Exercises

$h(\tau, \nu) = h_1 \delta(\tau - \tau_1)\delta(\nu - \nu_1)$. $(h \star\star x_{DD})(\tau, \nu) = \iint h_1\delta(\tau' - \tau_1)\delta(\nu' - \nu_1)\,x_{DD}(\tau - \tau', \nu - \nu')\,d\tau'd\nu'$.

The Dirac fixes $\tau' = \tau_1, \nu' = \nu_1$: $= h_1 x_{DD}(\tau - \tau_1, \nu - \nu_1)$. $\blacksquare$

$Y_{DD}[\ell, k] = h_1\,X_{DD}[(\ell-2)\bmod 8, (k-1)\bmod 8]$.

$Y_{DD}[\ell, k] = 1$ when $\ell = 2, k = 1$; zero elsewhere. The single input impulse becomes a single shifted output impulse.

Shifts by $(1, 0)$. Contributions: $Y_1[1, 0] = 1, Y_1[1, 2] = 1$.

Shifts by $(2, 1)$. Contributions: $Y_2[2, 1] = 0.7, Y_2[2, 3] = 0.7$.

$Y_{DD}[\ell, k]$ non-zero at four grid positions: $(1,0): 1, (1,2): 1, (2,1): 0.7, (2,3): 0.7$, zero elsewhere. The output support is the sum-set of input support and channel support, with cardinality at most $P \cdot |\text{supp}(X_{DD})| = 4$.

$(h \star\star x)(\tau, \nu) = \iint h(\tau', \nu')\,x(\tau - \tau', \nu - \nu')\,d\tau'd\nu'$.

Let $\tau'' = \tau - \tau', \nu'' = \nu - \nu'$. Jacobian $= 1$. $= \iint h(\tau - \tau'', \nu - \nu'')\,x(\tau'', \nu'')\,d\tau''d\nu''$ $= (x \star\star h)(\tau, \nu)$. $\blacksquare$

Each path is a shift-plus-phase matrix with one non-zero per row, so exactly $MN$ non-zeros per path.

$P$ paths in general position contribute disjoint non-zero patterns (different shifts). Total non-zeros: $P \cdot MN$.

Total entries: $(MN)^2 = 8192^2 = 67{,}108{,}864$. Non-zeros: $8 \cdot 8192 = 65{,}536$. Density: $65{,}536/67{,}108{,}864 \approx 10^{-3}$, or 0.1%. OTFS-scale grids admit essentially diagonal matrix-vector products in the sparse representation.

$l_{\max} = \lceil 8000 \cdot 5\,\text{MHz}\,\text{ns}/10^9\rceil$... Carefully: $l_{\max} = \lceil 8 \cdot 10^{-6} \cdot 5 \cdot 10^6\rceil = 40$. $k_{\max} = \lceil 200 \cdot 6.4 \cdot 10^{-3}\rceil = 2$.

$|\mathcal{S}| = (l_{\max} + 1)(2 k_{\max} + 1) = 41 \cdot 5 = 205$.

$|\mathcal{S}|/(MN) = 205/8192 \approx 2.5\%$. Even with a generous guard region, pilot overhead stays under 3% — much lower than typical OFDM pilot densities.

$\mathbf{H}_{DD} = h_1\,(\mathbf{I}_N\,\boldsymbol{\Delta}_N^1) \otimes \boldsymbol{\Pi}_M^1$. The $\boldsymbol{\Delta}_N$ and $\boldsymbol{\Pi}_M$ factors are each unitary; Kronecker products of unitaries are unitary.

Multiplying by $h_1$ with $|h_1| = 1$ preserves unitarity.

A unitary $\mathbf{H}_{DD}$ means no signal amplification or attenuation across DD cells. The MMSE detector reduces to a conjugate-transpose multiply: $\hat{\mathbf{x}} = \mathbf{H}_{DD}^{H}\,\mathbf{y}$. This is the simplest possible OTFS scenario — and it is already harder than OFDM because the shift structure couples multiple cells.

$\mathbf{H}_{DD} = \mathbf{F}\boldsymbol{\Lambda}\mathbf{F}^H$ with $\mathbf{F} = \mathbf{F}_N \otimes \mathbf{F}_M$ and $\boldsymbol{\Lambda}$ a diagonal matrix of eigenvalues.

