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# lec02 - 6.003 Signals and Systems Discrete-Time Systems...

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6.003: Signals and Systems Discrete-Time Systems September 15, 2009

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Discrete-Time Systems We 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 Growth
Multiple Representations of Discrete-Time 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.

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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 ; 0 , otherwise. 1 0 1 2 3 4 n x [ n ] = δ [ n ] We will use the unit sample as a “primitive” (building-block signal) to construct more complex signals.

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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 ; 0 , otherwise. 1 0 1 2 3 4 n x [ n ] = δ [ n ] We will use the unit sample as a “primitive” (building-block signal) to construct more complex signals.
Step-By-Step Solutions Difference equations are convenient for step-by-step 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 . . .

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Step-By-Step Solutions Difference equations are convenient for step-by-step 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 . . .
Step-By-Step Solutions Difference equations are convenient for step-by-step 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 . . .

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Step-By-Step Solutions Difference equations are convenient for step-by-step 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 . . .
Step-By-Step Solutions Difference equations are convenient for step-by-step analysis.

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lec02 - 6.003 Signals and Systems Discrete-Time Systems...

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