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Unformatted text preview: 6.003: Signals and Systems DiscreteTime Systems September 15, 2009 DiscreteTime Systems We start with discretetime systems because they are conceptually simpler than continuoustime systems illustrate the same important modes of thinking are increasingly important (digital electronics and computation) Example: Population Growth Multiple Representations of DiscreteTime Systems Systems can be represented in different ways to more easily address different types of issues. Verbal description: To reduce the number of bits needed to store a sequence of large numbers that are nearly equal, record the first number, and then record successive differences. Difference equation: y [ n ] = x [ n ] x [ n 1] Block diagram: 1 Delay + x [ n ] y [ n ] We will exploit particular strengths of each of these representations. Difference Equations Difference equations are mathematically precise and compact. Example: y [ n ] = x [ n ] x [ n 1] Difference Equations Difference equations are mathematically precise and compact. Example: y [ n ] = x [ n ] x [ n 1] Let x [ n ] equal the unit sample signal [ n ] , [ n ] = 1 , if n = 0 ; , otherwise. 1 0 1 2 3 4 n x [ n ] = [ n ] We will use the unit sample as a primitive (buildingblock signal) to construct more complex signals. Difference Equations Difference equations are mathematically precise and compact. Example: y [ n ] = x [ n ] x [ n 1] Let x [ n ] equal the unit sample signal [ n ] , [ n ] = 1 , if n = 0 ; , otherwise. 1 0 1 2 3 4 n x [ n ] = [ n ] We will use the unit sample as a primitive (buildingblock signal) to construct more complex signals. StepByStep Solutions Difference equations are convenient for stepbystep analysis. 1 0 1 2 3 4 n x [ n ] = [ n ] 1 0 1 2 3 4 n y [ n ] Find y [ n ] given x [ n ] = [ n ] : y [ n ] = x [ n ] x [ n 1] y [ 1] = x [ 1] x [ 2] = 0 0 = 0 y [0] = x [0] x [ 1] = 1 0 = 1 y [1] = x [1] x [0] = 0 1 = 1 y [2] = x [2] x [1] = 0 0 = 0 y [3] = x [3] x [2] = 0 0 = 0 ... StepByStep Solutions Difference equations are convenient for stepbystep analysis. 1 0 1 2 3 4 n x [ n ] = [ n ] 1 0 1 2 3 4 n y [ n ] Find y [ n ] given x [ n ] = [ n ] : y [ n ] = x [ n ] x [ n 1] y [ 1] = x [ 1] x [ 2] = 0 0 = 0 y [0] = x [0] x [ 1] = 1 0 = 1 y [1] = x [1] x [0] = 0 1 = 1 y [2] = x [2] x [1] = 0 0 = 0 y [3] = x [3] x [2] = 0 0 = 0 ... StepByStep Solutions Difference equations are convenient for stepbystep analysis. 1 0 1 2 3 4 n x [ n ] = [ n ] 1 0 1 2 3 4 n y [ n ] Find y [ n ] given x [ n ] = [ n ] : y [ n ] = x [ n ] x [ n 1] y [ 1] = x [ 1] x [ 2] = 0 0 = 0 y [0] = x [0] x [ 1] = 1 0 = 1 y [1] = x [1] x [0] = 0 1 = 1 y [2] = x [2] x [1] = 0 0 = 0 y [3] = x [3] x [2] = 0 0 = 0 ... StepByStep Solutions...
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This note was uploaded on 08/24/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.
 Spring '11
 DennisM.Freeman

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