A Doppler broadening spectrometer with the reduction of background based on standard electronics is presented. The peak to background ratio achieved for the annihilation line is about 470:1 at 530 keV. It allows us to clearly observe positron annihilation with core electrons. As examples of the work of the spectrometer preliminary measurements of positron annihilation with core electrons in Mg-Cd alloys, in uranium, and nickel are presented.