Widely Linear Models in Complex Signal Processing

• Widely linear models are augmented estimators that process both a complex observation and its conjugate, capturing the full second-order structure of improper or noncircular data.
• They leverage the real-valued structure in scenarios where parameters are real yet measurements are complex, leading to lower variance and improved performance over conventional BLUE or LS estimators.
• These models find applications in communications, MIMO systems, neural networks, and hypercomplex processing, offering measurement efficiency by using fewer complex measurements.

Widely linear models are complex-domain models that process an observation together with its complex conjugate, replacing the strictly linear form $\hat{x}=Wy$ by $\hat{x}=Wy+Vy^*$. They are the natural second-order models for improper or noncircular data, because covariance alone does not characterize such processes and the pseudo-covariance must also be retained. In classical estimation, this framework is especially important when real-valued parameter vectors are inferred from complex measurements, where conventional BLUE or LS estimators can return complex-valued estimates and fail to exploit the known real structure (Lang et al., 2016, Lang et al., 2017).

1. Statistical structure and the basic widely linear form

In complex-valued signal processing, an estimator or filter is usually called strictly linear if it has the form

$\hat{x} = Wy,$

where $y\in\mathbb{C}^{N_y}$ is a complex measurement vector. A widely linear estimator uses both $y$ and its complex conjugate: $\hat{x} = Wy + Vy^*.$ This is strictly more general, since $V=0$ recovers the conventional linear estimator. The construction is not complex-linear, but it is real-linear; equivalently, it is linear on the augmented pair $(y,y^*)$ (Lang et al., 2016).

The relevant second-order statistics are the covariance and pseudo-covariance. For a complex random vector $y$ with mean $m_y=E[y]$,

$\hat{x}=Wy+Vy^*$0

If $\hat{x}=Wy+Vy^*$1, $\hat{x}=Wy+Vy^*$2 is proper or circular; if $\hat{x}=Wy+Vy^*$3, $\hat{x}=Wy+Vy^*$4 is improper or noncircular. Real-valued vectors form the extreme case of impropriety in the complex embedding, since for $\hat{x}=Wy+Vy^*$5, covariance and complementary covariance coincide (Lang et al., 2017).

A compact way to encode these statistics is the augmented vector

$\hat{x}=Wy+Vy^*$6

with augmented covariance

$\hat{x}=Wy+Vy^*$7

Strictly linear processing uses only $\hat{x}=Wy+Vy^*$8, whereas widely linear processing accesses the full augmented covariance. This is why widely linear estimators can achieve lower MSE or lower variance than strictly linear estimators when the data are improper (Lang et al., 2016).

2. Classical linear estimation and the limitation of strictly linear models

A central model in the literature is the classical linear measurement equation

$\hat{x}=Wy+Vy^*$9

with $\hat{x} = Wy,$0 a deterministic but unknown real-valued parameter vector, $\hat{x} = Wy,$1 a complex measurement vector, $\hat{x} = Wy,$2 a complex measurement matrix, and $\hat{x} = Wy,$3 zero-mean complex noise (Lang et al., 2016).

If the estimator is restricted to be linear in $\hat{x} = Wy,$4 and unbiased, the standard BLUE is

$\hat{x} = Wy,$5

Under proper noise, the standard BWLUE for complex parameters collapses to the same expression. However, when $\hat{x} = Wy,$6 and $\hat{x} = Wy,$7 are complex while $\hat{x} = Wy,$8 is known to be real, this estimator is generally complex-valued. Its imaginary part cannot correspond to any true parameter value, and taking only $\hat{x} = Wy,$9 is only a heuristic correction; it is optimal only in special cases, such as when $y\in\mathbb{C}^{N_y}$0 is real (Lang et al., 2017).

The same structural issue appears for least squares. Treating $y\in\mathbb{C}^{N_y}$1 as complex yields the standard LS solution

$y\in\mathbb{C}^{N_y}$2

but this is not the true LS solution under the real-valued constraint on $y\in\mathbb{C}^{N_y}$3. A purely real composite model, obtained by stacking real and imaginary parts, is equivalent to the correct constrained problem, but it obscures the compact complex-domain structure. Widely linear models preserve that structure while enforcing the real-valued nature of the estimate (Lang et al., 2017).

3. Best widely linear unbiased estimation for real-valued parameter vectors

The main estimator derived in "Best Widely Linear Unbiased Estimator for Real Valued Parameter Vectors" addresses the case $y\in\mathbb{C}^{N_y}$4 directly. Starting from the componentwise widely linear form

$y\in\mathbb{C}^{N_y}$5

the condition for $y\in\mathbb{C}^{N_y}$6 to be real-valued for all realizations of $y\in\mathbb{C}^{N_y}$7 is

$y\in\mathbb{C}^{N_y}$8

Hence the estimator becomes

$y\in\mathbb{C}^{N_y}$9

Classical unbiasedness requires

$y$0

which yields the constraint

$y$1

where $y$2 is the $y$3-th canonical row vector (Lang et al., 2016).

