6 003 Signals and Systems Lecture 2 6 003 Signals and Systems September 15 2009 Discrete Time Systems We start with discrete time systems because they Discrete Time Systems are conceptually simpler than continuous time systems illustrate the same important modes of thinking are increasingly important digital electronics and computation Example Population Growth September 15 2009 Multiple Representations of Discrete Time Systems Difference Equations Systems can be represented in different ways to more easily address different types of issues Difference equations are mathematically precise and compact Example 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 y n x n x n 1 Let x n equal the unit sample signal n Difference equation y n x n x n 1 n Block diagram x n 1 y n 1 0 if n 0 otherwise x n n Delay 1 0 1 2 3 4 n We will exploit particular strengths of each of these representations We will use the unit sample as a primitive building block signal to construct more complex signals Step By Step Solutions Step By Step Solutions Difference equations are convenient for step by step analysis 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 Find y n given x n n x n n 1 0 1 2 3 4 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 1 x n n y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 1 Delay y n 0 y n n 1 0 1 2 3 4 n 6 003 Signals and Systems Lecture 2 Check Yourself September 15 2009 From Samples to Signals Lumping all of the possibly infinite samples into a single object the signal simplifies its manipulation DT systems can be described by difference equations and or block diagrams This lumping is an abstraction that is analogous to Difference equation y n x n x n 1 Block diagram x n 1 representing coordinates in three space as points representing lists of numbers as vectors in linear algebra creating an object in Python y n Delay In what ways are these representations different From Samples to Signals Operator Notation Operators manipulate signals rather than individual samples Symbols can now compactly represent diagrams x n 1 Let R represent the right shift operator y n Y R X RX Delay where X represents the whole input signal x n for all n and Y represents the whole output signal y n for all n Nodes represent whole signals e g X and Y The boxes operate on those signals Representing the difference machine Delay shift whole signal to right 1 time step Add sum two signals 1 multiply by 1 x n 1 Signals are the primitives Operators are the means of combination y n Delay with R leads to the equivalent representation Y X RX 1 R X Operator Notation Check Yourself Operator Representation of a Cascaded System System operations have simple operator representations Cascade systems multiply operator expressions Let Y RX Which of the following is are true 1 2 3 4 5 X y n x n for all n y n 1 x n for all n y n x n 1 for all n y n 1 x n for all n none of the above 1 Delay Using operator notation Y1 1 R X Y2 1 R Y1 Substituting for Y1 Y2 1 R 1 R X 2 Y1 1 Delay Y2 6 003 Signals and Systems Lecture 2 September 15 2009 Operator Algebra Operator Approach Operator expressions can be manipulated as polynomials Applies your existing expertise with polynomials to understand block diagrams and thereby understand systems Y1 X 1 1 Delay Y2 Delay Using difference equations y2 n y1 n y1 n 1 x n x n 1 x n 1 x n 2 x n 2x n 1 x n 2 Using operator notation Y2 1 R Y1 1 R 1 R X 1 R 2 X 1 2R R2 X Operator Algebra Check Yourself Operator notation facilitates seeing relations among systems Operator expressions for these equivalent systems if started at rest obey what mathematical property Equivalent block diagrams assuming both initially at rest Y1 Y2 X 1 1 Delay X X Delay 1 Y Delay Y Y Delay Delay X 2 Delay 1 Delay Delay Delay 1 commutate 2 associative 3 distributive 4 transitive 5 none of the above Equivalent operator expressions 1 R 1 R 1 2R R2 The operator equivalence is much easier to see Check Yourself Operator Algebra Explicit and Implicit Rules Recipes versus constraints How many of the following systems are equivalent to Y 4R2 4R 1 X Delay X 2 Delay 2 Recipe subtract a right shifted version of the input signal from a copy of the input signal Y X 1 Delay X Delay 4 Y Y 1 R X Delay Y Constraint the difference between Y and RY is X Delay X Delay 4 X Y Y Delay Y RY X 1 R Y X But how does one solve such a constraint 3 6 003 Signals and Systems Lecture 2 September 15 2009 Example Accumulator Example Accumulator Try step by step analysis it always works Start at rest x n y n The response of the accumulator system could also be generated by a system with infinitely many paths from input to output each with one unit of delay more than the previous Delay Find y n given x n n X y n x n y n 1 y 0 x 0 y 1 1 0 1 x n n y 1 x 1 y 0 0 1 1 y 2 x 2 y 1 y n 0 1 1 Delay Delay Delay Delay Delay Delay 1 0 1 2 3 4 n 1 0 1 2 3 4 Y n Y 1 R R2 R3 X Persistent response to a transient input Example Accumulator Example Accumulator These systems are equivalent in the sense that if each is initially at rest they will produce identical outputs from the same input The reciprocal of 1 R can also be evaluated using synthetic division 1 R Y1 X1 1 R R2 R3 1 R 1 1 R R R R2 R2 R2 R3 R3 R3 R4 Y2 1 R R2 R3 X2 Proof Assume X2 X1 Y2 1 R R2 R3 X2 1 R R2 R3 X1 1 R R2 R3 1 R Y1 1 R R2 R3 R R2 R3 Y1 Therefore Y1 1 1 R R2 R3 R4 1 R It follows that Y2 Y1 It also follows that 1 R and 1 R R2 R3 are reciprocals …
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