HW6 - input that will produce an unbounded output. Problem...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 310 Digital Signal Processing Problem Set 6 Fall 2003 Due: Wednesday, October 8, 2003 Problem 6.1 Find y [ n ] = h [ n ] * x [ n ] via the z-transform method if h [ n ] = - (1 / 4) n u [ - n - 1] and x [ n ] = (1 / 7) n u [ n ] - (7 / 4) 3 (1 / 7) n u [ n - 3]. Problem 6.2 The system transfer function for an LSI system is given by: H ( z ) = (1 + z - 1 )(1 + j 2 z - 1 )(1 - j 2 z - 1 ) (1 - 1 2 z - 1 )(1 + 1 4 z - 1 )(1 - 1+ j 4 z - 1 )(1 - 1 - j 4 z - 1 ) Draw a block diagram in Direct Form 2 of the implementation of two subsystems that cascaded together represent this transfer function. Problem 6.3 Two LSI systems with unit-pulse responses h 1 [ n ] and h 2 [ n ] are cascaded. Suppose h 1 [ n ] = (1 / 4) n u [ n ]. (a) Find h 2 [ n ] so that the overall cascaded system will have unit- pulse response h [ n ] = (1 / 2) n u [ n ]. (b) Draw a block diagram for the second system. Problem 6.4 Determine whether each of the following represents a BIBO stable system: 1. H ( z ) = z - 7 z 2 +1 / 9 ; causal. 2. H ( z ) = z ( z - 0 . 7)( z 2 + z +1) ; h [ n ] two sided. 3. H ( z ) = z +1 z 2 + j ; causal. 4. H ( z ) = z - 1 z +1 ; causal. 1
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For each case in which the system is determined to be unstable, find a bounded real
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Unformatted text preview: input that will produce an unbounded output. Problem 6.5 Determine for what values of real each of the following represents a BIBO stable system: (a) h [ n ] = n < 10 ( n + ) 2 n 1000 (0 . 1 + ) n n > 1000 ; (b) y [ n + 2] + y [ n ] = 10 x [ n ]. Problem 6.6 Recall that the step response of a system is its response to a unit step in the input, with zero initial conditions. The step response of an LSI system is s [ n ] = (1 / 3) n-2 u [ n + 2]. Is this system causal? Is it stable? Problem 6.7 The input x [ n ] = 2 n ( u [ n ]-3 u [ n-1]) to an unknown LSI system produces the output y [ n ] = (3 n-2 n ) u [ n ]. (a) Determine the unit-pulse response of the system. (b) Is the solution unique? What if the system is known to be unstable? What if it is causal? 2...
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HW6 - input that will produce an unbounded output. Problem...

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