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We discuss amplified spontaneous emission (ASE) generated in Raman amplifiers that are counter-pumped with trains of pulses. Our experimental and theoretical results show that if the peak power of the pump pulses is too high, the ASE output from the amplifier can be significantly higher than that from a continuous-wave pumped amplifier providing the same gain. This effect places a lower limit on the duty cycle of pump pulses one can use for a given level of Raman gain. Furthermore, we report an additional ASE enhancement if there is insufficient walkoff between the pump pulses and copropagating ASE to average the effects of higher frequency pump intensity noise. As a result, less pump intensity noise can be tolerated when pulse-pumping a fiber having a zero-dispersion wavelength located midway between the pump and signal wavelengths.