Frequency-Domain Linear Non-Redundant Precoding

• Frequency-domain non-redundant precoding is a set of linear transmit-processing strategies designed in the frequency domain to avoid redundant signal expansion.
• It encompasses various architectures such as per-tone beamforming, transform precoding, and compressed feedback schemes that optimize subcarrier resource allocation.
• These methods maintain full-rate transmission while enhancing system efficiency through minimized overhead and precise signal alignment across frequencies.

Searching arXiv for the papers on arXiv and closely related work to ground the article. Frequency-domain linear non-redundant precoding is best understood as an Editor’s term for a family of linear transmit-processing strategies in which precoding is designed in a frequency-domain representation—per subcarrier, per DFT bin, or through a frequency-domain transform—while avoiding unnecessary expansion of signaling dimensions. The literature does not use the expression as a uniform formal label. Instead, related works instantiate different meanings of “non-redundant”: minimal per-resource beamforming in OFDMA, implementation without explicit time-domain filtering, full-rate square transform precoding without transmission-rate loss, or compressed representation of a large family of subcarrier-wise precoders (Kibria et al., 2017, Kenney et al., 2021, Ge et al., 2024, Chou et al., 4 May 2026). The common thread is a structurally lean linear architecture in which the frequency domain is the design space, but redundancy is controlled rather than added indiscriminately.

1. Conceptual scope and terminology

In the surveyed literature, frequency-domain linear precoding appears in several distinct operational regimes. In coordinated OFDMA downlink, the natural object is a per-tone beamformer $\mathbf g_{kmn}\in\mathbb C^{N_t\times 1}$ assigned to one scheduled user on one subcarrier, so non-redundancy means one stream and one beamformer per active user-tone pair, with no frequency-domain spreading and no same-cell multiplexing on the same tone (Kibria et al., 2017). In wideband OFDM time-reversal, the precoder is diagonal in the subcarrier domain with weights $w_q^{(k)}=(H_q^k)^*$; there, the non-redundant aspect is primarily implementation-oriented, because the method avoids explicit time-domain TR filtering even though the back-off factor still reduces the data rate (Nguyen et al., 2019).

Other works make the rate-preserving meaning explicit. For OTFS over frequency-selective fading, the precoder $\mathbf V\in\mathbb C^{MN\times MN}$ is square, maps $MN$ input symbols to $MN$ transmitted symbols, and is stated to operate “without any transmission rate loss” (Ge et al., 2024). In iterative LMMSE-based block transmission, the precoder is an invertible or unitary reshaping such as $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$ or $$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$, again without adding parity dimensions; the outer code provides coding redundancy, not the linear precoder itself (Yuan et al., 2011). A third meaning is representational compression: in wideband mmWave MIMO-OFDM, the full set $\{\mathbf F[k]\}_{k=1}^K$ is not fed back independently, but represented by a frequency-flat analog stage plus anchor-subcarrier digital codewords and binary-search-based switching information, reducing feedback overhead from $\mathcal O(K)$ to $\mathcal O(K/M+\log M)$ (Chou et al., 4 May 2026).

This heterogeneity matters because “non-redundant” is not synonymous with “no diversity mechanism.” Some systems remain globally redundancy-bearing even when the precoder itself is structurally lean. Statistical STFBC precoding in MC-CDMA, for example, reuses one covariance-driven frequency-domain precoder across subcarriers, which is structurally compact, but the signaling still contains OSTBC and frequency-spreading redundancy by design (Chen et al., 2014). A precise reading therefore requires distinguishing structural minimality, rate preservation, and feedback compression.

2. Structural archetypes and mathematical forms

The frequency-domain precoders in this area fall into a small number of recurrent algebraic forms. Some are strictly per-tone and block diagonal across frequency; others deliberately mix symbols across tones through unitary transforms. The distinction is central to whether non-redundancy is interpreted as “one local beamformer per tone” or as “a full-rate invertible transform over many frequency-domain dimensions.”

Archetype	Representative form	Non-redundant interpretation
Per-tone beamforming	$w_q^{(k)}=(H_q^k)^*$0 or $w_q^{(k)}=(H_q^k)^*$1	One stream or one matrix per active tone
Diagonal matched filtering	$w_q^{(k)}=(H_q^k)^*$2	No explicit time-domain TR filter
Transform precoding	$w_q^{(k)}=(H_q^k)^*$3, $w_q^{(k)}=(H_q^k)^*$4, $w_q^{(k)}=(H_q^k)^*$5	Square or unitary full-rate mixing
Compressed precoder family	anchor codewords plus switching points	Avoid per-subcarrier feedback duplication

The first archetype appears in multicell MU-MISO OFDMA and in SCM with CP after circulant diagonalization. In both cases, the channel is rendered per-bin or per-subcarrier, and a separate linear map is optimized on each frequency component. The second archetype is frequency-domain time reversal, where diagonal weights are conjugate channel coefficients and any additional spreading is separated from the diagonal precoder itself (Nguyen et al., 2019).

