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Unformatted text preview: oints]: Let V (Ω) denote the DTFT of v [·]. Which of the following equations correctly shows
how to obtain V (Ω) from X (Ω)? Note, incidentally, that X (−Ω) = X ∗ (Ω), where the latter quantity
is the complex conjugate of X (Ω). (Circle the correct answer, and explain your reasoning.)
• V (Ω) = X (Ω)
• V (Ω) = 1/X (Ω)
• V (Ω) = 1/X (−Ω)
• V (Ω) = X (−Ω)
• V (Ω) = −X (Ω) Deﬁne the signal r[·] as the convolution of x[·] and v [·], so r[n] = (x ∗ v )[n]. More explicitly,
r[n] = ∞
0 x[m]v [n − m] = m=−∞ ∞
0 x[m]x[m − n] . (1) m=−∞ The signal r[·] is called the autocorrelation function of x[·], and r[n] is called the autocorrelation at lag n.
19. [4 points]: Denote the DTFT of r[·] by R(Ω). Write down an expression that shows how to
obtain R(Ω) from X (Ω); the result of the previous problem is likely to be helpful. If you’ve done
things correctly, you should ﬁnd that R(Ω) is entirely determined by the magnitude of X (Ω). Please
write your a...
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- Fall '12