Sampling and reconstruction of short pulses based on Gabor frames have been proved to be effective, which overcome the difficulties that finite rate of innovation (FRI) sampling is unable to reconstruct the pulse streams without the prior information of waveforms. However, the windows sequences of sampling scheme based on Gabor frames proposed at present show complicated structure and are hard to realize physically. The exponential reproducing windows are then introduced in this paper and the windows sequences can be simplified as a first-order analog Butterworth filter. At the same time, the compressed sensing (CS) measurement matrix is constructed for the recovery of Gabor coefficients. In order to satisfy the restricted isometry property (RIP) of the measurement matrices for perfect signal reconstruction, we select appropriate windows for support according to the energy accumulation property. A restricted condition is deduced for perfecting the signal reconstruction and the system robustness is analyzed. By numerical simulations the above analysis is verified. This novel scheme can be used to implement short pulses sampling and reconstruction in the field of instrumentation, condition monitoring, radar and the communication.