Frequency-Domain Spectral Shaping (FDSS)

• FDSS is a spectrum-engineering paradigm that applies deterministic frequency-domain filters to modify signal amplitude and phase, controlling PAPR, OOBE, and interference.
• It is widely utilized in 5G uplink systems and quantum-optical applications to optimize power efficiency, tailor spectral profiles, and improve modulation performance.
• Various design techniques—ranging from fixed windowing and polynomial-parameterized filters to cancellation-carrier methods—highlight tradeoffs between PAPR reduction, SER preservation, and spectral flatness.

Frequency-Domain Spectral Shaping (FDSS) denotes the deliberate modification of a signal’s spectral amplitude and, in some implementations, its spectral phase, in order to control downstream properties such as peak-to-average power ratio (PAPR), out-of-band emission (OOBE), temporal interference structure, comb-line occupancy, or nonlinear tolerance. In the canonical DFT-s-OFDM literature, FDSS is the multiplication of DFT-spread subcarriers by a deterministic frequency-domain window before OFDM modulation; in closely related quantum and photonic work, the same operative idea appears as direct engineering of optical, comb, or biphoton spectra in one- or two-dimensional frequency space (Carpi et al., 2024, Pitaval, 13 Oct 2025, Jin et al., 2018).

1. Formal definitions and common mathematical forms

In DFT-s-OFDM with spectral extension, FDSS is represented by a frequency-domain filter $F=[F_1,\dots,F_{sc}]$ applied elementwise to the extended spectrum,

$\tilde{X}=X^{\text{ext}}\odot F,$

after which the waveform is obtained by IFFT and transmitted; matched equalization at the receiver is likewise implemented by multiplication with $F^*$ (Carpi et al., 2024). In SC-FDMA/LFDMA analyses, the same operation is modeled as a transmit frequency response $H$, whose time-domain equivalent is a circular-convolution kernel $h(n)$, so that spectral shaping appears not as an ancillary heuristic but as part of the physical modulation chain (Benammar et al., 2014).

A second widely used formulation arises in OFDM spectral-containment work, where each subcarrier pulse is redesigned as a generalized pulse

$$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$

Here $\mathbf{p}_k$ is the conventional OFDM pulse, $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$ is a cancellation-carrier term, and $\mathbf{t}_k$ is a transition pulse active only near symbol edges. This unifies active interference cancellation (AIC) and time-domain pulse shaping within a single data-independent spectral-shaping framework (Díez et al., 2018).

A broader reading of the literature shows that FDSS is not restricted to classical multicarrier communications. In quantum-optical biphoton shaping, the engineered state is explicitly written in frequency space as

$$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$

so that controllable placement of spectral components and controllable relative phase become the essential shaping variables (Jin et al., 2018). This suggests that, across domains, FDSS is best understood as a spectrum-engineering paradigm rather than a single protocol.

2. Canonical FDSS in DFT-s-OFDM and SC-FDMA uplink systems

In 5G-style DFT-s-OFDM uplink systems, FDSS is primarily used to reduce PAPR and improve power-amplifier efficiency by tapering the subcarrier-domain power profile instead of leaving it flat. A representative implementation uses a deformed Hann window $\tilde{X}=X^{\text{ext}}\odot F,$0, denoted FDSS($\tilde{X}=X^{\text{ext}}\odot F,$1), with $\tilde{X}=X^{\text{ext}}\odot F,$2 controlling the shaping strength; the average transmitted spectrum is then weighted by $\tilde{X}=X^{\text{ext}}\odot F,$3, which pushes power away from band edges and reduces spectral splatter (Pitaval, 13 Oct 2025).

The best-established performance effect is PAPR reduction. For $\tilde{X}=X^{\text{ext}}\odot F,$4-BPSK in DFT-s-OFDM, FDSS$\tilde{X}=X^{\text{ext}}\odot F,$5 dB$\tilde{X}=X^{\text{ext}}\odot F,$6 yields a maximum PAPR slightly above $\tilde{X}=X^{\text{ext}}\odot F,$7 dB at $\tilde{X}=X^{\text{ext}}\odot F,$8 CCDF, while the same style of shaping lowers QPSK PAPR less aggressively. The same paper places FDSS in a satellite-access setting and shows that FDSS influences SINR through the mean and variance of the effective equalized gain $\tilde{X}=X^{\text{ext}}\odot F,$9, so its impact is not confined to envelope smoothing; in narrowband and moderately frequency-selective channels, spectral weighting interacts with channel selectivity and equalization (Pitaval, 13 Oct 2025).

