An estimate of the variation of a band-limited process

Abstract

Upper and lower bounds are established for the mean-square variation of a stationary process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(t)</tex> whose power spectrum is bounded by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{c}</tex> , in terms of its average power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{0}</tex> and the average power <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{1}</tex> of its derivative. It is shown that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\left( \frac{2}{\pi} \right)^{2} P_{1} \tau^{2} \leq E \{|X(t+\tau )-X(t)|^{2}\} \leq P_{1} \tau^{2} \leq \omega_{c}^{2}P_{0}\tau^{2}</tex> where the upper bounds are valid for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau</tex> and the lower bound for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau &lt; \pi / \omega_{c}</tex> . These estimates are applied to the mean-square variation of the envelope of a quasi-monochromatic process.