Null-Space-Based Transmission Scheme

• Null-space-based transmission is a design methodology that projects signals into a channel's null space to enable interference-free communication and secure transmissions.
• It relies on linear algebra techniques such as SVD and null-space decomposition to facilitate cognitive MIMO, beamforming, SWIPT, and radar waveform design.
• Practical applications include artificial-noise-aided secure transmission, blind null-space tracking, and successive precoding, balancing performance with computational efficiency.

A null-space-based transmission scheme is a design methodology in which a transmitter, beamformer, or waveform generator places selected signal components in the null space of a channel matrix or a structured linear operator so that protected receivers or protected dimensions experience zero or controlled interference. In wireless communications, this construction appears in cognitive MIMO, simultaneous wireless information and power transfer (SWIPT), artificial-noise-aided secure transmission, and downlink multi-user precoding; in radar, it appears in Doppler-resilient complementary waveform design; and in related inverse problems it appears as a range–null-space decomposition that separates observation-consistent and unobserved components (Sodagari, 2014, Manolakos et al., 2012, Luo et al., 12 Sep 2025, Ham, 2021).

1. Linear-algebraic foundation

In cognitive MIMO communications, the canonical formulation begins from the singular value decomposition

$$\mathbf{H}=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H,$$

where the null space corresponds to columns of $\mathbf{V}$ associated with near-zero singular values. A secondary transmitter can then project its signal into that null space so that transmission does not couple into the primary user’s receive subspace. The associated projection matrix is

$$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$

where $\breve{\mathbf{V}}$ collects the null-space right singular vectors. This operation is described as inverse waterfilling, because the secondary user transmits only in eigenmodes causing zero interference to the primary user (Sodagari, 2014).

The same structure generalizes beyond channel nulling. In OFDM channel estimation with sparse DMRS observations, the measurement operator $\mathbf{A}$ induces a range–null decomposition

$$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$

with the first term representing the observed or recoverable part and the second term the unobserved null-space component. Any observation-consistent estimate can then be written as

$$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$

which makes the null space an explicit design variable rather than merely a forbidden subspace (Qi et al., 3 Jul 2026).

This algebraic viewpoint unifies several otherwise different designs. In beamforming problems, the null space is usually defined by unintended users’ channel matrices; in waveform design, it is defined by a Vandermonde matrix that encodes Doppler samples; in inverse problems, it is defined by the measurement mask or compression operator. The central mechanism is the same: preserve the range-space component dictated by constraints, and use the null-space component to suppress interference, preserve secrecy, deliver energy, or reconstruct missing structure.

2. Underlay cognitive MIMO and blind null-space tracking

In underlay cognitive radio, the protected object is the channel from the secondary transmitter to the primary receiver. Let $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$ denote the SU-Tx to PU-Rx channel. The transmission objective is to find a precoder $\mathbf{T}(t)$ such that

$\mathbf{H}_{12}(t)\mathbf{T}(t)\approx \mathbf{0}.$

Blind Null Space Learning (BNSL) performs this acquisition through iterative Jacobi Eigenvalue Decomposition; after $\mathbf{V}$0 learning stages, one sweep produces an initial estimate of the null space. Blind Null Space Tracking (BNST) extends this by continuously updating the estimate in time-varying channels via modified Jacobi rotations, while keeping interference to the primary receiver lower than a threshold $\mathbf{V}$1 with probability $\mathbf{V}$2 and simultaneously transmitting information to the secondary receiver (Manolakos et al., 2012).

The interference criterion is expressed through the normalized maximum interference

$\mathbf{V}$3

and the success condition

$\mathbf{V}$4

After the initial sweep, BNST restricts the search for Jacobi rotation angles to a neighborhood around zero, because for low Doppler the channel evolves gradually and the required updates are small. The same tracking cycle also carries data through signal superposition

$\mathbf{V}$5

with $\mathbf{V}$6 chosen so that $\mathbf{V}$7 and $\mathbf{V}$8, ensuring that average energy-based interference measurements at the primary receiver are preserved. For a binary alphabet, the example $\mathbf{V}$9, $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$0, with $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$1, satisfies the stated constraints (Manolakos et al., 2012).

