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The enhancement of the efficiency in free-electron lasers (FELs) through the use of a tapered wiggler is well known. The physics of the tapered wiggler interaction has been studied in theory and simulation, and large efficiency enhancements have been observed in the laboratory in oscillators and seeded amplifiers. In this paper, we study the differences in the tapered wiggler interaction between seeded amplifiers and in FELs that start up from noise and grow to saturation in a single pass through the wiggler. This configuration is commonly referred to as self-amplified spontaneous emission (SASE). In comparison with seeded amplifiers, SASE FELs exhibit shot-to-shot fluctuations due to random phase noise in the electron bunches, and our purpose in this paper is to determine the effect of this phase noise on the tapered wiggler interaction. To this end, we study the interaction numerically using the MEDUSA simulation code for seeded and SASE FELs operating in the infrared regime. The results of the simulations indicate that the overall efficiencies of the seeded and SASE FELs are comparable for a uniform wiggler but that the output spectrum for the SASE FEL is much broader than for the seeded case. For a tapered wiggler, the efficiency enhancement in the SASE FEL is less than that found in the seeded example due to the broader excited spectrum that detunes the tapered wiggler interaction.