On the basis of transition radiation from a singe foil, the formulas of differential quantum spectrum of transition radiation from a stack of foils are derived with the superposition of radiation field and of radiation intensity. Using the formula of differential spectrum of incoherent transition radiation, we calculate the corresponding spectra produced by 54-MeV-electron respectively passing through the stack of beryllium foils (18 foils), magnesium foils (12 foils), and aluminium foils (12 foils). The results agree with the experiment.