Frequency-Invariant Differential Beamforming

• Frequency-invariant differential beamforming is a broadband array-processing method that synthesizes nearly constant angular responses across frequencies.
• It leverages low-order spatial derivatives from compact, closely spaced sensors to maintain performance and reduce spectral coloration.
• Recent extensions apply the method to arbitrary planar arrays and nearfield, time-domain formulations, balancing directivity with robustness.

Frequency-invariant differential beamforming is a class of broadband array-processing methods in which the spatial response is designed to remain approximately constant with frequency while preserving the compact-array, high-directivity behavior associated with differential processing. In the acoustic setting, the objective is to avoid spectral coloration and frequency-dependent null structure by synthesizing a desired angular response from low-order spatial information, typically using closely spaced sensors and modal or series-based beampattern matching. Recent work extends the classical planar, omnidirectional-sensor formulation to arbitrary planar arrays of first-order directional elements, and related developments cover steerable polynomial beamforming with HRTF models, null-constrained line arrays, dual-axis optimization on concentric circular arrays, and nearfield time-domain formulations in spherical harmonics (Miotello et al., 17 Aug 2025, Barfuss et al., 2016, Zhang et al., 25 Aug 2025, Ortigoso-Narro et al., 24 Nov 2025, Ma et al., 2021).

1. Broadband objective and differential-beamforming principle

A central requirement in broadband acoustic array processing is that the spatial response should remain approximately constant with frequency. If the beampattern changes significantly across frequency, the array colors the spectrum of the desired signal and its rejection/null structure becomes frequency dependent, which is undesirable in speech enhancement, teleconferencing, sound radiation, and related applications (Miotello et al., 17 Aug 2025). The same broad requirement appears in robot audition, where a beamformer should approximate several desired responses over the speech band while accounting for the acoustic distortion introduced by a humanoid robot head (Barfuss et al., 2016).

Differential beamforming is attractive because, when properly designed, it is inherently frequency invariant over a useful band: by exploiting spatial derivatives estimated from small intersensor spacing, the beamformer can realize a target directivity pattern whose angular shape does not strongly vary with frequency (Miotello et al., 17 Aug 2025). In the nearfield broadband setting, the same motivation persists, but standard far-field narrowband methods become insufficient because nearfield propagation is spherical, broadband processing requires frequency invariance, and FFT-based processing adds latency (Ma et al., 2021).

This broad principle has several concrete realizations. In compact planar arrays, it is commonly expressed as low-order angular-series matching. In line arrays, it may be enforced through null constraints and white-noise-gain constraints. In polynomial beamforming, it appears as a fixed bank of subfilters whose weighted combination preserves an approximately frequency-invariant family of beampatterns under steering. In concentric circular arrays, it can be formulated as a differentiable optimization problem that jointly regulates beamwidth, directivity, white-noise gain, and frequency consistency (Zhang et al., 25 Aug 2025, Barfuss et al., 2016, Ortigoso-Narro et al., 24 Nov 2025).

2. Classical planar formulation and modal matching

For the omnidirectional planar-array case, the standard model places the $m$-th element at

$\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$

and a far-field plane wave arriving from direction $\theta_s$ produces

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$

with steering term

$$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$

Beamforming uses a filter vector $\mathbf{h}(\omega)$ to produce

$Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$

and the beampattern is

$$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$

Frequency-invariant differential beamforming then proceeds by specifying a desired angular response and matching the actual beampattern to it (Miotello et al., 17 Aug 2025).

The classical target is written as a symmetric truncated circular-harmonic expansion,

$$\bar{\mathcal{B}(\mathbf{b}_{2N}, \theta, \theta_s)} = \sum_{n=0}^N a_n\cos\!\big(n(\theta-\theta_s)\big) = \sum_{n=-N}^{N} b_n e^{jn(\theta-\theta_s)}.$$

Here $\mathbf{b}_{2N}=[b_{-N},\dots,b_N]^T$ defines the desired pattern of order $\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$0, and steering is introduced through

$\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$1

The distortionless-response condition in the look direction is $\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$2.

In the omnidirectional setting, the Jacobi–Anger expansion,

$\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$3

permits a harmonic decomposition of the beampattern. After truncation to order $\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$4, the mode-by-mode condition becomes

$\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$5

This is written compactly as

$\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$6

where the rows of $\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$7 contain

$\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$8

Assuming at least $\mathbf{r}_m = r_m[\cos(\phi_m),\sin(\phi_m)],$9 sensors and full row rank, the minimum-norm solution is

$\theta_s$0

This is essentially the FIB-LSE idea: choose a desired beampattern in the angular domain and solve for the array weights by modal matching (Miotello et al., 17 Aug 2025).

3. Arbitrary planar arrays of first-order directional elements

A major extension replaces the omnidirectional element assumption with a first-order directional model. Each element $\theta_s$1 is described by the symmetric first-order directivity pattern

$\theta_s$2

where $\theta_s$3 sets the blend between an omnidirectional component and a steered dipole component, and $\theta_s$4 is the element orientation. In the simplified derivation $\theta_s$5 is taken independent of $\theta_s$6, but the initial formulation explicitly allows frequency dependence (Miotello et al., 17 Aug 2025).

