Exercises

$\log(\mathrm{BER}) \approx \log(C) - 4 \log(\mathrm{SNR})$. $d = -(-4) = 4$.

Diversity 4: each 3 dB of SNR reduces BER by 12 dB. Each decade of SNR reduces BER by 4 orders.

$d = P = 6$. Full DD diversity.

$\rho = 10^{1.5} = 31.6$, $4\rho = 126.5$. $\binom{11}{6} = 462$. $\mathrm{BER} \approx 462 \cdot (126.5)^{-6} = 462 \cdot 2.4 \times 10^{-13} \approx 1.1 \times 10^{-10}$.

At 15 dB, uncoded OTFS with $P = 6$ achieves BER $\sim 10^{-10}$ — essentially error-free. Compare with OFDM at same SNR: $\mathrm{BER} \sim 1/(4 \cdot 31.6) = 8 \times 10^{-3}$. OTFS is 8 orders of magnitude better.

$|\mathbf{H}\Delta|^2 = |h_1|^2 + |h_2|^2 = 1$ (for generic $\Delta\mathbf{x}$). (Depends on error vector; assume minimum-distance case.)

$\rho = 100$, so $|\mathbf{H}\Delta|^2 \cdot \rho = 100$. $\mathrm{PEP} = Q(\sqrt{100/2}) = Q(7.07) \approx 8 \times 10^{-13}$.

Averaging over Rayleigh $\chi^2_{2}$ (distribution of $|h_1|^2 + |h_2|^2$): $\mathrm{PEP}_{\text{avg}} \approx (1 + \rho)^{-2}$ (Rayleigh with diversity 2) $= 1/101^2 \approx 10^{-4}$.

Instantaneous PEP: $\sim 10^{-13}$ (this realization). Average PEP: $\sim 10^{-4}$. The average is dominated by bad channel realizations (deep fades), which is why averaging matters.

$35/(4\rho)^4 = 10^{-6}$. $(4\rho)^4 = 3.5 \times 10^7$. $4\rho = (3.5 \times 10^7)^{1/4} = 77$. $\rho = 19.3$ ($\approx 12.9$ dB).

OFDM for same BER: $\rho \sim 10^6/4 = 2.5 \times 10^5$ ($\approx 54$ dB). Totally impractical for radio.

OTFS saves $54 - 12.9 = 41$ dB at BER target $10^{-6}$ for $P = 4$. Spectacular on paper; halved in practice.

$\Pr(\log(1 + \rho \chi^2_{2P}/P) < R) = \Pr(\chi^2_{2P} < P(2^R - 1)/\rho) = F_{\chi^2, 2P}(P(2^R - 1)/\rho)$.

For small $\epsilon$: $F_{\chi^2, 2P}^{-1}(\epsilon) \approx \epsilon^{1/P}\cdot(P!)^{1/P}$.

$C_\epsilon \approx \log_2(1 + \rho \epsilon^{1/P}(P!)^{1/P}/P)$.

OFDM ($P = 1$): $C_\epsilon = \log_2(1 + \rho \epsilon)$. OTFS improvement factor at $\epsilon = 10^{-3}, P = 4$: $\epsilon^{1/P}/\epsilon = \epsilon^{-3/4} = 10^{9/4} = 178\times$. OTFS outage rate is 178× higher.

$d^*_{\text{OTFS}}(1.5) = 4 - 1.5 = 2.5$.

OFDM DMT: $d^*(1.5) = 1 - 1.5 = -0.5$ (negative, invalid — OFDM cannot support this rate reliability). OTFS can support $r = 1.5$ with $d = 2.5$ — well-defined.

OTFS can operate at higher rate-reliability regimes than OFDM. For 6G URLLC applications needing both high rate and high reliability, OTFS is structurally suited; OFDM cannot match.

$\mathbf{A}(\Delta\mathbf{x}) = \mathbf{B}^H \mathbf{B}$, where $\mathbf{B}$ has $P$ columns, each being a distinct shift of $\Delta\mathbf{x}$.

Suppose $\sum_i a_i \mathbf{B}_i = 0$ with $a_i \neq 0$. Then $\sum_i a_i \,\text{shift}_{(\ell_i, k_i)}(\Delta\mathbf{x}) = 0$. For distinct shifts and $\Delta\mathbf{x} \neq 0$, this is impossible (shifts commute with the group action; distinct shifts of a non-zero vector are linearly independent).

$\mathbf{B}$ has $P$ linearly independent columns, so rank $= P$. $\mathbf{A} = \mathbf{B}^H\mathbf{B}$ has rank $= P$. $\blacksquare$

Per-AP link: $\chi^2_{2P}$ fading. Per-AP diversity: $P = 3$.

$L$ independent APs: joint effective SNR is $\sum_\ell \chi^2_{2P, \ell}/P = \chi^2_{2LP}/LP$. Diversity order: $LP = 48$.

Cell-free OTFS with 16 APs × 3 paths = diversity 48 at the CPU. The combined BER decays as $\mathrm{SNR}^{-48}$ — effectively error-free at any reasonable SNR. This is the macro-diversity advantage of cell-free OTFS.

Diversity is only achieved if the AP channels are independent (realistic for different APs). Shadowing correlations reduce effective diversity.

$\rho_{\text{OFDM}} = 1/(4 \cdot 10^{-5}) = 2.5 \times 10^4$ (44 dB).

$\rho_{\text{OTFS}}^8 \cdot 4^8 = 6435/10^{-5} = 6.4 \times 10^8$. $\rho_{\text{OTFS}}^8 = 6.4 \times 10^8/65536 = 9.8 \times 10^3$. $\rho_{\text{OTFS}} = (9.8 \times 10^3)^{1/8} = 3.17$ (5 dB).

