HW2_ece220_2007_solution

HW2_ece220_2007_solution - Homework 2 Solution (ECE220 -...

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Unformatted text preview: Homework 2 Solution (ECE220 - Fall 2007) For all problems let H stand for the actions of a system such that: y [ n ] = H{ x [ n ] } or y ( t ) = H{ x ( t ) } . 1. (20 points) Provide both the definition as well as an example of an analog and digital system that is: (a) (5 points) Causal . Solution: A causal system is one which the output depends only on the present and past, not future, inputs. y [ n ] = nx [ n ]- x [ n- 1] y ( t ) = cos( x ( t )) If either of these systems was non-causal it would have a dependence on a future value of x [ n ] /x ( t ). Neither of these systems has any functions that will cause the output to rely on a future time (only on the current time or the past time). (b) (5 points) Stable . Solution: A stable system is one which any bounded input will result in the output being bounded (not tending to infinity). | x [ n ] | < B, y [ n ] = n 2 n 2 + 1 x £ n 3 / | y [ n ] | = fl fl fl fl n 2 n 2 + 1 x [ n 3 ] fl fl fl fl = fl fl fl fl n 2 n 2 + 1 fl fl fl fl fl fl x £ n 3 /fl fl < fl fl fl fl n 2 n 2 + 1 fl fl fl fl B ≤ 1 2 B | x ( t ) | < B, y ( t ) = Z t t- 5 x ( τ ) dτ | y ( t ) | = fl fl fl fl Z t t- 5 x ( τ ) dτ fl fl fl fl ≤ Z t t- 5 | x ( τ ) | dτ < Z t t- 5 Bdτ = 5 B (c) (10 points) Linear . In this case also compare the definition with that of Linear system pro- vided in Linear algebra (see e.g. http://mathworld.wolfram.com/LinearTransformation.html). Solution: A linear system is one which preserves input signal addition and input signal scalar mul- tiplication. This property is what allows superposition to work. y [ n ] = x [ n ] + x [ n + 2] H{ αx 1 [ n ] + βx 2 [ n ] } = αx 1 [ n ] + βx 2 [ n ] + αx 1 [ n + 2] + βx 2 [ n + 2] = α ( x 1 [ n ] + x 1 [ n + 2]) + β ( x 2 [ n ] + x 2 [ n + 2]) = α H{ x 1 [ n ] } + β H{ x 2 [ n ] } y ( t ) = x ( t )cos(3 t ) H{ αx 1 ( t ) + βx 2 ( t ) } = ( αx 1 ( t ) + βx 2 ( t ))cos(3 t ) = αx 1 ( t )cos(3 t ) + βx 2 ( t )cos(3 t ) = α H{ x 1 ( t ) } + β H{ x 2 ( t ) } 2. (40 points) Moving average Consider a the following moving average system with input x [ n ] and output y [ n ] related by y [ n ] = 1 2 n + 1 n + n X k = n- n x [ k ] where n is a positive integer. (a) (5 points) Is this system linear? Support your claim....
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This note was uploaded on 05/31/2008 for the course ECE 2200 taught by Professor Johnson during the Fall '05 term at Cornell.

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HW2_ece220_2007_solution - Homework 2 Solution (ECE220 -...

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