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 x n 1 y n Delay 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 0 if n 0 otherwise x n n n 1 0 1 2 3 4 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 0 if n 0 otherwise x n n n 1 0 1 2 3 4 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 Find y n given x n n 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 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Step By Step Solutions Difference equations are convenient for step by step analysis Find y n given x n n 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 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Step By Step Solutions Difference equations are convenient for step by step analysis Find y n given x n n 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 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Step By Step Solutions Difference equations are convenient for step by step analysis Find y n given x n n 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 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Step By Step Solutions Difference equations are convenient for step by step analysis Find y n given x n n 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 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Step By Step Solutions Difference equations are convenient for step by step analysis Find y n given x n n 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 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Step By Step Solutions Difference equations are convenient for step by step analysis Find y n given x n n 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 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 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 x n 1 x n n y n Delay y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 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 x n y n 1 x n n Delay 0 y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 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 1 Delay 1 0 1 x n n y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 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 0 1 Delay 0 1 1 x n n y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 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 0 1 Delay 1 0 1 0 x n n y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 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 0 1 …
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