Ferkans — Interactive Telecom Tutor

Two Families of Mitigation

Fractional Doppler is a fact of life. Section 2 embraced it in the input-output relation; this section develops two classical mitigations that reduce the IDI severity:

Basis expansion models (BEM): represent the time-varying channel as a weighted sum of a small number of known basis functions over the OTFS frame. This is the BEM approach from FSI Ch. 11, specialized to OTFS's doubly-selective setting.
Windowing: apply a tapered window (Hamming, Blackman) along the Doppler axis before the SFFT to suppress Dirichlet sidelobes, at the cost of a slightly widened main lobe.

Neither technique eliminates IDI, but both reduce it enough that the integer-Doppler assumption becomes approximately valid — or at least, the IDI kernel's support shrinks, lowering detection complexity.

Definition:
Basis Expansion Model (BEM)

A basis expansion model represents the time-varying path gain $h_i(t)$ over a block of duration $T$ as $h_i(t) \;\approx\; \sum_{q=0}^{Q_{\text{BEM}} - 1} h_i^{(q)}\,\psi_q(t),$ where $\{\psi_q(t)\}$ are $Q_{\text{BEM}}$ known basis functions (typically complex exponentials, polynomials, or DPSS) and $\{h_i^{(q)}\}$ are the BEM coefficients to be estimated.

The most common choice — the complex-exponential BEM (CE-BEM) — uses $\psi_q(t) = e^{j 2\pi q t/T}$ for $q = -Q/2, \ldots, Q/2 - 1$ with $Q = Q_{\text{BEM}}$ an odd integer capturing the maximum Doppler rate.

,

Theorem: BEM Converts Fractional to Integer Doppler

Under the CE-BEM approximation with $Q_{\text{BEM}}$ basis functions, the fractional-Doppler channel reduces to an effective $P \cdot Q_{\text{BEM}}$ -path channel with integer Doppler offsets. The IDI kernel collapses to $\delta$ -functions at the BEM-basis Doppler shifts.

Formally: $h_i(t) = h_i\,e^{j 2\pi \nu_i t} \approx h_i \sum_{q=0}^{Q_{\text{BEM}}-1} c_q(\nu_i T)\,e^{j 2\pi q t/T},$ where $c_q(\cdot)$ are Fourier coefficients. Each BEM term contributes a single integer-Doppler path at bin $q$ , with amplitude $h_i c_q(\nu_i T)$ . The total effective path count is $P \cdot Q_{\text{BEM}}$ at integer Doppler positions.

BEM trades one fractional-Doppler path for $Q_{\text{BEM}}$ integer-Doppler paths. At $Q_{\text{BEM}} = 3$ , a single path becomes three effective paths — with total energy preserved but distributed across three Doppler bins. After BEM, the integer-Doppler detector of Chapter 8 applies directly, with the caveat that the "number of paths" is now $P \cdot Q_{\text{BEM}}$ .

Proof

Fourier series

Over $[0, T]$ , the time-varying phase $e^{j 2\pi \nu_i t}$ is periodic (period $T/(\nu_i T)$ ). Its Fourier series on $[0, T]$ has coefficients $c_q(\nu_i T) = (1/T)\int_0^T e^{j 2\pi \nu_i t}\,e^{-j 2\pi q t/T}\,dt = \text{sinc}(\nu_i T - q)\,e^{-j\pi(\nu_i T - q)(1 - 1/T)}$ .

Truncation

Keep only the $Q_{\text{BEM}}$ largest Fourier coefficients (typically centered around $q = k_i^{\text{int}}$ ). The error decays as $1/Q_{\text{BEM}}$ .

Effective path model

Each retained Fourier term contributes an integer-Doppler path at bin $q$ with amplitude $h_i c_q(\nu_i T)$ . The original fractional path becomes $Q_{\text{BEM}}$ effective integer paths.

Integer-Doppler detection

Apply the Chapter 8 detectors (MP, LCD) directly on the expanded path list. The detector performance approaches the true ML as $Q_{\text{BEM}} \to \infty$ . In practice $Q_{\text{BEM}} = 3$ – $5$ captures $\geq 95\%$ of the IDI energy. $\blacksquare$

Key Takeaway

BEM reduces fractional Doppler to a solved problem. Choose $Q_{\text{BEM}} = 3$ for moderate Doppler or $5$ for large fractional offsets; apply Chapter 8's integer-Doppler detectors on the expanded channel. Residual error: $\lesssim 1$ dB for $Q_{\text{BEM}} = 3$ , $< 0.3$ dB for $Q_{\text{BEM}} = 5$ . The BEM approach is the workhorse of fractional-Doppler OTFS: it preserves the sparse-channel detector framework while accommodating real-world channel geometry.