SC-FDMA analyses reach the same structural conclusion from a different angle. By recasting the localized SC-FDMA transmitter as a multirate circular-convolution system, spectral shaping is absorbed into the equivalent response $F^*$0, which then governs both the analytical PSD and the SINR. In this formulation, rectangular mapping is only the unshaped limiting case; root-raised-cosine spectral shaping widens support, changes interference structure, and can require frequency combining when duplicated bins are used (Benammar et al., 2014).

FDSS has also been reinterpreted as a waveform-synthesis mechanism rather than only a PAPR tool. In chirp-based DFT-s-OFDM, a carefully designed FDSS filter equal to the Fourier-series coefficients of a target chirp converts the nominally single-carrier-like waveform into a linear combination of circularly shifted chirps. Closed-form FDSS coefficients are derived for sinusoidal and linear chirps, and the paper shows that low-ripple chirps incur less noise enhancement and therefore better BER after MMSE equalization (Sahin et al., 2020).

3. Window design, learning, and spectrum extension

A major line of work treats FDSS design itself as an optimization problem. In learned FDSS for DFT-s-OFDM, the filter is polynomially parameterized as

$F^*$1

and optimized end-to-end under the multi-objective loss

$F^*$2

where $F^*$3 is SER-related, $F^*$4 penalizes PAPR through both $F^*$5 and AUCCDF, and $F^*$6 enforces spectral flatness. The framework explicitly incorporates spectral extension, matched FDSS equalization, and constrained zero-ISI designs based on Nyquist structure (Carpi et al., 2024).

The numerical results make the tradeoff structure unusually explicit. Relative to an RRC baseline at $F^*$7 and $F^*$8, non-flat learned filters range from Learned-A, with $F^*$9 dB SNR loss and $H$0 dB PAPR gain, to Learned-D, with $H$1 dB SNR loss and $H$2 dB PAPR gain. Under stricter link-budget constraints, Almost Flat gives $H$3 dB PAPR gain, Flat gives $H$4 dB, and Zero-ISI gives $H$5 dB with essentially no SNR loss. The same work reports that increasing the polynomial degree beyond $H$6 did not improve performance and that resampling a learned filter from $H$7, $H$8 to $H$9, $h(n)$0 preserves part of the gain but does not eliminate configuration dependence (Carpi et al., 2024).

Spectrum extension (SE) provides a complementary design axis. In FDSS-SE, the DFT-spread sequence is periodically repeated and circularly shifted before application of a real symmetric FDSS window $h(n)$1. The central result is non-monotonicity: there exists a PAPR-optimal SE size $h(n)$2 and a distinct rate-optimal SE size $h(n)$3, and larger SE is not always beneficial. For QPSK with Hann FDSS, SE gives a roughly $h(n)$4 dB to $h(n)$5 dB PAPR gain over FDSS alone; for Kaiser FDSS the gain is about $h(n)$6 dB; for 16-QAM with Hann it is about $h(n)$7 dB; and for $h(n)$8-BPSK the SE benefit shrinks from about $h(n)$9 dB to $$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$0 dB as shaping increases (Pitaval et al., 23 Sep 2025).

The same design logic appears in alternative modulation formats. Repeated-and-offset QPSK (RO-QPSK) is constructed so that its intrinsic subcarrier covariance has Hann weight

$$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$1

which means the waveform is already spectrally shaped before any external FDSS is applied. The reported consequence is about $$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$2 dB PAPR for RO-QPSK alone, comparable to $$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$3-BPSK with aggressive FDSS$$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$4 dB$$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$5, and around $$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$6 dB when moderate FDSS is added; the same study argues that this is especially favorable in narrowband and moderately frequency-selective satellite channels (Pitaval, 13 Oct 2025).

