20091ee114_1_PS4%20_Sol

# 20091ee114_1_PS4%20_Sol - UCLA Dept of Electrical...

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Dept. of Electrical Engineering EE 114, Winter 2009 Problem Set 4 Solutions 1. Consider the signal: x ( n ) = β n for n 0 (1) (a) Using the autocorrelation method of linear prediction analysis, ﬁnd the 1 st order predic- tion coeﬃcient, i.e. a 1 . Find the corresponding minimum error, E min . (b) Using the covariance method of linear prediction analysis, ﬁnd the 1 st order predic- tion coeﬃcient, i.e. a 1 . Find the corresponding minimum error, E min . (c) What can you say about the accuracy of each of these linear prediction methods. Which method worked best for this example signal? Hint: N n =0 ρ n = 1 - ρ N +1 1 - ρ Answer : (a) For the 1 st -order autocorrelation case: a 1 = R (1) R (0) R (0) = N - 1 X n =0 x 2 ( n ) = N - 1 X n =0 β 2 n R (0) = 1 - β 2 N 1 - β 2 R (1) = N - 1 X n =1 x ( n ) x ( n - 1) = N - 1 X n =1 β n β n - 1 = 1 β N - 1 X n =1 β 2 n = 1 β " β 2 - β 2 N 1 - β 2 # R (1) = β " 1 - β 2 N - 2 1 - β 2 # a 1 = β " 1 - β 2 N - 2 1 - β 2 # × " 1

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20091ee114_1_PS4%20_Sol - UCLA Dept of Electrical...

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