For proper noise, minimizing the estimator variance under this constraint produces

$y$4

and therefore

$y$5

The most transparent expression is

$y$6

This estimator is real-valued by construction and unbiased in the classical sense (Lang et al., 2016).

Its covariance matrix simplifies to

$y$7

By contrast, the BLUE covariance is

$y$8

The proposed estimator therefore exploits the real-valued parameter structure that BLUE and the standard proper-noise BWLUE do not use; the paper states that it in general outperforms BLUE and BWLUE in terms of the variances of the vector estimator’s elements (Lang et al., 2016).

A broader formulation in (Lang et al., 2017) allows improper noise by replacing the ordinary covariance with the augmented noise covariance

$y$9

and writing

$\hat{x} = Wy + Vy^*.$0

with $\hat{x} = Wy + Vy^*.$1. Under proper noise this reduces to the compact real-part formula above (Lang et al., 2017).

4. Widely linear least squares and measurement efficiency

The least-squares analogue of the real-valued constrained estimator is the widely linear least squares estimator. For the real-valued parameter vector $\hat{x} = Wy + Vy^*.$2, minimizing

$\hat{x} = Wy + Vy^*.$3

over $\hat{x} = Wy + Vy^*.$4 gives

$\hat{x} = Wy + Vy^*.$5

Using $\hat{x} = Wy + Vy^*.$6 and $\hat{x} = Wy + Vy^*.$7, this becomes

$\hat{x} = Wy + Vy^*.$8

Thus WLLS is the LS solution that respects the real-valued structure without leaving complex notation (Lang et al., 2017).

The weighted version is

$\hat{x} = Wy + Vy^*.$9

with $V=0$0. If $V=0$1 is real, then

$V=0$2

but otherwise WLLS strictly improves upon the real-part heuristic (Lang et al., 2017).

The same paper emphasizes a measurement-efficiency consequence. In the equivalent real-composite model, full column rank requires

$V=0$3

Hence only half as many complex measurements as real parameters are needed: $V=0$4 This is a structural gain: widely linear estimators tailored to real-valued $V=0$5 use $V=0$6 real equations, whereas standard estimators designed for complex $V=0$7 effectively estimate $V=0$8 real unknowns (Lang et al., 2017).

5. Applications in communication and signal processing

Widely linear models are used whenever data or impairments create noncircularity. The core signal-processing examples listed in the literature include communications with noncircular constellations, IQ imbalance and mixer imperfections, array processing with noncircular sources, adaptive filtering, and widely linear MMSE filtering in MIMO systems (Lang et al., 2016).

A recurrent estimation example is the recovery of a real-valued impulse response from complex spectral measurements. In (Lang et al., 2016), the proposed BWLUE for real parameter vectors was evaluated for a discrete-time impulse response $V=0$9 using 20 equidistant frequency-response measurements. The new estimator provided a significant performance improvement, almost matching WLMMSE for moderate noise variances, and outperformed BLUE/BWLUE by about two orders of magnitude in average BMSE across the full range of $(y,y^*)$0 considered. In (Lang et al., 2017), the same problem was revisited in a more involved setting with noisy magnitude and phase measurements, where a two-step "WLLS + BWLUE-real" approach nearly attained an oracle bound obtained from true noise statistics.

In full-duplex direct-conversion radio, transmitter and receiver IQ imbalance make the dominant digital self-interference waveform widely linear in the transmitted data, rather than purely linear. The paper "Widely-Linear Digital Self-Interference Cancellation in Direct-Conversion Full-Duplex Transceiver" models the residual self-interference as

$(y,y^*)$1

and shows that classical linear cancellation leaves the conjugate component essentially untouched, whereas widely linear digital cancellation suppresses both terms (Korpi et al., 2014).

In downlink precoding, widely linear processing is particularly effective when the transmitted symbols are real-valued. "Low-Complexity Widely-Linear Precoding for Downlink Large-Scale MU-MISO Systems" formulates WL-MMSE precoding in an augmented real-valued model and reports that, in overloaded systems, a polynomial-expansion WL-MMSE precoder with only a few terms achieves a substantially higher sum rate than conventional MMSE precoding (Zarei et al., 2015). Closely related work on large-scale MIMO with transmitter IQ imbalance develops WL-ZF, WL-MF, WL-MMSE, and WL-BD precoders, and shows that WL-ZF preserves the same multiplexing gain as ideal ZF while incurring only a minor power loss related to the system scale and IQ parameters (Zhang et al., 2017).

