Fourth-Order Difference Co-Array Analysis

• FODCA is a virtual array constructed from fourth-order cumulants, extending aperture and degrees of freedom in sparse array processing.
• It employs dual formulation forms—(a+b)-(c+d) and (a-b)+(c-d)—to exploit non-Gaussian signal properties and mitigate Gaussian noise.
• Implementations like FOGNA and FOHA demonstrate enhanced DOA resolution, reduced mutual coupling, and robust performance in challenging conditions.

Fourth-Order Difference Co-Array (FODCA) is a virtual-array construct in sparse array processing obtained from fourth-order cumulants of sensor outputs. For a linear array with physical sensor positions $\mathbb{P}=\{p_1,p_2,\dots,p_N\}$, a formal definition used in recent work is

$$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$

with $k_1,k_2,k_3,k_4\in[1,N]$. The elements of $\Delta_4(\mathbb{P})$ act as virtual sensor locations generated by fourth-order statistics, thereby enlarging the virtual aperture and the degrees of freedom (DOFs) available for direction-of-arrival (DOA) estimation. In the same line of work, FODCA is contrasted with second-order difference co-arrays and interpreted as part of a broader family of fourth-order co-array constructions, including extended formulations that combine multiple cumulant cases. Because all cumulants of order $>2$ vanish for Gaussian processes, FODCA-based processing naturally suppresses Gaussian noise while exploiting the non-Gaussian structure of the sources (Wang et al., 2024, Wang et al., 27 Aug 2025).

1. Statistical basis and formal definition

The standard narrowband array model used in the recent FODCA literature is

$$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$

or, when mutual coupling is included,

$$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$

where the sources are non-Gaussian and mutually independent, the additive noise is Gaussian and independent of the sources, and the steering term at sensor $n$ is

$$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$

or equivalently with the sign convention used in the corresponding paper. Under these assumptions, the circular fourth-order cumulants reduce to array-manifold expressions whose exponents are linear combinations of four sensor positions. Two forms emphasized in the recent formulation are

$$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$

and

$$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$0

Each exponent defines a virtual sensor location in the fourth-order co-array (Wang et al., 27 Aug 2025).

In the broader co-array hierarchy summarized in the same literature, the conventional second-order difference co-array is

$$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$1

while fourth-order constructions generate lags of the form $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$2 or related sign patterns. With $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$3 physical sensors, second-order DCA provides $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$4 lags, third-order DCA gives $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$5 lags, and FODCA can yield $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$6 lags. The useful DOFs for subspace DOA estimation are tied to the number of virtual lags, especially the longest hole-free contiguous segment of the virtual array (Wang et al., 2024).

2. Expression forms, virtual manifolds, and extended formulations

Recent FODCA research distinguishes multiple fourth-order index patterns rather than treating the co-array as a single fixed expression. One line of work starts from three fourth-order cumulant tensors,

$$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$7

$$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$8

$$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$9

and associates them with three fourth-order co-arrays: $k_1,k_2,k_3,k_4\in[1,N]$0

$k_1,k_2,k_3,k_4\in[1,N]$1

$k_1,k_2,k_3,k_4\in[1,N]$2

In that framework, $k_1,k_2,k_3,k_4\in[1,N]$3 is the classical FODCA form, symmetric around zero, while $k_1,k_2,k_3,k_4\in[1,N]$4 is essentially the negation of $k_1,k_2,k_3,k_4\in[1,N]$5. The corresponding equivalent steering vectors are obtained from Kronecker and Khatri-Rao combinations of the physical steering vector $k_1,k_2,k_3,k_4\in[1,N]$6, and the vectorized cumulant tensors satisfy

$k_1,k_2,k_3,k_4\in[1,N]$7

with $k_1,k_2,k_3,k_4\in[1,N]$8 (Wang et al., 2024).

On that basis, the fourth-order extend co-array (FOECA) is defined by the union

$k_1,k_2,k_3,k_4\in[1,N]$9

with the combined cumulant vector

$\Delta_4(\mathbb{P})$0

FOECA is described as equivalent to FODCA in the sense that it retains the standard fourth-order difference lags while adding extra virtual positions from other fourth-order index patterns, thereby enlarging the usable virtual aperture (Wang et al., 2024).