$\hat{\mathbf{x}} = (\mathbf{H}_{DD}^{H}\mathbf{H}_{DD} + \sigma^2\mathbf{I})^{-1}\mathbf{H}_{DD}^{H}\mathbf{y}$. In the eigenbasis: $\hat{\mathbf{x}} = \mathbf{F}(|\boldsymbol{\Lambda}|^2 + \sigma^2)^{-1}\boldsymbol{\Lambda}^H\mathbf{F}^H\mathbf{y}$.

$\mathbf{F}^H\mathbf{y}$ is the SFFT; diagonal multiply by $\boldsymbol{\Lambda}^H/(|\boldsymbol{\Lambda}|^2 + \sigma^2)$; then $\mathbf{F} = \text{ISFFT}$. Total: $O(MN\log(MN))$ operations once $\boldsymbol{\Lambda}$ is known. $\blacksquare$

The CP of length $L_{CP}$ prepends the last $L_{CP}$ samples of an OFDM symbol before transmission. At the receiver, after CP removal, the channel's linear convolution becomes a circular convolution — provided $L_{CP}$ absorbs the entire channel memory.

The delay-axis channel memory is $l_{\max}$. If $L_{CP} < l_{\max}$, samples from the previous symbol "leak" into the current one (inter-block interference). The circular convolution identity fails for the first $l_{\max} - L_{CP}$ delay bins.

When $L_{CP} < l_{\max}$, the DD channel matrix $\mathbf{H}_{DD}$ acquires off-diagonal "leakage" terms and is no longer exactly block-circulant. The ISFFT no longer diagonalizes it exactly, and detection accuracy degrades. Reduced-CP OTFS (Chapter 20) handles this with iterative leakage compensation.

The circular shift $\boldsymbol{\Pi}_M^{\ell_1}$ has eigenvalues $e^{-j 2\pi m \ell_1 / M}$ for $m = 0, \ldots, M - 1$. $\boldsymbol{\Pi}_N^{k_1}$ has eigenvalues $e^{-j 2\pi n k_1 / N}$. $\boldsymbol{\Delta}_N^{\ell_1}$ has eigenvalues $e^{j 2\pi n \ell_1 / N}$.

The eigenvalues of a Kronecker product are products of the factor eigenvalues. Combining: $\lambda_{n, m} = h_1\,e^{-j 2\pi (n k_1/N - m \ell_1/M)}\,e^{j 2\pi n \ell_1/N}$.

$|\lambda_{n, m}| = |h_1|$ for all $(n, m)$ — the matrix is a unitary-times-scalar. Every "eigenvalue" has the same magnitude. This is why single-path channels give flat fading in the DD domain — no cell is particularly bad.

For $P > 1$, the eigenvalues sum coherently with different phases, producing a non-flat $|\lambda_{n, m}|$ distribution — some cells are better than others. The diversity order of Chapter 9 measures how this distribution concentrates.

The noise at the DD-domain receiver output is $W_{DD} = \text{SFFT}(W_{TF})$ where $W_{TF}$ is the TF-domain noise after OFDM demodulation.

The OFDM demodulator applies a (unitary) $M$-point DFT per symbol to the time-domain noise; the result is i.i.d. $\mathcal{CN}(0, \sigma^2)$ across subcarriers and symbols.

The SFFT is unitary (Theorem TSFFT and ISFFT Are Inverses). Applying a unitary to an i.i.d. $\mathcal{CN}$ vector yields another i.i.d. $\mathcal{CN}$ vector of the same variance. Thus $W_{DD}[\ell, k] \sim \mathcal{CN}(0, \sigma^2)$, i.i.d. $\blacksquare$

$Y_{DD}[\ell, k] = h_1\,X_{DD}[\ell - \ell_1, k - k_1] + W_{DD}[\ell, k]$. Signal power: $|h_1|^2\,\mathbb{E}|X_{DD}|^2$. Noise power: $\sigma^2$. Effective SNR: $|h_1|^2\,P_{\text{data}}/\sigma^2$.

Single subcarrier in OFDM: $Y_{TF} = H(f_0, t_0)\,X_{TF} + W_{TF}$. Effective SNR: $|H(f_0, t_0)|^2\,P_{\text{data}}/\sigma^2$.

For single-path, $|H(f_0, t_0)| = |h_1|$ (unit-modulus eigenvalue), so OTFS and OFDM have the same per-cell SNR. The difference appears for $P > 1$: OFDM SNR fluctuates across cells (deep fades are possible), OTFS SNR is spread more evenly because each DD symbol experiences all paths via the convolution. Chapter 9's diversity result quantifies this evenness as a reduction in outage probability.