The reported simulations show that the null space of i.i.d. Rayleigh fading MIMO channels can change much faster than the per-element coherence time of the channel matrix, which motivates tracking rather than one-time acquisition. BNST maintains low interference more consistently than BNSL, especially at low Doppler frequencies; the $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$2 interference reduction can be several dB better; and superimposed data transmission during BNST learning achieves low BER at moderate SNR without increasing interference to the PU-Rx, with average interference increase only $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$3 dB (Manolakos et al., 2012).

The main caveat is CSI sensitivity. If the secondary user uses an estimated interference channel $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$4, singular values and singular vectors are perturbed. Weyl’s theorem yields

$$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$5

Mirsky’s theorem gives the Frobenius-norm aggregate bound

$$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$6

and Wedin’s theorem bounds null-space misalignment through canonical angles. The resulting interference leakage raises the PU BER and can degrade performance to that of open loop MIMO; the same conclusions apply to null-space-based MIMO radar waveform design operating in the same or adjacent bands (Sodagari, 2014).

3. Multiuser SWIPT and interference-free beamforming

In multiuser SWIPT, a multi-antenna Hybrid Access Point serves information users (IUs) and energy users (EUs) simultaneously, while suppressing cross-interference between wireless information transmission and wireless energy transmission. The null-space-based construction uses the HAP’s spatial degrees of freedom so that information beams for IU $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$7, $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$8, lie in the null space of the other IUs’ channels, and energy beams for EU $$P_{\breve{\mathbf{V}}} = \breve{\mathbf{V}} (\breve{\mathbf{V}}^H\breve{\mathbf{V}})^{-1} \breve{\mathbf{V}}^H,$$9, $\breve{\mathbf{V}}$0, lie in the null space of all IUs’ channels: $\breve{\mathbf{V}}$1 This yields interference-free intended reception for IUs, while EUs can harvest from both information beams and energy beams unless waveform separation is required (Luo et al., 12 Sep 2025).

The SWIPT literature in the supplied sources makes a sharp distinction between Gaussian signaling and deterministic sinusoidal waveforms. When both information and energy signals use i.i.d. Gaussian waveforms, the information beams themselves carry substantial energy suitable for wireless energy transfer, and dedicated energy beams are unnecessary and potentially harmful due to the spatial cost of suppressing EB interference at IUs. When energy signals use deterministic sinusoidal waveforms and IBs remain Gaussian, the two signal types must be separated, and dedicated EBs are transmitted in the IUs’ null space. The benefit of dedicated EBs grows when the received RF power lies in the energy harvester’s high-efficiency region under the nonlinear EH model (Luo et al., 12 Sep 2025).

The corresponding optimization maximizes total harvested DC energy under IU capacity constraints and a transmit power budget. After SVD-based null-space construction, beamformers are expressed as linear combinations of null-space basis vectors, yielding tractable convex SDP formulations. For Gaussian signaling, the solution sets all EBs to zero; for deterministic sinusoidal signaling, power is split between IBs and EBs according to the extra benefit captured by the reward factor $\breve{\mathbf{V}}$2. A low-complexity algorithm further ignores the WET contribution of IBs during EB optimization, decoupling the multiuser design into a point-to-point WIT problem and a WET beamforming problem. The reported computational complexity reductions are $\breve{\mathbf{V}}$3 and $\breve{\mathbf{V}}$4 for the cases $\breve{\mathbf{V}}$5 and $\breve{\mathbf{V}}$6, respectively, with negligible performance loss (Luo et al., 12 Sep 2025).

