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20091ee114_1_PS2_Sol

20091ee114_1_PS2_Sol - UCLA Dept of Electrical Engineering...

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UCLA Dept. of Electrical Engineering EE 114, Winter 2009 Problem Set 2 Solutions 1. During spectral analysis of speech, a pre-emphasis filter is generally used to boost higher frequencies, since they tend to contain discriminative information. A common form of the pre-emphasis filter is: H ( z ) = 1 G 1 - az - 1 · (1) (a) If we want to design the filter to have 5 times the gain at the Nyquist frequency relative to DC, i.e: | H ( π ) | = 5 | H (0) | , (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 filter be unity, i.e.: X n = -∞ | h ( n ) | 2 = 1 , (3) what should the parameter G be set to? Answer: (a) The frequency response of the pre-emphasis filter is determined as: | H ( ω ) | 2 = H ( ω ) H * ( ω ) = 1 G 2 1 - ae - jωn · ‡ 1 - ae jωn · = 1 G 2 1 + a 2 - 2 a cos ( ω ) · ⇒ | H ( ω ) | = 1 G q 1 + a 2 - 2 a cos ( ω ) Using this expression: | H ( π ) | | H (0) | = 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 filter is: h ( n ) = 1 G [1 , - a ]. To find the correct value of G : X n = -∞ | h ( n ) | 2 = 1 G 2 1 + a 2 · = 1 G = p 1 + a 2 G = r 13 9 2. A 2 nd -order vocal tract transfer function is given by:
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