4. OFDM spectral containment, emission masks, and implementation complexity

In OFDM spectral-containment work, FDSS is often directed less at PAPR than at OOBE suppression and mask compliance. The generalized-pulse method replaces each data-carrier pulse by a conventional pulse plus cancellation-carrier and transition terms, making the optimization data-independent and permitting analytical PSD calculation. In a stringent EN 50561-1 scenario, conventional raised-cosine pulse shaping requires about $$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$7 data-carrier loss, generalized pulses with cancellation carriers only reduce this to $$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$8, and generalized pulses with cancellation carriers plus harmonically designed transition pulses reduce it further to $$\mathbf{h}_k=\mathbf{p}_k+\mathbf{P}_{\mathcal C}\pmb{\alpha}_k+\mathbf{t}_k.$$9; the reported complexity increases relative to the raised-cosine reference are $\mathbf{p}_k$0 and $\mathbf{p}_k$1, respectively, with PAPR changes of $\mathbf{p}_k$2 dB and approximately no penalty (Díez et al., 2018).

Dynamic-spectrum environments motivate reusable shaping rather than mask-specific redesign. A low-complexity method based on preoptimized generalized pulses combines AIC terms and adaptive symbol transition (AST) terms, parameterized by relative position to the nearest passband edge. By applying closed-form frequency shifts, frequency reversal, and conjugation, the same coefficient sets can be reused when the passband location or width changes. The method reports about $\mathbf{p}_k$3 dB maximum normalized PSD in a wide-passband notch, about $\mathbf{p}_k$4 dB in a narrow-passband case with the bandwidth-adaptive method, and about $\mathbf{p}_k$5 dB in a challenging notch against prior literature baselines, while keeping PAPR similar to the ad hoc method and avoiding online re-optimization (Giménez et al., 30 Dec 2025).

Another implementation-oriented development modifies the OFDM pulse itself so that its Fourier transform becomes real-valued: $\mathbf{p}_k$6 Because many spectral-shaping coefficients then become real-valued or Hermitian-symmetric, the number of optimization variables and real products can be reduced by up to $\mathbf{p}_k$7 while preserving PSD and PAPR behavior. The letter gives explicit reductions from $\mathbf{p}_k$8 to $\mathbf{p}_k$9 real products for AIC and from $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$0 to $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$1 for regular transition-pulse shaping (Giménez et al., 5 Nov 2025).

A closely related but distinct application is spectrum-skirt filling in fixed microwave backhaul. There, a non-Nyquist SSF filter is designed to match the regulatory spectral mask and exploit inactive adjacent channels. The benchmark RRC case uses $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$2 and $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$3, whereas SSF uses $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$4 and $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$5. The paper reports that conventional RRC with $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$6-QAM gives about $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$7 Mbit/s, while SSF reaches up to $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$8 Mbit/s in one comparison and can achieve $\mathbf{P}_{\mathcal C}\pmb{\alpha}_k$9 Mbit/s with only $\mathbf{t}_k$0-QAM, whereas conventional RRC would need $\mathbf{t}_k$1-QAM (Dobre et al., 2020).

5. Quantum and ultrafast-optical FDSS

In quantum optics, FDSS-like control appears as direct engineering of joint spectral amplitude, relative spectral phase, and frequency-bin structure. A prominent example is the bidirectionally pumped type-II SPDC source based on group-velocity-matched PPKTP, where the biphoton spectrum is shaped jointly in a two-dimensional frequency space. Crystal temperature tunes spectral-mode placement, producing two-mode separations of $\mathbf{t}_k$2 at $\mathbf{t}_k$3 and $\mathbf{t}_k$4 at $\mathbf{t}_k$5, with separation tunable up to $\mathbf{t}_k$6 by heating to $\mathbf{t}_k$7; a tilted glass plate controls the relative phase $\mathbf{t}_k$8 continuously over $\mathbf{t}_k$9 to $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$0. Two-photon spectral intensity verifies mode placement, while Hong-Ou-Mandel interference verifies coherence and phase control, with visibilities of $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$1 at $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$2 and $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$3 at $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$4, and beating periods of $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$5 and $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$6, consistent with $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$7 (Jin et al., 2018).

