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Unformatted text preview: nswer in terms of X (Ω). In terms of X (Ω) we can write R(Ω) = Page 13 of 17 6.02 Fall 2012, Quiz 2 20. [3 points]: State whether each of the following statements is True or False, with a very short
explanation in each case.
– R(Ω) is real. True / False – R(Ω) is an even function of Ω. True / False – R(Ω) is nonnegative at all Ω. True / False We know from the inverse DTFT that
r[n] = 1
2π from which
r[0] = 1
2π π R(Ω)ej Ωn dΩ ,
−π
π R(Ω) dΩ .
−π 21. [2 points]: In the latter equation, express r[0] in terms of x[·] using Eq. (1), and express R(Ω)
in terms of X (Ω) using the result of Problem 19. Write down the resulting equation; this equality is
known as Parseval’s theorem for a discretetime signal. Parseval’s theorem: (From what you’ve proved above, the following chain of reasoning is justiﬁed:
r[n] =
≤
=
=
= π
1
R(Ω)ej Ωn dΩ
2π −π
π
1
R(Ω)ej Ωn dΩ
2π −π
π
1
R(Ω) dΩ
2π −π
π
1
R(Ω) dΩ
2π −π
r[0] . This establishes that the autocorrelation function has its maximum magnitude when the lag is 0. You’ve
used similar ideas for preamble detection.) Page 14 of 17 6.02 Fall 2012, Quiz 2 V Q to the Rescue James Bond at the casino in Macau wants ur...
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This document was uploaded on 03/17/2014 for the course ELECTRICAL 6.02 at MIT.
 Fall '12
 GeorgeVerghese

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