For DEcay SPECtroscopy (DESPEC) experiments with implanted radioactive beams a high resolution gamma tracking array is being designed at GSI. Exploiting the imaging capability of the array, a background suppression method is developed based on a separation of events originating from the target and from localized background sources. Monte-Carlo simulations show that the suppression of the background by a factor from 10 up to 100 or even more can be achieved. The Peak/Total ratio can be as well significantly increased. The dependence of the suppression efficiency on the detector properties is discussed.