Spectral Codebook: Concepts and Applications

• Spectral Codebook is a family of constructions that utilize spectral representations to design efficient beam management, neural audio processing, and array configurations.
• It leverages physical and statistical properties such as spatial DFT transforms, eigenmodes, and sparse sampling to optimize spectral efficiency and reduce feedback overhead in wireless systems.
• Applications span diverse domains from beamspace management in massive MIMO to discrete spectrogram quantization in neural speech synthesis, demonstrating the versatility of spectral codebooks.

to=arxiv_search 大发快三大小单双ി with_escalated_permissions: false, "query": "\"Spectral Codebook\" OR beamspace codebook OR codebook design spectral efficiency", "max_results": 10 } A spectral codebook is a codebook whose structure, indexing, or evaluation is tied to a spectral representation of the underlying signal space. In the literature surveyed here, that representation is not uniform: in large-array wireless systems it is often a beamspace or angular-spectrum representation induced by a spatial DFT, a lens, or a near-field propagation operator; in standards-oriented beam management it is also tied to effective spectral efficiency after training and feedback overhead; and in neural speech synthesis it denotes a discrete codebook over spectrogram-derived latents rather than waveform latents (Shen et al., 2017, Dreifuerst et al., 2023, Langman et al., 2024, Liu et al., 2023). This suggests that “spectral codebook” is best understood as a family of related constructions rather than a single standardized object.

Sense of the term	Core object	Representative papers
Beamspace or angular-spectrum codebook	Codewords shaped by spatial DFT, lens, or beamspace transforms	(Shen et al., 2017, Dreifuerst et al., 2023, Dreifuerst et al., 2024)
Spectral-efficiency-oriented codebook	Codebook optimized for throughput after training, feedback, or channel-use effects	(Dreifuerst et al., 2023, Zhang et al., 6 Aug 2025, Dreifuerst et al., 2024)
Near-field spectral basis	Eigenmodes or sparse DFT patterns matched to spherical-wave propagation	(Liu et al., 2023, Zhou et al., 2024, Carlson et al., 2023)
Spectrogram-domain codec	Discrete latent codebook over mel-spectrogram features	(Langman et al., 2024)

1. Terminological scope

The wireless literature often uses “spectral” in a spatial sense rather than a frequency-waveform sense. In lens-based mmWave massive MIMO, the closest notion is a beamspace-aware equivalent-channel codebook: the lens is modeled as a spatial DFT transform $\mathbf U$, the beamspace channel is $\mathbf h_k^b = \mathbf U \mathbf h_k$, and the beam selector produces the equivalent channel $\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$. The proposed codewords are generated as

$$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$

with $\mathbf c_{k,i}$ constrained to the physical channel subspace spanned by the path steering matrix $\mathbf A_k$ (Shen et al., 2017). In this usage, “spectral codebook” is effectively a beamspace or transform-domain codebook.

A second usage emphasizes throughput rather than representation. In sub-6GHz 5G NR beam management, codebooks are judged not only by beamforming gain but by their effect on SSB detection, CSI-RS estimation, feedback quantization, reconstructed CSI, MU-MIMO precoding, and finally effective spectral efficiency after accounting for beam-management overhead (Dreifuerst et al., 2023). In task-oriented semantic communications, the phrase “spectral efficiency-aware codebook” refers to a discrete semantic codebook whose activation distribution is regularized toward a channel-favorable input distribution, because sparse activation can collapse onto only a small subset of constellation symbols (Zhang et al., 6 Aug 2025).

A third usage is specific to neural audio. “Spectral codec” refers to a codec that quantizes mel-spectrogram-derived encoder outputs rather than waveform-derived latents. In that setting the codebook is spectral because the discrete bottleneck is applied in the spectrogram domain, and because the resulting token distribution is intended to be easier for non-autoregressive TTS models to predict (Langman et al., 2024).

2. Beamspace and lens-domain codebooks in large-array wireless systems

In lens-based mmWave massive MIMO, the key observation is that sparse physical propagation induces a low-dimensional channel subspace before and after the lens transform. The downlink user channel is modeled as

$$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$

where $P_k\ll M$ because of limited scattering. The lens acts as a spatial Fourier transformer,

$\mathbf h_k^b=\mathbf U\mathbf h_k,$

and the beam selector produces the reduced-dimensional equivalent channel

$$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$

The codebook is then generated by drawing isotropic vectors $\mathbf h_k^b = \mathbf U \mathbf h_k$0 on the $\mathbf h_k^b = \mathbf U \mathbf h_k$1-dimensional unit sphere, forming $\mathbf h_k^b = \mathbf U \mathbf h_k$2, and mapping them through the same lens-and-selection pipeline: $\mathbf h_k^b = \mathbf U \mathbf h_k$3 This matches the structural distribution of the equivalent channel direction much better than RVQ or Grassmannian codebooks defined in the ambient $\mathbf h_k^b = \mathbf U \mathbf h_k$4-dimensional space (Shen et al., 2017).

