Exercises

$\delta = 5/15{,}000 = 3.3 \times 10^{-4}$; $\rho \approx 3.6 \times 10^{-7}$.

$\delta = 500/15{,}000 = 0.033$; $\rho \approx 3.6 \times 10^{-3}$.

$\delta = 2000/15{,}000 = 0.133$; $\rho \approx 0.058$ (5.8%). HST at 15 kHz subcarrier spacing is marginal for OFDM.

$|\Phi_{m m'}(\delta)|^2 = \text{sinc}^2(\delta + m - m')$.

$\sum_{m'}\text{sinc}^2(\delta + m - m') = 1$ by the partition-of-unity property of sinc-squared.

Power conservation: the OFDM subcarrier's total transmitted energy is preserved, only redistributed. Doppler is lossless; it just mis-routes energy to neighboring bins. $\blacksquare$

$\delta = 6/60 = 0.1$.

$\rho_{\text{ICI}} \approx (\pi \cdot 0.1)^2/3 = 0.0329$.

$\text{SINR}_{\max} \approx 1/\rho_{\text{ICI}} - 1 = 29.4$ (15 dB). This is below the 24 dB target for 64-QAM: at this $\delta$, OFDM cannot support 64-QAM regardless of transmit power. Must drop to 16-QAM or 4-QAM, or use OTFS.

Windowing softens the leakage to neighbors by reducing side-lobe heights. For a raised-cosine window with roll-off $\alpha$, the ICI scales as $(1 - \alpha/2)^2 \cdot (\pi\delta)^2/3$ for small $\delta$.

At $\delta \sim 1$, the Taylor expansion fails entirely. ICI ≈ 1 (all energy spills outside the transmitted bin). No windowing can save this — the problem is fundamental, not a matter of pulse-shape tuning.

Windowed OFDM helps in the moderate-Doppler regime (0.01 ≤ δ ≤ 0.2) but cannot rescue LEO satellite operation. OTFS, which does not rely on subcarrier orthogonality at all, is the more fundamental fix.

$B_c \approx 1/\tau_{\max} = 500$ kHz; $T_c \approx 1/f_D = 2$ ms. Product: $B_c T_c = 10^3$ (dimensionless, product of samples).

$T_{\text{frame}} = N T_s = 14 \cdot 33.3\,\mu\text{s} = 467\,\mu$s. $T W = 14 \cdot 1 = 14$ (with $W/\Delta f = M$, but here $TW = T \cdot W$ needs concrete bandwidth; assume $W = 10$ MHz for clarity: $TW = 4670$). Cells per frame: $4670/1000 = 4.67$.

Five coherence cells per frame — OFDM needs five pilots. Very modest. OTFS needs one pilot — also modest. The pilot savings become dramatic at higher bandwidth and frame duration, where cells-per-frame grows but the $P$-sparse OTFS estimator scales only with $P$.

For BPSK, $P_e = \mathbb{E}[Q(\sqrt{2|h|^2\,\mathrm{SNR}})]$ with $|h|^2 \sim$ Exponential(1).

The integral gives $P_e = (1 - \sqrt{\mathrm{SNR}/(1 + \mathrm{SNR})})/2 \approx 1/(4\,\mathrm{SNR})$ for large SNR.

SNR = 100; $P_e \approx 1/400 = 2.5 \times 10^{-3}$. Decent but far from the $10^{-5}$ to $10^{-6}$ targets of reliable links. OTFS with diversity order $P = 4$ would scale as $1/\mathrm{SNR}^4$, giving BER $\sim 10^{-8}$ at the same 20 dB — a many-order-of-magnitude reliability advantage.

$X_{DD}[\ell, k] = (1/\sqrt{MN})\sum_{n, m} X_{TF}[n, m]\,e^{-j 2\pi(k n/N - m \ell/M)}$ $= (1/\sqrt{MN})\,e^{-j 2\pi(k n_0/N - m_0 \ell/M)}$.

$|X_{DD}[\ell, k]| = 1/\sqrt{MN}$ — uniform across all $(\ell, k)$. This is a full-grid uniform pattern, not yet "one column."

To match the "one-column" picture (Theorem TDD-Domain Image of an OFDM Subcarrier), we need to account for the rectangular-pulse periodic structure and the modular delay identification $\ell = -m_0 \bmod M$. Within the single-symbol context (and using the $M$-point DFT embedded in the SFFT), the symbol identifies with a single delay.

The "single-column" picture applies when we view the DD image of a subcarrier that is transmitted across all OFDM symbols. For one isolated $X_{TF}[n_0, m_0]$, the DD-plane image is a cell, not a column — the distinction is about what fraction of the lattice the data occupies in the OFDM vs OTFS case.

Fix $f_D$; scale $\Delta f \to k\Delta f$. $\delta \to \delta/k$. $\rho_{\text{ICI}} \to \rho_{\text{ICI}}/k^2$.

$T_s = 1/\Delta f \to T_s/k$.