A closely related formulation projects the interference signals from both intra-wireless information transfer and inter-wireless energy transfer into the null space, simplifying the system into a point-to-point WIT and WET problem. In that formulation, information beams for IU $\breve{\mathbf{V}}$7 and energy beams satisfy

$\breve{\mathbf{V}}$8

and the beamformers are parameterized as

$\breve{\mathbf{V}}$9

The analysis confirms that dedicated energy beamforming is unnecessary; in the optimal null-space-based design, the EB component is always zero. The low-complexity algorithm based on MRT in the appropriate null spaces reduces computational complexity by $\mathbf{A}$0 and $\mathbf{A}$1 for the cases $\mathbf{A}$2 and $\mathbf{A}$3, respectively (Luo et al., 11 Mar 2025).

4. Secure transmission, artificial noise, and phase-only zero-forcing

In physical-layer security, null-space-based transmission most often appears as artificial-noise projection. In a directional modulation network, the null-space projection baseline chooses the confidential-message beamforming vector and the artificial-noise projection matrix so that the AN lies entirely in the null space of Bob’s channel,

$\mathbf{A}$4

which totally eliminates AN interference at Bob. This is computationally simple, typically requiring an SVD or similar computation with complexity $\mathbf{A}$5, but it restricts the AN structure and can reduce its effect against Eve when the channels are highly correlated (Yu et al., 2017).

The secrecy-rate-maximization alternative replaces strict null-space projection by joint optimization of the useful precoding vector $\mathbf{A}$6 and the artificial-noise projection matrix $\mathbf{A}$7. The transmit signal is

$\mathbf{A}$8

and the secrecy rate is

$\mathbf{A}$9

An alternatively iterative structure and a General Power Iterative method are used to optimize the two variables. With leakage-based initialization, the method can readily double its convergence speed compared to a random initial value; with only four iterations, it may rapidly converge to its rate ceil; and its secrecy-rate performance is much better than those of conventional leakage-based and null-space projection methods in the medium and large SNR regions (Yu et al., 2017).

A more restrictive but hardware-relevant setting is phase-only zero-forcing for AN transmission with a single RF chain and analog beamforming. There the beamforming vector has constant magnitude,

$$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$0

and must satisfy

$$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$1

For a single user, existence is characterized by the polygon inequality

$$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$2

For multiple users, the successive partition zero-forcing (SPZF) scheme transforms the multi-user zero-forcing task into optimizing channel partitioning to minimize outage probability, recursively reducing the problem to successive single-user polygon constructions. Theoretical analysis shows that the proposed SPZF scheme can attain arbitrarily low outage probability in the limit of large number of transmit antenna. Three partition algorithms—random, iterative, and genetic—are presented, and the iterative and genetic algorithms achieve higher secrecy rates than the random algorithm, particularly under conditions of high SNR, large number of eavesdroppers, or small number of transmit antennas (Hong et al., 26 Mar 2025).

These two secure-transmission lines reveal a recurring tradeoff. Strict nulling guarantees interference suppression at protected users, but the feasible null space can be too restrictive for secrecy-rate maximization; relaxed or structured use of the available null-space degrees of freedom can improve secrecy performance at the cost of more complex optimization.

5. Radar waveform construction by null-space constraints

In radar, null-space-based transmission is formulated at the waveform-sequence level rather than the spatial-beamforming level. For Doppler-resilient complementary waveforms, the transmitted Golay pulse train is controlled by a characteristic vector $$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$3,

$$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$4

and the receive filter uses coefficients $$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$5,

$$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$6

The cross-ambiguity function contains a range-sidelobe term controlled by

$$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$7

The design goal is to force this term to zero over a specified Doppler interval $$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$8 (Wang et al., 2021).

