Discrete time signals systems ppt

discrete time systems and signal processing ppt and state space representation of discrete time systems ppt
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Prof.EvanBaros,United Kingdom,Teacher
Published Date:26-07-2017
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Multiple Representations of Discrete-Time Systems Discrete-Time (DT) systems can be represented in different ways to more easily address different types of issues. Verbal descriptions: preserve the rationale. “Next year, your account will contain p times your balance from this year plus the money that you added this year.” Difference equations: mathematically compact. yn +1 = xn+ pyn Block diagrams: illustrate signal flow paths. xn + Delay yn p Operator representations: analyze systems as polynomials. (1 − pR) Y = RX 2Multiple Representations of Continuous-Time Systems Similar representations for Continuous-Time (CT) systems. Verbal descriptions: preserve the rationale. “Your account will grow in proportion to your balance plus the rate at which you deposit.” Differential equations: mathematically compact. dy(t) = x(t)+ py(t) dt Block diagrams: illustrate signal flow paths. Z t x(t) + y(t) (·)dt −∞ p Operator representations: analyze systems as polynomials. (1 − pA)Y = AX 3Differential Equations Differential equations are mathematically precise and compact. r (t) 0 h (t) 1 r (t) 1 We can represent the tank system with a differential equation. dr (t) r (t) − r (t) 1 0 1 = dt τ You already know lots of methods to solve differential equations: • general methods (separation of variables; integrating factors) • homogeneous and particular solutions • inspection Today: new methods based on block diagrams and operators, which provide new ways to think about systems’ behaviors. 4Block Diagrams Block diagrams illustrate signal flow paths. DT: adders, scalers, and delays – represent systems described by linear difference equations with constant coefficents. xn + yn Delay p CT: adders, scalers, and integrators – represent systems described by a linear differential equations with constant coefficients. Z t x(t) + (·)dt y(t) −∞ p Delays in DT are replaced by integrators in CT. 5Operator Representation CT Block diagrams are concisely represented with the A operator. Applying A to a CT signal generates a new signal that is equal to the integral of the first signal at all points in time. Y = AX is equivalent to t y(t) = x(τ) dτ −∞ for all time t. 6Check Yourself + y˙(t) =x˙(t)+py(t) X A Y p p + y˙(t) =x(t)+py(t) X A Y + y˙(t) =px(t)+py(t) X Y p A Which block diagrams correspond to which equations? 1 1. 2. 3. 4. 5. none 7Check Yourself + y˙(t) =x˙(t)+py(t) X A Y p p + y˙(t) =x(t)+py(t) X A Y + y˙(t) =px(t)+py(t) X Y p A Which block diagrams correspond to which equations? 1 1. 2. 3. 4. 5. none 8Evaluating Operator Expressions As with R, A expressions can be manipulated as polynomials. Example: W + + X Y A A Z t w(t) = x(t) + x(τ)dτ −∞ Z t y(t) = w(t)+ w(τ )dτ −∞ Z Z Z Z t t t τ 2 y(t) = x(t)+ x(τ)dτ + x(τ)dτ + x(τ )dτ dτ 1 1 2 −∞ −∞ −∞ −∞ W = (1+ A) X 2 Y = (1+ A) W = (1+ A)(1 + A) X = (1+2A + A ) X 9Evaluating Operator Expressions Expressions in A can be manipulated using rules for polynomials. • Commutativity: A(1 −A)X = (1 −A)AX 2 • Distributivity: A(1 −A)X = (A−A )X � � � � • Associativity: (1 −A)A (2 −A)X = (1 −A) A(2 −A) X 10Check Yourself Determine k so that these systems are “equivalent.” 1 + + X A A Y −0.7 −0.9 + X A A Y k 1 k 2 1. 0.7 2. 0.9 3. 1.6 4. 0.63 5. none of these 11Check Yourself Write operator expressions for each system. W + + X A A Y −0.7 −0.9 2 W = A(X −0.7W ) (1+0.7A)W = AX (1+0.7A)(1+0.9A)Y = A X → → 2 2 Y = A(W −0.9Y ) (1+0.9A)Y = AW (1+1.6A+0.63A )Y = A X W + X A A Y k 1 k 2 2 2 W = A(X +k W +k Y ) Y = A X +kAY +kA Y 1 2 1 2 → 2 2 Y = AW (1−kA−kA )Y = A X 1 2 k = −1.6 1 12Check Yourself Determine k so that these systems are “equivalent.” 1 + + X A A Y −0.7 −0.9 + X A A Y k 1 k 2 1. 0.7 2. 0.9 3. 1.6 4. 0.63 5. none of these 13Elementary Building-Block Signals Elementary DT signal: δn. 1, if n = 0; δn = 0, otherwise δn 1 n 0 • simplest non-trivial signal (only one non-zero value) • shortest possible duration (most “transient”) • useful for constructing more complex signals What CT signal serves the same purpose? 14Elementary CT Building-Block Signal Consider the analogous CT signal: w(t) is non-zero only at t = 0. ⎧ 0 t 0 ⎨ w(t) = 1 t = 0 ⎩ 0 t 0 w(t) 1 t 0 Is this a good choice as a building-block signal? No Z t w(t) (·)dt 0 −∞ The integral of w(t) is zero 15Unit-Impulse Signal The unit-impulse signal acts as a pulse with unit area but zero width. p (t)  1 unit area 2 δ(t) = lim p (t)  →0 t  − p (t) p (t) p (t) 1/2 1/4 1/8 4 2 1 t t t 1 1 1 1 1 1 − − − 2 2 4 4 8 8 16Unit-Impulse Signal The unit-impulse function is represented by an arrow with the num­ ber 1, which represents its area or “weight.” δ(t) 1 t It has two seemingly contradictory properties: • it is nonzero only at t = 0, and • its definite integral (−∞, ∞) is one Both of these properties follow from thinking about δ(t) as a limit: p (t)  1 unit area 2 δ(t) = lim p (t)  →0 t  − 17Unit-Impulse and Unit-Step Signals The indefinite integral of the unit-impulse is the unit-step.  Z t 1; t ≥ 0 u(t) = δ(λ) dλ = 0; otherwise −∞ u(t) 1 t Equivalently δ(t) u(t) A 18Impulse Response of Acyclic CT System If the block diagram of a CT system has no feedback (i.e., no cycles), then the corresponding operator expression is “imperative.” + + X Y A A 2 Y = (1+ A)(1 + A) X = (1+2A + A ) X If x(t) = δ(t) then 2 y(t) = (1+2A + A ) δ(t) = δ(t)+2u(t)+ tu(t) 19CT Feedback Find the impulse response of this CT system with feedback. x(t) + y(t) A p Method 1: find differential equation and solve it. y˙(t) = x(t)+ py(t) Linear, first-order difference equation with constant coefficients. αt Try y(t) = Ce u(t). αt αt αt Then y˙(t) = αCe u(t)+ Ce δ(t) = αCe u(t)+ Cδ(t). αt αt Substituting, we find that αCe u(t)+ Cδ(t) = δ(t)+ pCe u(t). pt Therefore α = p and C = 1 → y(t) = e u(t). 20CT Feedback Find the impulse response of this CT system with feedback. x(t) + y(t) A p Method 2: use operators. Y = A (X + pY ) Y A = X 1 − pA Now expand in ascending series in A: Y 2 2 3 3 = A(1 + pA + p A + p A + ···) X If x(t) = δ(t) then 2 2 3 3 y(t) = A(1 + pA + p A + p A + ···) δ(t) 1 1 2 2 3 3 pt = (1+ pt + p t + p t + ···) u(t) = e u(t) . 2 6 21

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