The spectral content of a stochastic process X t can be found from the power spectral density, formally defined as S ( ω ω ) = lim T → → ∞ ∞ 1 T E { | ∫ ∫ 0 T e i ω ω t X t d t | 2 } , where E stands for the expected value. The power spectral density of Brownian motion is found to be S B M ( ω ω ) = 4 D ω ω 2 . where D is the diffusion coefficient of X t. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e., S ( 1 ) ( ω ω , T ) = 1 T | ∫ ∫ 0 T e i ω ω t X t d t | 2 , which for an individual realization of a Brownian motion trajectory, it is found to have expected value μ μ B M ( ω ω , T ) μ μ BM ( ω ω , T ) = 4 D ω ω 2 [ 1 − − sin ⁡ ⁡ ( ω ω T ) ω ω T ] and variance σ σ BM 2 ( ω ω , T ) σ σ S 2 ( f , T ) = E { ( S T ( j ) ( f ) ) 2 } − − μ μ S 2 ( f , T ) = 20 D 2 f 4 [ 1 − − ( 6 − − cos ⁡ ⁡ ( f T ) ) 2 sin ⁡ ⁡ ( f T ) 5 f T + ( 17 − − cos ⁡ ⁡ ( 2 f T ) − − 16 cos ⁡ ⁡ ( f T ) ) 10 f 2 T 2 ] .