Ferkans — Interactive Telecom Tutor

The IDI Effect on the Input-Output Relation

Section 1 established that a single fractional-Doppler path smears its DD response across multiple bins via the IDI kernel. This section extends the full $P$ -path input-output relation to account for fractional offsets. The result is a sparse-but-extended channel model: instead of $P$ path taps, we have $P \times (2k_{\text{IDI}} + 1)$ effective taps, where $k_{\text{IDI}}$ is the number of neighboring Doppler bins with significant leakage.

The point is that the input-output relation remains linear — the IDI kernel is a fixed function of $(k, \epsilon)$ , so the DD output is still a sparse 2D convolution. The convolution kernel is just richer (multi-tap instead of single-tap per path).

Theorem: DD Input-Output Under Fractional Doppler

For a channel with $P$ paths where each path $i$ has integer delay $\ell_i$ , integer Doppler $k_i$ , and fractional offset $\epsilon_i$ , the discrete DD input-output relation is $Y_{DD}[\ell, k] \;=\; \sum_{i=1}^{P}\sum_{q \in \mathcal{N}_i} h_i\,K_{\text{IDI}}(q, \epsilon_i)\,X_{DD}[(\ell-\ell_i)\bmod M,\,(k-k_i-q)\bmod N] \;+\; W_{DD}[\ell, k],$ where $\mathcal{N}_i = \{-k_{\text{IDI}}, \ldots, k_{\text{IDI}}\}$ is the effective IDI support per path, typically $k_{\text{IDI}} = 2$ for $\epsilon \leq 0.5$ .

The equivalent extended channel matrix $\mathbf{H}_{DD}^{\text{ext}}$ is sparse with $P \cdot (2 k_{\text{IDI}} + 1)$ taps per row — a modest expansion from the integer-case $P$ taps.

Each physical path contributes not a single DD-convolution tap but a small cluster of taps (the IDI kernel). The total number of taps is $P \cdot (2 k_{\text{IDI}} + 1)$ ; for $P = 10$ and $k_{\text{IDI}} = 2$ , this is 50 taps — still far less than the dense $MN$ taps of OFDM. OTFS sparsity is preserved, just at a larger effective $P$ .

Proof

Per-path contribution

From Theorem TDD Response Under Fractional Doppler, path $i$ contributes to cell $(\ell, k)$ in proportion to $K_{\text{IDI}}(k - k_i, \epsilon_i)$ , shifted by the integer indices $(\ell_i, k_i)$ .

Linearity

Multiple paths sum linearly: $Y[\ell, k] = \sum_i h_i \cdot \text{(path$ i $'s smeared contribution)}$ . The path $i$ response at shift $q$ is $K_{\text{IDI}}(q, \epsilon_i)$ scaled.

Convolution form

Truncating to $|q| \leq k_{\text{IDI}}$ (essentially zero contribution beyond this range), the sum is a 2D convolution of $X_{DD}$ with the extended kernel.

Sparsity

The extended kernel is sparse: $P (2 k_{\text{IDI}} + 1)$ non-zero taps out of $MN$ . At $P = 10, k_{\text{IDI}} = 2$ : 50 taps; density $\sim 10^{-3}$ on typical grids — still orders of magnitude sparser than OFDM's dense TF matrix. $\blacksquare$

,

Key Takeaway

Fractional Doppler widens the channel, not its density class. The OTFS channel under fractional offsets has $P(2k_{\text{IDI}} + 1)$ effective taps, typically $5P \sim 50$ . Still far sparser than OFDM's $MN \sim 10^4$ dense matrix. The detectors of Chapter 8 (MP, LCD) remain applicable; they just operate on a slightly richer sparse factor graph.

IDI Power vs Fractional Offset $\epsilon$

Plot the IDI power fraction — energy leaked to neighboring Doppler bins — as a function of $\epsilon \in [0, 0.5]$ . At $\epsilon = 0$ : zero leakage (integer case). At $\epsilon = 0.5$ : worst case, ~60% of energy in the main lobe, 40% leaked to neighbors. Quantifies when fractional effects matter.

Parameters

Doppler bins

N

32

IDI support

k_{\text{IDI}}

3

Theorem: Extended Channel Matrix Structure

The extended channel matrix $\mathbf{H}_{DD}^{\text{ext}} \in \mathbb{C}^{MN \times MN}$ can be written as $\mathbf{H}_{DD}^{\text{ext}} \;=\; \sum_{i=1}^{P}\sum_{q=-k_{\text{IDI}}}^{k_{\text{IDI}}} h_i\,K_{\text{IDI}}(q, \epsilon_i)\,\boldsymbol{\Psi}_{\ell_i, k_i + q},$ where $\boldsymbol{\Psi}_{\ell, k}$ is a sparse shift-phase matrix (one non-zero per row), so the total matrix has at most $P(2k_{\text{IDI}} + 1) \cdot MN$ non-zeros. Like the integer-case matrix, $\mathbf{H}_{DD}^{\text{ext}}$ is block-circulant with circulant blocks (with fractional-shifted eigenvalues).

The extended matrix has the same block-circulant structure as the integer case but with more non-zero entries per row. The 2D DFT still diagonalizes it (providing efficient MMSE and LCD), and message passing still works (factor-graph degree bounded by $(2k_{\text{IDI}}+1)P$ ).

