Ferkans — Interactive Telecom Tutor

The Cost of Ignoring DD Sparsity

We have seen that OFDM under Doppler develops ICI, and that ICI is the DD-space manifestation of using the wrong signaling lattice. We now quantify what OFDM wastes by ignoring the DD structure of the channel. The answer comes in three parts:

Diversity order 1 per cell: OFDM's per-subcarrier capacity is bottlenecked by the worst-case fading realization at that subcarrier. OTFS with the same symbols averages across all DD cells — diversity order $P$ .
Pilot overhead $\sim 10\%$ : OFDM requires a dense pilot grid (demodulation reference symbols, DMRS). OTFS needs a single embedded pilot — overhead $\sim 1\%$ .
Equalization complexity: OFDM's per-subcarrier equalization works at low Doppler but requires dense linear systems ( $O((MN)^3)$ ) under high Doppler. OTFS inherits the sparse channel matrix structure of Chapter 4 — $O(MN \log MN)$ or $O(PMN)$ .

The DD lens makes all three losses structural, not incidental.

Theorem: OFDM Has Diversity Order 1 per Subcarrier

Consider a flat-fading OFDM subcarrier with channel coefficient $H \sim \mathcal{CN}(0, 1)$ . The bit-error probability with uncoded QPSK behaves as $P_e \sim \frac{1}{4\,\mathrm{SNR}}, \qquad \mathrm{SNR} \to \infty.$ That is, the BER scales as $\mathrm{SNR}^{-1}$ — diversity order 1.

A single fading coefficient determines the subcarrier's reliability. A deep fade (rare but not implausible for complex Gaussian) leaves the subcarrier undetectable regardless of transmit power. This is the textbook single-tap-fading result.

OTFS, by spreading each data symbol across all TF cells (Chapter 3), implicitly averages across $P$ independent fading realizations — one per path — which lifts the diversity order to $P$ (Chapter 9). This is a direct consequence of DD sparsity.

Proof

Conditional BER

Given $|H|^2 = \gamma$ , BER = $Q(\sqrt{2\gamma\,\mathrm{SNR}})$ .

Average over $\gamma$

$\gamma \sim$ Exponential(1) for $H \sim \mathcal{CN}(0, 1)$ . $P_e = \int_0^\infty Q(\sqrt{2\gamma\,\mathrm{SNR}})\,e^{-\gamma}\,d\gamma$ .

High-SNR asymptotic

Using the asymptotic expansion $P_e \approx 1/(4\,\mathrm{SNR})$ at large SNR. This is the diversity-order-1 BER floor. $\blacksquare$

Why OTFS Achieves Diversity $P$ (Preview)

In OTFS, a single data symbol $X_{DD}[\ell, k]$ is spread by the ISFFT across the entire TF grid, and the channel acts on this spread form via 2D convolution with a $P$ -tap kernel. From the detector's perspective, each data symbol is "heard" through $P$ independent fading realizations (the $P$ paths). The received statistic is a weighted sum, so the effective SNR at the detector is $\sum_i |h_i|^2 \cdot \mathrm{SNR}$ , which is chi-squared with $2P$ degrees of freedom — diversity order $P$ .

Chapter 9 makes this rigorous. The key point here is that OFDM, by signaling on the TF grid, receives each data symbol through only the TF cell it was transmitted on — a single fading realization. The diversity-order-1 penalty is the cost of ignoring the DD spreading that is built into the channel by the physics.

BER vs SNR: OFDM Diversity 1 vs OTFS Diversity $P$

Plot BER curves for (a) OFDM with single-tap Rayleigh fading per subcarrier, and (b) OTFS with $P$ -tap DD channel using a matched-filter detector. Observe that OTFS's curves tilt steeper — diversity order $P$ — while OFDM has the classical diversity-1 slope. Vary $P$ to see the slope change. The gap at BER $10^{-3}$ is typically 5-15 dB in OTFS's favor for $P = 4$ .

Parameters

DD channel paths

P

4

Min SNR (dB)0

Max SNR (dB)30

SNR grid points15

Theorem: Pilot-Overhead Gap Between OFDM and OTFS

For a channel with $\tau_{\max}, f_D$ over a TF grid of $M$ subcarriers and $N$ symbols:

OFDM pilot overhead to track the TF channel at the coherence-cell Nyquist rate: $\sim (\tau_{\max} W + f_D T)/MN$ cells per period, which for 5G NR scales as $5$ – $10\%$ of resource elements.
OTFS pilot overhead with embedded single pilot + minimum guard region: $(l_{\max} + 1)(2 k_{\max} + 1)/(MN) \sim 1$ – $3\%$ .

The ratio of OTFS to OFDM pilot overhead is typically $1/5$ to $1/10$ , representing a $5$ – $10\%$ spectral-efficiency gain for the OTFS design.

OFDM must estimate $H[n, m]$ at each of $MN$ TF cells at the Nyquist rate of the coherence cell. Essentially, it samples a dense 2D function. OTFS must estimate only $P$ path parameters — a $P$ -sparse problem. The ratio is $MN/P$ on the theoretical lower bound, with practical overhead reduced by reasonable guard-region sizes. At typical $MN = 10^4$ and $P = 10$ , the theoretical ratio is 1000; real implementations lose roughly a factor of 100 to guard-region padding. Still an order of magnitude gap.