Required CP length $L_{CP} \geq \tau_{\max}$ is fixed by the physical channel, not by $\Delta f$. CP fraction of symbol: $L_{CP}/(T_s + L_{CP}) \to kL_{CP}/(T_s + kL_{CP})$. Grows with $k$.

Wider subcarriers reduce ICI (quadratically) but inflate CP overhead (linearly-with-bandwidth). There is an optimal $\Delta f$ for each deployment. 5G NR numerologies are chosen precisely to balance this trade-off across mobility classes.

$\text{sinc}(0.1) = \sin(0.1\pi)/(0.1\pi) = 0.9836$. $|\Phi_{m m}|^2 = 0.9835^2 \approx 0.9672$.

$\rho_{\text{ICI}} = 1 - 0.9672 = 0.0328$.

$\text{SINR}_{\max} = 0.9672/0.0328 = 29.5 \approx 14.7$ dB, close to the approximate $3/(\pi\delta)^2 = 30.4 \approx 14.8$ dB.

Per OFDM symbol: $O(M^3)$ for matrix inversion. Per frame: $O(N M^3) = O(M^3 N)$.

Single DD-domain operation: $O(PMN)$ (matched filter), or $O(MN \log MN)$ via 2D FFT.

OFDM/OTFS = $O(M^3 N)/O(PMN) = O(M^2/P)$. At $M = 512, P = 10$: ratio $\sim 26{,}000$. OFDM with linear MMSE is 4-5 orders of magnitude slower than OTFS with matched filter at realistic frame sizes. This is the complexity cost of OFDM's ICI mitigation.

$\rho_{\text{ICI}} \approx (\pi \cdot 0.1)^2/3 \approx 0.033$.

Both polarizations contribute: each sees its own ICI plus cross-talk from the other polarization through the same leakage mechanism.

Total interference per polarization $\approx 2 \cdot 0.033 = 0.066$, SINR ceiling $\approx 1/0.066 - 1 = 14.2$ ($\approx 11.5$ dB). Dual polarization under Doppler is significantly worse than single polarization — MIMO-OFDM inherits and amplifies the ICI problem. MIMO-OTFS (Chapter 16) avoids this by preserving DD sparsity across spatial streams.

DFT-s-OFDM precodes data with a DFT before OFDM modulation. Effectively: $s = \text{OFDM}(\text{DFT}(\mathbf{data}))$.

Applying a DFT in the data domain is a time-shift in the TF domain. It does not change the DD-plane footprint of the signaling lattice.

DFT-s-OFDM still places each data symbol in a TF cell (modulo the DFT permutation). Under Doppler, it still suffers from ICI — the DFT precoder does not create DD sparsity. OTFS does, through the ISFFT precoder (the key difference: ISFFT is a 2D DFT, not a 1D one, and it is symplectic).

$H(f, t) \approx H(f, t_0) + (t - t_0)\partial H/\partial t$ over one OFDM symbol.

The drift term $(t - t_0)\partial H/\partial t$ produces ICI proportional to $\int_0^{T_s}(t - t_0/2)\cdot e^{-j 2\pi(m-m')\Delta ft/T_s}\,dt$.

Taking PSDs: $\mathbb{E}[|\text{ICI}|^2](f) \propto \mathbb{E}[|\partial H/\partial t|^2](f)$, which is the time-derivative PSD of the channel.

ICI is a PSD projection of the channel's rate of change. For block-fading (constant over the frame), ICI is zero. For time-varying channels, ICI PSD is a filtered version of the channel's time-derivative spectrum. The Jakes Doppler spectrum (isotropic) produces a specific ICI PSD shape.

$\rho_{\text{ICI}} \approx (\pi \cdot 0.05)^2/3 = 8.2 \times 10^{-3}$. $\text{SINR} = (1 - \rho)/\rho \approx 120$ ($\approx 21$ dB).

$\text{BER} = Q(\sqrt{2 \cdot 120}) = Q(15.5) \approx 2 \times 10^{-53}$.

At moderate vehicular mobility, QPSK is essentially uncoded error-free. The OFDM problem is not at the QPSK level — it is at 16-QAM and above, where the constellation density demands higher SINR than the ICI ceiling allows.

Normalize: $\rho_{\text{ICI}}(\Delta f) + \eta(\Delta f)$ with $\eta$ the CP overhead fraction. $\rho_{\text{ICI}} \propto (f_D/\Delta f)^2$; $\eta \propto L_{CP}\,\Delta f$.

$d/d\Delta f(\alpha/\Delta f^{2} + \beta\Delta f) = 0$ gives $\Delta f^* \propto (\alpha/\beta)^{1/3}$. Inserting: loss $\sim (\alpha\beta^2)^{1/3}$.

OTFS loss does not depend on $\Delta f$ in the same way — it depends only on the DD-grid alignment with the channel. At deep-Doppler ($\delta \sim 1$ for LEO), no OFDM numerology satisfies both $\rho_{\text{ICI}} < 0.01$ and $\eta < 0.1$. OTFS operates with $\eta \sim 0.05$ and effectively zero ICI. The gap is structural, not a matter of tuning.