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On the spectral representation of the sampling cardinal series expansion of weakly stationary stochastic processes. (English) Zbl 0770.60043

Let \(\{X(t): t\in\mathbb{R}\}\) be a weakly stationary mean-square continuous process. If \(X(t)\) is band-limited, i.e. if there exists \(\omega>0\) such that the spectral measure \(\varphi\) satisfies the condition \(\varphi((- \infty,\Omega])=\varphi([\omega,\infty))=0\), then the stochastic sampling theorem [e.g. E. Wong and B. Hajek, Stochastic processes in engineering systems (1985; Zbl 0545.60003)] says that for all \(\tilde\omega\geq\omega>0\) \[ X(t)=\lim_{n\to\infty}\sum^ n_{k=- n}X(k\pi/\tilde\omega)\text{sinc} (\tilde\omega t-k\pi)\quad\text{(in }L_ 2) \] with \(\text{sinc}:=\sin x/x\). Now let \(X(t)\) be non-band- limited and for \(\omega>0\) let \[ X_ a(t):=\sum^ \infty_{k=- \infty}X(k\pi/\omega)\text{sinc}(\omega t-k\pi) \] be the sampling cardinal series expansion. The authors derive a spectral representation of \(X_ a(t)\) and thus obtain an easy proof of a result of J. L. Brown jun. [IEEE Trans. Inform. Theory IT-24, 254-256 (1978; Zbl 0393.94002)], \[ E(| X(t)-X_ a(t)|^ 2)\leq 4(\sigma^ 2- F(\omega)+F(-\omega)), \] where \(F\) denotes the spectral distribution and \(\sigma^ 2=E(| X(t)|^ 2)\). These results are also extended to vector stochastic processes.
Reviewer: W.Schmid (Ulm)

MSC:

60G10 Stationary stochastic processes
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