With this inclusion, the signal model becomes

$\theta_s$7

or, in vector form,

$\theta_s$8

where $\theta_s$9. The beampattern changes accordingly to

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$0

while white-noise gain and directivity factor retain their standard definitions but now reflect the actual directional element responses.

The key technical issue is that $$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$1 multiplies the propagation factor, so the Jacobi–Anger expansion alone no longer separates the angular dependence. The generalized modal expansion is

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$2

with Fourier-projection coefficient

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$3

The derived closed form is

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$4

This expression exposes two contributions: the first from the omnidirectional part and the second from the dipole part. It is the mathematical core of the generalization because it embeds both sensor directivity and sensor orientation into a harmonic-domain description that remains compatible with modal matching (Miotello et al., 17 Aug 2025).

After truncation to order $$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$5, one obtains the per-mode equality

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$6

or compactly

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$7

with solution

$$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$8

When $$Y_m(\omega)=D_m(\omega,\theta_s)X(\omega)+V_m(\omega),$$9 for all elements, $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$0 reduces to $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$1, so the formulation collapses exactly to the classical FIB-LSE method. The design therefore remains a strict generalization of frequency-invariant differential beamforming to arbitrary planar arrays with first-order directional sensors (Miotello et al., 17 Aug 2025).

This framework is explicitly unconstrained with respect to planar geometry: elements may be arbitrarily placed in the plane, with arbitrary orientations $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$2 and first-order directivities $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$3. Because the target is encoded by the coefficients $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$4, any symmetric directivity shape that can be approximated by a truncated circular-harmonic series can be synthesized, including cardioid, hypercardioid, supercardioid, and higher-order patterns. Since the framework is reciprocity-based, it applies equally to differential microphone arrays and differential loudspeaker arrays (Miotello et al., 17 Aug 2025).

4. Robustness measures, order-dependent tradeoffs, and simulation evidence

The standard robustness and selectivity measures are the white-noise gain (WNG) and the directivity factor (DF). In the generalized planar formulation, these retain their standard definitions but depend on the actual directional element responses (Miotello et al., 17 Aug 2025). In line-array formulations, the WNG is

$$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$5

and under the distortionless constraint $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$6, this simplifies to $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$7. The corresponding DF in a cylindrically isotropic noise field is

$$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$8

with pseudo-coherence matrix $$D_m(\omega,\theta_s)=e^{j\frac{\omega}{c}r_m\cos(\theta_s-\phi_m)}.$$9 given in closed form for mixed omnidirectional and directional microphones (Zhang et al., 25 Aug 2025).

The simulation evidence in the planar first-order setting is explicit. In a fixed nine-element random planar array with a 2 cm aperture and at least 8 mm spacing, the method accurately reproduced both a first-order supercardioid and a fourth-order hypercardioid steered to $\mathbf{h}(\omega)$0, with strong frequency invariance over the full band and only minor deviations at higher frequencies. In large-scale Monte Carlo tests, when the array size was set to the theoretical minimum $\mathbf{h}(\omega)$1, the average beampatterns remained close to the targets, but deviations and variability were larger, especially at lower order. As the order increased, the directivity factor rose and the white-noise gain fell, matching classical differential-array tradeoffs. Increasing the number of microphones improved beampattern accuracy and substantially increased white-noise gain, while leaving the directivity factor essentially unchanged when the target response was fixed. When $\mathbf{h}(\omega)$2 was held constant and the order was varied from 1 to 3, the higher element count yielded precise approximations across all orders, reduced variability, and more stable performance. The smallest viable sensor count can yield white-noise gain below $\mathbf{h}(\omega)$3 dB, indicating amplification of self-noise; adding sensors mitigates this problem (Miotello et al., 17 Aug 2025).

Related behavior appears in null-constrained line arrays. In the 11-microphone line-array study, the NC design had higher WNG but larger MSE, whereas the improved null-constrained design lowered WNG somewhat while significantly improving MSE and frequency invariance. The paper identifies $\mathbf{h}(\omega)$4 dB as a strong compromise, often keeping MSE below $\mathbf{h}(\omega)$5 dB while maintaining usable WNG. The same study reports that bidirectional microphones provide the best robustness among the tested directional types (Zhang et al., 25 Aug 2025).

These results reinforce a persistent design fact: higher selectivity, stronger null control, and broader steering flexibility are normally purchased with reduced robustness. Frequency invariance does not remove that tradeoff; it reorganizes it around modal accuracy, sensor count, element directivity, and conditioning of the design equations. This suggests that practical performance depends at least as much on model fidelity and array redundancy as on nominal beamformer order.

5. Alternative architectures and problem settings

Several recent formulations extend the frequency-invariant differential-beamforming idea beyond the classical planar modal-matching setting.

