandwidth or, equivalently, transmits at a rate faster than the Nyquist rate associated with the occupied bandwidth. In the formulation developed for fractional cosine transform-based NOFDM (FrCT-NOFDM), FTN operation is obtained by compressing the subcarrier spacing with a bandwidth compression factor $\alpha<1$, which increases spectral efficiency but destroys subcarrier orthogonality and introduces inter-carrier interference (ICI) (Zhou et al., 2016). The topic sits at the intersection of classical faster-than-Nyquist signaling, spectrally efficient multicarrier modulation, and practical receiver design for non-orthogonal systems. In this literature, FTN-NOFDM is motivated both by bandwidth-limited channels and by applications such as visible light communications and coherent optical transmission, where moderate bandwidth compression can trade manageable ICI for improved spectral compactness (Zhou et al., 2017, Wang et al., 24 Aug 2025).

1. Definition and conceptual scope

FTN signaling was originally framed as transmitting symbols at intervals shorter than the Nyquist interval, thereby introducing intentional inter-symbol interference. In the multicarrier setting, the analogous mechanism is frequency-domain packing: instead of reducing pulse intervals in time, one narrows the subcarrier spacing below the orthogonal spacing used by conventional OFDM, which produces a non-orthogonal frequency-division multiplexing waveform (Zhou et al., 2016). The defining FTN criterion in this context is explicit: only when the compressed subcarrier spacing is sufficiently small that the resulting symbol rate exceeds the Nyquist rate associated with the occupied bandwidth does NOFDM become FTN-NOFDM (Zhou et al., 2016).

A useful contrast with related terminology is important. Some papers use FTN in a broad sense to describe any intentional violation of orthogonality for spectral-efficiency gain, while others reserve FTN-NOFDM for frequency-domain subcarrier compression specifically. The latter is the meaning developed in the FrCT-NOFDM capacity work, where the dominant impairment is ICI rather than the inter-symbol interference typical of time-packed single-carrier FTN (Zhou et al., 2016). This suggests that FTN-NOFDM is best understood as the frequency-domain analogue of classical FTN, whereas time-packed OFDM-based FTN systems belong to a related but distinct branch of the literature (Chadaga, 23 Sep 2025).

The broader FTN literature repeatedly warns that denser packing alone does not guarantee a genuine increase in information rate per unit time or bandwidth unless localization, power normalization, and the geometry of the non-orthogonal signal set are treated carefully (Gattami et al., 2015, Hefnawy et al., 2014). That caution is directly relevant to FTN-NOFDM, because many of its apparent gains depend on how bandwidth and power are defined and on whether interference is treated as noise or is actively mitigated.

2. Signal construction and the FrCT-NOFDM model

A canonical FTN-NOFDM realization is fractional cosine transform-based NOFDM. In this construction, the $N$-point inverse fractional cosine transform generates the time-domain signal as

$x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$

and the forward FrCT is

$X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$

with

$$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$

These equations define the FrCT-NOFDM modulator/demodulator pair, and the basis functions are

$\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$

When $\alpha=1$, the transform reduces to the ordinary Type-II DCT and the system is orthogonal; when $\alpha<1$, the subcarrier frequencies are compressed and the basis is no longer orthogonal (Zhou et al., 2016).

The bandwidth compression factor $\alpha$ directly scales the subcarrier spacing. For DCT-OFDM,

$\Delta f_{\text{DCT-OFDM}}=\frac{1}{2T},$

while for FrCT-NOFDM,

$N$0

Thus, $N$1 compresses the spacing below the orthogonal value and is the source of the non-orthogonality (Zhou et al., 2016). For large $N$2, the baseband bandwidth is approximated as

$N$3

the Nyquist rate for that occupied bandwidth is

$N$4

and the FrCT-NOFDM symbol rate is

$N$5

Hence

$N$6

so the multicarrier transmission rate is accelerated by the factor $N$7 relative to the Nyquist rate associated with the compressed bandwidth (Zhou et al., 2016). The authors state that when $N$8, the subcarrier spacing is equal to $N$9 of the symbol rate per subcarrier and the transmission rate is about $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$0 faster than Nyquist rate (Zhou et al., 2016).

