6.003: Signals and Systems Lecture 2 September 15, 200916.003: Signals and SystemsDiscrete-Time SystemsSeptember 15, 2009Discrete-Time SystemsWe 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 GrowthMultiple Representations of Discrete-Time SystemsSystems can be represented in different ways to more easily addressdifferent types of issues.Verbal description: ‘To reduce the number of bits needed to storea sequence of large numbers that are nearly equal, record the firstnumber, and then record successive differences.’Difference equation:y[n] = x[n] − x[n − 1]Block diagram:−1Delay+x[n] y[n]We will exploit particular strengths of each of these representations.Difference EquationsDifference equations are mathematically precise and compact.Example:y[n] = x[n] − x[n − 1]Let x[n] equal the “unit sample” signal δ[n],δ[n] =1, if n = 0;0, otherwise.−1 0 1 2 34nx[n] = δ[n]We will use the unit sample as a “primitive” (building-block signal)to construct more complex signals.Step-By-Step SolutionsDifference equations are convenient for step-by-step analysis.−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] − x[n − 1]y[−1] = x[−1] − x[−2]= 0 − 0 = 0y[0] = x[0] − x[−1]= 1 − 0 = 1y[1] = x[1] − x[0]= 0 − 1 = −1y[2] = x[2] − x[1]= 0 − 0 = 0y[3] = x[3] − x[2]= 0 − 0 = 0. . .Step-By-Step SolutionsBlock diagrams are also useful for step-by-step analysis.Represent y[n] = x[n] − x[n − 1] with a block diagram: start “at rest”−1Delay+x[n] y[n]0−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]6.003: Signals and Systems Lecture 2 September 15, 20092Check YourselfDT systems can be described by difference equations and/orblock diagrams.Difference equation:y[n] = x[n] − x[n − 1]Block diagram:−1Delay+x[n] y[n]In what ways are these representations different?From Samples to SignalsLumping all of the (possibly infinite) samples into a single object —the signal — simplifies its manipulation.This lumping is an abstraction that is analogous to• representing coordinates in three-space as points• representing lists of numbers as vectors in linear algebra• creating an object in PythonFrom Samples to SignalsOperators manipulate signals rather than individual samples.−1Delay+x[n] y[n]Nodes represent whole signals (e.g., X and Y ).The boxes operate on those signals:• Delay = shift whole signal to right 1 time step• Add = sum two signals• −1: multiply by −1Signals are the primitives.Operators are the means of combination.Operator NotationSymbols can now compactly represent diagrams.Let R represent the right-shift operator:Y = R{X} ≡ RXwhere X represents the whole input signal (x[n] for all n) and Yrepresents the whole output signal (y[n] for all n)Representing the difference machine−1Delay+x[n] y[n]with R leads to the equivalent representationY = X − RX = (1 − R) XOperator Notation: Check YourselfLet Y = RX. Which of the following is/are true:1. y[n] = x[n] for all n2. y[n + 1] = x[n] for all n3. y[n] = x[n + 1] for all n4. y[n − 1] = x[n] for all n5. none of the aboveOperator Representation of a Cascaded SystemSystem operations have simple operator representations.Cascade systems → multiply operator expressions.−1Delay+−1Delay+XY1Y2Using operator notation:Y1= (1 − R) XY2= (1 − R) Y1Substituting for Y1:Y2= (1 − R)(1 − R) X6.003: Signals and Systems Lecture 2 September 15, 20093Operator AlgebraOperator expressions can be manipulated as polynomials.−1Delay+−1Delay+XY1Y2Using 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)2X= (1 − 2R + R2) XOperator ApproachApplies your existing expertise with polynomials to understand blockdiagrams, and thereby understand systems.Operator AlgebraOperator notation facilitates seeing relations among systems.“Equivalent” block diagrams (assuming both initially at rest):−1Delay+−1Delay+XY1Y2DelayDelay−2+X YEquivalent operator expressions:(1 − R)(1 − R) = 1 − 2R + R2The operator equivalence is much easier to see.Check YourselfOperator expressions for these “equivalent” systems(if started “at rest”) obey what mathematical property?Delay−1+DelayX YDelayDelay Delay−1+X Y1. commutate 2. associative3. distributive 4. transitive5. none of the aboveCheck YourselfHow many of the following systems are equivalent toY = (4R2+ 4R + 1) X ?Delay2+Delay2+X YDelay+Delay4+X YDelay4+Delay+X YOperator Algebra: Explicit and Implicit RulesRecipes versus constraints.Recipe: subtract a right-shifted version of the input signal from acopy of the input signal.−1Delay+X YY = (1 − R) XConstraint: the difference between Y and RY is X.Delay+X YY = RY + X(1 − R) Y = XBut how does one solve such a constraint?6.003: Signals and Systems Lecture 2 September 15, 20094Example: AccumulatorTry step-by-step analysis: it always works. Start “at rest.”+Delayx[n] y[n]−1 0 1 2 34nx[n] = δ[n]−1 0 1 2 34ny[n]Find y[n] given x[n] = δ[n]: y[n] = x[n] + y[n − 1]y[0] = x[0] + y[−1]= 1 + 0 = 1y[1] = x[1] + y[0]= 0 + 1 = 1y[2] = x[2] + y[1]= 0 + 1 = 1. . .Persistent response to a transient input!Example: AccumulatorThe response of the accumulator system could also be generated bya system with infinitely many paths from input to output, each withone unit of delay more than the previous.DelayDelay DelayDelay Delay Delay+... ...X YY = (1 + R + R2+ R3+ · · ·) XExample: AccumulatorThese systems are equivalent in the sense that if each is initially atrest, they will produce identical outputs from the same input.(1 − R) Y1= X1⇔ ? Y2= (1 + R + R2+ R3+ · · ·) X2Proof: 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= Y1It follows that Y2= Y1.It also follows that (1 − R) and (1 + R + R2+ R3+ · · ·) are reciprocals.Example: AccumulatorThe reciprocal of 1−R can also be evaluated using synthetic division.1+R+R2+R3+ · · ·1 − R11−RRR−R2R2R2−R3R3R3−R4· · ·Therefore11 − R= 1 + R + R2+ R3+ R4+ · · ·FeedbackSystems with signals that depend
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