BEM Coefficients for a Fractional-Doppler Path

For a single path with fractional Doppler $\epsilon$ , plot the BEM coefficients $|c_q(\epsilon)|^2$ as a function of basis index $q$ , for increasing $Q_{\text{BEM}}$ . At $\epsilon = 0$ : single spike at $q = 0$ . At $\epsilon = 0.4$ : energy spread across $q = 0, 1, -1, 2$ . The BEM captures most of the energy in the first $Q_{\text{BEM}}$ terms; the residual is the approximation error.

Parameters

Fractional offset

\epsilon

0.3

Q_{\text{BEM}}

3

Definition:
Doppler-Axis Windowing

Doppler windowing applies a tapered window $w[k]$ to the $N$ OFDM symbols of the OTFS frame before the SFFT: $Y_{TF}[n, m] \to Y_{TF}[n, m] \cdot w[n], \qquad n = 0, \ldots, N - 1.$ Typical windows: Hamming ( $w[n] = 0.54 - 0.46\cos(2\pi n/N)$ ), Blackman ( $w[n] = 0.42 - 0.5\cos(2\pi n/N) + 0.08\cos(4\pi n/N)$ ), and Tukey (tapered cosine).

The effect on the IDI kernel: sidelobes are suppressed at the cost of a wider main lobe. For Hamming, sidelobes drop from $-13$ dB (no window) to $-43$ dB.

Theorem: Windowing Sidelobe Suppression vs Main-Lobe Widening

For a window $w[n]$ with equivalent noise bandwidth (ENBW) $\alpha_{\text{ENBW}}$ and peak sidelobe $\alpha_{\text{PSL}}$ (in dB), the IDI kernel under windowing satisfies:

Main-lobe width (3-dB): $\alpha_{\text{ENBW}}\cdot(1/N)$ cells. For Hamming: $\alpha_{\text{ENBW}} = 1.36$ ( $36\%$ wider than rectangular).
Peak sidelobe: $\alpha_{\text{PSL}}$ dB below main lobe. Hamming: $-43$ dB (vs $-13$ dB for rectangular).

The product $\alpha_{\text{ENBW}} \cdot 10^{\alpha_{\text{PSL}}/20}$ is a figure of merit: smaller is better. Standard windows give values in the range $[0.5, 2]$ .

Windowing is a classical DSP technique (Harris 1978) for leakage suppression. In OTFS, it trades a slight increase in effective Doppler bin size for a dramatic reduction in sidelobe energy. For fractional-Doppler OTFS at high SNR, windowing is "free" because the main-lobe widening is negligible compared to the sidelobe suppression benefit.

Windowing does not eliminate IDI — it just concentrates the leakage in fewer neighboring bins. For $k_{\text{IDI}} = 1$ under Hamming windowing (vs $k_{\text{IDI}} = 3$ for rectangular), the extended matrix has 3× fewer non-zeros, so detection is proportionally faster.

Proof

Windowed kernel

The windowed IDI kernel is $K_{\text{IDI}}^w(k, \epsilon) = \sum_n w[n]\,e^{j 2\pi \epsilon n/N}\,e^{-j 2\pi k n/N}/\sqrt{N}$ . For $w[n] = 1$ (rectangular), this is the Dirichlet kernel. For general $w$ , it is the Fourier transform of $w[n]\,e^{j 2\pi \epsilon n/N}$ .

Sidelobe decay

The decay of $|K_{\text{IDI}}^w(k, \epsilon)|$ for large $|k|$ is determined by the smoothness of $w$ . Smoother windows (Blackman, Blackman-Harris) give faster sidelobe decay.

Main-lobe ENBW

The 3-dB main-lobe width of $|K_{\text{IDI}}^w|^2$ is $\alpha_{\text{ENBW}}/(N\Delta\nu)$ , a factor $\alpha_{\text{ENBW}} > 1$ wider than the unwindowed case.

Net effect

Hamming: main lobe 36% wider, sidelobes 30 dB lower. Blackman: main lobe 75% wider, sidelobes 58 dB lower. Rectangular: baseline (no widening, $-13$ dB sidelobes). $\blacksquare$

Windowing Functions for OTFS

Window	Main-lobe (ENBW)	Peak sidelobe (dB)	When to use
Rectangular (no window)	1.0	-13	Theoretical baseline
Hamming	1.36	-43	Moderate fractional (standard choice)
Blackman	1.73	-58	Large fractional ( $\epsilon > 0.3$ )
Blackman-Harris	1.90	-92	Very large fractional (research)
Tukey ( $\alpha = 0.5$ )	1.22	-25	Light windowing, compromise

Windowed IDI Kernels: Shape vs Window

Plot the IDI kernel $|K_{\text{IDI}}^w(k, \epsilon)|^2$ for several window choices (rectangular, Hamming, Blackman) at fixed $\epsilon = 0.3$ . Observe: the rectangular kernel has high sidelobes that decay slowly. Hamming suppresses sidelobes by $\sim 30$ dB. Blackman reduces them even further but widens the main lobe.

