Question

1. Auto- and Cross-Correlation. For each of the following, compute the cross correlation T/2 Rry(,) = E[drpd, + n-linx t-Tax(ry(, + rdr . Hint: Use trigonometric identities (see HW 1), 27T such as sin a sin b-2 [cos(a-b)-cos(a + b)] . Also use the fact that j cos(ont-б unless co-0 x(t) = sin(2n/r), y(t)-sin(2nft) (here x and y are the same, so Rry-Rrr is the a. autocorrelation of x). x(t) = sin(2nft), y(t) = sin(2nf(t-to)) c. x(t)-n(), y()2x(t) +n2(t) where n,(1), n2(t) are both Gaussian white noise with RMS b. e 1. Assume n1(t),n2(t) are independent and uncorrelated, i.e. ETn1(t)n2(r)] 0 for any t,r. J x(t')h(t-r)dt, where h is a convolution kernel function, and ni (t) d. h x(t) ni(t), y-x 0 is Gaussian white noise (as in part c). h + n,(,) = j x(r)h(,-r,dr, + n2(1) where h is a convolution kernel x(t) = ni (1), y = x e. 0 function, and n(), n2(t) are Gaussian white noise (as in part c)

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