A distinct but related construction appears in optical FTN-NOFDM for coherent transmission, where the compression factor is written as a rational number

$x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$1

and the signal is generated through an inverse fractional Fourier transform representation that can be rewritten as a sparse $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$2-point IFFT. In that formulation, FTN-NOFDM is implemented by placing active subcarriers every $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$3 bins inside a $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$4-point grid, then exploiting pruning to reduce the FFT cost (Wang et al., 24 Aug 2025). This suggests an architectural split in the literature between transform-defined FrCT systems and FFT-compatible rational-compression implementations.

3. Orthogonality loss, interference structure, and detection

The mathematical signature of FTN-NOFDM is the loss of subcarrier orthogonality. In FrCT-NOFDM this is characterized by the $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$5 correlation matrix

$x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$6

This matrix is the Gram matrix of the FrCT basis. When $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$7, $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$8; when $x_n=\sqrt{\frac{2}{N}\sum_{k=0}^{N-1}W_k X_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$9, off-diagonal terms become nonzero, so distinct subcarriers interfere (Zhou et al., 2016). The literature notes that adjacent subcarriers contribute more interference than distant ones and that ICI worsens as $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$0 decreases (Zhou et al., 2016).

The corresponding discrete input-output model can be written as

$X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$1

where $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$2 is the transmitted symbol vector, $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$3 is the received transform output, and $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$4 is AWGN (Guo et al., 2018). This formal resemblance to an $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$5 MIMO channel motivates receiver designs imported from lattice detection and MIMO search methods.

A widely studied detector is iterative interference cancellation based on the correlation matrix. In one formulation, the update is

$X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$6

where $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$7 is the received symbol vector, $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$8 is the correlation matrix, $X_k=\sqrt{\frac{2}{N}\,W_k\sum_{n=0}^{N-1}x_n \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right),$9 is the identity matrix, and $$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$0 is the estimate at iteration $$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$1. For 2-PAM, the detector initializes $$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$2, applies thresholded decisions each iteration, and updates the threshold as

$$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$3

with $$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$4 the total number of iterations (Zhou et al., 2017). The same basic cancellation idea reappears in coherent optical FTN-NOFDM, where the equalized symbol vector is refined by subtracting the off-diagonal interference term $$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$5, and an LDPC-assisted version replaces the hard-decision reconstruction with a remapped vector from LDPC decoding (Wang et al., 24 Aug 2025).

For stronger compression or higher-order modulation, sphere decoding becomes relevant. In FrCT-based FTN-NOFDM, maximum-likelihood detection is posed as

$$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$6

and sphere decoding reduces the search to a hypersphere (Guo et al., 2018). A sphere decoder with box optimization tightens pruning by adding a lower-bound term to the partial metric and is reported to preserve the BER performance of conventional sphere decoding while significantly reducing the average number of expanded nodes, especially for high-order modulation (Guo et al., 2018). This suggests that FTN-NOFDM detection has evolved along the same path as overloaded MIMO detection: exact ML formulations exist, but practical value depends on structured complexity reduction.

The interference model also matters for information-theoretic treatment. The FrCT capacity paper observes that the aggregate ICI is approximately Gaussian and independent of AWGN, which supports treating residual ICI as an additional noise term in the capacity bound (Zhou et al., 2016). That approximation underlies several later interpretations of achievable performance.

4. Capacity limit and information-theoretic interpretation

A central theoretical result for FTN-NOFDM is the explicit capacity-limit expression derived for FrCT-NOFDM. Starting from Shannon’s AWGN benchmark,

$$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$7

the derivation counts the number of independent amplitudes in bandwidth $$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$8 over duration $$W_k= \begin{cases} \frac{1}{\sqrt{2}}, & k=0,\ 1, & k=1,2,\dots,N-1. \end{cases}$$9 using the compressed subcarrier spacing

$\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$0

This yields

$\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$1

signal-space dimensions, leading through a sphere-packing argument to the idealized FTN-NOFDM limit

$\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$2

When residual ICI is modeled as Gaussian and independent of AWGN, the practical expression becomes

$\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$3

The $\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$4 prefactor is the mathematical expression of the denser packing: the number of signaling dimensions in the same bandwidth-duration product increases by the factor $\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$5 (Zhou et al., 2016).

This result is theoretically important because it makes explicit the condition under which FTN-NOFDM can exceed the conventional Nyquist benchmark. Comparing

$\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$6

with the FTN-NOFDM expression shows that the gain from $\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$7 must outweigh the penalty from $\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$8. In the ideal case $\phi_k(n)=\sqrt{\frac{2}{N}\,W_k \cos\!\left(\frac{\pi \alpha (2n+1)k}{2N}\right).$9, any $\alpha=1$0 gives a larger upper bound; in practice, the useful operating region is limited by how well interference is canceled (Zhou et al., 2016).

