OTFS as Precoded OFDM

OTFS = Precoded OFDM

The two-stage Hadani-Rakib OTFS transmitter — ISFFT then OFDM Heisenberg — admits a clean reinterpretation. Everything except the ISFFT is OFDM. The ISFFT is a precoder: a linear transform applied to the data symbols before handing them to an OFDM modulator. This reinterpretation is not merely cosmetic; it is the basis for the OTFS deployment strategy ("software precoder on top of 5G NR OFDM") and the operational equivalence between OTFS and Zak-OTFS.

The point is that OTFS is not a new waveform — it is OFDM with a specific 2D-DFT precoder. Every OFDM detector, every OFDM pilot scheme, every OFDM synchronization procedure still works; what changes is how the detector interprets the received TF symbols.

Theorem: OTFS as OFDM With ISFFT Precoder

The OTFS transmit waveform equals the OFDM transmit waveform applied to the ISFFT-precoded data: $s_{\text{OTFS}}(t; X_{DD}) \;=\; s_{\text{OFDM}}(t;\, X_{TF} = \text{ISFFT}(X_{DD})).$ Conversely, given an OFDM receiver that outputs TF-grid estimates $\hat{Y}_{TF}$ , the SFFT recovers the DD-grid estimate: $\hat{X}_{DD} = \text{SFFT}(\hat{Y}_{TF})$ .

OTFS is therefore precoded OFDM with the ISFFT as precoder and the SFFT as its inverse.

The Hadani-Rakib construction is manifestly this: stage 2 (ISFFT) produces the TF grid; stage 3 (Heisenberg) is OFDM. The composition is "ISFFT then OFDM," i.e., OFDM with a precoder.

This framing has two engineering consequences: (i) OTFS can be deployed on existing OFDM silicon as a firmware update, (ii) every feature of OFDM (synchronization, PAPR control, MIMO extensions) is inherited by OTFS with minimal modification. The DD-grid semantics is a lens on the OFDM signaling, not a distinct waveform.

Proof

Transmit direction

OTFS transmitter: $X_{DD} \to \text{ISFFT} \to X_{TF} \to \text{OFDM mod.} \to s(t)$ . OFDM transmitter on data $X_{TF}$ : same block from TF grid onwards. The ISFFT is the only extra block.

Receiver direction

OTFS receiver: $r(t) \to \text{OFDM demod.} \to \hat{Y}_{TF} \to \text{SFFT} \to \hat{X}_{DD}$ . An OFDM receiver computes $\hat{Y}_{TF}$ and would stop there. OTFS adds the SFFT post-processing.

Equivalence under channel

The channel acts on the time-domain waveform identically in both cases — it does not "know" whether the transmit data came from ISFFT precoding or not. The DD-grid perspective emerges only after the SFFT at the receiver.

Conclusion

The OTFS signal chain is OFDM-with-precoder-and-postcoder. The DD-grid structure is the space in which the data is authored, while the TF-grid structure is the space in which it is transmitted. $\blacksquare$

Key Takeaway

OTFS deploys as a precoder on top of OFDM. The ISFFT and SFFT are the only blocks beyond 5G NR OFDM; both are software-level additions implementable in firmware. No silicon changes, no RF front-end redesign, no synchronization rework are required. This concrete deployability — not just performance gains under mobility — is the second argument for OTFS's candidacy as a 6G waveform. The first (performance) is in Chapters 5 and 9; the second (deployment) is this theorem.

OTFS vs OFDM: Transceiver Blocks

Block	OFDM	OTFS	Extra work for OTFS
Data mapping	TF-grid QAM	DD-grid QAM	Different symbol table
Precoding	None	ISFFT	Added: 2D FFT, software
OFDM modulator (Heisenberg)	Standard IFFT+CP+DAC	Same	None
Channel	Same	Same	None
OFDM demodulator (Wigner)	Standard FFT	Same	None
Postcoding	None	SFFT	Added: 2D FFT, software
Detection	Per-subcarrier 1-tap ZF	DD-domain MMSE/MP (Ch 8)	Different algorithm
Pilots	Per-subcarrier DMRS	Embedded pilot (Ch 7)	Different pilot pattern
Hardware	Standard 5G NR silicon	Same silicon + firmware	Firmware update only

The Path to Standardization

The "precoded OFDM" framing of OTFS is what makes its 6G standardization path plausible. 3GPP does not need to adopt a new waveform — it needs to adopt a new signaling mode within the existing OFDM framework. Proposals under discussion include:

Mode-based precoding: signal in the TF mode (OFDM) or DD mode (OTFS) per slot or per UE.
Multi-domain multiple access (MDMA): different UEs on the same frame use different precoders (some OFDM, some OTFS) based on their mobility.
Hybrid precoders: per-UE adaptive mixing of OFDM and OTFS based on estimated Doppler.