The analytical consequence is dimension reduction in the quantization problem itself. With

$\mathbf h_k^b = \mathbf U \mathbf h_k$5

the paper states $\mathbf h_k^b = \mathbf U \mathbf h_k$6 asymptotically, so the normalized equivalent channel can be written as $\mathbf h_k^b = \mathbf U \mathbf h_k$7. The quantization error bound becomes

$\mathbf h_k^b = \mathbf U \mathbf h_k$8

and the feedback-bit scaling for a rate gap no larger than $\mathbf h_k^b = \mathbf U \mathbf h_k$9 bps/Hz is

$\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$0

The essential point is that the exponent depends on $\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$1, not $\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$2, so the codebook behaves like a $\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$3-dimensional problem rather than an $\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$4-dimensional one (Shen et al., 2017).

A related beamspace construction appears in ML-based NR beam management. There, arbitrary beamformers are mapped into a common angular basis through a beamspace projection

$\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$5

which converts dynamic codebooks into a consistent spatial-spectral representation that can be learned by a neural network (Dreifuerst et al., 2023). This suggests that beamspace regularization and beamspace generation are now central recurring mechanisms whenever “spectral codebook” is used for large-array wireless design.

3. Spectral efficiency as the governing codebook criterion

In 5G NR beam management, the codebook is part of an end-to-end chain rather than an isolated beam set. The sub-6GHz study distinguishes three codebooks with separate roles: the SSB codebook for initial access, the CSI-RS codebook for beam refinement and CSI acquisition, and the FB codebook for CSI Type-II feedback quantization (Dreifuerst et al., 2023). The most explicit performance criterion is the overhead-aware effective sum spectral efficiency

$\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$6

which excludes beam-management resources from useful data transmission. Under this criterion, a larger or more refined codebook is not automatically better, because additional pilot or feedback overhead can offset beam-gain improvements.

The same paper proposes Beamspace-Codex (BSC), a fully connected neural network with 6 dense layers and dropout that learns the SSB codebook while leaving the CSI-RS and feedback formats compatible with Releases 15–17. The learned SSB codebooks improve SSB RSRP by about $\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$7–$\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$8 dB on average over traditional DFT codebooks, even in wideband, dispersive FR1 channels. At the same time, the work shows that feedback configuration can dominate codebook gains: increasing the Type-II feedback resolution parameter $\mathbf h_k^e = \mathbf S^H \mathbf h_k^b$9 from $$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$0 to $$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$1 can yield more than $$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$2 effective data rate, whereas CSI-RS codebook enlargement shows diminishing returns after about $$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$3 (Dreifuerst et al., 2023). The paper’s central conclusion is therefore not just that beamspace codebooks matter, but that codebook design and feedback design are coupled spectral-efficiency problems.

The network-level version of the same idea is “network beamspace learning” (NBL). NBL learns SSB and CSI-RS codebooks jointly across multiple cells using beamspace inputs derived from previous codebooks and SSB feedback. Its training loss is the mean-squared mismatch between an idealized SVD-based target rate

$$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$4

and the rate obtained after beam management with the learned codebooks. In the reported results, the learned codebooks outperform traditional codebooks by over $$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$5 dB in beam alignment and achieve more than $$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$6 improvements in network spectral efficiency (Dreifuerst et al., 2024). This suggests that once interference, association, and beam training are coupled across cells, a “spectral codebook” becomes an explicitly network-level object rather than a per-link beam dictionary.

Task-oriented semantic communications uses “spectral efficiency-aware codebook” in a different but conceptually related sense. The codebook

$$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$7

quantizes semantic features by nearest-neighbor lookup, and the activation probabilities

$$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$8

are optimized rather than merely observed. The method WS-DC regularizes the induced input distribution toward a hybrid target

$$\mathbf d_{k,i}=\mathbf S^H \mathbf U \mathbf c_{k,i},$$9

using a Wasserstein objective

$\mathbf c_{k,i}$0

Here the codebook is “spectral” because its activation distribution determines how efficiently the discrete semantic indices occupy the finite-alphabet channel signal space (Zhang et al., 6 Aug 2025).

