## 1 Introduction  
Directional loudspeaker arrays have attracted the attention of researchers because of their promising applications in spatial audio reproduction [1, 2], public address systems [3, 4], and personal sound zone creation [5, 6]. The most commonly used directional loudspeaker arrays are additive arrays. Additive arrays generate highly directional beam-patterns by controlling the interference of the waves generated by multiple loudspeakers. However, due to the diffraction limit, small-sized additive arrays cannot radiate highly directional beampatterns at low frequencies [7]. In contrast, differential arrays with small aperture sizes can overcome the diffraction limit to generate and radiate a narrow beampattern. Due to their compact size, frequency-invariant beampattern, and high spatial directivity, differential arrays have been studied intensively in microphone array applications over the past two decades. Many different 'geometries of differential microphone arrays have been developed and studied, such as linear [8, 9, 10, 11, 12, 13, 14], planar [15, 16, 17, 18, 19], and volumetric [20, 21, 22, 23]. Differential beamforming of linear arrays has been extensively studied, but most research sets the main lobe at the end-fire direction [8, 9, 11, 13]. Although some studies investigate beam steering for linear arrays, their beam steering capability is quite limited [10, 14]. Planar arrays, such as circular arrays [17, 19] or concentric circular arrays [15, 16, 18], can effectively solve the beam steering issue of linear arrays, allowing for arbitrary steering within the plane of the array. Volumetric arrays, such as cube arrays [20; 21; 23] and spherical arrays [22], can produce consistent sound acquisition performance in the three-dimensional space. However, the required large number of microphones and huge array volume hinders its applications in some scenarios. Because of the reciprocity principle of the acoustic field, the differential beamformers have also been applied to loudspeaker arrays to produce highly directional patterns [7; 24; 25; 26; 27; 28; 29]. The differential loudspeaker array is significant for the field of acoustics and sound technology, offering a novel approach to improving directional sound quality in various applications such as sound reinforcement systems [24; 26; 29], spatial sound reproduction [27], and personal sound zones [25; 28]. [PERSON] used a second-order broadside differential loudspeaker array with three elements to create a near-field sound zone in a car cabin [25]. [PERSON] et al. devised a null-constrained method to generate a higher-order broadside differential beampattern [26]. This approach is convenient for designing differential beamformers in real applications. However, the frequency-invariant beampattern may only be maintained over part of the frequency range due to the constraints imposed only at several discrete directions. [PERSON] et al. designed a steerable first-order differential beampattern using a small-size loudspeaker line array with monopole and dipole elements [28]. Our recent research studied the series expansion method for designing frequency-invariant beampatterns using a loudspeaker array [27; 29]. In [27]. we design steerable beampattern using a circular loudspeaker array. This method requires prior knowledge of the target beampattern, which may not be feasible in real-world applications. We also proposed a method to design broadside beampattern using a differential loudspeaker line array [29]. We first derived the modal domain target to beampattern the nulls' information. Then, it employed modal matching with Jacobi-Anger expansion and distortionless constraint (MM-JAD) to approximate the target beampattern, which led to better-preserved frequency-invariant beampatterns across the frequency range of interest, such as the speech frequency range. This paper proposed a robust broadband differential beamforming design to allow the loudspeaker line array to radiate broadside frequency-invariant radiation patterns with robustness.