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20111ee114_1_HW2_Sol

20111ee114_1_HW2_Sol - UCLA Dept of Electrical Engineering...

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UCLA Dept. of Electrical Engineering EE 114, Winter 2011 Problem Set 2 Solutions 1. During spectral analysis of speech, a pre-emphasis fllter is generally used to boost higher frequencies, since they tend to contain discriminative information. A common form of the pre-emphasis fllter is: H ( z ) = 1 G 1 ¡ az ¡ 1 · (1) (a) If we want to design the fllter to have 5 times the gain at the Nyquist frequency relative to DC, i.e: j H ( ) j = 5 j H (0) j ; (2) what should the parameter a be set to? (Hint: a should lie in the range (0 ; 1)). (b) If it is desired that the total energy of the fllter be unity, i.e.: 1 X n = ¡1 j h ( n ) j 2 = 1 ; (3) what should the parameter G be set to? Answer: (a) The frequency response of the pre-emphasis fllter is determined as: j H ( ! ) j 2 = H ( ! ) H / ( ! ) = 1 G 2 1 ¡ ae ¡ j!n · ‡ 1 ¡ ae j!n · = 1 G 2 1 + a 2 ¡ 2 a cos ( ! ) · ) j H ( ! ) j = 1 G q 1 + a 2 ¡ 2 a cos ( ! ) Using this expression: j H ( ) j j H (0) j = s 1 + a 2 + 2 a 1 + a 2 ¡ 2 a = 5 1 + a 2 + 2 a 1 + a 2 ¡ 2 a = 25 24 a 2 ¡ 52 a + 24 = 0 ) a = 2 3 1
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(b) The inverse Z-Transform of the given fllter is: h ( n ) = 1
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