The broader FTN literature complicates the interpretation. A study of time-domain FTN emphasizes that more densely packed non-orthogonal dimensions do not by themselves imply a real increase in information rate per unit time or bandwidth unless localization and signal geometry are handled carefully (Gattami et al., 2015). A spectrum-sharing study reaches a related conclusion in a multi-access setting: when spectral efficiency is defined as sum rate per total occupied bandwidth, Shannon sinc pulses maximize efficiency, and for root-raised-cosine spectra the efficiency decreases monotonically with roll-off once adjacent-band interference is included (Hefnawy et al., 2014). These results do not invalidate the FTN-NOFDM capacity derivation, but they indicate that comparisons are sensitive to bandwidth accounting, interference modeling, and whether interference is treated as structured or as noise.

A later FTN survey frames the issue more generally: FTN should be understood as deliberately violating orthogonality to increase packing density, whether in time or in alternative domains such as spectrally efficient frequency division multiplexing (Li et al., 29 Dec 2025). This suggests that the FTN-NOFDM capacity expression is best read as a model-specific upper limit for non-orthogonal multicarrier packing under its stated assumptions, not as a universal guarantee that any compressed-subcarrier implementation outperforms orthogonal OFDM.

5. Implementations and application domains

Visible light communications

FrCT-NOFDM was proposed for visible light communications precisely because it is real-valued and can therefore be directly applied to IM/DD systems without the expensive upconversion required by common FrFT-based NOFDM signals (Zhou et al., 2017). In that work, the transmitted electrical waveform is made unipolar through single-sided clipping with DC bias,

$\alpha=1$1

with

$\alpha=1$2

The signal is modeled over diffuse VLC channels such as the ceiling-bounce model, and the paper reports that for a 100 Mbit/s system the occupied bandwidth decreases from $\alpha=1$3 MHz for DCT-OFDM to $\alpha=1$4, $\alpha=1$5, and $\alpha=1$6 MHz for $\alpha=1$7, $\alpha=1$8, and $\alpha=1$9, respectively (Zhou et al., 2017). In simulation, FrCT-NOFDM with $\alpha<1$0 and $\alpha<1$1 reaches the $\alpha<1$2 FEC limit at 22 dB SNR, whereas DCT-OFDM reaches it at 24.2 dB, a reported $\alpha<1$3 dB improvement attributed to reduced high-frequency distortion in the bandwidth-limited VLC channel (Zhou et al., 2017). The same study reports that when the transmitter uses $\alpha<1$4, BER becomes worse than the $\alpha<1$5 FEC limit if the receiver mismatch exceeds $\alpha<1$6, which is interpreted as a physical-layer security sensitivity to the compression parameter (Zhou et al., 2017).

Coherent optical transmission

In long-haul coherent optics, FTN-NOFDM is proposed as a way to improve spectral efficiency for single-wavelength 400G transmission while keeping low-order modulation. The subcarrier number is set to eight to enable low-complexity signal generation using a pruned inverse FFT and manageable ICI cancellation (Wang et al., 24 Aug 2025). A frequency tone-based timing recovery is introduced because conventional timing recovery fails on FTN-NOFDM; the timing-error metric is derived from spectral components around the inserted tone, and a notch filter removes the tone afterward (Wang et al., 24 Aug 2025). The receiver then uses a $\alpha<1$7 time-domain MIMO equalizer, conventional iterative detection, and LDPC-assisted iterative detection to refine the ICI reconstruction (Wang et al., 24 Aug 2025).

Experimentally, FTN-NOFDM, PCS-OFDM, and QPSK-OFDM are compared in a 400G coherent optical system over 11 cascaded 125-GHz WSSs and 2000 km transmission. The FTN-NOFDM configuration uses $\alpha<1$8, a net spectral efficiency of about $\alpha<1$9, and eight subcarriers (Wang et al., 24 Aug 2025). The study reports that FTN-NOFDM exhibits comparable WSS filtering tolerance to PCS-OFDM and superior nonlinearity tolerance, while PCS-OFDM achieves the best BER performance (Wang et al., 24 Aug 2025). This suggests that FTN-NOFDM is practically attractive in coherent optical links when spectral compactness and nonlinear tolerance are prioritized over minimum BER at a fixed OSNR.

