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HW3_ece220_2007_solution

# HW3_ece220_2007_solution - Homework 3 Solution(ECE220 Fall...

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Homework 3 Solution (ECE220 - Fall 2007) 1. (40 points) A bit more on transformations (a) (10 points) Suppose you are betting at the roulette in Las Vegas - red or black. You start with y [0] = \$50 and every time you win you get 2 x [ n ] of your original bet x [ n ] while if you loose, you subtract the amount x [ n ] from your funds. Your funds amount is the output, the game is the system and the bet you make is the input. To describe this mathematically let h [ n ] = 2 , if you win at game n ; - 1 , if you loose at game n . Clearly the money you have at the n th game iteration is: y [ n ] = y [ n - 1] + h [ n ] x [ n ] Prove that this transformation is linear and causal. Solution: y [ n ] = y [ n - 1] + h [ n ] x [ n ] = y [0] + n - 1 X k =1 h [ k ] x [ k ] + h [ n ] x [ n ] = y [0] + n X k =1 h [ k ] x [ k ] From that, the output at time n clearly only depends on the input at time n or before, so the system is causal. By strict definition the system is not linear, but with the hint posted on blackboard αy 1 [0] + βy 2 [0] = y [0] = 50. The system can be treated as linear. Basically what we splitting the initial condition (the nonlinear part) between the summed signals. This makes for this system because if you were to have two people betting they would each need a portion of they money to bet with.

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HW3_ece220_2007_solution - Homework 3 Solution(ECE220 Fall...

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