Ferkans — Interactive Telecom Tutor

Sidelobes and the Window

Section 2 fixed the transmit and receive pulses for bi-orthogonality. A complementary knob is the window — a smooth function multiplied into the DD-domain signal before ISFFT, or into the received signal before SFFT. Windows control the spectral sidelobes of the OTFS symbol, at the cost of main-lobe widening. This is the same trade-off every DFT-based analysis faces, here applied in the OTFS context.

Definition:
DD-Domain Windowing

An OTFS window $w[\ell, m]$ is a 2D function applied to the DD-domain symbol matrix $s[\ell, m]$ : $\tilde{s}[\ell, m] \;=\; w[\ell, m] \cdot s[\ell, m].$ The windowed symbols are then passed through the ISFFT + Heisenberg modulator.

Typical window choices:

Rectangular ( $w = 1$ ): no windowing. Sharp edges $\to$ high sidelobes.
Hamming: moderate suppression, 3 dB main lobe widening.
Blackman: strong suppression, 4 dB widening.
Kaiser: $\beta$ -parameterized. $\beta = 6$ : -40 dB sidelobes.
Nuttall: 4-term minimum. -98 dB sidelobes.

Effect on OTFS: sidelobe reduction in the time-frequency output spectrum. Important for:

Out-of-band emission (OOBE): regulatory compliance.
Adjacent-channel interference: coexistence with other services.
PAPR: window shapes affect peak power.

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Theorem: Window Sidelobe-Widening Trade-off

For a window function $w(t)$ of duration $T_s$ , the main-lobe 3-dB width $B_{\text{ML}}$ and sidelobe level SLL satisfy approximately $B_{\text{ML}} \cdot \mathrm{SLL} \;\geq\; \mathrm{const}.$ Concretely:

Rectangular: $B_{\text{ML}} = 0.89/T_s$ , SLL = -13 dB.
Hamming: $B_{\text{ML}} = 1.30/T_s$ , SLL = -43 dB.
Blackman: $B_{\text{ML}} = 1.68/T_s$ , SLL = -58 dB.
Nuttall: $B_{\text{ML}} = 1.98/T_s$ , SLL = -98 dB.

Trade-off: doubling the main lobe (widening by 2) cuts sidelobes by $\sim 30$ dB.

Windows redistribute energy between the main lobe and sidelobes. A sharper window (rectangular) has the narrowest main lobe but pays 13 dB sidelobes. Smoother windows have wider main lobes but dramatically suppress sidelobes. The engineer's choice: narrow main lobe for spectral efficiency, low sidelobes for coexistence. Most OTFS implementations use Hamming or Blackman — a compromise.

Proof

Fourier pair

Window $w(t)$ , response $W(f) = \int w(t) e^{-j2\pi ft} dt$ . Main lobe: $|W(f)|$ near $f = 0$ . Sidelobes: further out.

Energy conservation

$\int |w(t)|^2 dt = \int |W(f)|^2 df$ (Parseval). Fixed energy; redistribution between main lobe and sidelobes.

Widening and SLL

Narrower time support (sharper cutoff) $\Rightarrow$ wider frequency content, higher sidelobes. Smoothing the edges narrows the frequency content.

Empirical constant

Harris 1978 catalogue: empirical main-lobe × SLL product is $\sim -15$ dB cycles/sec. $\blacksquare$

Definition:
OTFS Window Types

Four OTFS-specific windowing approaches:

Rectangular (no window): baseline. Used when sidelobes don't matter (isolated band).

TX-only windowing: $w[\ell, m]$ applied at transmitter. Shapes emitted spectrum.

RX-only windowing: $w[\ell, m]$ applied at receiver. Improves SINR at cost of noise filtering.

TX+RX dual window: windows at both ends. Combined effect. Most aggressive sidelobe suppression.

Separable DD: $w[\ell, m] = w_\tau[\ell] \cdot w_\nu[m]$ . Simplifies implementation. Most common.

Non-separable: $w[\ell, m]$ full 2D. More flexible but complex. Used for tight OOBE constraints.

OTFS Window Spectral Comparison

Plot OTFS symbol spectrum for different window choices (rectangular, Hamming, Blackman, Kaiser, Nuttall). Shows main-lobe and sidelobe trade-offs. Sliders: window type, Kaiser $\beta$ .

Parameters

Window type

Kaiser

\beta

6

Samples256

Theorem: OTFS Out-of-Band Emission

For OTFS with windowed Tx pulse, the out-of-band emission (OOBE) at offset $f_{\text{OOB}}$ from band edge is $\mathrm{OOBE}(f_{\text{OOB}}) \;\approx\; \mathrm{SLL}(f_{\text{OOB}}) + 10 \log_{10}(M),$ where SLL is the window's single-tone sidelobe and $M$ is the number of OTFS subcarriers.

