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Sampling and Random Sequences

Module by: Don Johnson. E-mail the author

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The usual Sampling Theorem applies to random processes, with the spectrum of interest beign the power spectrum. If stationary process Xt X t is bandlimited - 𝒮 X ω=0 𝒮 X ω 0 , |ω|>W ω W , as long as the sampling interval TT satisfies the classic constraint T<πW T W the sequence XlT X l T represents the original process. A sampled process is itself a random process defined over discrete time. Hence, all of the random process notions introduced in the previous section apply to the random sequence X ∼ l≡XlT X ∼ l X l T . The correlation functions of these two processes are related as R X ∼ k=E X ∼ l X ∼ l+k= R X kT R X ∼ k X ∼ l X ∼ l k R X k T

We note especially that for distinct samples of a random process to be uncorrelated, the correlation function R X kT R X k T must equal zero for all non-zero kk. This requirement places severe restrictions on the correlation function (hence the power spectrum) of the original process. One correlation function satisfying this property is derived from the random process which has a bandlimited, constant-valued power spectrum over precisely the frequency region needed to satisfy the sampling criterion. No other power spectrum satisfying the sampling criterion has this property. Hence, sampling does not normally yield uncorrelated amplitudes, meaning that discrete-time white noise is a rarity. White noise has a correlation function given by R X ∼ k=σ2δk R X ∼ k σ 2 δ k , where δ· δ · is the unit sample. The power spectrum of white noise is a constant: 𝒮 X ∼ ω=σ2 𝒮 X ∼ ω σ 2 .

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