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In this paper, we derive an expression for the signal to interference-plus-noise ratio of a linear multistage parallel interference cancellation receiver. We focus on a linear multistage receiver which converges to the linear minimum mean-squared error receiver as the number of stages increases. The signal to interference-plus-noise ratio is given in terms of the system loading, the partial cancellation factor, the number of stages, and the signal-to. noise ratio. Our expression also allows a simple approximation for the bit error rate at each stage. Finally, we perform a numerical optimization to maximize the signal to interference-plus-noise ratio expression with respect to the partial cancellation factor of the resulting linear multistage receiver.