6 003 Signals and Systems Feedback Poles and Fundamental Modes February 9 2010 Last Time Multiple Representations of DT Systems Verbal descriptions preserve the rationale 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 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 Example The accumulator system has feedback X Y Delay By contrast the difference machine does not have feedback X 1 Delay Y Last Time Feedback Cyclic Signal Paths and Modes The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths X Y p0 x n n Delay y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Each cycle creates another sample in the output Last Time Feedback Cyclic Signal Paths and Modes The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths X Y p0 x n n Delay y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Each cycle creates another sample in the output Last Time Feedback Cyclic Signal Paths and Modes The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths X Y p0 x n n Delay y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Each cycle creates another sample in the output Last Time Feedback Cyclic Signal Paths and Modes The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths X Y p0 x n n Delay y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Each cycle creates another sample in the output Last Time Feedback Cyclic Signal Paths and Modes The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths X Y p0 x n n Delay y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Each cycle creates another sample in the output Last Time Feedback Cyclic Signal Paths and Modes The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths X Y p0 x n n Delay y n n 1 0 1 2 3 4 n 1 0 1 2 3 4 Each cycle creates another sample in the output The response will persist even though the input is transient Geometric Growth Poles These unit sample responses can be characterized by a single number the pole which is the base of the geometric sequence X Y p0 n p0 y n 0 y n Delay if n 0 otherwise y n y n n 1 0 1 2 3 4 p0 0 5 n 1 0 1 2 3 4 p0 1 n 1 0 1 2 3 4 p0 1 2 Check Yourself How many of the following unit sample responses can be represented by a single pole n n n n n Check Yourself How many of the following unit sample responses can be represented by a single pole 3 n n n n n Geometric Growth The value of p0 determines the rate of growth y n y n y n y n z 1 p0 1 1 p0 0 0 p0 1 p0 1 magnitude magnitude magnitude magnitude 0 1 diverges alternating sign converges alternating sign converges monotonically diverges monotonically Second Order Systems The unit sample responses of more complicated cyclic systems are more complicated X Y R 1 6 R 0 63 y n n 1 0 1 2 3 4 5 6 7 8 Not geometric This response grows then decays Factoring Second Order Systems Factor the operator expression to break the system into two simpler systems divide and conquer X Y R 1 6 R 0 63 Y X 1 6RY 0 63R2 Y 1 1 6R 0 63R2 Y X 1 0 7R 1 0 9R Y X Factoring Second Order Systems The factored form corresponds to a cascade of simpler systems 1 0 7R 1 0 9R Y X X Y2 0 7 R 0 9 1 0 7R Y2 X X Y1 0 9 Y R 1 0 9R Y1 X R 1 0 9R Y Y2 Y 0 7 R 1 0 7R Y Y1 The order doesn t matter if systems are initially at rest Factoring Second Order Systems The unit sample response of the cascaded system can be found by multiplying the polynomial representations of the subsystems 1 1 1 Y X 1 0 7R 1 0 9R 1 0 7R 1 0 9R z z z z 1 0 7R 0 72 R2 0 73 R3 1 0 9R 0 92 R2 0 93 R3 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 Multiplying Polynomial Graphical representation of polynomial multiplication Y 1 aR a2 R2 a3 R3 1 bR b2 R2 b3 R3 X 1 1 a X 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 Multiplying Polynomials Tabular representation of polynomial multiplication 1 aR a2 R2 a3 R3 1 bR b2 R2 b3 R3 1 aR 2 a R2 a3 R3 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 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 n 1 0 1 2 3 4 5 6 7 8 Partial Fractions Use partial fractions to rewrite as a sum of simpler parts X Y R 1 6 R 0 63 Y 1 1 4 5 3 5 2 X 1 0 9R 1 0 7R 1 0 9R 1 0 7R 1 1 6R 0 63R Second Order Systems Equivalent Forms The sum of simpler parts suggests a parallel implementation Y 4 5 3 5 X 1 0 9R 1 0 7R X Y1 0 9 R Y2 0 7 4 5 3 5 R 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 Y Partial Fractions Graphical representation of the sum of geometric sequences y1 n 0 9n for n 0 n 1 0 1 2 …
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