For a single path, $\mathbf{H}_{DD} = h_1\,(\boldsymbol{\Pi}_N^{k_1}\boldsymbol{\Delta}_N^{\ell_1}) \otimes \boldsymbol{\Pi}_M^{\ell_1}$. $\boldsymbol{\Pi}_N \otimes \boldsymbol{\Pi}_M$ is a permutation (product of permutations). $\boldsymbol{\Delta}_N^{\ell_1} \otimes \mathbf{I}_M$ is a diagonal unitary. Write $\mathbf{H}_{DD} = h_1\,\mathbf{P}\,\mathbf{D}$ where $\mathbf{P}$ is a permutation and $\mathbf{D}$ is diagonal.

$\mathbf{H}_{DD}^{-1} = h_1^{-1}\,\mathbf{D}^{-1}\,\mathbf{P}^{-1}$ — inverse of diagonal (element-wise reciprocal) and inverse of permutation (reverse permutation). Both operations are $O(MN)$.

Single-path OTFS detection is perfect zero-forcing at $O(MN)$ complexity — the simplest possible wireless receiver. Every $P > 1$ complication in OTFS detection derives from the sum of multiple shift-phase matrices no longer being a permutation. $\blacksquare$

$MN = 65{,}536$. Matrix entries: $MN^2 = 4.3 \times 10^9$ complex. At 16 bytes each: $68.7$ GB. Untenable.

$P = 16$ triples, each $\sim 16$ bytes total = 256 bytes.

Sparse storage is 2.7 × 10^8 times smaller — a difference between trivial and infeasible. This is why every practical OTFS implementation stores the channel as its path parameters, not as an explicit matrix.

The path produces a signal with instantaneous frequency $\nu_i(t)$. In the DD grid view, the Doppler index $k_i = T \nu_i$ is no longer an integer constant — it drifts.

$Y_{DD}[\ell, k] = \sum_i h_i\sum_{\Delta k} c_{i,\Delta k}\,X_{DD}[(\ell - \ell_i)\bmod M, (k - k_i^{(0)} - \Delta k)\bmod N] + W_{DD}[\ell, k]$, where $c_{i, \Delta k}$ is a spreading coefficient that decays away from $\Delta k = 0$ (Bessel-like for linear drift).

A drifting path contributes not a single shifted copy of the input but a smeared contribution across neighboring Doppler bins. The sparsity degrades from $P$ to roughly $P \cdot W_{\text{drift}}$, where $W_{\text{drift}}$ is the Doppler drift width.

Drift becomes significant when $\alpha_i T > 1/T$, i.e., the Doppler changes by more than one bin over the frame. For accelerating vehicles and LEO satellites, this is a real effect; Chapter 18 (LEO) treats it with basis-expansion models. For typical terrestrial mobility, drift over 10 ms is well within one Doppler bin — block fading is accurate.

Given $\mathbf{H}_{DD}$, the channel $\mathbf{y} = \mathbf{H}_{DD}\mathbf{x} + \mathbf{w}$ is a linear Gaussian vector channel with input power $P/MN$ per symbol. Standard MIMO capacity: $C_{\text{inst}} = \log_2\det(\mathbf{I} + (\mathrm{SNR}/MN)\mathbf{H}_{DD}^{H}\mathbf{H}_{DD})$.

Average over channel realizations: $C = \mathbb{E}[C_{\text{inst}}]$.

OFDM per-subcarrier capacity: $C_k = \log_2(1 + \text{SNR}|H_k|^2)$. Sum over subcarriers: $C_{\text{OFDM}} = \mathbb{E}\left[\sum_k \log_2(1 + \text{SNR}|H_k|^2)\right]$. For a deterministic (non-fading) channel, equal water-filling over OFDM subcarriers and DD cells yields the same capacity (same total degrees of freedom).

For high-mobility fading, OFDM's per-subcarrier outage probability is higher than OTFS's per-cell outage (due to OTFS's diversity). The ergodic capacity is nearly identical; the outage capacity (fraction of realizations with rate $\geq R$) favors OTFS at high mobility. Chapter 9 quantifies this rigorously.

The full information-theoretic comparison between OTFS and OFDM under mobility is an active research area — the diversity result of Chapter 9 is a first-order characterization, but the gap between achievable rates and capacity in finite blocklength (ITA Ch. 26) for OTFS remains open.