hw8 - -2] = 2 x [ n ] . i. Find the impulse response if the...

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ECE130B February 22, 2010 Home work 8 Due Date : March 1, 2010. Reading assignment : Chapter 10 of the text book. Exercise 1. Find the impulse response of a causal system whose input x [ n ] and output y [ n ] are related by the equation y [ n + 1] + y [ n ] + y [ n - 1] = x [ n ] + x [ n - 1] + x [ n - 2] . Bounded signals . A signal x [ n ] is said to be bounded is there exists a ±nite number M such that | x [ n ] | <M for -∞ <n < . Stable system . A system is said to be (bounded-input-bounded-output) stable if for all bounded input signals the output signal is also bounded. Exercise 2. Show that an LTI system is stable if, and only if, its impulse response h [ n ] is abso- lutely summable ; that is s n = -∞ | h [ n ] | < . Exercise 3. System A has impulse response h A [ n ] = ( - 3/7) n u [ n ] , while system B has impulse response h B [ n ] = ( - 3/7) n u [ - n ] . Which of the systems, if any, is stable? Exercise 4. The input x [ n ] , and output y [ n ] , of an LTI system satisfy the equation 2 y [ n ] - 5 y [ n - 1] + 2 y [ n
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Unformatted text preview: -2] = 2 x [ n ] . i. Find the impulse response if the system is known to be causal. ii. Find the impulse response if the system is known to be stable. iii. Can the system be both causal and stable at the same time? Justify your answer. Exercise 5. Find the z-transforms of the following functions along with their region of conver-gence. i. a n u [ n ] ii. a n u [-n-1] iii. n a n u [ n ] iv. z n (careful) Exercise 6. If X ( z ) is the z-transform of x [ n ] with ROC a &lt; | z | &lt; b , what are the z-transforms and ROCs of the following functions i. x [ n-n ] ii. x [-n ] iii. z n x [ n ] iv. n x [ n ] Justify all your answers. Exercise 7. Find the discrete signal x [ n ] , whose z-tranform is given by X ( z ) = ( z 4 + 1)-1 , with ROC | z | &lt; 1 . 1...
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