SNR gain: 44 - 5 = 39 dB.

Theoretical 39 dB; practical $\sim 20$ dB after accounting for channel estimation, suboptimal detection, fractional Doppler. Still enormous.

Detector (MP): $d = 4$. Code ($d_{\min} \approx 10$): additional $d_{\text{code}} = 10$. Combined effective: some product-like aggregation, roughly $d \sim 4 \cdot 10 / (4 + 10) \approx 3$ (with overlap). Full joint: up to $d = P \cdot d_{\text{code}} = 40$ in best case.

With effective $d = 20$: $\rho^{20} \sim 10^5$, $\rho \sim 1.2$ (1 dB). Real LDPC-OTFS operating points: 5-8 dB for $10^{-5}$ — aligned.

LDPC + OTFS achieves excellent operating points. Real 5G NR link budgets: LDPC + OTFS at 5-8 dB SNR for $10^{-5}$ BER. This is the "coded" performance that matters for deployment.

All paths equally strong. Coding gain: $C \propto \prod (|h_i|^2)^{-1} = P^P$ — moderate.

Paths: $|h_1|^2 = 2/(2^P - 1) \approx 0.5, |h_P|^2 = 1/(2^P - 1) \approx 2^{-P}$. Weakest path ($i = P$): very weak. Coding gain: $\prod|h_i|^2 \propto 2^{-P(P+1)/2}$ — large penalty.

Same diversity slope $P$ in both cases. But the exponential profile has dramatically worse coding gain — the weakest path is the bottleneck. At finite SNR, the equal-power channel gives 5-10 dB better BER than the exponential-profile channel.

To realize OTFS's full advantage, ensure all $P$ paths have comparable strength. In practice: position APs to avoid dominant LOS + weak scattering configurations. Cell-free OTFS (Chapter 17) naturally provides this by pooling paths from many APs.

$\rho = 10^{1.2} = 15.85$, $4\rho = 63.4$. $\binom{9}{5} = 126$. $\mathrm{BER} = 126/63.4^5 = 126/10^9 \approx 10^{-7}$.

Effective diversity after LDPC: roughly matched to uncoded OTFS diversity; code primarily extracts coding gain ($\sim 7$ dB). Coded BER at SNR = 5 dB with OTFS: $\sim 10^{-7}$.

Uncoded: 12 dB for $10^{-7}$. Coded: $\sim 5$ dB for $10^{-7}$. Channel code saves $\sim 7$ dB — standard rate-1/2 LDPC gain.

After MMSE/ML detection: per-cell effective channel is $\lambda_{n, m}$ with $|\lambda|^2 \sim$ chi-squared (sum of $P$ paths). Mean: 1.

$C_{\text{OTFS}} = \mathbb{E}[\log_2(1 + \rho|\lambda|^2)] = \mathbb{E}_{\chi^2}[\log_2(1 + \rho\chi^2_{2P}/P)]$. For large $P$: $\chi^2_{2P}/P \to 1$, so $C \to \log_2(1 + \rho)$ — AWGN capacity.

OFDM per-cell: single Rayleigh tap. Ergodic: $\mathbb{E}_{\chi^2_2}[\log_2(1 + \rho \chi^2_2)]$. For Rayleigh, this evaluates to $(1/\ln 2) e^{1/\rho} E_1(1/\rho)$.

For large $P$, OTFS per-cell fading averages out → AWGN capacity. For $P = 1$, OTFS = OFDM (single tap). For intermediate $P$, OTFS is slightly higher than OFDM at the same SNR (diversity helps at moderate SNR). The gap is modest — OTFS's advantage is in outage, not ergodic.

$5 \leq \log_2(1 + \rho\,10^{-5/6})$. $2^5 = 32 \leq 1 + \rho \cdot 0.464$. $\rho \geq 31/0.464 = 67 = 18.3$ dB.

OFDM: $5 \leq \log_2(1 + \rho\,10^{-5})$. $\rho \geq 31/10^{-5} = 3.1 \times 10^6 = 65$ dB. Impossible.

OTFS: 18 dB. OFDM: 65 dB. Gap: 47 dB. URLLC is a canonical regime where OTFS is decisive.

Each physical path has Doppler index $k_i = k_i^{\text{int}} + \epsilon_i$ with $|\epsilon_i| < 0.5$. The path's DD response spreads over multiple bins, with the non-integer part decaying as sinc.

Diversity is determined by the dimension of the effective channel subspace. Fractional leakage creates $k_{\max}$-bounded effective dimension. If $k_{\max} < P$, the effective diversity collapses to $k_{\max}$.

To preserve full $P$ diversity under fractional Doppler, the receiver must estimate the fractional offsets and apply super-resolution (Chapter 10). This is the motivation for Chapter 10's fractional-Doppler processing.

The exact interaction between fractional Doppler and diversity remains an active research topic. Preliminary results suggest the gap can be mostly closed by careful detector design — see Raviteja et al. (2020) on fractional-Doppler OTFS.

After MP / LCD with full iteration, each symbol sees effective SNR $\rho_{\text{eff}} = \rho \sum_i |h_i|^2/P$. For equal-power paths: $\rho\chi^2_{2P}/P$.

Per-cell achievable rate: $\log_2(1 + \rho_{\text{eff}})$.

$\Pr(\log_2(1 + \rho\chi^2_{2P}/P) < R) = \epsilon$ is the outage condition. OTFS's per-cell channel matches this.

With asymptotically long channel codes that achieve the per-cell Shannon capacity, OTFS achieves the outage capacity. In practice, strong LDPC at 5G NR block lengths (1024-2048) is within 1 dB of this bound. $\blacksquare$