The third archetype uses a global transform. In OTFS over frequency-selective fading, the proposed design is

$w_q^{(k)}=(H_q^k)^*$6

where $w_q^{(k)}=(H_q^k)^*$7 is an algebraically constructed Vandermonde or unitary matrix chosen so that every nonzero codeword-difference vector excites all diversity branches (Ge et al., 2024). In the iterative LMMSE framework, the DFT matrix $w_q^{(k)}=(H_q^k)^*$8 spreads coded symbols across many channel uses or eigenmodes, while $w_q^{(k)}=(H_q^k)^*$9, $\mathbf V\in\mathbb C^{MN\times MN}$0, or $\mathbf V\in\mathbb C^{MN\times MN}$1 align and load the resulting directions (Yuan et al., 2011). The fourth archetype is not a different physical precoder but a different representation of the same family of subcarrier-wise linear maps: only anchors and switch points are conveyed, rather than a fresh index for every tone (Chou et al., 4 May 2026).

A related adjacent model is GMUD precoding for single-carrier frequency-selective channels, where the multipath channel is lifted into a block Toeplitz matrix and factorized as $\mathbf V\in\mathbb C^{MN\times MN}$2. This is not OFDM-style per-subcarrier precoding, but it is still linear, frequency-selective, and non-redundancy-adding in the sense that $\mathbf V\in\mathbb C^{MN\times MN}$3 introduces no explicit coding overhead (0812.4334).

3. Tone-wise minimal beamforming in coordinated OFDMA systems

A clear instance of structural non-redundancy is coordinated linear precoding in downlink multicell MU-MISO OFDMA (Kibria et al., 2017). The model has $\mathbf V\in\mathbb C^{MN\times MN}$4 coordinated base stations, $\mathbf V\in\mathbb C^{MN\times MN}$5 users per cell, $\mathbf V\in\mathbb C^{MN\times MN}$6 OFDMA subcarriers, $\mathbf V\in\mathbb C^{MN\times MN}$7 transmit antennas at each BS, and single-antenna users. Subcarrier assignments are non-overlapping within each cell, expressed by $\mathbf V\in\mathbb C^{MN\times MN}$8, so each BS serves at most one in-cell user on each subcarrier. Because of this OFDMA exclusivity, the paper states that there is no intra-cell interference; interference arises only from other BSs transmitting on the same tone.

For user $\mathbf V\in\mathbb C^{MN\times MN}$9, the received signal is

$MN$0

The corresponding SINR is

$MN$1

and the instantaneous rate is $MN$2. Each tone has its own beamforming vector, interference is only same-subcarrier inter-cell interference, and the main coupling across tones is the per-BS sum-power constraint. This is therefore a subcarrier-wise coordinated beamforming problem, not a joint wideband precoder that mixes symbols across subcarriers.

The optimization is weighted sum-rate maximization with per-BS transmit-power constraints. The paper reformulates the nonconvex WSRM problem through sequential parametric convex approximation and solves a sequence of SOCPs. Hyperbolic constraints are transformed into SOC form, a phase-fixing condition $MN$3 is imposed, and the difficult product term is handled either by geometric mean or by recursive hyperbolic reformulation (Kibria et al., 2017). The method is described as less-complex, fast, provably convergent, and convergent to a local optimal or suboptimal solution rather than to a global optimum.

The relevance to frequency-domain linear non-redundant precoding is not terminological but structural. The paper does not formalize a non-redundant-precoder manifold or reduced-coordinate parameterization. Nevertheless, the design variable on each active resource element is just one vector $MN$4, with one stream per scheduled user-tone, no same-cell multiuser superposition on a tone, and no coding redundancy across tones. A plausible implication is that this is one of the leanest possible frequency-domain linear precoder structures for OFDMA multicell coordination.

4. Spreading, focusing, and compressed representation across frequency

Frequency-domain time-reversal precoding in wideband MISO-OFDM provides a different interpretation of lean design (Nguyen et al., 2019). The diagonal precoder on antenna $MN$5 is

$MN$6

so the per-subcarrier weight is $MN$7. The paper shows how a back-off factor $MN$8, usually realized by time-domain upsampling in classical TR, can be implemented entirely in the OFDM frequency-domain chain by spreading $MN$9 data symbols across $MN$0 subcarriers through a $MN$1 matrix $MN$2 satisfying $MN$3. This realizes TR focusing without reverting to explicit time-domain TR filtering. At the intended position, increasing $MN$4 or $MN$5 reduces NMSE; at the unintended position, increasing $MN$6 worsens NMSE while increasing $MN$7 helps somewhat (Nguyen et al., 2019). The key caveat is conceptual: the method is non-redundant mainly in implementation structure, not in coding-rate sense, because BOF still reduces the number of independent symbols per OFDM symbol.

