## 1 Introduction
However, the MV method of beamforming involves lots of matrix operation which augment the computation complexity, due to the calculation of correlation matrix. [PERSON] et al. proposed a compressive sensing-based approach combined with the combination of the adaptive minimum variance (MV) algorithm to reduce computational burden [14]. Deep learning is an emerging method for beamforming [15]. Different neural networks are used to improve image quality or to augment imaging speed in beamforming [16; 17]. However, this algorithm often requires a lot of data for training or has a high computational complexity, which is not suitable for high -frequency imaging in practical settings. Compared with algorithms based on MV, DAS requires a relatively small amount memory in the temporal domain because there is no matrix computation. [PERSON] [18] introduced the plane wave imaging process to the Fourier domain, which provided a new direction for beamforming. Compared with beamforming methods in the temporal domain, beamforming in the Fourier domain might augment computational efficiency by applying fast Fourier transform (FFT) [19]. [PERSON] [18] introduced the mapping method to coordinate the echo signal from temporal spectrum to frequency-wavenumber space, which achieved to beamform in wavenumber-space. [PERSON] et al. [20] proposed a novel remapping method related to the steered angle in reception. After that, [PERSON] et al. [21] proposed a beamforming process in the Fourier domain based on the exploding reflector model (ERM), which is widely used in seismology [22]. Compared with [PERSON]'s method, this model assumes that the echo waves of the backscatters only propagate in the direction of the sensor, so that the principle of [PERSON]'s migration is different from the model proposed by [PERSON]. This model is closer to the reality, thus improving imaging quality. Then [PERSON] et al. [23] modified the structure of the algorithm by aligning coordinates before the axial inverse fast Fourier transform (IFFT) and the lateral IFFT, which enables direct coherent compounding in the Fourier domain. Based on f-k migration, a circular statistics vector (CSV) was used as a weighted factor to achieve better image quality [24]. High imaging frame rate will lead to the problem of large data load and memory storage. Moreover, a large amount of data also reduces the transfer speed which limits transmission speed in telemedicine imaging. Compressed Sensing (CS) is a new sampling theory, which allows the signal to be sampled under the Nyquist sampling rate [25]. This theory is based on the signal's sparsity, which means the original signal can be randomly sampled to obtain discrete samples. There is only a small amount of non-zero information in this discrete sample. The sample can be reconstructed to the original signal by a recovery algorithm with high quality. Compressed sensing has been widely used in data acquisition [26], radar [27] and the communication field. [PERSON] et al. [28] and [PERSON] et al. [29] considered the transmitted ultrasonic signal as an impulse to form a sparse measurement matrix with a large number of zero-elements to reduce memory usage. Then, wavelet sparsity was introduced to the recovery algorithm to reconstruct the image. [PERSON] et al. [30; 31] proposed a CS-based Fourier beamforming with a Xampling scheme for undersampling. [PERSON] et al. [32] firstly joint compressed sensing with beamforming in the wavenumber domain. Although compressed sensing allows the signal to be sampled at a sampling rate lower than the Nyquist sampling rate, the measurement matrix occupies lots of memory as well. Reducing transducer elements is an effective method in DAS to reduce the data load with a small measurement matrix [33]. In this paper, we propose a new framework based on [PERSON]'s f-k and compressed sensing theory, which is a compressive beamformer in the wavenumber domain, to obtain a smaller amount of data load during signal transmission and better image quality in telemedicine.