Proof

OFDM pilot density

Coherence cell area: $B_c T_c \approx 1/(\tau_{\max} f_D)$ . Cells per frame: $MN/(B_c T_c)$ . Pilot density per cell: 1. Overhead: $MN \cdot \tau_{\max} f_D/(TW) = \tau_{\max} f_D \cdot MN/(TW) \approx \tau_{\max} f_D$ (since $TW = MN$ ). For typical numerology, this is $\sim 5$ – $10\%$ .

OTFS pilot density

Guard region $|\mathcal{G}| = (l_{\max} + 1)(2 k_{\max} + 1)$ . Overhead: $|\mathcal{G}|/(MN)$ . For realistic channels, $l_{\max} \ll M$ and $k_{\max} \ll N$ , so overhead is $\sim 1$ – $3\%$ .

Ratio

OTFS/OFDM ratio: roughly $1/5$ to $1/10$ . The efficiency advantage is a free $5$ – $10\%$ of spectrum. $\blacksquare$

Example: Bandwidth Saved by OTFS Pilot Design

A 5G cell operates at 100 MHz carrier bandwidth with OFDM. If 8% of resource elements are allocated to pilots (typical), and OTFS could reduce this to 2%, how much throughput is gained per user?

Solution

Pilot savings

8% - 2% = 6% freed resource elements per frame.

Throughput translation

At a target spectral efficiency of, say, 4 bits/s/Hz, 6% of 100 MHz translates to $\approx 24$ Mbps additional throughput per cell.

Per-user impact

For 10 concurrent users per cell, this is $\sim 2.4$ Mbps per user — a meaningful gain at 5G throughput targets. In cell-free massive MIMO (Chapter 17 of this book), the cell-free OTFS studies report similar $5$ – $8\%$ net throughput gains after accounting for pilot overhead — empirical confirmation of the analysis here.

🎓CommIT Contribution(2023)

Pilot-Overhead Reduction in Cell-Free Massive MIMO with OTFS

M. Mohammadi, H. Q. Ngo, M. Matthaiou, G. Caire — IEEE Trans. Wireless Communications

In their 2023 paper on cell-free massive MIMO with OTFS, Mohammadi, Ngo, Matthaiou, and Caire provided the first rigorous performance evaluation of OTFS in a distributed-access-point setting. One of their headline findings — which this chapter derives at the single-link level — is that the pilot overhead reduction translates into a measurable spectral-efficiency gain: up to 35% improvement in 95%-likely per-user throughput at vehicular speeds, driven largely by the lower pilot burden.

The CommIT contribution refines the naïve pilot-overhead comparison we make here in three ways: (i) it accounts for the distributed AP structure (where pilots must identify the channel to each AP separately), (ii) it incorporates realistic OFDM pilot patterns (not the theoretical minimum), and (iii) it deals with fractional Doppler. The net result confirms the intuition of this section: OTFS's pilot efficiency is a concrete, measurable advantage in realistic networks. Chapter 17 of this book develops the full cell-free OTFS analysis.

cell-freepilot-overheadcaire-otfs

Key Takeaway

OFDM wastes the DD sparsity. Three concrete losses relative to OTFS: (i) diversity order 1 per cell instead of $P$ , so deep fades destroy symbols that OTFS would average over; (ii) pilot overhead $5$ – $10\%$ instead of $1$ – $3\%$ , a direct spectral-efficiency tax; (iii) ICI-limited SINR ceiling under high Doppler, vs OTFS's essentially ICI-free operation. All three follow from the same structural fact: OFDM signals on the TF grid, which ignores the channel's sparse DD structure. OTFS signals on the DD grid, which exploits it.

Why This Matters: Building the OTFS Modulator

With the OFDM weaknesses catalogued, Chapter 6 builds the full OTFS modulator from the tools already assembled. The construction has two conceptually distinct versions: (a) the Hadani-Rakib two-step (ISFFT + OFDM Heisenberg transform), which is the originally proposed and most-deployed form, and (b) the Zak-OTFS direct construction, which is conceptually cleaner and preferred in the most recent CommIT tutorial by Mohammed-Hadani-Chockalingam-Caire (2022). Both produce mathematically identical transmit signals under idealized bi-orthogonal pulses; Chapter 20 treats the pulse-shape corrections.

Common Mistake: OTFS Does Not "Replace" OFDM

Mistake:

Concluding from this chapter that OTFS should replace OFDM everywhere.

Correction:

The analysis of this chapter shows OFDM's failure mode under high mobility. For low-mobility scenarios (indoor, pedestrian, fixed access), OFDM remains optimal — ICI is negligible, pilot overhead is modest, and the per-subcarrier equalization is extremely cheap. The correct reading is that OTFS is the right waveform for the high-mobility subset of 6G use cases: V2X, HST, LEO satellite, UAV-BS. For stationary users, OFDM stays. This is why 5G NR will not replace OFDM with OTFS wholesale; it will add OTFS-like signaling for specific traffic classes. See the standards discussion in Chapter 19.

What OFDM Wastes: The Case for OTFS