Chapter 19 covers the standards perspective in detail. The key message here: OTFS is deployable as an enhancement, not a replacement — which is exactly what makes it viable.

Example: Verifying the Equivalence: $M = N = 4$

For $M = N = 4$ , show numerically that $s_{\text{OTFS}}(t; X_{DD}) = s_{\text{OFDM}}(t; \text{ISFFT}(X_{DD}))$ for a random $X_{DD}$ .

Solution

Compute ISFFT

Take $X_{DD} \in \mathbb{C}^{4 \times 4}$ random. Apply ISFFT to get $X_{TF} \in \mathbb{C}^{4 \times 4}$ .

OFDM transmit of $X_{TF}$

Per-symbol IFFT of each column of $X_{TF}$ , concatenate with CP. This is the OFDM transmit.

Full OTFS transmit

DD → ISFFT → TF (same $X_{TF}$ ) → Heisenberg → $s(t)$ . The Heisenberg transform and OFDM modulator are identical — both are IFFT + CP + concatenation.

Verify

Numerically, the two waveforms are identical to floating-point precision. This is the content of Theorem TOTFS as OFDM With ISFFT Precoder: OTFS = OFDM with ISFFT precoding.

OFDM vs OTFS: Waveform Comparison

For the same data symbols, compare the time-domain OFDM waveform (symbols placed directly on TF grid) with the OTFS waveform (symbols placed on DD grid, ISFFT-precoded, then OFDM-modulated). Observe that under i.i.d. QPSK data, the OTFS waveform has more uniform magnitude envelope (spreading property) than OFDM. PAPR is similar; time-domain structure is different.

Parameters

Subcarriers

M

Symbols

N

Random seed7

Why This Matters: Cell-Free OTFS Connection

The "precoded OFDM" framing is particularly important for cell-free massive MIMO deployments (Chapter 17). In a distributed AP architecture, the central processing unit handles the ISFFT/SFFT (software), while the access points run standard OFDM DACs/ADCs (hardware). The CommIT work by Mohammadi-Ngo-Matthaiou-Caire (2023) leverages this separation: no custom AP hardware is needed for cell-free OTFS — only firmware-level ISFFT coordination at the CPU. This is what makes cell-free OTFS deployable in existing O-RAN architectures.

🔧Engineering Note

Software Stack for OTFS Deployment

A reference OTFS software stack on top of 5G NR OFDM hardware:

PHY layer (baseband processor):
- MAC layer (unchanged): delivers QAM data to PHY
- New: DD-grid mapper (organizes data into $M \times N$ grid)
- New: ISFFT precoder ( $O(MN\log(MN))$ 2D FFT)
- Existing: OFDM modulator (IFFT + CP)
- Existing: RF DACs, PAs, antenna
RF layer (unchanged): identical to 5G NR
Receiver baseband:
- Existing: RF ADCs, OFDM demodulator (FFT + CP removal)
- New: SFFT postcoder ( $O(MN\log(MN))$ 2D FFT)
- New: DD-domain detector (MMSE or MP, Chapter 8)
- Existing: decoder, MAC layer

The new blocks total $\sim 3-5$ kLoC in typical implementations — a small addition to a $\sim 100$ kLoC 5G modem. Memory requirements scale as $O(MN \cdot \text{samples per symbol})$ — standard OFDM buffer sizes.

Practical Constraints

•
ISFFT/SFFT: $O(MN \log(MN))$ additional compute per frame
•
Memory: one additional $MN$ -sized complex buffer
•
Firmware update for modems, no silicon changes

📋 Ref: 3GPP TR 38.912 (6G study item, 2024+)

The OTFS Receiver Chapter Summary