A compressed wideband interpretation appears in reduced-feedback hybrid precoding for mmWave MIMO-OFDM (Chou et al., 4 May 2026). There the full family of digital precoders $MN$8 is not treated as $MN$9 independent objects. A frequency-flat analog precoder $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$0 is first constructed from dominant AoA and AoD directions shared across subcarriers, and the digital stage is quantized on pilot-spaced anchor subcarriers. Within each interval, a binary-search-based hierarchical interpolation algorithm determines where the preferred codeword switches. The result is a feedback-efficient representation whose overhead scales as $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$1 rather than $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$2, with the explicit numerical example $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$3, $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$4, $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$5 giving $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$6 bits instead of $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$7 bits, reported as more than $$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$8 feedback reduction (Chou et al., 4 May 2026). Here non-redundancy means avoiding duplicate subcarrier reports when neighboring tones share nearly identical digital precoders.

Statistical precoder design for space-time-frequency block codes in MISO-MC-CDMA gives yet another compressed interpretation (Chen et al., 2014). Starting from a time-domain multipath correlation model, the paper derives the frequency-domain covariance

$$\tilde{\mathbf x}=\mathbf V\mathbf W^{1/2}\mathbf F\mathbf x$$9

Under independent taps, $$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$0, hence $$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$1 for all subcarriers. This makes it possible to reuse one statistical precoder on all selected tones. For OSTBC, the optimal transmit-direction matrix aligns with the eigenvectors of the transmit covariance, $$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$2, while the diagonal loading matrix $$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$3 follows a water-filling solution; in high effective SINR, equal power allocation $$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$4 is near-optimal, whereas in low SINR the solution collapses to single-beam allocation (Chen et al., 2014). The signaling itself remains diversity-redundant because of STFBC spreading, but the precoder representation is frequency-compact and mode-efficient.

5. Full-rate transform precoding and single-carrier block transmission

Single-carrier modulation with CP in multi-user massive MIMO illustrates frequency-domain precoding without conversion to a multicarrier waveform (Kenney et al., 2021). After CP removal, the circulant channel matrices are diagonalized by the DFT, yielding a per-bin model

$$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$5

The downlink RZF precoder is

$$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$6

and its central implementation advantage is that the inverse $$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$7 is reused from FD MMSE uplink detection. The method remains single-carrier in time domain, with the DFT serving only as a computational diagonalization tool, and the paper states that the performance of the RZF precoder exceeds that of the ZF precoder for low and moderate input SNR conditions, while being equal at high input SNR (Kenney et al., 2021). In this setting, non-redundancy means that, apart from the CP needed for circularization, no OFDM-style multicarrier redundancy is introduced.

A more general full-rate transform viewpoint is developed for iterative LMMSE detection over coded linear systems (Yuan et al., 2011). Under full CSIT, the transmitter forms

$$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$8

and under partial CSIT,

$$\tilde{\mathbf x}=\mathbf P\mathbf\Pi\mathbf F\mathbf x$$9

The DFT matrix $\{\mathbf F[k]\}_{k=1}^K$0 spreads each coded symbol over many channel uses, while $\{\mathbf F[k]\}_{k=1}^K$1, $\{\mathbf F[k]\}_{k=1}^K$2, or $\{\mathbf F[k]\}_{k=1}^K$3 align and load the available eigenmodes. The stated purpose is to asymptotically ensure the Gaussianness of messages in iterative detection; with matched transfer functions, the proposed LP-LMMSE scheme is described as potentially information lossless (Yuan et al., 2011). This is one of the clearest formal examples in which non-redundancy refers to a unitary or invertible linear reshaping that adds no extra transmission dimensions.

For OTFS in frequency-selective fading, the full-rate interpretation is explicit (Ge et al., 2024). The proposed precoder

$\{\mathbf F[k]\}_{k=1}^K$4

is square, channel-independent at the transmitter, and designed so that every nonzero codeword-difference vector produces nonzero transformed coordinates. The paper states that the proposed linear precoded OTFS systems guarantee the maximal diversity and potential coding gains in time or frequency selective fading channels without any transmission rate loss and do not require CSI at the transmitter. In the frequency-selective case, the achieved diversity order is exactly $\{\mathbf F[k]\}_{k=1}^K$5, where $\{\mathbf F[k]\}_{k=1}^K$6 is the number of resolvable paths (Ge et al., 2024). Among the surveyed works, this is the most direct example of frequency-domain linear non-redundant precoding in the strict full-rate sense.