Another biphoton setting uses multiplexed cold atomic ensembles with frequency-shifted cascade emissions. Under Gaussian pumping of width $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$8, the single-ensemble joint spectrum is a Gaussian ridge enforcing $$S(\omega_1,\omega_2)=s(\omega_1,\omega_2)+e^{i\phi}s(\omega_2,\omega_1),$$9, modulated by a Lorentzian superradiant linewidth. Multiplexing yields

$\tilde{X}=X^{\text{ext}}\odot F,$00

so spectral shaping is effected by controlled shifts $\tilde{X}=X^{\text{ext}}\odot F,$01 and $\tilde{X}=X^{\text{ext}}\odot F,$02. For symmetric spectral shaping, Schmidt eigenvalues become pairwise degenerate, mode densities show interference fringes, and the entanglement entropy grows approximately as $\tilde{X}=X^{\text{ext}}\odot F,$03 plus the continuous-frequency excess entropy of a single ensemble (Jen et al., 2015).

Ultrafast spectro-temporal waveform conversion provides a third quantum-compatible manifestation. By imposing a designed temporal phase $\tilde{X}=X^{\text{ext}}\odot F,$04 on a strong $\tilde{X}=X^{\text{ext}}\odot F,$05 nm pump in a PPLN waveguide, a weak $\tilde{X}=X^{\text{ext}}\odot F,$06 nm exponentially decaying pulse is upconverted to $\tilde{X}=X^{\text{ext}}\odot F,$07 nm while its spectrum is broadened and reshaped according to

$\tilde{X}=X^{\text{ext}}\odot F,$08

The experiment reports spectral overlap $\tilde{X}=X^{\text{ext}}\odot F,$09 between theory and the measured $\tilde{X}=X^{\text{ext}}\odot F,$10 nm output, and a numerical follow-up shows that spectral phase correction can compress a $\tilde{X}=X^{\text{ext}}\odot F,$11 ns mono-exponentially decaying pulse toward a $\tilde{X}=X^{\text{ext}}\odot F,$12 ps Lorentzian target, yielding about $\tilde{X}=X^{\text{ext}}\odot F,$13 ps FWHM with a smoothed correction; the conversion platform is reported to operate with $\tilde{X}=X^{\text{ext}}\odot F,$14 signal-to-noise ratio and to be compatible with single-photon states (Agha et al., 2014).

6. Comb shaping, metrology, and nonlinear spectral-temporal design

FDSS also describes direct manipulation of comb-line structure. In terahertz quantum-cascade lasers with an external frequency-dependent reflector, selective reinjection of chosen longitudinal modes shapes harmonic frequency combs by controlling both line spacing and frequency offset. With reflector peaks aligned to modes $\tilde{X}=X^{\text{ext}}\odot F,$15 and $\tilde{X}=X^{\text{ext}}\odot F,$16, the selected separation is $\tilde{X}=X^{\text{ext}}\odot F,$17 FSR and a $\tilde{X}=X^{\text{ext}}\odot F,$18th-order harmonic comb is obtained; the same method yields an $\tilde{X}=X^{\text{ext}}\odot F,$19th-order HFC with $\tilde{X}=X^{\text{ext}}\odot F,$20 ps pulses and, in a wider-gain device, a $\tilde{X}=X^{\text{ext}}\odot F,$21th-order HFC with $\tilde{X}=X^{\text{ext}}\odot F,$22 ps FWHM pulses, while also permitting comb triggering from initially unlocked dynamics (Silvestri et al., 2024).

A related source-level approach uses a fast-gain ring QCL operated as a synthetic frequency lattice. Modulation at $\tilde{X}=X^{\text{ext}}\odot F,$23 GHz and $\tilde{X}=X^{\text{ext}}\odot F,$24 GHz creates nearest-neighbor and next-nearest-neighbor couplings between cavity modes, with relative phase $\tilde{X}=X^{\text{ext}}\odot F,$25 generating a staggered phase flux in a triangular ladder. The resulting dispersion

$\tilde{X}=X^{\text{ext}}\odot F,$26

becomes asymmetric when $\tilde{X}=X^{\text{ext}}\odot F,$27, and the steady-state spectrum $\tilde{X}=X^{\text{ext}}\odot F,$28 develops a tunable central lobe that can be shifted across the full measured bandwidth of about $\tilde{X}=X^{\text{ext}}\odot F,$29 (Piciocchi et al., 14 Mar 2025).

