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# ex6 - difference equation y[n – y[n-1 – y[n-2 = x[n(1...

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Digital Signal Processing (Winter 2005) Exercise No. 6 (due to 22/12) 1. Determine the z-transform, including the ROC of the following sequences: a. x[n] = (½) n (u[n] – u[n-10]) b. x[n] = 2 n u[-n] 2. Determine the inverse z-transform of the following functions: a. a z az z X - = - ) 1 log( ) ( 1 (try to define the logarithm using series) b. a z a z az z X / 1 1 ) ( 1 1 - - = - - c. 2 / 1 1 1 ) ( 2 8 1 1 4 3 1 2 1 + + - = - - - z z z z z X 3. The unilateral z-transform is defined as (The lower limit in the sum is zero instead of - ° ). We denote the relation between a discrete-time signal x[n] and its unilateral z-transform X + (z) by a. Prove the following time shifting property for the unilateral z-transform b. Determine the unilateral z-transform of the following signal c. Consider a system for which the input and output satisfy the following

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Unformatted text preview: difference equation y[n] – y[n-1] – y[n-2] = x[n] (1) Assume that the input for the system is x[n] = ± [n] and that the initial conditions are y[-1] = y[-2] = 0. What will be the output of the system? & ∞ =-+ = ] [ ) ( n n z n x x X ) ( ] [ z X n x u + → ← ) ( ] 1 [ ] [ 1 1 z X z z m k x m n x m m k k u +-= +-+--→ ←-& , ] [ ≥ = n p n x n d. Apply the unilateral Z transform to Eq. (1) and use the linearity property, the time-shifting property and the initial condition in order to find Y + (z). e. Find an explicit expression for y[n] by calculating the inverse unilateral z-transform of Y + (z)....
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ex6 - difference equation y[n – y[n-1 – y[n-2 = x[n(1...

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