GMUD precoding for single-carrier transmission over frequency-selective channels remains adjacent rather than direct (0812.4334). It does not use OFDM-style per-tone weights, but exploits a block Toeplitz representation of the multipath channel and factorizes $\{\mathbf F[k]\}_{k=1}^K$7, using the first column of $\{\mathbf F[k]\}_{k=1}^K$8 as the transmission beam. The method is linear and does not add explicit coding redundancy, but it is more accurately described as block-domain precoding for a frequency-selective channel than as a conventional frequency-domain precoder.

6. No-CP multicarrier, diversity theory, and conceptual boundaries

The strongest non-redundant multicarrier case arises in MU-MIMO FBMC-OQAM under strong channel frequency selectivity (Rottenberg et al., 2016). Because FBMC-OQAM has no cyclic prefix, there is no redundancy available to absorb channel dispersion, and the usual flat-per-subcarrier approximation is insufficient. The transmitter and receiver still use one matrix per subcarrier, $\{\mathbf F[k]\}_{k=1}^K$9 and $\mathcal O(K)$0, but the asymptotic MSE includes not only the classical mismatch term $\mathcal O(K)$1 but also terms involving $\mathcal O(K)$2, $\mathcal O(K)$3, and, in general, $\mathcal O(K)$4. The resulting derivative-aware ZF and MMSE designs remain single-tap and purely frequency-domain, yet explicitly account for strong selectivity (Rottenberg et al., 2016).

A central structural result is that excess BS antennas can be used not only for user separation but also for first-order selectivity compensation. In the MU SDMA specialization with $\mathcal O(K)$5 BS antennas and $\mathcal O(K)$6 single-antenna users, the optimized ZF decoder or precoder takes the form pseudo-inverse plus null-space correction. The paper states that if $\mathcal O(K)$7, then the first-order asymptotic distortion can be completely canceled at high SNR; if $\mathcal O(K)$8, only partial compensation is possible (Rottenberg et al., 2016). This sharply distinguishes single-tap non-redundant FBMC from CP-based OFDM, where circularization handles the frequency selectivity before per-tone processing.

The diversity behavior of linear precoders is clarified from a different angle by high-SNR analysis of flat-fading MIMO linear precoding (Mehana et al., 2012). Although this work is only indirectly frequency-domain, its conclusions are informative for per-tone deployment. ZF precoding has diversity $\mathcal O(K)$9; fixed-regularization RZF and matched-filter precoding have full diversity $\mathcal O(K/M+\log M)$0 below the threshold

$\mathcal O(K/M+\log M)$1

and zero diversity above it in the fixed-rate regime; Wiener precoding has a rate-dependent diversity law rather than a single threshold (Mehana et al., 2012). A plausible implication is that per-tone frequency-domain designs inherit strong sensitivity to how regularization is chosen and to whether residual interference creates an error floor.

Several common misconceptions are resolved by reading these works together. First, frequency-domain processing does not imply multicarrier signaling: CP-SCM can be precoded binwise in the DFT domain while remaining single-carrier in time (Kenney et al., 2021). Second, avoiding explicit time-domain filtering does not imply absence of rate back-off: frequency-domain TR still uses BOF and therefore fewer independent symbols per OFDM block (Nguyen et al., 2019). Third, “single-tap” does not imply “locally flat channel assumption”: in FBMC-OQAM, single-tap matrices can be optimized against $\mathcal O(K/M+\log M)$2 and $\mathcal O(K/M+\log M)$3 to mitigate strong selectivity (Rottenberg et al., 2016). Fourth, non-redundancy is not a universal algebraic property but an architectural one whose meaning shifts across OFDMA beamforming, transform precoding, hybrid codebook compression, and full-rate OTFS design.

Taken together, these results define frequency-domain linear non-redundant precoding as a broad design philosophy rather than a single standardized technique. Its most minimal form is one beamformer per scheduled tone (Kibria et al., 2017). Its most compressed form is a sparse description of a wideband precoder family across correlated subcarriers (Chou et al., 4 May 2026). Its strongest rate-preserving form is a square transform designed to guarantee maximal diversity without transmission-rate loss (Ge et al., 2024). Its most demanding no-redundancy operating point is single-tap FBMC without CP, where all compensation must come from frequency-domain linear design and spatial null-space exploitation (Rottenberg et al., 2016).