The HRTF-based robust least-squares frequency-invariant polynomial beamformer for robot audition uses a filter-and-sum architecture followed by a polynomial postfilter,

$\mathbf{h}(\omega)$6

Its beamformer response is

$\mathbf{h}(\omega)$7

and for the HRTF-based design the sensor response is taken as the measured HRTF,

$\mathbf{h}(\omega)$8

The steering parameter is

$\mathbf{h}(\omega)$9

Beam steering is therefore performed by changing only $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$0, while the FIR filters remain fixed after design. The optimization is solved independently for each frequency $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$1 using a fixed frequency-independent desired response, with WNG and distortionless constraints, and the frequency samples are converted into FIR filters using Matlab’s fir2. The method is limited to the horizontal dimension; two-dimensional steering is left for future work (Barfuss et al., 2016).

A different extension appears in steerable-invariant beamforming for a line array of alternating omnidirectional and directional microphones. There, each microphone has directivity

$Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$2

which becomes $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$3 for $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$4. Instead of requiring an analytic target beam pattern and a truncation order as in Jacobi–Anger-based methods, the null-constraint-based framework uses only the desired steering direction and the null directions. The reference filter is obtained from unit response at the look direction and zeros at the null angles; the final beamformer solves a convex program that balances beampattern fidelity and WNG robustness (Zhang et al., 25 Aug 2025).

For dual-axis steering, the autograd-based concentric circular microphone-array formulation treats frequency-invariant beamforming as a differentiable optimization problem. The array uses multiple rings with radii $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$5, and the beamformer coefficients are parameterized by ring-level weights $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$6, an intra-ring Gaussian window $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$7, and the steering phase term. The optimization constrains azimuth and elevation beamwidth and includes losses that can combine directivity, WNG, and frequency-variation penalties. The reported comparison includes delay-and-sum, modified delay-and-sum, a Jacobi–Anger expansion-based method, and a Gaussian window-based gradient descent approach; the proposed method is particularly strong in elevation control at lower frequencies (Ortigoso-Narro et al., 24 Nov 2025).

6. Nearfield, time-domain, and adjacent differential-frequency formulations

Frequency-invariant differential-beamforming ideas also extend into nearfield and time-domain processing. The time-domain nearfield frequency-invariant beamforming method is built in the spherical-harmonic domain, where the beamformer response is

$Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$8

The designed coefficient includes a radial focusing term $Z(\omega)=\mathbf{h}^H(\omega)\mathbf{y}(\omega),$9, a spectral factor $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$0, a causal delay term, and an angular steering factor $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$1. After inverse Fourier transform, the beamformer admits a sample-by-sample time-domain implementation using pressure coefficients, velocity coefficients, their time derivatives, and convolutions with $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$2 and $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$3. In the reported simulations, the time-domain implementation had higher multiplication count but zero latency, while the frequency-domain version had lower multiplication count but significant latency due to frame processing (Ma et al., 2021).

The same work frames the method as mathematically aligned with higher-order modal/differential array processing rather than as a purely classical planar differential beamformer. The beampattern coefficients $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$4 control the spatial shape, the radial term provides nearfield focusing, and the spectral factor supports approximate frequency invariance over a broad band. Reported results include a frequency-invariant mainlobe over 400–4000 Hz, sidelobe level mostly below $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$5 dB, and magnitude-squared coherence between target and beamformer output close to $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$6 over the same band (Ma et al., 2021).

An adjacent but distinct line of work uses frequency-difference processing for ambiguity-free broadband DOA estimation in sparse arrays. There, the effective steering depends on a difference frequency $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$7 rather than the absolute carrier, and the method is described as differential in spirit because it relies on

$$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$8

The proposed PTFT-based compressive frequency-difference beamforming retains ambiguity-free behavior when $$\mathcal{B}[\mathbf{h}(\omega),\theta]=\mathbf{h}^H(\omega)\mathbf{d}(\omega,\theta).$$9 and adds a coarse-to-fine histogram statistics stage to suppress artifact DOAs from cross terms (Wang et al., 5 Mar 2025). Although this is a DOA-estimation framework rather than a beampattern-synthesis framework, it occupies a closely related conceptual space in which differential manipulation is used to stabilize broadband spatial processing.

Two common misconceptions follow from these broader developments. First, frequency invariance is not identical to perfect equality of beampattern at all frequencies; the reported results repeatedly note high-frequency deviations, interpolation degradation between prototype look directions, or sensitivity to beamwidth-estimation and hyperparameter choices (Miotello et al., 17 Aug 2025, Barfuss et al., 2016, Ortigoso-Narro et al., 24 Nov 2025). Second, differential beamforming is not restricted to omnidirectional sensors, rigid analytic targets, or a single steering axis: recent formulations explicitly accommodate arbitrary planar first-order elements, line arrays with mixed microphone types and null constraints, HRTF-based runtime steering, and dual-axis optimization on concentric circular arrays (Miotello et al., 17 Aug 2025, Zhang et al., 25 Aug 2025, Barfuss et al., 2016, Ortigoso-Narro et al., 24 Nov 2025).