Sampling the Doppler interval at $$\mathbf{h} = \mathbf{A}^\dagger \mathbf{A}\mathbf{h} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\mathbf{h},$$9 leads to the Vandermonde matrix

$$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$0

and the null-space constraint

$$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$1

Any nonzero $$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$2 can be used to extract the transmit and receive coefficients, for example

$$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$3

with more general constructions available for complex values. The interval $$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$4 is fully selectable; if $$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$5, a nontrivial null space exists (Wang et al., 2021).

Because an arbitrary null-space vector may yield poor receiver SNR, SNR maximization is performed within the null space. The SNR is

$$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$6

and with a null-space basis $$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$7, the search becomes

$$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$8

The paper studies basis selection and heuristic coordinate descent, both fully within the null space, so Doppler resilience is retained while SNR is improved (Wang et al., 2021).

The same framework extends to fully polarimetric radar, where the coefficients of the receiver filter and the characteristic vector can be applied to achieve nearly perfect Doppler resilient performance and fully suppress the inter-antenna interferences. The reported examples include $$\hat{\mathbf{h}} = \mathbf{A}^\dagger \mathbf{y} + (\mathbf{I}-\mathbf{A}^\dagger \mathbf{A})\tilde{\mathbf{h}},$$9, length-64 Golay pairs, and $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$0 rad, where sidelobes are suppressed to below $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$1 dB within the interval; for $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$2, sidelobes are suppressed over the entire interval, outperforming previous binomial design methods (Wang et al., 2021).

6. Extensions, generalizations, and robustness limits

Null-space-based transmission has also been adapted to more flexible multi-user precoding architectures. In downlink MIMO-RSMA with successive null-space precoding, the private precoder for user $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$3 is built from the null space of the successively augmented channel matrix of users $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$4 through $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$5. With

$\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$6

and $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$7 an orthonormal basis for $\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$8, the private precoder is

$\mathbf{H}_{12}(t)\in\mathbb{C}^{N_r\times N_t}$9

This construction allows signals intended for user $\mathbf{T}(t)$0 to avoid interference to previously indexed users while allowing controlled interference to later users. The associated weighted-sum-rate maximization is non-convex and is handled by successive convex approximation. The reported simulations show gains of $\mathbf{T}(t)$1 to $\mathbf{T}(t)$2 WSR over baselines, especially when CSI is significantly imperfect (Krishnamoorthy et al., 2021).

The broader range–null perspective also appears in recent estimation and reconstruction models. In OFDM channel estimation, DANCE reconstructs the unobserved null-space component through a learned diffusion prior and uses a noise-adaptive posterior correction

$\mathbf{T}(t)$3

with $\mathbf{T}(t)$4 calibrated according to the observation noise level. It achieves consistently lower NMSE than conventional estimators and diffusion-based posterior sampling methods under different signal-to-noise ratios, DMRS configurations, Doppler frequency shifts, and train-test distribution mismatches, and reaches near-optimal accuracy with as few as $\mathbf{T}(t)$5 sampling steps (Qi et al., 3 Jul 2026).

In neural vocoding, the reconstruction of the target spectrogram is formulated as the superimposition between range-space and null-space: $\mathbf{T}(t)$6 The range-space term projects the mel representation into the target linear-scale domain, and the null-space term is instantiated via neural networks to infill spectral details. The paper explicitly notes a transmission perspective in which null-space-based transmission schemes enable lossless transmission of observed features and only transmit or generate null-space details (Li et al., 9 Mar 2026).

The main limitation that recurs across these domains is null-space fragility under model mismatch. In cognitive MIMO, imperfect CSI causes null-space misalignment and interference leakage (Sodagari, 2014). In successive null-space precoding, the computed basis vectors become sensitive to channel perturbations, although derivative-based analysis shows that additional inter-user interference is small when estimation errors are small (Krishnamoorthy et al., 2021). This suggests that null-space-based transmission is best understood not as a single algorithm, but as a design pattern whose success depends on how accurately the protected subspace is identified, how efficiently the remaining degrees of freedom are exploited, and how robustly the null-space constraint is maintained as the system evolves.