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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 δ [ n ] , δ [ n ] = 1 , if n = 0 ; , otherwise. − 1 0 1 2 3 4 n x [ n ] = δ [ 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 δ [ n ] , δ [ n ] = 1 , if n = 0 ; , otherwise. − 1 0 1 2 3 4 n x [ n ] = δ [ 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 ] = δ [ n ] − 1 0 1 2 3 4 n y [ n ] Find y [ n ] given x [ n ] = δ [ 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 ] = δ [ n ] − 1 0 1 2 3 4 n y [ n ] Find y [ n ] given x [ n ] = δ [ 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 ] = δ [ n ] − 1 0 1 2 3 4 n y [ n ] Find y [ n ] given x [ n ] = δ [ 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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 Spring '11
 DennisM.Freeman
 Delay, discretetime systems

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