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In some applications of multi-target tracking, appearing targets are suitably modeled as spawning from existing targets. However, in the original formulation of the cardinalized probability hypothesis density (CPHD) filter, this type of model is not supported; instead appearing targets are modeled by spontaneous birth only. In this paper we derive the necessary equations for a CPHD filter for the case when the process model also includes target spawning. For this generalized filter, the cardinality prediction formula might become computationally intractable for general spawning models. However, when the cardinality distribution of the spawning targets is either Bernoulli or Poisson, we derive expressions that are practical and computationally efficient. Simulations show that the proposed filter responds faster to a change in target number due to spawned targets than the original CPHD filter. In addition, the performance of the filter, considering the optimal subpattern assignment (OSPA), is improved when having an explicit spawning model.