Proof

Per-path decomposition

Path $i$ 's contribution is a sum of $(2k_{\text{IDI}} + 1)$ shift-phase matrices weighted by IDI kernel values $K_{\text{IDI}}(q, \epsilon_i)$ .

Block-circulant preserved

Each $\boldsymbol{\Psi}_{\ell_i, k_i + q}$ is a block-circulant shift matrix. Linear combinations are block-circulant. The full matrix $\mathbf{H}_{DD}^{\text{ext}}$ retains this structure.

Eigenvalues

2D DFT diagonalizes $\mathbf{H}_{DD}^{\text{ext}}$ with eigenvalues $\lambda[n, m] = \sum_{i, q} h_i K_{\text{IDI}}(q, \epsilon_i)\,e^{j 2\pi(k_i + q)n/N}\,e^{-j 2\pi \ell_i m/M}$ . Same structural form as integer case, just with $(2k_{\text{IDI}}+1)$ -fold more terms per path. $\blacksquare$

Example: SNR Degradation From IDI

An OTFS system with $N = 32$ Doppler bins receives a single path with $\epsilon = 0.4$ . Using a detector that assumes integer Doppler (no IDI handling), compute the effective SNR loss.

Solution

Main-lobe energy

$|K_{\text{IDI}}(0, 0.4)|^2 \approx \text{sinc}^2(0.4) = 0.68$ . The integer-Doppler detector assigns all path energy to bin 0; it captures only 68% of the actual energy.

IDI leakage (treated as noise)

Leaked energy: $1 - 0.68 = 32\%$ . At bins $\pm 1, \pm 2$ : $\sim 30\%$ leakage. Treated as noise by integer-Doppler detector.

Effective SNR

$\text{SNR}_{\text{eff}} = \text{SNR}_{\text{actual}} \cdot 0.68 / (1 + 0.32 \cdot \text{SNR}_{\text{actual}})$ . At true SNR = 20 dB: $\text{SNR}_{\text{eff}} = 100 \cdot 0.68/(1 + 32) = 2.06$ (3.1 dB). 17 dB penalty from ignoring IDI.

Fractional-aware detector

With IDI-aware detection (e.g., LCD extended to include IDI kernel), the leakage becomes signal rather than noise: recovered SNR back toward 20 dB. Typical penalty: 0.5-1 dB from detector sub-optimality, vs. 17 dB from ignoring IDI entirely.

Extended DD Channel Matrix Sparsity Pattern — Sparsity pattern of the DD channel matrix under integer-Doppler (left) and fractional-Doppler (right) assumptions, for $P = 4$ paths on an $(M, N) = (16, 8)$ grid. Integer case: $P \cdot MN = 512$ nonzeros, density $1.6\%$ . Fractional case ( $k_{\text{IDI}} = 2$ ): $5P \cdot MN = 2560$ nonzeros, density $7.8\%$ . Still far sparser than OFDM's dense $MN \times MN$ structure under high Doppler.

Fractional-Aware MMSE via Extended Matrix

Complexity:

O(MN\log(MN))

Input: Received

\mathbf{y}_{DD}

, path parameters

\{(h_i, \ell_i, k_i, \epsilon_i)\}_{i=1}^P

,

IDI support

k_{\text{IDI}}

, noise variance

\sigma^2

Output: MMSE estimate

\hat{X}_{DD}

1. Compute extended eigenvalues:

\lambda[n, m] = \sum_{i=1}^P \sum_{q=-k_{\text{IDI}}}^{k_{\text{IDI}}} h_i\,K_{\text{IDI}}(q, \epsilon_i)\,e^{j 2\pi(k_i + q)n/N}\,e^{-j 2\pi \ell_i m/M}

.

2. 2D FFT of received grid:

\tilde{\mathbf{y}} = \mathbf{F}\mathbf{y}_{DD}

.

3. Element-wise Wiener:

\tilde{x}[n, m] = \lambda^*[n, m]\,\tilde{y}[n, m]/(|\lambda[n, m]|^2 + \sigma^2)

.

4. Inverse 2D FFT:

\hat{X}_{DD} = \mathbf{F}^H\tilde{\mathbf{x}}

.

5. Quantize to QAM alphabet.

6. Return

\hat{X}_{DD}

.

The algorithm is structurally identical to integer-Doppler MMSE (Algorithm ADD-Domain MMSE Detection via 2D FFT) — only the eigenvalues $\lambda[n, m]$ change. Fractional offsets affect only the pre-computed eigenvalue sum, not the runtime path. Total complexity: same $O(MN\log(MN))$ as integer case.

Why This Matters: Message Passing With Fractional Doppler

The MP-OTFS algorithm of Chapter 8 extends to fractional Doppler with modest modifications: each factor node now has $(2k_{\text{IDI}}+1)\cdot P$ incident variables (instead of $P$ ). Messages still propagate; BP still converges in $O(\log(MN))$ iterations. Complexity goes up from $O(P \cdot MN)$ to $O((2k_{\text{IDI}}+1) P \cdot MN)$ — a constant factor ( $\sim 5$ ) for typical $k_{\text{IDI}} = 2$ . Practical impact: 5× more compute but same algorithmic framework.

Inter-Doppler Interference (IDI)