MIT6_003S10_lec02

# MIT6_003S10_lec02 - 6.003 Signals and Systems Discrete-Time...

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6.003: Signals and Systems Discrete-Time Systems February 4, 2010

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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 increasingly important (digital electronics and computation)
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: x [ n ] 1 Delay + 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]
unit sample as a “primitive” (building-blo 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 0 123 4 n

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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 ] , 1 , if n =0 ; δ [ n 0 , otherwise. x [ n δ [ n ] 1 0 123 4 n We will use the unit sample as a “primitive” (building-block signal) to construct more complex signals.
y [ 1] = x [ 1] x [ 2] =0 0=0 y [0] = x [0] x [ 1] =1 0=1 y [1] = x [1] x [0] 1= 1 y [2] = x [2] x [1] y [3] = x [3] x [2] ... 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] x [ n δ [ n ] y [ n ] 1 0 123 4 1 0 4 n n

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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 y [3] = x [3] x [2] ... 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] x [ n δ [ n ] y [ n ] 1 0 123 4 1 0 4 n n
y [1] = x [1] x [0] =0 1= 1 y [2] = x [2] x [1] 0=0 y [3] = x [3] x [2] ... 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] y [0] = x [0] x [ 1] =1 0=1 x [ n δ [ n ] y [ n ] 1 0 123 4 1 0 4 n n

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y [2] = x [2] x [1] =0 0=0 y [3] = x [3] x [2] ... 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] y [0] = x [0] x [ 1] =1 0=1 y [1] = x [1] x [0] 1= 1 x [ n δ [ n ] y [ n ] 1 0 123 4 1 0 4 n n
y [3] = x [3] x [2] =0 0=0 ...

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## This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Fall '11 term at MIT.

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MIT6_003S10_lec02 - 6.003 Signals and Systems Discrete-Time...

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