Gaussian-Sinc Pulse Shape

• The Gaussian-Sinc (GS) pulse shape is a composite function that combines the sinc function's orthogonality with the Gaussian's time-frequency localization for optimal signal processing.
• It balances inherent trade-offs by providing strong ISI suppression through nulls at sampling instants while minimizing spectral leakage, ensuring high channel estimation accuracy.
• Its implementation in systems like Zak–OTFS achieves significant SNR gains and inspires further optimizations, such as Hermite-based expansions, for improved communication performance.

The Gaussian-Sinc (GS) pulse shape is a composite pulse that combines the orthogonality and zero-interference properties of the sinc function with the strong time-frequency localization of the Gaussian. It plays a pivotal role in modern communication and sensing systems, particularly within frameworks such as Zak–OTFS, where simultaneous requirements of channel estimation accuracy, data detection performance, and spectral efficiency must be balanced.

1. Definition and Mathematical Formulation

The GS pulse is defined, in its canonical form for delay-Doppler (DD) processing (e.g., Zak–OTFS), as a separable product of sinc and Gaussian functions in each dimension: $$w_{tx}(\tau, \nu) = \Omega_\tau \Omega_\nu \sqrt{BT} \, \mathrm{sinc}(B\tau) \, \mathrm{sinc}(T\nu) \, \exp(-\alpha_\tau B^2 \tau^2) \exp(-\alpha_\nu T^2 \nu^2)$$ where:

• $B$ and $T$ are the prescribed bandwidth and time-duration (without expansion);
• $\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$ enforces nulls at the Nyquist sampling points, promoting orthogonality;
• $\exp(-\alpha B^2\tau^2)$ (and analogously in $\nu$) is the Gaussian window, providing energy localization and sidelobe suppression;
• $\Omega_\tau, \Omega_\nu$ are normalization factors to enforce unit energy;
• $\alpha_\tau, \alpha_\nu$ are roll-off parameters chosen (e.g., $0.044$) to achieve no excess bandwidth or time expansion (Das et al., 6 Feb 2025, Mehrotra et al., 16 Oct 2025, Jesbin et al., 20 Oct 2025).

This construction achieves joint benefits: the sinc component cancels inter-symbol interference (ISI) at sampling instants; the Gaussian envelope suppresses spectral/temporal leakage, facilitating robust channel estimation and practical implementation.

2. Trade-offs in Pulse Shape Design

Pulse shaping in the DD domain is subject to the Balian–Low Theorem, which enforces a trade-off between perfect time–frequency localization, or "concentration", and strict lattice orthogonality for critically sampled systems (Jesbin et al., 20 Oct 2025). Table 1 summarizes the core trade-offs:

Pulse	Orthogonal on DD Lattice	Localized (Low Side Lobes)	Time/Bandwidth Expansion
Sinc	✔	✗ (High Sidelobes)	✗
Gaussian	✗	✔	✗
RRC	✔	✔	✔ (Expansion Required)
GS (Composite)	~ (Good Nulls, Not Exact)	✔ (Lower Sidelobes)	✗
IOTA	✔	✔	✗

This table is drawn directly from cited works; see (Das et al., 6 Feb 2025, Mehrotra et al., 16 Oct 2025), and (Jesbin et al., 20 Oct 2025) for detailed filter characterizations.

The GS pulse is engineered as a compromise, balancing the need for lattice-orthogonality (sinc nulls) and energy localization (Gaussian window), avoiding time-frequency expansion that would reduce spectral efficiency.

3. Implementation in Zak–OTFS and Communication Systems

In Zak–OTFS, the pulse shaping filter determines both the system's resistance to ISI and its ability to yield accurate, sparse representations of the physical channel. The GS filter is implemented directly as a transmitter pulse shape and is matched or near-matched at the receiver (Das et al., 6 Feb 2025). Explicit closed-form expressions for the effective channel and noise covariance are available, simplifying both theoretical analysis and simulation.

When transmitted, the GS pulse enables:

• Lower side lobes than the sinc pulse, improving channel/input-output (I/O) relation estimation.
• Preserved nulls at DD sampling points, reducing ISI and aiding data detection.
• No throughput loss, as the time and bandwidth dimensions are not expanded—unlike RRC filters.

