Ferkans — Interactive Telecom Tutor

The Bottom Line

Chapter 5 gave the qualitative case for OTFS over OFDM. This section gives the quantitative one. We compile head-to-head comparisons across the key deployment scenarios — pedestrian, vehicular, HST, mmWave, LEO satellite — and report the SNR gains, BER advantages, and throughput improvements of OTFS relative to OFDM. These are the numbers that appear in operator RFPs for 6G waveforms.

Theorem: SNR Gain of OTFS Over OFDM at BER Target

Consider OTFS with $P$ -path Rayleigh channel and OFDM on the same channel (both with ML detection). At target uncoded BER $\epsilon$ , the SNR gain (in dB) of OTFS over OFDM is $\Delta \mathrm{SNR}\, [\text{dB}] \;=\; \frac{10(P - 1)}{\log_{10}(1/\epsilon)}\,\log_{10}\left(\frac{\epsilon\,(2P - 1)!/(P! P!)}{4^{P - 1}}\right).$ For $\epsilon = 10^{-5}$ , $P = 4$ : $\Delta\mathrm{SNR} \approx 10$ dB. For $\epsilon = 10^{-5}$ , $P = 8$ : $\Delta\mathrm{SNR} \approx 18$ dB. For $\epsilon = 10^{-3}$ , $P = 4$ : $\Delta\mathrm{SNR} \approx 7$ dB.

Larger $P$ means more diversity, bigger gain. Stricter BER target means more diversity pays off (the $P$ -th order of SNR is more valuable at smaller error rates). For operational 5G deployment targets (BER $10^{-5}$ after channel coding), OTFS with $P = 4$ saves ~10 dB of transmit power vs OFDM — a dramatic advantage.

Proof

Equate BERs

Set $\mathrm{BER}_{\text{OTFS}} = \epsilon$ and $\mathrm{BER}_{\text{OFDM}} = \epsilon$ at their respective SNRs $\rho_{\text{OTFS}}$ and $\rho_{\text{OFDM}}$ .

Ratio

$\epsilon \approx \binom{2P-1}{P}/(4\rho_{\text{OTFS}})^P$ . $\epsilon \approx 1/(4\rho_{\text{OFDM}})$ . Dividing: $\rho_{\text{OTFS}}^P/\rho_{\text{OFDM}} = (\text{const})/\epsilon^{P-1}$ .

In dB

Taking logs: $P \log \rho_{\text{OTFS}} - \log \rho_{\text{OFDM}} = \log(1/\epsilon^{P-1}) + \log(\text{const})$ . Gain $\Delta\rho = \rho_{\text{OFDM}}/\rho_{\text{OTFS}}$ in dB follows. $\blacksquare$

OTFS-OFDM SNR Gain at Target BER

Plot the SNR gain (OFDM SNR minus OTFS SNR in dB) required to achieve a target BER, as a function of target BER for various $P$ . The gain grows with $P$ (more diversity) and with 1/BER (stricter reliability benefits more from diversity). At the BER target of a URLLC link (1e-5), OTFS with $P = 8$ saves ~18 dB — enough to quadruple the operational cell radius.

Parameters

Max

P

to plot8

Min BER shown0.000001

Max BER shown0.1

Quantitative OTFS vs OFDM Comparison by Scenario

Scenario	$P$	SNR gain (dB)	BER gap at 20 dB SNR	Outage advantage
Pedestrian (indoor, 5 GHz)	3	7	$10^{-5}$ vs $10^{-3}$	10x
Urban vehicular (3.5 GHz)	4-6	10-12	$10^{-8}$ vs $10^{-3}$	100x
HST (mmWave)	4	10	$10^{-7}$ vs $10^{-3}$ (coded $10^{-2}$ )	100x
V2X (5.9 GHz)	6	14	$10^{-9}$ vs $10^{-3}$	1000x
LEO satellite (10 GHz)	2	4	$10^{-4}$ vs 0.5 (fail)	$\infty$ (OFDM floor)
Cell-free (10+ APs)	many	20+ (aggregate)	N/A (system)	$\gg 100$ x

Example: Cell-Radius Gain From OTFS Diversity

An OFDM link at 20 dB SNR achieves BER = $10^{-3}$ at cell edge. An OTFS link with $P = 4$ needs what SNR for the same BER? And what cell-radius advantage does this give?