4. Near-field and physically matched spatial-spectral codebooks

Near-field XL-MIMO shifts spectral codebook design from beamspace sparsity to propagation-matched eigenmodes. In the DPSS-based design, the codebook is derived from the transmit and receive autocorrelation matrices

$\mathbf c_{k,i}$1

induced by the near-field Green’s function. Under a paraxial approximation, the correlation becomes a sinc-kernel Toeplitz matrix after quadratic phase compensation,

$\mathbf c_{k,i}$2

so the eigenvectors are DPSS/Slepian sequences up to a deterministic chirp: $\mathbf c_{k,i}$3 The resulting codebook

$\mathbf c_{k,i}$4

is orthogonal, matched to near-field physics, and fixed in size at $\mathbf c_{k,i}$5. For target NMSEs $\mathbf c_{k,i}$6 dB, the minimum required codebook sizes reported are $\mathbf c_{k,i}$7 for DFT, $\mathbf c_{k,i}$8 for the spherical codebook, and $\mathbf c_{k,i}$9 for the proposed DPSS codebook (Liu et al., 2023).

A different near-field construction uses a sparse DFT codebook. Uniform sparse antenna activation with interval $\mathbf A_k$0 converts the dense ULA into a sparse linear array and induces angular periodicity in the received beam pattern: $\mathbf A_k$1 That periodicity enables a three-phase training scheme: sweep only one angular period with a sparse DFT codebook, resolve the resulting ambiguity with a central sub-array, and then search range with a polar-domain codebook. The beam-training overhead becomes

$\mathbf A_k$2

with relaxed optimum $\mathbf A_k$3, so the overhead scales as $\mathbf A_k$4. The reported overhead reduction relative to exhaustive search is $\mathbf A_k$5, without compromising rate performance in the high-SNR regime (Zhou et al., 2024).

Dynamic metasurface antennas impose an even stronger hardware constraint. Feasible element weights lie on the Lorentzian set

$\mathbf A_k$6

and the waveguide feed introduces a phase profile $\mathbf A_k$7. A practical hierarchical codebook is therefore constructed by starting from a binary hierarchical DFT codebook, compensating the waveguide phase, and mapping each desired weight to $\mathbf A_k$8 via Euclidean or Lorentzian-constrained modulation. Because global phase rotation changes the projected DMA weights, the paper optimizes

$\mathbf A_k$9

for each target steering angle. The resulting EM:BF method yields about $$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$0 dB spectral-efficiency improvement over unoptimized DMA mapping; the active phased array has a $$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$1 dB input-power advantage over the best DMA method in spectral efficiency, while the best DMA method has a $$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$2 dB input-power advantage over the passive phased array and achieves about $$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$3 the peak energy efficiency of the active phased array (Carlson et al., 2023).

Near-field XL-RIS generalizes the codeword from a point focus to a spatial region. For a codeword region $$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$4, the design objective is to maximize average regional spectral efficiency

$$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$5

while minimizing the outage probability

$$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$6

The adaptive-weight block coordinate descent rule increases the weights of points with below-average spectral efficiency,

$$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$7

thereby broadening or reshaping the beam as required by the target region. The method supports arbitrary codeword-region shapes and joint multi-XL-RIS codewords, and the simulations report higher spectral efficiency, lower outage probability, and better robustness to codeword-region location and area variations than existing methods (Zhang et al., 15 Aug 2025).

5. Code-domain and array-configuration interpretations

In code-domain multiple access, the codebook is often “spectral” because it determines how information is embedded across shared multicarrier resources. SCMA is the canonical example. Bits are mapped directly to sparse multidimensional codewords, not to QAM symbols followed by spreading, and the overload factor is

$$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$8

The design problem is formulated as

$$\mathbf h_k=\sum_{i=1}^{P_k} g_{k,i}\mathbf a(\psi_{k,i})=\mathbf A_k\mathbf g_k,$$9

with factor-node degree $P_k\ll M$0 controlling MPA complexity $P_k\ll M$1. Real/imaginary shuffling can reduce this to $P_k\ll M$2, and reducing the number of projection points per dimension yields complexity $P_k\ll M$3 with $P_k\ll M$4. In the reported uplink fading experiment, SCMA outperforms LDS, OFDMA, and SC-FDMA by more than $P_k\ll M$5 dB at the tested spectral efficiency (Taherzadeh et al., 2014).

