Many communication systems involve high bandwidth, while sparse, radio frequency (RF) signals. Working with high frequency signals requires appropriate system-level components such as high-speed analog-to-digital converters (ADC). In particular, an analog signal should be sampled at rates that meet the Nyquist requirements to avoid aliasing. However, implementing high-speed ADC devices can be a limiting factor as well as expensive. To mitigate the caveats with high-speed ADC, the solution space can be explored in several dimensions such as utilizing the compressive sensing (CS) framework in order to reduce the sampling rate to the order of information rate of the signal rather than a rate dictated by the Nyquist. In this note, we review the compressive sensing structure and its extensions for continuous-time signals, which is ultimately used to reduce the sampling rate of high-speed ADC devices. Moreover, we consider the application of the compressive sensing framework in wireless sensor networks to save power by reducing the transmission rate of sensor nodes. We propose an alternative solution for the CS minimization problem that can be solved using gradient descent methods. The modified minimization problem is potentially faster and simpler to implement at the hardware level.