A related but distinct formulation defines FODCA itself as the union of both $\Delta_4(\mathbb{P})$1 and $\Delta_4(\mathbb{P})$2 forms. Taken together, these two formulations show that current research has moved away from the earlier practice of using only one fourth-order expression form. This suggests that the main conceptual shift in the field is from a single-case difference-only viewpoint toward a multi-form fourth-order virtual-array framework (Wang et al., 27 Aug 2025).

3. Sparse-array realizations: FOGNA and FOHA

A major recent FODCA-oriented design is the fourth-order generalized nested array (FOGNA). It consists of three subarrays placed side-by-side: a concatenated nested array (CNA) with $\Delta_4(\mathbb{P})$3 sensors and small spacings, and two sparse linear subarrays with large inter-sensor spacings. The total number of physical sensors is

$\Delta_4(\mathbb{P})$4

With

$\Delta_4(\mathbb{P})$5

and

$\Delta_4(\mathbb{P})$6

the physical sensor set $\Delta_4(\mathbb{P})$7 is given in closed form by

$\Delta_4(\mathbb{P})$8

$\Delta_4(\mathbb{P})$9

and, with

$>2$0

$>2$1

The purpose of this construction is to obtain closed-form sensor locations, maximize DOF under the proposed structure, and reduce mutual coupling by concentrating the dense spacing in only one subarray (Wang et al., 2024).

Another recent design family is the fourth-order hierarchical array (FOHA), which uses three disjoint subarrays

$>2$2

Here $>2$3 is a dense generator, while $>2$4 and $>2$5 are sparse arithmetic progressions: $>2$6 with

$>2$7

$>2$8

The generator $>2$9 can be instantiated either as a nested array, yielding FOHA(NA), or as a concatenated nested array, yielding FOHA(CNA). In both cases, the design is hierarchical because the second-order co-arrays of $$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$0 determine $$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$1, the addition of $$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$2 creates third-order difference sets, and $$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$3 completes the fourth-order coverage (Wang et al., 27 Aug 2025).

4. Hole-free structure, degrees of freedom, and redundancy

Hole-freeness is central because spatial smoothing and subspace DOA methods are simplest when the virtual lags form a contiguous interval. For FOGNA, the recent analysis proves that the FOECA is hole-free. The proof is built in stages: the sum co-array of the CNA core produces

$$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$4

adding the second subarray yields a contiguous difference set over

$$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$5

and adding the third subarray extends this to

$$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$6

Accordingly, the FOECA DOF of FOGNA is

$$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$7

For $$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$8, the asymptotic behavior reported for FOGNA is

$$\boldsymbol{x}(t)=\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$9

This is compared against $$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$0 for FL-NA, $$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$1 for SE-FL-NA, and $$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$2 for FO-Fractal(NA) (Wang et al., 2024).

For FOHA, hole-freeness is guaranteed by a simple set of conditions on the generator parameters: $$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$3 Under these conditions, the FODCA contains a contiguous segment

$$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$4

with

$$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$5

so that

$$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$6

The same work also defines fourth-order redundancy by

$$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$7

and gives, for FOHA(NA)/FOHA(CNA),

$$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$8

A general lower bound attributed there to Chen and Liu is

$$\boldsymbol{x}(t)=\boldsymbol{C}\boldsymbol{A}(\theta)\boldsymbol{s}(t)+\boldsymbol{v}(t),$$9

The stated consequence is that FOHA designs aim not only at larger DOFs but also at lower redundancy than existing FODCA-based arrays (Wang et al., 27 Aug 2025).

Design	Reported DOF evidence
FOGNA	$n$0, $n$1, $n$2
FOHA(NA)	$n$3
FOHA(CNA)	$n$4

These values come from separate comparative studies and therefore reflect different benchmark sets. Even so, both design lines support the same general conclusion: exploiting more than one fourth-order expression form enlarges the longest contiguous fourth-order co-array segment (Wang et al., 2024, Wang et al., 27 Aug 2025).

5. DOA estimation, spatial smoothing, and empirical performance

The recent FODCA literature uses fourth-order cumulant-based DOA estimation with spatial smoothing and MUSIC. A typical workflow is: estimate fourth-order cumulant tensors from the array data; vectorize and stack the cumulant entries; map them to the virtual lag domain defined by FODCA or FOECA; construct a virtual covariance or cumulant matrix; apply spatial smoothing on the contiguous virtual aperture; and then run MUSIC by computing the noise subspace and evaluating a pseudospectrum over candidate angles. In FOGNA, the hole-free FOECA removes the need for interpolation or compressive-sensing hole filling. In FOHA, cumulant entries associated with the same lag are coherently averaged on the virtual array (Wang et al., 2024, Wang et al., 27 Aug 2025).