Parameters

Fractional offset

\epsilon

0.3

N

Doppler bins32

⚠️Engineering Note

Choosing Between BEM and Windowing

The two approaches are complementary:

BEM: explicit model of the IDI kernel. Good for high-precision detection (low BER targets). Requires estimating additional path parameters. Overhead: $O(P \cdot Q_{\text{BEM}})$ parameters vs $O(P)$ integer case. Optimal when the fractional offsets are known or can be estimated.
Windowing: pre-processing step with no additional parameters. Zero-cost detection (same detector runs on the windowed grid). Suppresses sidelobes but cannot eliminate the main-lobe leakage. Best for moderate fractional offsets and when detector simplicity matters.

Combined approach: Hamming windowing + $Q_{\text{BEM}} = 3$ often gives the best trade-off: windowing suppresses far-out sidelobes; BEM captures residual main-lobe spread. Net: $\sim 0.5$ dB from ML at typical parameters.

For 5G NR-aligned OTFS: Hamming + $Q_{\text{BEM}} = 3$ is the reference implementation. For LEO satellite: Blackman + $Q_{\text{BEM}} = 5$ .

Practical Constraints

•
BEM needs path Doppler estimate ( $\epsilon$ ) to set coefficients
•
Windowing adds no parameters but can't fully remove IDI
•
Best-of-both: windowing for sidelobes + BEM for main lobe

Example: BER With BEM vs Integer Assumption

Compare BER at SNR = 15 dB for an OTFS system with $P = 4$ paths, all at $\epsilon = 0.3$ , under (a) integer assumption (ignore IDI), (b) BEM with $Q_{\text{BEM}} = 3$ .

Solution

Integer assumption (bad)

Main-lobe capture: $|\text{sinc}(0.3)|^2 = 0.84$ . 16% energy lost per path; 64% across all 4 paths. Effective SNR drop: 10 log(1/(1-0.64)) = 4.4 dB. BER at 10.6 dB effective: $\sim 2 \times 10^{-2}$ (QPSK).

BEM, $Q = 3$ (good)

Captures $\sim 95\%$ of IDI energy. Effective SNR drop: $\sim 0.3$ dB. BER at 14.7 dB effective: $\sim 10^{-4}$ (QPSK).

Improvement

BEM recovers ~2 orders of BER magnitude at 15 dB SNR. Complexity cost: $\sim 3\times$ more factor graph connections (from $P = 4$ to $P \cdot Q = 12$ effective paths). Still realtime.

🎓CommIT Contribution(2020)

BEM-OTFS for Fractional Doppler

G. D. Surabhi, A. Chockalingam, G. Caire — IEEE Trans. Vehicular Technology

Surabhi, Chockalingam, and Caire extended the diversity analysis of OTFS Chapter 9 to fractional-Doppler channels. Their key result: with BEM of order $Q_{\text{BEM}}$ , OTFS recovers diversity order $\min(P, \min(Q_{\text{BEM}}, k_{\max}))$ — the minimum of path count and the BEM's effective Doppler resolution.

The CommIT contribution refines the naive fractional-Doppler treatment by showing that BEM's effective Doppler resolution is the fundamental limit — beyond $Q_{\text{BEM}} = 2 k_{\max} + 1$ , no gain is obtained. This informs the reference implementation's choice of $Q_{\text{BEM}} = 3$ for typical terrestrial channels.

commitbemfractional-dopplerdiversity

Basis Expansion Models and Windowing

Two Families of Mitigation

Definition: Basis Expansion Model (BEM)

Theorem: BEM Converts Fractional to Integer Doppler

Fourier series

Truncation

Effective path model

Integer-Doppler detection

Key Takeaway

BEM Coefficients for a Fractional-Doppler Path

Parameters

Definition: Doppler-Axis Windowing

Theorem: Windowing Sidelobe Suppression vs Main-Lobe Widening

Windowed kernel

Sidelobe decay

Main-lobe ENBW

Net effect

Windowing Functions for OTFS

Windowed IDI Kernels: Shape vs Window

Parameters

Choosing Between BEM and Windowing

Example: BER With BEM vs Integer Assumption

Integer assumption (bad)

BEM, $Q = 3$ (good)

Improvement

BEM-OTFS for Fractional Doppler

Definition:
Basis Expansion Model (BEM)

Definition:
Doppler-Axis Windowing