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114_1_sol_midterm_2009_Part2

# 114_1_sol_midterm_2009_Part2 - 10 Magnitude(dB 5 0 5 0 1 2...

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0 1 2 3 4 5 6 -5 0 5 10 frequency (radians) Magnitude (dB) Figure 3: Frequency response for j H ( ! ) j 2 . 6

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3. An unknown signal, x ( n ) is found to have the autocorrelation function: R x ( m ) = [1 ; fl; fl 2 ; fl 3 ; : : : ] (9) for j fl j < 1 (a) Find the 1 st order prediction coe–cient a 1 and the corresponding minimum error, E min . (b) Find the approximated transfer function H ( z ) for the signal x ( n ) based on 1 st order linear predictive analysis. (c) Find the magnitude spectrum j H ( ! ) j 2 corresponding to your answer from part b . (d) Let fl = ¡ 0 : 9. Sketch the pole-zero plot corresponding to H ( z ). Can you comment regarding the spectral composition of the unknown signal x ( n )? 7
Answer : (a) The 1 st order linear prediction coe–cient is determined as: a 1 = R x (1) R x (0) = fl (10) The minimum error is determined as: E min = R x (0) ¡ a 1 R x (1) = 1 ¡ fl 2 (11) (b) The approximated transfer function is determined as: H ( z ) = p 1 ¡ fl 2 1 ¡ flz ¡ 1 : (12) (c) The magnitude spectrum is determined from the answer in part

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114_1_sol_midterm_2009_Part2 - 10 Magnitude(dB 5 0 5 0 1 2...

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