Performance-wise, the GS filter achieves an SNR gain of about 4 dB at $10^{-2}$ uncoded BER over both pure Gaussian and sinc filters, increasing to over 6 dB at coded BER $10^{-4}$ (Das et al., 6 Feb 2025).

4. Orthogonality, Localization, and the IOTA Procedure

Although the GS pulse achieves a favorable compromise, it is not strictly orthogonal on the DD lattice, leading to residual ISI under practical non-idealities (Mehrotra et al., 16 Oct 2025). The Isotropic Orthogonal Transform Algorithm (IOTA) is applied to generate maximally localized, strictly orthogonal pulses within the original time-bandwidth limits:

• A prototype pulse (often Gaussian or prolate spheroidal) is mapped onto the DD lattice, forming a matrix of shifts.
• The Gram matrix of shifted pulses is "whitened" (inverse square root), resulting in an orthogonal set.
• The resultant IOTA pulse maintains localization comparable to Gaussian/GS prototypes, but gains strict orthogonality and no time-bandwidth expansion (Mehrotra et al., 16 Oct 2025).

This approach provides a true simultaneous optimization of three objectives: localization (for I/O estimation), orthogonality (for detection), and spectral efficiency (no expansion).

5. Extensions: Hermite Basis and Systematic Design

Generalizing beyond the ad hoc GS construction, the pulse can be expanded as a linear combination of scaled Hermite basis functions: $$w_1(\tau) = \sum_{n=0}^{N_c-1} c_{2n} \varphi_{2n}(\sigma_\tau, \tau)$$ with $\varphi_{2n}$ denoting scaled Hermite functions, even orders chosen for symmetry (Jesbin et al., 20 Oct 2025). The coefficients $c_{2n}$ are optimized (e.g., via singular value decomposition) to minimize ISI energy at sampling points while enforcing a unit-norm constraint.

Simulation results demonstrate Hermite-based designs can match or slightly outperform the GS pulse in bit error rate (BER), achieving near-ideal ISI suppression and strong localization.

6. Broader Context: Analogous Constructions in Signal Processing and Physics

The Gaussian-sinc functional form—or close analogs—arise in diverse settings:

• Interpolating ISI-free kernels: In shift-invariant spaces with Gaussian generators, spectral factorization produces "root" pulses analogous to the GS pulse in communication (0907.2412).
• Bayesian inference for band-limited signals: The Whittaker–Shannon interpolation formula, central to band-limited Gaussian processes with sinc kernels, achieves exact recovery at Nyquist rates; combining with Gaussian priors yields GPs with both localization and band-limitation (Tobar, 2019).
• Ultrafast optics: Sinc and Gaussian envelopes are employed for coherent control and attosecond pulse generation; composite pulses exploit broad spectra (sinc) and temporally localized excitation (Gaussian) to achieve robust control and bandwidth broadening (Kumar et al., 2012, Rajpoot et al., 2019).

A plausible implication is that the GS pulse's formulation leverages structurally optimal properties from both domains—vanishing ISI (sinc) and minimal out-of-band/inter-symbol leakage (Gaussian)—and finds utility wherever strict trade-offs between infinity-localization and orthogonality must be navigated.

7. Significance and Practical Impact

The GS pulse is a critical enabler for high-mobility, doubly selective channels and integrated sensing/communication (ISAC) scenarios, as it allows:

• Efficient, practical channel estimation due to lower side lobe energy (fewer aliasing artifacts);
• High data detection accuracy through strong ISI suppression without sacrificing throughput;
• Adaptability to different channel and system constraints due to parametric (bandwidth, roll-off) control.

Overall, the GS pulse serves as a foundation for advanced pulse shape optimization, and its design approach—integration of nulling, localization, and strict resource (time/bandwidth) constraints—is central to the development of next-generation communication and radar systems (Das et al., 6 Feb 2025, Mehrotra et al., 16 Oct 2025, Jesbin et al., 20 Oct 2025).