One-dimensional signalling provides another canonical use case. For multiuser MISO with PAM symbols and widely linear estimation at the receivers, the real-part constraints in WL-ZF and WL-MMSE allow the transmitter to exploit $(y,y^*)$2 real spatial degrees of freedom. The paper "User Selection and Widely Linear Multiuser Precoding for One-dimensional Signalling" states that widely linear processing can potentially double the number of simultaneous users compared to linear processing of one-dimensionally modulated signals, and proposes a user-selection algorithm that can likewise double the number of simultaneously selected users (Bavand et al., 2017).

Receiver design for noncircular modulation follows the same logic. In DS-CDMA with BPSK and jamming, a widely linear augmented MMSE receiver combined with multiple-candidate SIC and vector space projection improves both MAI suppression and jamming suppression, because the receiver processes both $(y,y^*)$3 and $(y,y^*)$4 and explicitly includes the complementary covariance $(y,y^*)$5 (Yang et al., 2014). In DFT-precoded OFDM over wideband frequency-selective channels, the WL front-end for real constellations combines $(y,y^*)$6 and $(y,y^*)$7, leading to post-SNR expressions that are closer to the matched-filter bound than those of conventional linear equalizers and DFEs (Kuchi, 2013).

6. Extensions to machine learning and hypercomplex domains

The same augmented-variable principle extends beyond classical estimators. In complex-valued neural networks, a widely linear layer replaces the strictly linear map $(y,y^*)$8 with

$(y,y^*)$9

The paper "Widely Linear Complex-valued Autoencoder: Dealing with Noncircularity in Generative-Discriminative Models" uses this transform in every layer of an autoencoder and develops the corresponding backpropagation with $y$0 calculus. Its motivation is explicit: strictly linear complex autoencoders are optimal only when the outputs of each layer are independent of the conjugate of the inputs, whereas the widely linear model allows the network to consider all second-order statistics of the inputs (Yu et al., 2019).

Hypercomplex generalizations replace conjugation by a larger augmented set. In quaternion processing, a widely linear estimator uses the quaternion regressor together with its three involutions: $y$1 The augmented quaternion vector

$y$2

plays the same role as $y$3 in the complex case: it provides a sufficient representation of second-order structure for quaternion-valued random processes and leads to widely linear MMSE and QLMS constructions (Talebi et al., 12 Mar 2026).

A different extension appears in low-bit quantization for LLMs. "FAIRY2I" proves a lossless mathematical equivalence between a real linear map

$y$4

and a widely linear complex map

$y$5

with a unique pair $y$6. This equivalence is used to transform pre-trained real-valued Transformer layers into widely linear complex form before phase-aware quantization (Wang et al., 2 Dec 2025).

A communications-oriented machine-learning example is "Widely Linear Augmented Extreme Learning Machine Based Impairments Compensation for Satellite Communications," where a complex ELM with augmented hidden layer is combined with a tailored widely linear least-squares output stage. In the reported experiments, CELM-WLLS improved BER performance by approximately $y$7 dB over CELMAH while also achieving a two-thirds reduction in computational complexity (Luo et al., 17 Jun 2025).

7. Scope, misconceptions, and terminological variation

A common misconception is that widely linear processing is only relevant when the noise is improper. The classical estimators in (Lang et al., 2016) and (Lang et al., 2017) show otherwise: even with proper noise, widely linear processing becomes useful when the parameter vector is known to be real-valued but the observation model is complex-valued. In that setting, the extra structure comes from the parameter space rather than from an improper disturbance.

Another misconception is that widely linear estimation is equivalent to computing a conventional complex estimator and then taking its real part. This is true only in special cases, such as when $y$8 or $y$9 is real. In general, WLLS and BWLUE-real impose different optimality conditions and deliver lower variance than $m_y=E[y]$0 or $m_y=E[y]$1 (Lang et al., 2017).

The term itself also has a distinct usage outside complex statistical signal processing. In "Transition to Linearity of Wide Neural Networks is an Emerging Property of Assembling Weak Models," "widely linear models" refers to wide neural networks that become approximately linear in parameter space in an $m_y=E[y]$2 neighborhood of initialization, with a nearly constant NTK along training. That usage concerns an emergent parameter-space linearization of wide networks, rather than the augmented $m_y=E[y]$3 framework of complex-valued estimation (Liu et al., 2022).

In the signal-processing sense, however, the unifying idea is stable across domains: once the full second-order structure is not captured by a strictly complex-linear map, the model must be augmented. In the complex case this means $m_y=E[y]$4 and $m_y=E[y]$5; in the quaternion case it means the signal and its involutions; in structured estimation with real-valued parameters it means matching the estimator class to the real subspace while retaining compact complex notation. Widely linear models are therefore best understood as the second-order complete linear models for improper or structurally constrained complex data.