In frequency-comb metrology, programmable spectral shaping is used to improve the precision of comb-mode resolved spectral-domain interferometry. Hardware shaping with a waveshaper plus software normalization flattens the source within $\tilde{X}=X^{\text{ext}}\odot F,$30 dB, transforms a $\tilde{X}=X^{\text{ext}}\odot F,$31 THz 3-dB bandwidth into an effectively square spectrum of $\tilde{X}=X^{\text{ext}}\odot F,$32 THz 3-dB width, and exploits $\tilde{X}=X^{\text{ext}}\odot F,$33 comb modes from a $\tilde{X}=X^{\text{ext}}\odot F,$34 GHz electro-optic comb. The practical outcome is a precision improvement from about $\tilde{X}=X^{\text{ext}}\odot F,$35 nm to about $\tilde{X}=X^{\text{ext}}\odot F,$36 nm in CORE-SDI, corroborated by simulations and by a $\tilde{X}=X^{\text{ext}}\odot F,$37 kHz, $\tilde{X}=X^{\text{ext}}\odot F,$38 nm vibration measurement (Jang et al., 2023).

Nonlinear fiber systems introduce a further generalization: spectral shaping of the intensity-fluctuation PSD rather than the field spectrum itself. A semi-analytical model expresses the intensity-fluctuation PSD $\tilde{X}=X^{\text{ext}}\odot F,$39 as the sum of self-beating, inter-symbol beating, and block-induced correction terms, making explicit that the low-frequency region near $\tilde{X}=X^{\text{ext}}\odot F,$40 is the primary driver of XPM. CCDM is characterized by $\tilde{X}=X^{\text{ext}}\odot F,$41, yielding effectively zero energy fluctuations at $\tilde{X}=X^{\text{ext}}\odot F,$42, whereas ESS exhibits a finite low-frequency pedestal at moderate block length. The same framework derives the dip-width law $\tilde{X}=X^{\text{ext}}\odot F,$43 and the shaped-system optimum symbol rate

$\tilde{X}=X^{\text{ext}}\odot F,$44

linking block length, pulse shape, symbol rate, and dispersion in a single FDSS-style design rule (Prasad et al., 5 Mar 2026).

7. Recurring tradeoffs, limits, and conceptual boundaries

A consistent result across the literature is that FDSS does not offer a free simultaneous optimum in all performance dimensions. In learned DFT-s-OFDM, PAPR reduction, SER preservation, and spectral flatness cannot all be maximized simultaneously; stronger PAPR optimization drives filters toward bell-shaped spectra and larger ISI, while structural constraints such as flat passbands or zero-ISI reduce available degrees of freedom and therefore reduce achievable PAPR gain (Carpi et al., 2024).

An analogous non-monotonicity appears in FDSS-SE, where increasing spectrum extension initially improves PAPR and can improve rate, but eventually worsens both. The distinction between $\tilde{X}=X^{\text{ext}}\odot F,$45 and $\tilde{X}=X^{\text{ext}}\odot F,$46, together with the SNR dependence of the rate optimum, implies that SE size should be configured per window and per link condition rather than fixed globally (Pitaval et al., 23 Sep 2025).

In OFDM spectral-containment systems, the main tradeoff is between mask compliance, memory, and online complexity. Preoptimized adaptive-mask methods avoid online optimization but introduce scheme-dependent memory and per-symbol product counts; modified Hermitian-symmetric pulse frameworks cut arithmetic cost but assume compatibility with specific optimization structures and, when $\tilde{X}=X^{\text{ext}}\odot F,$47 is even, may require one extra sample in the pulse (Giménez et al., 30 Dec 2025, Giménez et al., 5 Nov 2025).

Quantum and photonic realizations expose a different boundary: “arbitrary shaping” is often a directional claim rather than a completed capability. The PPKTP biphoton demonstration establishes joint amplitude and phase control, but the realized state remains effectively a two-mode frequency superposition; the cold-ensemble multiplexing work shows entropy growth and mode-pair degeneracy, but the engineered spectrum remains constrained by the Gaussian-plus-Lorentzian structure of the underlying cascade emission (Jin et al., 2018, Jen et al., 2015).

Taken together, these results indicate that FDSS is not a single field-specific recipe. In communications it is usually a deterministic subcarrier-domain window or an equivalent generalized pulse; in quantum and photonic systems it can mean coherent synthesis of discrete spectral modes, comb-line selection, or suppression of low-frequency intensity fluctuations. The unifying principle is deliberate spectral-domain control to induce a desired system-level behavior, with the exact optimization variables, constraints, and observables determined by the underlying physical platform.