The empirical results reported for FOGNA emphasize resolution, source capacity, and robustness to mutual coupling. With 7 sensors and three sources at $n$5, $n$6, and $n$7, only FOGNA resolves the two close sources separated by $n$8. With 9 sensors and 40 sources, FOGNA resolves all sources with sharper peaks than FL-NA, SE-FL-NA, FO-Fractal(NA), and SD-FODC(NA), while FL-NA fails to resolve all sources. Over SNR from $n$9 dB to $$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$0 dB, snapshots from 10,000 to 16,000, and source counts up to 50, the reported root-mean-square error (RMSE),

$$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$1

is consistently smallest for FOGNA in the presented simulations (Wang et al., 2024).

The FOHA study reports analogous behavior. Across comparisons with FL-NA, SE-FL-NA, FO-Fractal(NA), FO-SDE(NA), SD-FOSA(NA), and SD-FOSA(CNA-NA), FOHA(CNA) has the lowest RMSE and FOHA(NA) the second lowest in the no-coupling setting. The same ranking is reported for RMSE versus SNR, snapshots, and number of sources, and FOHA(CNA) is described as having the highest DOFs for fixed $$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$2. The study also states that higher DOFs translate into the ability to resolve 20 or more non-Gaussian sources with $$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$3 and to maintain low RMSE at angular separations $$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$4–$$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$5 (Wang et al., 27 Aug 2025).

6. Mutual coupling, assumptions, and scope

Mutual coupling is modeled in the recent FODCA literature by a $$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$6-banded symmetric Toeplitz matrix $$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$7, with entries

$$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$8

and coupling leakage

$$a_n(\theta_i)=e^{j\frac{2\pi p_n d}{\lambda}\sin(\theta_i)}$$9

In one simulation model,

$$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$0

FOGNA reduces mutual coupling by confining dense spacing to the CNA core and placing the remaining subarrays far apart; for $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$1, its leakage is reported to be smaller than that of FL-NA, SE-FL-NA, FO-Fractal, and SD-FODC, and for $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$2 the leakage values are $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$3, $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$4, $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$5, $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$6, and $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$7 for FL-NA, SE-FL-NA, FO-Fractal(NA), SD-FODC(NA), and FOGNA, respectively (Wang et al., 2024).

FOHA provides an analytic leakage relation between the dense generator $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$8 and the full three-subarray array. When $$\text{cum}(x_{k_1},x_{k_2},x_{k_3}^*,x_{k_4}^*) =\sum_{i=1}^D e^{-j\frac{2\pi d}{\lambda}(p_{k_1}+p_{k_2}-p_{k_3}-p_{k_4})\sin\theta_i}\,c_{4,s_i}^1,$$9, the sparse branches $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$00 and $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$01 effectively have identity coupling matrices, the overall coupling matrix becomes block-diagonal, and the resulting leakage satisfies $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$02, where $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$03 is the leakage of the generator. This formalizes the intuition that adding widely spaced sparse subarrays dilutes the coupling contribution of the dense core. The same study reports that for $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$04, FOHA(NA) and FOHA(CNA) have lower leakage than all baselines considered (Wang et al., 27 Aug 2025).

The scope of FODCA-based processing is constrained by the assumptions underlying fourth-order cumulants. The source signals must be non-Gaussian and mutually independent; the additive noise is assumed Gaussian and independent of the sources; the sources are far-field and narrowband; and reliable estimation requires stationarity and sufficient snapshots because fourth-order cumulant estimates have higher variance than covariance estimates. One paper additionally derives necessary and sufficient conditions for signal reconstruction using least-common-multiple constraints on the physical sensor positions, and specializes these conditions to FOHA(NA) and FOHA(CNA) through $$\Delta_4(\mathbb{P}) := \{(p_{k_1}+p_{k_2})-(p_{k_3}+p_{k_4})\} \cup \{(p_{k_1}-p_{k_2})+(p_{k_3}-p_{k_4})\},$$05. A common misconception in earlier FODCA practice is that a single fourth-order expression form is fundamental. The recent literature instead shows that multi-form constructions—whether framed as an extended co-array or as a broader FODCA definition—can increase DOFs, reduce redundancy, and improve robustness to mutual coupling without changing the underlying fourth-order statistical principle (Wang et al., 2024, Wang et al., 27 Aug 2025).