HW3_ece220_2007

HW3_ece220_2007 - Homework 3(ECE220 Fall 2007 Due Thursday...

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Homework 3 (ECE220 - Fall 2007) Due: Thursday, September 27 at the beginning of lecture. Reasoning and work must be shown to gain full/partial credit. WRITE YOUR NAME AND NET ID ON ALL PAGES HANDED IN! 1. (40 points) A bit more on transformations – General digital and analog transformations (indicated usually x [ n ] 7→ y [ n ] for digital and x ( t ) 7→ y ( t ) for analog transformations) map input signals in output signals in an arbitrary way. Careful because the notation 7→ or sometimes y [ n ] = F ( x [ n ]) may lead to confusion because only two specific values at time n appear in the expression; transformations, however, construct the value of y [ n ] at each n using not only the value x [ n ] at the same index n , but also on values before and after it x [ n - k ] for k 6 = n for several (if not all) values of k Z . The same is true for analog transformations as well where y ( t ) depends not only on x ( t ) but on values before and after it x ( t - τ ), τ R . The only exception to this rule are memoryless transformations. Let x denote the whole signal x [ n ], by performing the association that we discussed in class of viewing all values of the signal ( ...,x [ n - 2] ,x [ n - 1] ,x [ n ] ,x [ n + 1] ,x [ n + 2] ,... ) as coordinates of a vector on an orthogonal reference system in an very large, say infinite, dimensional space. The notation x 7→ y or y = F ( x ) is more precise now, because it does not leave out any component in both sides of the equation, and this is how most transformations are operating and, when possible, are also specified mathematically. Look at an example: (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
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This note was uploaded on 05/31/2008 for the course ORIE 310 taught by Professor Callister during the Fall '07 term at Cornell.

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HW3_ece220_2007 - Homework 3(ECE220 Fall 2007 Due Thursday...

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