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In this paper, we present an analysis for the noise due to finite word length effects for digital signal power processors using Welch's power spectrum estimation technique to measure the power of Gaussian random signals over a frequency band of interest. The input of the digital signal processor contains a finite-length time interval in which the true Gaussian signal is contaminated by Gaussian noise. We analytically derive the roundoff noise-to-signal ratio in the measurement of the signal power. We also present computer simulations which validate the analytical results. These results can be used in tradeoff studies of hardware design such as number of bits required at each processing stage. The results presented in this paper are currently being used in the design of a digital Doppler processor ,  for a radar scatterometer -.