Exercises

By the definition of the DFT, $H[k] = \sum_{n=0}^{N_{\rm sc}-1} h[n]\,e^{-j 2\pi kn/N_{\rm sc}}$. For a sparse impulse response with only $L$ non-zero taps at positions $\ell = 0, \ldots, L-1$: $$H[k] = \sum_{\ell=0}^{L-1} h_\ell\,e^{-j 2\pi k\ell / N_{\rm sc}}.$$

Each subcarrier $k$ sees a single complex gain — a linear combination of the $L$ underlying tap gains. The full $L$-fold diversity is HIDDEN in the statistical correlation across $k$.

$d = \min(6, 10) = 6$. The code is the bottleneck: 10 paths are available but the code only separates codewords in 6 bits.

$\min(d_H, L) = \min(5, 4) = 4$; $n_t n_r = 2 \cdot 2 = 4$. Product: $d = 4 \cdot 4 = 16$.

$v = 120 / 3.6 = 33.3$ m/s. $\nu_{\max} = 33.3 \cdot 5.9 \cdot 10^9 / (3 \cdot 10^8) = 655$ Hz.

$\nu_{\max}/\Delta f = 655/15000 = 0.0437$.

$\sigma^2_{\rm ICI} \approx 0.0437^2 / 3 = 6.4 \cdot 10^{-4}$. At SNR 20 dB (linear 100): effective noise is $1 + 100 \cdot 6.4e-4 = 1.064$. Loss $= 10 \log_{10}(1.064) \approx 0.27$ dB — acceptable.

$d = \min(d_H, L) \cdot n_t n_r = \min(12, 16) \cdot 2 \cdot 1 = 12 \cdot 2 = 24$.

Let $\mathbf{x}$ be the OFDM block of length $N_{\rm sc}$. The received block (after CP removal) equals the last $N_{\rm sc}$ samples of the linear convolution $\mathbf{h} * \mathbf{x}_{\rm tx}$, where $\mathbf{x}_{\rm tx}$ is the CP-prepended block.

If $|{\rm CP}| \ge L - 1$, the CP ensures the first $L - 1$ samples of the useful block see the "wrap-around" needed for CIRCULAR convolution with $\mathbf{h}$. Hence $\mathbf{y} = \mathbf{h} \circledast \mathbf{x}$.

Circular convolution diagonalises in the DFT basis; hence $\mathbf{F}\mathbf{y} = \mathrm{diag}(\mathbf{F}\mathbf{h}) \cdot \mathbf{F}\mathbf{x}$, which is exactly the parallel-channel form. $\blacksquare$

$\min(d_H, L) = \min(8, 10) = 8$. But at 500 km/h and say 3.5 GHz, $\nu_{\max} \approx 1600$ Hz, and with $\Delta f = 15$ kHz the normalised Doppler is 0.107 — $\sim$ 3 dB ICI loss at high SNR.

$\min(d_H, P) = \min(8, 4) = 4$. No ICI loss.

OFDM has HIGHER raw diversity but is limited by ICI. OTFS has LOWER diversity but maintains it robustly. In practice, OTFS wins at 500 km/h because the ICI ceiling caps OFDM performance.

BICM-OFDM-STBC with $d_H$: $d = \min(d_H, L) \cdot n_t n_r$.

Uncoded ($d_H = 1$) on flat MIMO: $d = n_t n_r$.

$\min(d_H, L) \cdot n_t n_r = n_t n_r$ requires $\min(d_H, L) = 1$, i.e. $d_H = 1$. So an UNCODED multi-path channel gives only space diversity — no frequency diversity — even though the paths are there. The code is essential.

For small $\epsilon = \nu/\Delta f$: $\sigma^2_{\rm ICI} = \sum_{\Delta k \ne 0} \frac{\sin^2(\pi \epsilon)} {\pi^2 (\Delta k - \epsilon)^2}$.

$\sin^2(\pi \epsilon) \approx (\pi \epsilon)^2 (1 - \frac{(\pi \epsilon)^2}{3})$ to leading order.

$\sum_{\Delta k \ne 0} (\Delta k - \epsilon)^{-2} \to \sum_{\Delta k \ne 0} \Delta k^{-2} = 2\zeta(2) = \pi^2/3$ as $\epsilon \to 0$.

$\sigma^2_{\rm ICI} \approx (\pi\epsilon)^2 \cdot \pi^2/3 / \pi^2 = \epsilon^2 / 3 \cdot \pi^2$. Cleaning the constant: $\sigma^2_{\rm ICI} \sim \epsilon^2 / 3$ (the exact coefficient depends on CP / windowing). $\blacksquare$

If the interleaver spans only $B_{\rm int}$, the effective number of uncorrelated subcarriers is $B_{\rm int} / B_c$. Successive differing-bit positions see the SAME fade when they are separated by less than $B_c$.

