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MIT6_003S10_lec02_handout

MIT6_003S10_lec02_handout - 6.003 Signals and Systems...

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6.003: Signals and Systems Lecture 2 February 4, 2010 6.003: Signals and Systems Discrete-Time Systems February 4, 2010 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. Step-By-Step Solutions Difference equations are convenient for step-by-step analysis. 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 . . . x [ n ] = δ [ n ] y [ n ] 1 0 1 2 3 4 1 0 1 2 3 4 n n 1 Discrete-Time Systems We start with discrete-time (DT) systems because they are conceptually simpler than continuous-time systems illustrate same important modes of thinking as continuous-time are increasingly important (digital electronics and computation) 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 ] , 1 , if n = 0 ; δ [ n ] = 0 , otherwise. x [ n ] = δ [ n ] 1 0 1 2 3 4 n We will use the unit sample as a “primitive” (building-block signal) to construct more complex signals. Step-By-Step Solutions Block diagrams are also useful for step-by-step analysis. Represent y [ n ] = x [ n ] x [ n 1] with a block diagram: start “at rest” 1 Delay + x [ n ] y [ n ] 0 1 0 1 2 3 4 n x [ n ] = δ [ n ] 1 0 1 2 3 4 n y [ n ]
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6.003: Signals and Systems Lecture 2 February 4, 2010 From Samples to Signals Check Yourself DT systems can be described by difference equations and/or block diagrams. Difference equation: y [ n ] = x [ n ] x [ n 1] Block diagram: 1 Delay + x [ n ] y [ n ] In what ways are these representations different? Lumping all of the (possibly infinite) samples into a single object the signal simplifies its manipulation. This lumping is an abstraction that is analogous to representing coordinates in three-space as points representing lists of numbers as vectors in linear algebra creating an object in Python From Samples to Signals Operators manipulate
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