A more explicit resource-level spectral construction is the UDCG-based SCMA codebook. For each resource shared by $P_k\ll M$6 users, the design assigns $P_k\ll M$7 sub-constellations whose sum is uniquely decomposable. If

$P_k\ll M$8

with $P_k\ll M$9, then $\mathbf h_k^b=\mathbf U\mathbf h_k,$0 such constellations can form a UDCG, each with $\mathbf h_k^b=\mathbf U\mathbf h_k,$1 points. The superimposed constellation MED becomes a direct design target: $\mathbf h_k^b=\mathbf U\mathbf h_k,$2 The reported superimposed MEDs remain nonzero and uniform across resources for $\mathbf h_k^b=\mathbf U\mathbf h_k,$3, and BER gains are especially strong for large-size codebooks, including about $\mathbf h_k^b=\mathbf U\mathbf h_k,$4 dB over GAMCB and $\mathbf h_k^b=\mathbf U\mathbf h_k,$5 dB over LCRCB at BER $\mathbf h_k^b=\mathbf U\mathbf h_k,$6 for $\mathbf h_k^b=\mathbf U\mathbf h_k,$7 on AWGN (Zhang et al., 2021).

For sporadic mMTC, “spectral codebook” can also mean a spreading-sequence codebook over multicarrier spectral resources. In the codebook-based MC-CDMA scheme, each user maps bits directly to one of $\mathbf h_k^b=\mathbf U\mathbf h_k,$8 circularly shifted spreading sequences rather than to a PSK symbol followed by spreading. The receiver uses a modified group matching pursuit algorithm, and the reported gain is about $\mathbf h_k^b=\mathbf U\mathbf h_k,$9 dB at BER $$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$0 for modulation order $$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$1 relative to the conventional scheme (Alam et al., 2018). Sparse-Encoded Codebook Index Modulation extends the same idea by using the codebook index itself as an index-modulation dimension. With $$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$2 codebooks, the total number of bits per OFDM symbol becomes

$$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$3

and the spectral efficiency is

$$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$4

Here the codebook is no longer only a sensing matrix; it is an information-bearing object (Arslan et al., 2020).

A learned variant appears in CD-NOMA. The MU-MDM autoencoder represents each user codeword directly in $$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$5, adopts dense resource mapping, and replaces per-user normalization by a Level-3 global sum-power constraint,

$$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$6

Its BER-oriented loss

$$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$7

jointly shapes geometry and bit labeling. The best learned dense codebook comes within $$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$8 dB of the equivalent single-user benchmark at BER $$\mathbf h_k^e=\mathbf S^H\mathbf h_k^b=\mathbf S^H\mathbf U\mathbf A_k\mathbf g_k.$$9, and dense versus sparse mapping differs by only about $\mathbf h_k^b = \mathbf U \mathbf h_k$00 dB under Level-3 normalization (Han et al., 2021).

Array Configuration Codebook (ACC) moves the codebook from waveform or beam weights to array geometry itself. A codeword is an $\mathbf h_k^b = \mathbf U \mathbf h_k$01 vector of activated antenna-pixel positions chosen from structured families such as CA, USA, MoA, NA, and CPA. For $\mathbf h_k^b = \mathbf U \mathbf h_k$02 candidate pixels and $\mathbf h_k^b = \mathbf U \mathbf h_k$03 RF chains,

$\mathbf h_k^b = \mathbf U \mathbf h_k$04

whereas the instantiated ACC contains $\mathbf h_k^b = \mathbf U \mathbf h_k$05 codewords. Two-stage scanning reduces worst-case overhead to $\mathbf h_k^b = \mathbf U \mathbf h_k$06. In the reported communication setting, USA is best for $\mathbf h_k^b = \mathbf U \mathbf h_k$07 users, CA becomes slightly superior for $\mathbf h_k^b = \mathbf U \mathbf h_k$08, and for localization NA achieves the best RMSE because of its large continuous virtual aperture (Lu et al., 28 Aug 2025). This suggests that a spectral codebook can also be a codebook over spatial sampling patterns and induced array spectra, not merely over beamforming vectors.

