Question

. Consider a moving average MA(2) model: y(t) = e(t)+b, e(t-1) +b,elt - 2) Assume that the noise e(t) has is i.i.d. with variance o = 1.

Answer 1

Power Spectral Density (PSD) power spectral density (PSD), Szelf) given by Fourier Transform of a given function. 14 Self) - =2jaft eht ylt) e j= 1 given, It) = eit +b, et-1) + b, elt-2) elt) = noise Selfl= ectie ² jaft out de to fect- dt fb, ect- lé Dezinftit + b2 feet- & inft dit exponent In

too e(t) = S(t) Sao otherwise fourier transferri of $(+) s srce S(t) x 2-273 ft S. (f) If 2 enjft dit е = I = Jelt) [e(f) - AS 2 I Sect-1) = 2n jt tar lelf-i)- 2 j Alf- lelt-2). Pength & /elf-2) 2 Alf-2)

Sxlf) elf) b I Af + felf- 1 T4-1) b2 + n (f-2) Self SFD elfde step function elf) is assumed a step function because we have proved (above) the the error is elt) is white Proof is attached below gaussian noise

@ Definition . e(t) is said to be white noise vector, if in addition to being ind every variable in elt) also has a normal distribution with zero mean and the variance o²=1, e lt) is a white noise Il M=0 (which can be deduced by compare ing two ego) 0 Ytt) = ett) +b, e(t => tbelt-3) given. ② y(t) = M + Ex + b ect-) + bzelt-2) at bn elt-n) general eg" for mm u o

I step function If U=0 => el) is a while Gaussian noise # For white Gaussian noise, u=0 the SFD is given by No Sxlf) = = the at & (t) = 7²5 No? 2 (t) o SCT) 2 S(x) - to otherwise e(t) = Sit) FS 2 No xo