3GPP OOBE mask (5G NR): $\leq -30$ dBc at 1 MHz offset, $\leq -50$ dBc at 10 MHz.

For $M = 256$ subcarriers: $M$ -factor adds 24 dB. Window SLL requirement: -30 dBc + 24 dB = -54 dB sidelobes. Hamming achieves -43 dB (insufficient). Blackman: -58 dB (OK). Blackman is the minimum for 5G NR compliance.

The factor $10 \log(M)$ is the "sum of many subcarriers" boost. Each subcarrier contributes its own sidelobe; they add up. For $M = 256$ , the boost is 24 dB — requiring more aggressive windows than a single-tone analysis would suggest. This is why Blackman or Nuttall windows are preferred in 5G NR and likely in 6G OTFS.

Proof

Single-tone sidelobe

One subcarrier at edge: $|W(f - f_{\text{edge}})| \leq \mathrm{SLL}$ .

Many subcarriers

Sum of $M$ subcarriers with random phases. Coherent sum of $M$ sidelobes: $\sqrt{M}$ amplitude, or $10 \log M$ in power.

Total OOBE

$\mathrm{OOBE} = \mathrm{SLL} + 10 \log M$ dB.

Regulatory

3GPP mask -30 dBc → window SLL must be ≤ -30 - 10 log 256 = -54 dB. Blackman (-58 dB) passes; Hamming (-43 dB) fails. $\blacksquare$

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Key Takeaway

Blackman or Nuttall windows for 5G/6G OTFS. Hamming is insufficient at typical subcarrier counts ( $M \geq 256$ ) due to the $10 \log M$ sidelobe-sum penalty. Commercial OTFS implementations should use Blackman baseline, Nuttall for tight OOBE constraints.

Example: Window Selection for Enterprise 6G OTFS

A 6G OTFS deployment in a dense enterprise environment must comply with 3GPP OOBE mask (-30 dBc at 1 MHz, -50 dBc at 10 MHz) and $M = 256$ subcarriers, $W_s = 15$ kHz. Choose window.

Solution

Requirement

Window SLL $\leq -30 - 10 \log 256 = -54$ dB at 1 MHz.

Options

Hamming: -43 dB. Fails.
Kaiser $\beta = 10$ : -70 dB. Passes.
Blackman: -58 dB. Passes with margin.
Nuttall: -98 dB. Over-engineered.

Recommendation

Blackman. Meets regulation with 4 dB margin. Main-lobe widening ~1.7× — acceptable.

PAPR impact

Blackman window reduces PAPR by ~1 dB vs rectangular (smoother envelope). Net gain: regulatory compliance + PAPR improvement.

🔧Engineering Note

Window Choices in Commercial OTFS

Commercial OTFS window choices:

Cohere reference design: Kaiser $\beta = 6$ . Balance of sidelobes and main-lobe width.
5G NR-compatible OTFS: Blackman. Standard regulatory compliance.
LEO-OTFS (Buzzi-Caire): Nuttall 4-term. Tight OOBE for coexistence with terrestrial.
Low-cost OTFS (IoT): Hamming. Skips regulatory compliance, uses out-of-band masking via filtering.

Hardware: window coefficients pre-computed and stored in ROM. $M = 256$ window: 1 KB of storage. Multiplication per-symbol: modest compute.

Dynamic switching: scheduler can pick window per-slot based on coexistence requirements. E.g., indoor dense: Blackman. Remote outdoor: Hamming.

Practical Constraints

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Default: Blackman (5G NR compliant)
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Tight OOBE: Nuttall
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Low-cost IoT: Hamming + filter
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Hardware cost: 1 KB ROM + 1 mult per sample

Common Mistake: Windowed OTFS Breaks Bi-Orthogonality

Mistake:

Applying a Tx window without a corresponding Rx window. The bi-orthogonality condition of §2 assumed no windowing. With a Tx window only, the ambiguity function is distorted, and the DD grid is no longer orthogonal.

Correction:

For bi-orthogonal operation with windowing: apply matched windows at Tx and Rx. Their product in the ambiguity function preserves bi-orthogonality approximately. At high SNR and good window choice, the degradation is $< 0.5$ dB. Alternative: use Tx window for OOBE compliance and accept $\sim 0.5$ dB detector penalty. Commercial implementations typically window only at Tx (for regulatory) and compensate with slightly wider Rx filter.

Window Design for Sidelobe Control