Check with reasons whether the functions fit, t + 1) listed below represent in each case a valid autocorrelation function (ACF); if so, why? Case (1) (iv) (v) (a x t + (1 - [TI) (2 - ITD) (AITI+B)/(CIT|+D) sin(/TD) + 1 a x sin(t[) + b x T (-1 sT s + 1) fit, t + 1) (-2 s T s+2) and for all t (A, B, C, D): (-pi/2 s T s (- pi/2 s T s + Constants and +pi/2) pi/2) 0 otherwise 0 otherwise B/D > A/C 0 otherwise 0 otherwise a: constant pi = 3.1416 Valid ACF? YES/NO Choices Reason(s)? (1) (iv) (v) No: No; Yes; Yes; No: It is not Not maximum at Maximum at T Maximum at T = 0 1 It is a function T =0 an even = 0 with It is an even of time function Odd function Even symmetry function Yes; Valid Yes; Yes; Valid Yes; Valid Yes; Valid ACF Maximum at T = E[y] is Max at T = 0 2 0 independent Even symmetry Not a function of Ever at the ordinate time. Even function Even symmetry of time symmetry Yes; No: No: No; No: Maximum at T = 0 Minimum It is not function Maximum It is a function It has odd at T =0 of time at T =0 of time Symmetry Yes; Yes; Even No; Minimum No: function at T = 0 No; It is a function Maximum at T = It is a function O with even Maximum It is an even of time symmetry of time at T =0 function Yes; It is an odd Yes; function No: No; Yes; It is not a Maximum at T = It is a function It is a function It is not a function function 0 of time of time of time of time USEFUL FORMULAS Properties of valid Autocorrelation function (ACF) The ACF of a signal is defined as the measure of similarity or coherence between a signal and its time delayed version. ACF of a real function x(t): Rxx(T) = E[x(t) x x(t + T)], where T is a variable called time-shift (delay) parameter ACF is an even function ACF reaches its peak at the origin: [RIT)| s R(O) ACF of a periodic function is, itself, periodic with the same period Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all) is the sum of the autocorrelations of each function separately ACF is related to the power spectral density via the Fourier transform Rxx (T - 0) = m2(x); Rxx (T - Infinity) = [m,(x)]2 O Choice 1 O Choice 2 O Choice 3 O Choice 4 O Choice 5