HW2_ece220_2007_solution

# HW2_ece220_2007_solution - Homework 2 Solution (ECE220 -...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## This note was uploaded on 05/31/2008 for the course ECE 2200 taught by Professor Johnson during the Fall '05 term at Cornell.

### Page1 / 6

HW2_ece220_2007_solution - Homework 2 Solution (ECE220 -...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online