Solution

OTFS SNR

$\mathrm{BER}_{\text{OTFS}} = \binom{7}{4}/(4\rho)^4 = 10^{-3}$ . $(4\rho)^4 = 35/10^{-3} = 3.5 \times 10^4$ . $4\rho = (3.5 \times 10^4)^{1/4} = 13.7$ . $\rho = 3.4 \approx 5.3$ dB.

SNR gain

OFDM: 20 dB. OTFS: 5.3 dB. Gain: 14.7 dB.

Cell-radius gain

Assuming power-law path loss $\sim d^{-4}$ : $10 \cdot 4 \cdot \log_{10}(d_{\text{OTFS}}/d_{\text{OFDM}}) = 14.7$ . $d_{\text{OTFS}}/d_{\text{OFDM}} = 10^{14.7/40} = 2.3$ .

Interpretation

OTFS extends the cell radius by $\sim 2.3\times$ for the same BER target. Equivalently, the area coverage of a single cell grows by $\sim 5\times$ . Direct network-level spectral-efficiency benefit for operators.

Theorem: Capacity Gap: OTFS vs OFDM

Under the same channel and same total transmit power, OTFS and OFDM have the same ergodic capacity (assuming ideal ML/MAP detection and channel coding): $C_{\text{erg}}^{\text{OTFS}} = C_{\text{erg}}^{\text{OFDM}} = \mathbb{E}_h[\log_2(1 + \rho|h|^2)].$ OTFS is not superior in ergodic capacity. The differences are:

Outage capacity: OTFS > OFDM (strictly) because of $P$ -fold diversity averaging.
Finite-blocklength capacity: OTFS > OFDM when blocklength is small (URLLC); roughly equal at long blocklength.
Practical detector complexity: OTFS requires MP or LCD vs OFDM's trivial per-subcarrier equalization (at low mobility).

Both waveforms transmit the same number of bits per channel use on the same average channel. The difference is not "bandwidth efficiency" but robustness: OTFS distributes bits across paths, OFDM concentrates them per coherence cell. Under fluctuating channels, the robust approach delivers more reliable rates.

Proof

Both unitary

ISFFT (OTFS precoder) and OFDM modulator are both unitary transforms. They preserve the frame's total spectral efficiency.

Same channel, same capacity

Capacity is a property of the channel, not the signaling scheme (as long as the signaling is optimal). Both waveforms achieve the same capacity with the right codes.

Outage-specific advantage

Under realistic constraints (finite blocklength, target outage rate), OTFS's DMT dominance translates to real throughput gains that appear in the finite-blocklength formulas. This is why OTFS wins at URLLC but matches OFDM at eMBB (where long codes average out fades). $\blacksquare$

Key Takeaway

OTFS's advantage is reliability, not capacity. For the same average channel, OTFS and OFDM have the same ergodic capacity. OTFS wins on outage capacity (diversity-driven), BER at fixed SNR (uncoded), and finite-blocklength throughput (URLLC). The advantages are decisive for 6G use cases with strict reliability targets — V2X, autonomous vehicles, URLLC industrial IoT, and LEO satellite links.

For eMBB (best-effort high throughput) at low mobility, the practical choice remains OFDM because of its simpler detector and mature silicon. Where OTFS's mobility / reliability advantage is needed, it is decisive; where OFDM works, it continues to work.

The Caveats

The comparisons in this section assume:

Perfect channel estimation: real systems lose 0.5-2 dB to estimation error (Chapter 7).
ML detection: real systems use MP or LCD, losing 1-2 dB.
Integer Doppler: fractional Doppler reduces effective $P$ (Chapter 10), typically by 20-40%.
Ideal pulse shapes: practical pulses (raised cosine, etc.) introduce small cross-talk.

Aggregate real-world gap: typically half the theoretical. At $\epsilon = 10^{-5}$ and $P = 4$ , expected real SNR gap is 4-6 dB (theoretical: 10 dB). Still significant, but not as dramatic as the clean numbers suggest.

Why This Matters: Where the Gains Compound

The single-link advantages analyzed here compound in larger systems:

Cell-free OTFS (Chapter 17): per-UE diversity + pilot savings + distributed processing = 25-35% net throughput gain vs OFDM cell-free.
ISAC (Chapter 12-14): OTFS's DD structure enables joint sensing + comms with single waveform; OFDM-ISAC requires compromise. The CommIT contributions in Chapter 12 quantify this.
LEO satellite (Chapter 18): OFDM fails (overspread); OTFS works. Infinite advantage, not a percentage.

This section gives the single-link baseline; the next chapters build on it.

OTFS vs OFDM: Quantitative Comparison