The classical Shannon sampling theorem is suitable for reconstructing a band-limited signal from its sampled values taken at regular instances with equal step by using the well-known sinc function. However, due to the inertia of the measurement apparatus, it is impossible to measure the value of a signal precisely at such discrete time. In practice, only unsymmetrically local averages of signal near the regular instances can be measured and used as the inputs for a signal reconstruction method. In addition, when implemented in hardware, the traditional sinc function cannot be directly used for signal reconstruction. We propose using the Taylor expansion of sinc function to reconstruct signal sampled from unsymmetrically local averages and give the upper bound of the reconstruction error (i.e., truncation error). The convergency of the reconstruction method is also presented.