# Approximation for SQRT( I^{2} + Q^{2} ) with Gaussian Noise Present

When a complex signal is represented in terms of in-phase (I) and quadrature-phase (Q) components, a commonly used approximation for the signal root-mean-square (RMS) value is

Assuming a large number of OFDM subcarriers in the intended application, I and Q appear to both be Gaussian thereby making r a Rayleigh-distributed random variable where

We are interested in the ratio

From symmetry, it suffices to consider a limited range for θ as π/4 . Letting

leads to

and

where γ = 0.375. It is worthwhile evaluating just how good this choice for γ really is. We seek then to minimize the mean-square error given by

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