Generalisations: SC-FDMA, FBMC, OTFS

SC-FDMA, also called DFT-spread OFDM (DFT-s-OFDM), adds a DFT precoding stage before the standard OFDM IDFT. The transmitter architecture is:

1. Map $M$ data symbols $\{d[0], \ldots, d[M-1]\}$ to a block.
2. Apply an $M$-point DFT: $D[k] = \text{DFT}_M\{d[m]\}$.
3. Map the $M$ DFT outputs to $M$ of the $N$ OFDM subcarriers (localised or distributed mapping).
4. Apply the $N$-point IDFT to generate time-domain samples.
5. Add cyclic prefix and transmit.

Because the transmitted signal passes through a DFT before the IDFT, the time-domain waveform resembles a single-carrier signal with much lower PAPR than conventional OFDM. SC-FDMA is used for the uplink in LTE and remains an option in 5G NR.

FBMC replaces the rectangular pulse shape of each OFDM subcarrier with a well-localised prototype filter $p(t)$ that has much better frequency confinement. The most common variant is FBMC/OQAM (offset QAM), which transmits real-valued symbols staggered by half a symbol period to maintain orthogonality:

$$x(t) = \sum_{k=0}^{N-1} \sum_{n} a_{k,n}\, p(t - nT/2)\, e^{j2\pi k \Delta f\, t}\, e^{j\phi_{k,n}}$$

where $a_{k,n} \in \mathbb{R}$ and $\phi_{k,n}$ provides a phase offset pattern. Key advantages:

• No cyclic prefix needed — the filter provides sufficient time-domain containment.
• Better spectral containment — dramatically lower out-of-band emissions compared to OFDM.
• Higher spectral efficiency — no CP overhead.

However, FBMC's real-orthogonality (instead of complex) complicates MIMO and channel estimation.

OTFS modulates data symbols in the delay-Doppler domain rather than the time-frequency domain used by OFDM. Each data symbol $x[\ell, k]$ (at delay index $\ell$ and Doppler index $k$) is spread across all time-frequency resources via a 2D transform:

$$X[n, m] = \frac{1}{\sqrt{NM}} \sum_{\ell=0}^{M-1} \sum_{k=0}^{N-1} x[\ell, k]\, e^{j2\pi\left(\frac{nk}{N} - \frac{m\ell}{M}\right)}$$

Key properties:

• Each data symbol experiences the full channel diversity in both time and frequency.
• The channel in the delay-Doppler domain is sparse and quasi-static (changes slowly compared to time-frequency).
• Naturally suited for high-mobility scenarios (vehicular, high-speed rail, LEO satellite).