IM-DD optical links

A different optical branch considers adaptive multi-band modulation in NOM-p-based FTN-NOFDM for IM-DD links. There, the single FTN band is divided into $\alpha$0 sub-bands, each compressed by

$\alpha$1

and assigned descending QAM orders according to the low-pass-like channel response,

$\alpha$2

In a 32.23-Gb/s, 20-km IM-DD experiment, the proposed three-band design reduces BER from $\alpha$3 to $\alpha$4 at $\alpha$5 dBm received optical power and cuts complexity by $\alpha$6 relative to the conventional single-band scheme (Song et al., 2023). This suggests that FTN-NOFDM can benefit from coarse channel-aware modulation partitioning as well as from pure bandwidth compression.

6. Performance tradeoffs, misconceptions, and research directions

The literature is consistent on one point: FTN-NOFDM is governed by a bandwidth-saving versus interference tradeoff. Moderate compression can improve spectral efficiency and, in bandwidth-limited channels, even BER; excessive compression produces too much residual ICI. This pattern appears in multiple domains. In VLC, $\alpha$7 and $\alpha$8 outperform orthogonal DCT-OFDM, while $\alpha$9 cannot reach the $\Delta f_{\text{DCT-OFDM}}=\frac{1}{2T},$0 FEC limit even at 30 dB SNR (Zhou et al., 2017). In FrCT-based FTN-NOFDM with sphere decoding, QPSK at $\Delta f_{\text{DCT-OFDM}}=\frac{1}{2T},$1 is reported to be $\Delta f_{\text{DCT-OFDM}}=\frac{1}{2T},$2 faster than the Nyquist rate and to have almost the same performance as OFDM, while more aggressive compression is harder to detect (Guo et al., 2018). In long-haul coherent optics, $\Delta f_{\text{DCT-OFDM}}=\frac{1}{2T},$3 emerges as an experimentally favorable operating point (Wang et al., 24 Aug 2025).

A common misconception is that FTN-NOFDM necessarily “breaks Shannon.” The available work does not support that claim. The capacity-limit expression for FrCT-NOFDM is derived within Shannon-style sphere packing and includes the interference penalty explicitly (Zhou et al., 2016). At the same time, other FTN analyses show that careless reasoning about densely packed signals can appear to violate Shannon bounds when localization or total occupied bandwidth are not treated correctly (Gattami et al., 2015). A plausible implication is that FTN-NOFDM gains are real only under fair comparisons that include interference mitigation cost, true occupied bandwidth, and consistent power definitions.

Another misconception is that the only issue is detection complexity. Complexity is central, but not sufficient as a summary. The general FTN literature shows that acceleration or compression beyond a threshold may saturate spectral-efficiency gains under fixed transmit-power constraints and may cause severe peak-to-average power or effective SNR problems under alternative SNR definitions (Zhang et al., 2024, Zhang et al., 17 Jun 2026). Although those results are derived for time-packed FTN rather than compressed-subcarrier NOFDM, the underlying warning is directly relevant: extreme non-orthogonal packing can become useless or impractical once power normalization and implementation constraints are accounted for.

The most active technical directions implied by the current literature are receiver design, channel estimation, and architecture adaptation. Sphere decoding with box optimization addresses high-order detection complexity in AWGN FrCT-NOFDM (Guo et al., 2018). LDPC-assisted iterative detection improves optical ICI mitigation (Wang et al., 24 Aug 2025). For FTN more broadly, superimposed-pilot channel estimation, joint channel estimation and data detection, and spectral interference alignment have been developed in single-carrier FTN systems and appear conceptually adaptable to FTN-NOFDM, especially in doubly selective or low-overhead regimes (Keykhosravi et al., 21 Mar 2025, Wu et al., 20 Mar 2026). This suggests that future FTN-NOFDM research will likely converge on joint estimation-detection architectures rather than standalone interference cancellers.

A final structural point is that FTN-NOFDM is no longer a purely theoretical curiosity. The literature now spans explicit capacity expressions (Zhou et al., 2016), complexity-reduced near-ML detection (Guo et al., 2018), VLC-oriented real-valued transform implementations (Zhou et al., 2017), IM-DD optical adaptation (Song et al., 2023), and long-haul coherent optical experiments (Wang et al., 24 Aug 2025). The open question is not whether compressed-subcarrier non-orthogonal multicarrier signaling is possible, but under which channel, power, and DSP constraints it provides a better end-to-end tradeoff than orthogonal OFDM or shaped OFDM alternatives.