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

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Unformatted text preview: 6.003: Signals and Systems Discrete-Time Systems September 15, 2009 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. 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 (building-block 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 (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 ] = [ 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 ... Step-By-Step Solutions Difference equations are convenient for step-by-step 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 ... Step-By-Step Solutions Difference equations are convenient for step-by-step 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 ... Step-By-Step 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.

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

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