6 003 Signals and Systems Lecture 3 6 003 Signals and Systems September 17 2009 Last Time Multiple Representations of DT Systems Verbal descriptions preserve the rationale Feedback Poles and Fundamental Modes 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 equations mathematically compact y n x n x n 1 Block diagrams illustrate signal flow paths x n 1 y n Delay Operator representations analyze systems as polynomials Y 1 R X September 17 2009 Last Time Feedback Cyclic Signal Paths and Modes Last Time Feedback Cyclic Signal Paths and Modes Systems with signals that depend on previous values of the same signal are said to have feedback The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths X Y Example The accumulator system has feedback X p0 Y Delay x n n By contrast the difference machine does not have feedback X 1 Delay y n 1 0 1 2 3 4 n 1 0 1 2 3 4 n Y Each cycle creates another sample in the output Delay The response will persist even though the input is transient Geometric Growth Poles Check Yourself These system responses can be characterized by a single number the pole which is the base of the geometric sequence X What value of p0 represents the signal below Y y n p0 y n y n 1 0 1 2 3 4 p0 0 5 n p0 0 Delay y n n n if n 0 otherwise 1 2 3 4 y n 1 0 1 2 3 4 p0 1 n 1 0 1 2 3 4 n p0 1 2 1 p0 0 8 p0 0 8 p0 0 64 interspersed with p0 0 64 none of the above 6 003 Signals and Systems Lecture 3 September 17 2009 Geometric Growth Second Order Systems The value of p0 determines the rate of growth The unit sample responses of more complicated cyclic systems are more complicated y n y n y n X y n Y R 1 6 R 1 p0 1 1 p0 0 0 p0 1 p0 1 0 magnitude magnitude magnitude magnitude 1 0 63 z y n diverges alternating sign converges alternating sign converges monotonically diverges monotonically 1 0 1 2 3 4 5 6 7 8 n Not geometric This response grows then decays Factoring Second Order Systems Factoring Second Order Systems Factor the operator expression to break the system into two simpler systems divide and conquer The factored form corresponds to a cascade of simpler systems X 1 0 7R 1 0 9R Y X X Y 0 7 R 1 6 Y2 Y 0 9 R 1 0 7R Y2 X R 1 0 9R Y Y2 R 0 63 Y1 X 0 9 Y X 1 6RY 0 63R2 Y 1 1 6R 0 63R2 Y X Y 0 7 R 1 0 9R Y1 X R 1 0 7R Y Y1 1 0 7R 1 0 9R Y X The order doesn t matter if systems are initially at rest Factoring Second Order Systems Multiplying Polynomial The unit sample response of the cascaded system can be found by multiplying the polynomial representations of the subsystems Graphical representation of polynomial multiplication Y 1 aR a2 R2 a3 R3 1 bR b2 R2 b3 R3 X Y 1 1 1 X 1 0 7R 1 0 9R 1 0 7R 1 0 9R z z 1 a z z 1 0 7R 0 72 R2 0 73 R3 1 0 9R 0 92 R2 0 93 R3 X Multiply then collect terms of equal order Y 1 0 7 0 9 R 0 72 0 7 0 9 0 92 R2 X 0 73 0 72 0 9 0 7 0 92 0 93 R3 1 R a2 R2 a3 R3 b R b2 R2 b3 R3 Y Collect terms of equal order Y 1 a b R a2 ab b2 R2 a3 a2 b ab2 b3 R3 X 2 6 003 Signals and Systems Lecture 3 Multiplying Polynomials Partial Fractions Tabular representation of polynomial multiplication 2 2 3 3 2 2 3 Use partial fractions to rewrite as a sum of simpler parts 3 1 aR a R a R 1 bR b R b R 1 aR a2 R2 a3 R3 September 17 2009 1 bR b2 R2 b3 R3 1 aR a2 R2 a3 R3 bR abR2 a2 bR3 a3 bR4 b2 R2 ab2 R3 a2 b2 R4 a3 b2 R5 b3 R3 ab3 R4 a2 b3 R5 a3 b3 R6 X Y R 1 6 R 0 63 Group same powers of R by following reverse diagonals Y 1 a b R a2 ab b2 R2 a3 a2 b ab2 b3 R3 X y n 1 0 1 2 3 4 5 6 7 8 1 4 5 3 5 Y 1 X 1 0 9R 1 0 7R 1 0 9R 1 0 7R 1 1 6R 0 63R2 n Second Order Systems Equivalent Forms Partial Fractions The sum of simpler parts suggests a parallel implementation Graphical representation of the sum of geometric sequences y1 n 0 9n for n 0 Y 4 5 3 5 X 1 0 9R 1 0 7R X Y1 0 9 Y Y2 n y2 n 0 7n for n 0 R 0 7 4 5 1 0 1 2 3 4 5 6 7 8 3 5 1 0 1 2 3 4 5 6 7 8 R n y n 4 5 0 9 n 3 5 0 7 2 for n 0 If x n n then y1 n 0 9n and y2 n 0 7n for n 0 Thus y n 4 5 0 9 n 3 5 0 7 n for n 0 1 0 1 2 3 4 5 6 7 8 n Partial Fractions Poles Partial fractions provides a remarkable equivalence X Y The key to simplifying a higher order system is identifying its poles Poles are the roots of the denominator of the system functional when R z1 R 1 6 Start with system functional Y 1 1 1 X 1 p0 R 1 p1 R 1 0 7R 1 0 9R 1 1 6R 0 63R2 z z R 0 63 X Y1 0 9 4 5 Y 1 and find roots of denominator z z2 Y 1 z2 2 1 6 0 63 X z 0 7 z 0 9 z 1 6z 0 63 2 1 z z z z z 0 7 z 0 9 Substitute R R Y2 0 7 p0 0 7 3 5 R 0 The poles are at 0 7 and 0 9 follows from thinking about system as polynomial factoring 3 1 p1 0 9 6 003 Signals and Systems Lecture 3 Check Yourself Population Growth Consider the system described by 1 1 1 y n y n 1 y n 2 x n 1 x n 2 4 8 2 How many of the following are true 1 2 …
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