## 1 Introduction  
\"Big Data\" is an increasingly ubiquitous term that refers to the growing size and complexity of datasets due to the variety of sources from which they can be acquired. Big Data is transforming the fields of science, engineering, medicine, healthcare, finance, business, and society in general. As the sizes of datasets increase, efficient compression techniques are desperately required for acquisition, storage, and reliable retrieval of data. Compressed sensing (CS) has emerged as an alternative signal-processing solution that addresses the requirements for reductions in data storage and acquisition time, since, assuming they are sufficiently sparse, original signals can be reconstructed using fewer data samples acquired at frequencies lower than those dictated by the Shannon-Nyquist theory [1; 2; 3]. Accordingly, the CS technique has been applied in many fields, including imaging, communications, pattern recognition, sound processing, and video processing [4; 5; 6; 7; 8; 9; 10]. The CS technique also offers promising benefits for medical image processing applications [7; 11; 12; 13]. For instance, the relatively long data acquisition times for MRI systems, which make patients uncomfortable as they are required to remain motionless for prolonged periods, is dramatically reduced with CS [14]. Similarly, many recent studies have focused on reducing the size of the hardware and data storage of portable systems used for medical ultrasound, by lowering the sampling rate using the CS theory [15]. In general, when CS signals are processed for additional analysis, reconstruction is required to convert them to their original forms, as most traditional methods for signal analysis utilize uncompressed signals acquired in the time domain. In particular, with medical imaging analysis, the envelope of the received signal in the time domain contains important anatomical information, while, in the frequency domain, spectral information provides fundamental background for quantitative analysis of tissue properties. For the applications of nuclear magnetic resonance spectroscopy, the CS reconstruction contains the spectral information since the sparse transformation is the Fourier transform itself [16; 17]. Similarly, [PERSON] et al. showed that the traditional processing framework for frequency-domain analysis of compressed medical ultrasound data contains a redundant transformation from the time domain to the frequency domain [18]. We have observed that the same redundancy exists with time-domain analyses such as envelope detection for time-varying signals for B-mode ultrasound images. In this paper, we propose a signal-processing framework for ultrasound CS data that is simpler and faster in execution than the traditional framework, and produces more accurate results in both time- and frequency-domain analyses. With the traditional framework, CS signals first need to be recovered to the uncompressed forms and then transformed from the time domain into the frequency domain using a technique such as the fast Fourier transform (FFT), to estimate the spectral information required for frequency-domain analysis. Here, when an appropriate basis matrix is used, this spectral information is directly extracted from the CS data, avoiding reconstruction of the original signal and the redundant transformation of the reconstructed time-domain signal to the frequency domain. In addition, time-domain properties like the envelope of the original signal are computed during the reconstruction process without fully converting them into their original forms. More specifically, because the envelope of a signal is defined by its magnitude in a complex analytic form, the traditional framework uses the Hilbert transform to obtain analytic time-domain signals for envelope detection. In contrast, with the proposed method, the analytic signal is calculated directly from the frequency information obtained during the reconstruction process, using the inverse Fourier transform (IFFT).