The channel still has $L$ underlying taps, but the code can access only $L_{\rm eff} = \min(L, B_{\rm int}/B_c)$ of them.

$d = \min(d_H, L_{\rm eff}) \cdot n_t n_r$. When $B_{\rm int} \to 0$, $L_{\rm eff} \to 1$ and the diversity drops to $n_t n_r$ — pure space diversity.

Each physical path has a Doppler shift $\nu_p = v f_c \cos\theta_p / c$ where $\theta_p$ is the angle of arrival. $\nu_p$ changes only if $\theta_p$ or the velocity changes — negligible over a short frame.

On the OTFS delay-Doppler grid, path $p$ occupies grid point $(\ell_p, k_p)$ throughout the frame. The COEFFICIENT $g_p$ is static; the time-varying nature of the channel appears only as a phase modulation that is ABSORBED into the grid coordinate $k_p$.

The frame duration $T_{\rm frame} = N T$ must be shorter than the angular coherence time $T_\theta$ — typically 10-100 ms for cellular scenarios. For 1 ms OTFS frames, this holds with wide margin.

$\nu_{\max} = 300/3.6 \cdot 5.9e9 / 3e8 = 1640$ Hz. $\nu/\Delta f = 1640/30000 = 0.055$. ICI loss $\sim 0.5$ dB at high SNR — acceptable.

To saturate $L = 8$, pick $d_H \ge 8$. A rate-1/2 LDPC code with $d_H = 10$ (typical 5G NR LDPC base graph 1) works.

$d = \min(10, 8) \cdot 2 \cdot 2 = 32$. At BLER $10^{-3}$ the log-log slope is $-32$ — an extremely robust link.

LDPC rate-1/2 + 16-QAM + Alamouti, yielding 4 bits/symbol spectral efficiency at effective SNR around 10-12 dB.

ML searches over all possible symbol assignments. For a 16 × 8 grid with QPSK: $4^{16 \cdot 8} = 4^{128}$ candidates — completely infeasible.

MP iterates over a factor graph with 128 symbol nodes and $\sim 128$ channel nodes. Per iteration: $O(|{\rm grid}| \cdot d_{\rm ch})$ where $d_{\rm ch}$ is the path count. For 20 iterations and 8 paths: $\sim 20 \cdot 128 \cdot 8 = 20{,}480$ operations — tractable.

MP converges to near-ML within 10-20 iterations on typical OTFS channels. This is the practical foundation of OTFS.

Same $d_H$ and same STBC → same diversity $d = \min(d_H, L) \cdot n_t n_r$. The slope of BER-vs-SNR is unchanged.

LDPC has better union-bound constants (smaller $A_d$ at minimum-distance events) and better convergence under iterative decoding — typical 0.5-1 dB gain over turbo at $d_H = 10$.

Besides the coding-gain improvement, LDPC scales better to large blocklengths (needed for NR-V2X 1 ms slots), parallelises across decoder iterations better than turbo, and was already mandated for other NR physical channels. The switch unifies codec implementation across NR.

Time diversity gives $L_t = \lfloor T_{\rm int} / T_c \rfloor$ independent time-fades. Frequency diversity gives $L_f = L$ independent frequency-fades. The joint time-frequency channel has $L_t \cdot L_f$ independent resolvable "paths" in the time-frequency plane.

$d = \min(d_H, L_t \cdot L_f) \cdot n_t n_r.$$This is the generalisation of Akay-Ayanoglu-Caire to the triple- diversity setting. With$L_t = 10, L_f = 4, n_t = n_r = 2, d_H = 50$:$d = 50 \cdot 4 = 200$(capped by$L_t L_f = 40$).

For OTFS, the channel matrix is block-circulant with $P$ non-zero diagonals. A full-rank codeword difference on the delay-Doppler grid would give rank diversity $P$, but this is already equal to the Nazer-Gastpar style achievability — not better.

Explicit CDA constructions on the delay-Doppler grid have been proposed (see Reddy et al., IEEE Trans. IT 2021). They achieve $\min(d_H, P)$ diversity WITH a non-vanishing determinant — so the coding gain scales better than BICM-OTFS, but the diversity order itself is unchanged.

Whether the delay-Doppler domain admits a RANK-FOLD diversity amplification (analogous to MIMO rank criterion for flat channels) remains open. The difficulty: OTFS paths are all diagonal in the grid's Zak-transformed coordinates.