# White Gaussian Noise

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## Abstract

Connexions module: m11281 1 White Gaussian Noise ∗ Don Johnson This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † By far the easiest detection problem to solve occurs when the noise vector consists of statistically inde- pendent, identically distributed, Gaussian random variables. In this book, a white sequence consists of statistically independent random variables. The white sequence's mean is usually taken to be zero 1 and each component's variance is σ2. The equal-variance assumption implies the noise characteristics are unchanging throughout the entire set of observations. The probability density of the zero-mean noise vector evaluated at r − si equals that of Gaussian random vector having independent components ( K = σ2I) with mean si. p n (r − si ) = ( 1 2piσ2 )L 2 e−( 1 2σ2 (r−si)T (r−si)) The resulting detection problem is similar to the Gaussian example examined so frequently in the hypothesis testing sections, with the distinction here being a non-zero mean under both models. The logarithm of the likelihood ratio becomes (r − s0)T (r − s0)− (r − s1)T (r − s1) M1 ≷ M0 2σ2ln (η) and the usual simplifications yield in rT s1 − s1 T s1 2 − ( rT s0 − s0 T s0 2 )M1 ≷ M0 σ2ln (η) The quantities in parentheses express the signal processing operations for each model. If more than two signals were assumed possible, quantities such as these would need to be computed for each signal and the largest selected. This decision rule is optimum for the additive, white Gaussian noise problem. Each term in the computations for the optimum detector has a signal processing interpretation. When expanded, the term si T si equals ∑L−1 l=0 si 2 (l), which is the signal energyEi. The remaining term - rT si - is the only one involving the observations and hence constitutes the sufficient statistic Υi (r) for the additive white Gaussian noise detection problem. Υi (r) = rT si An abs

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