6.003: Signals and SystemsDiscrete-Time SystemsSeptember 15, 2009Discrete-Time SystemsWe start with discrete-time systems because they• are conceptually simpler than continuous-time systems• illustrate the same important modes of thinking• are increasingly important (digital electronics and computation)Example: Population GrowthMultiple Representations of Discrete-Time SystemsSystems can be represented in different ways to more easily addressdifferent types of issues.Verbal description: ‘To reduce the number of bits needed to storea sequence of large numbers that are nearly equal, record the firstnumber, and then record successive differences.’Difference equation:y[n] = x[n] − x[n − 1]Block diagram:−1Delay+x[n] y[n]We will exploit particular strengths of each of these representations.Difference EquationsDifference equations are mathematically precise and compact.Example:y[n] = x[n] − x[n − 1]Difference EquationsDifference 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;0, otherwise.−1 0 1 2 34nx[n] = δ[n]We will use the unit sample as a “primitive” (building-block signal)to construct more complex signals.Difference EquationsDifference 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;0, otherwise.−1 0 1 2 34nx[n] = δ[n]We will use the unit sample as a “primitive” (building-block signal)to construct more complex signals.Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram:−1Delay+x[n] y[n]−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+x[n] y[n]0−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+1 1−10−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+1 → 0−10 → −1−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+1 → 0−100 → −1−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+0−10−1−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+0−10−1 → 0−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+0 00−1 → 0−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+0 000−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+0 000−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with
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