Fiber Bragg Grating (FBG) sensor systems have been widely implemented in many application areas for the real time measurement and monitoring of various physical parameters, such as pressure, stress, temperature. This paper reports a cost-effective design and implementation of a Fiber Bragg Grating (FBG) sensor system based on a narrow band Distributed Bragg Reflector (DBR) laser and the Field Programmable Gate Array (FPGA) demodulation technology. The FPGA circuit is employed to control a narrow band DBR laser for continuous wavelength scanning, which enables multichannel data collection and processing at a high speed. Several aspects of this FBG sensor system have been investigated, including the design of a FPGA demodulation circuit and the creation of a high-speed peak detection algorithm. In addition, a wavelength shifting compensation scheme is implemented in the FBG sensor system, which has reduced the measured wavelength shifting error of a narrow band DBR laser by about one order of magnitude at a wide ambient temperature range of -35°C ~ +35°C. A cost effective FBG sensor system has been accomplished at a demodulation speed of 1kHz with 8-channels. This has paved a way towards a simple and affordable multi-channel FBG sensor system for a wide range of applications.