# H13 - H13 HW6(due Mar 6 Exercises 6.1 6.2 from the textbook...

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H13 HW6 (due Mar. 6). Exercises 6.1, 6.2 from the textbook. The spectral density function or simply the spectrum of a stationary time series, 1 () 2 ih h h fe ω ωγ π =−∞ = , (1) is the counterpart of the covariance function h γ in frequency domain. Here h is assumed to satisfy || h −∞ < ∞ (i.e., h is absolutely summable ). Since h is an even function, 1 = h , (1) can also be written as 0 1 1 2 c o s 2 = ⎛⎞ =+ ⎝⎠ h h f h . (2) The sequence h can be recovered from f through the inverse Fourier transform h f ed ωω = . (3) Setting in (3), we have 0 h = 2 0 σω == X f d , i.e., the total variance of the process can be decomposed into contributions from different frequencies, so that f d represents the contribution to the total variance of the components in the frequency range (, ) d + . Example 1 The spectrum of a White Noise 2 2 1 ,0 , 22 0, 0 Z Z hh h h h σ ππ =−∞ = =⇒ = = The spectrum f

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H13 - H13 HW6(due Mar 6 Exercises 6.1 6.2 from the textbook...

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