Near-sensor convolution engines have many applications in Internet-of-Things. Pulsed unary processing has been recently proposed for high-performance and energy-efficient processing of data using simple digital logic. In this work, we propose a low-cost, high-performance, and energy-efficient near-sensor convolution engine based on pulsed unary processing. The proposed near-sensor engine removes the necessity of using costly analog-to-digital converters. Synthesis results show that the proposed pulse-based design significantly improves the hardware cost and energy consumption compared to the conventional fixed-point binary-radix and also to stochastic computing-based designs.