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20111eeM214A_1_ps511_sol

20111eeM214A_1_ps511_sol - UCLA Dept of Electrical...

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Unformatted text preview: UCLA Dept. of Electrical Engineering EE M214A, Winter 2011 Problem Set 5 Solution February 16, 2011 1. Answer: 2. Answer: 1 3. Answer: 4. Answer: (a) The autocorrelation function is given by: R ( i ) = N- 1 summationdisplay n =0 x ( n ) x ( n- i ) R (0) = 3 2 + 2 2 + (- 1) 2 + 1 2 = 15 R (1) = 3 × 2 + 2 ×- 1 +- 1 × 1 = 3 R (2) = 3 ×- 1 + 2 × 1 =- 1 R (3) = 3 × 1 = 3 Thus: R ( i ) = [15 , 3 ,- 1 , 3] For order N = 2, the normal equations can be expressed in matrix form as: 2 R = bracketleftbigg R (0) R (1) R (1) R (0) bracketrightbigg = bracketleftbigg 15 3 3 15 bracketrightbigg r = bracketleftbigg R (1) R (2) bracketrightbigg = bracketleftbigg 3- 1 bracketrightbigg a = R- 1 r = bracketleftbigg 2 9- 1 9 bracketrightbigg G = radicalbig E min ⇒ H ( z ) = G 1- 2 9 z- 1 + 1 9 z- 2 (b) E min = R (0)- 2 summationdisplay k =1 a k R ( k ) (-12) = 15- parenleftbigg 2 9 parenrightbigg (3)- parenleftbigg- 1 9 parenrightbigg (- 1) (-11) = 14 . 22 (-10) (-9) Since LPC assumes an all-pole system, the minimum error occurs at the poles of the transfer...
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20111eeM214A_1_ps511_sol - UCLA Dept of Electrical...

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