ps1_fall09

ps1_fall09 - Department of Electrical and Computer...

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Unformatted text preview: Department of Electrical and Computer Engineering Digital Speech Processing Problem Set No. 1 Problem 1 Consider the sequence of values described by the difference equation: y [ n ] = 2 y [ n- 1]- . 5 y [ n- 2] with initial conditions: y [- 1] = 0 (1) y [- 2] =- 2 (a) Solve for y [0] ,y [1] ,y [2] ,y [3] (b) Determine an analytic solution for y [ n ] for all n (c) If the limit of sequential terms is given by the factor G , i.e., lim n →∞ y [ n ] y [ n- 1] = G determine G . Problem 2 An input signal, x [ n ], defined over the region-∞ < n < ∞ , is passed through a cubic non-linearity yielding the output signal y [ n ] = ( x [ n ]) 3 (a) If the input to the system is of the form x 1 [ n ] = cos( ω n ) ,-∞ < n < ∞ determine the output spectrum, Y 1 ( e jω ), and plot the magnitude of Y 1 ( e jω ) (assuming ω << ω S , the radian sampling frequency of the digital system). Hint: recall the trigonometric relationships cos 2 ( x ) = 1 2 + cos(2 x ) 2 cos 3 ( x ) = cos(3 x ) 4 +...
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ps1_fall09 - Department of Electrical and Computer...

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