6. Spectrogram-domain spectral codebooks in neural speech synthesis

In neural audio, a spectral codebook is a discrete bottleneck over spectral features rather than over waveform samples. The “spectral codec” takes an 80-dimensional mel-spectrogram as input, encodes it with a HiFi-GAN-based encoder, discretizes the encoder output, and reconstructs the time-domain waveform with a HiFi-GAN decoder (Langman et al., 2024). For the 44.1 kHz setup, the window size is $\mathbf h_k^b = \mathbf U \mathbf h_k$09 samples and the hop length is $\mathbf h_k^b = \mathbf U \mathbf h_k$10, giving a frame rate of about

$\mathbf h_k^b = \mathbf U \mathbf h_k$11

frames per second.

The key quantizer is FSQ, adopted from Mentzer et al. Rather than learning a vector dictionary, FSQ uses an implicit Cartesian grid. In the reported system there are 8 codebooks/groups, each with levels $\mathbf h_k^b = \mathbf U \mathbf h_k$12, so the per-group codebook size is

$\mathbf h_k^b = \mathbf U \mathbf h_k$13

the total quantized dimensionality is $\mathbf h_k^b = \mathbf U \mathbf h_k$14, and the bitrate is about $\mathbf h_k^b = \mathbf U \mathbf h_k$15 kbps (Langman et al., 2024). A multi-band variant splits the 80 mel bins into 8 groups of 10 bands each and encodes them separately.

The paper’s main claim is not that the spectral codec dominates waveform codecs on all metrics, but that it produces a simpler target distribution for non-autoregressive TTS. Reconstruction quality is comparable to equivalent waveform codecs, while TTS quality improves markedly. Among codec-based systems, “Mel-spectrogram + FSQ” and “Multi-band mel + FSQ” achieve the best TTS metrics, and in ASR evaluation the synthesized speech yields

$\mathbf h_k^b = \mathbf U \mathbf h_k$16

for “Mel + FSQ,” close to ground truth

$\mathbf h_k^b = \mathbf U \mathbf h_k$17

The paper’s qualitative conclusion is that only the spectral codec with FSQ gives high-quality audio accurate to ground truth in non-autoregressive FastPitch (Langman et al., 2024). In this domain, then, a spectral codebook is not a beamspace object at all; it is an implicit product-structured codebook over mel-spectrogram-derived latents.

7. Limitations, assumptions, and recurrent misconceptions

One recurring misunderstanding is to equate “spectral codebook” exclusively with frequency-domain waveform processing. The cited wireless papers instead use the term, or its closest analogues, for beamspace, angular-spectrum, spatial-sampling, or spectral-efficiency-aware constructions (Shen et al., 2017, Liu et al., 2023, Dreifuerst et al., 2024). This suggests that the adjective “spectral” is domain-dependent: in array processing it usually refers to spatial spectra, whereas in neural speech it refers to spectrograms.

Another recurring issue is that spectral matching does not remove physical assumptions. The lens-based subspace codebook assumes limited scattering, slowly varying AoDs, asymptotic orthogonality, dominant-beam capture by the selector, and perfect norm feedback (Shen et al., 2017). The 5G NR learned codebook is intentionally constrained by standard compatibility: no change to feedback format, no extra side information, and no change to beam-training timing (Dreifuerst et al., 2023). The semantic-communication formulation is derived under AWGN and average-power arguments even though the practical transmitter uses finite constellations (Zhang et al., 6 Aug 2025). The XL-RIS region-based codebook assumes near-field LOS-dominated propagation, geometry knowledge for offline design, and target regions known in advance (Zhang et al., 15 Aug 2025).

A further misconception is that spectral codebooks are always learned dictionaries. Several of the strongest examples are deterministic or eigen-analytic: the lens-domain equivalent-channel codebook is generated from an AoD-defined subspace (Shen et al., 2017), the DPSS codebook is the eigenbasis of a near-field covariance operator (Liu et al., 2023), and the sparse DFT near-field codebook exploits periodicity induced by structured sparse sampling (Zhou et al., 2024). Conversely, some learned systems are spectral not because they learn arbitrary vectors, but because they learn activation distributions, feedback-aware beamspace images, or spectrogram-derived discrete latents (Zhang et al., 6 Aug 2025, Langman et al., 2024).

Taken together, the literature shows that a spectral codebook is best characterized by the spectral object it is matched to: a beamspace transform, a near-field propagation operator, a resource-domain superposition, a channel-input distribution, or a spectrogram manifold. The common design principle is structural matching. Codebooks perform well when they live on, or are evaluated with respect to, the same low-dimensional or physically meaningful manifold that governs the true signal.