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Properties of LTI Systems

Properties of LTI Systems. Commutative Property x[n]*h[n]=h[n]*x[n] Distributive Property x[n]*(h 1 [n]+ h 2 [n])= x[n]*h 1 [n]+x[n]* h 2 [n] Associative Property x[n]*(h 1 [n]* h 2 [n])= (x[n]*h 1 [n])* h 2 [n] Memoryless If h[n]=0 for n not equal 0. I.e. h[n]=K d [n].

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Properties of LTI Systems

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  1. Properties of LTI Systems • Commutative Property • x[n]*h[n]=h[n]*x[n] • Distributive Property • x[n]*(h1[n]+ h2[n])= x[n]*h1[n]+x[n]* h2[n] • Associative Property • x[n]*(h1[n]* h2[n])= (x[n]*h1[n])* h2[n] • Memoryless • If h[n]=0 for n not equal 0. I.e. h[n]=Kd[n].

  2. Properties of LTI Systems • Invertibility w(t)=x(t) h1(t) x(t) h(t) y(t) Identity System d(t) x(t)

  3. Properties of LTI Systems • Causality • h[n]=0 for n<0 • or h(t)=0 for t<0. • Stability

  4. Unit Step Response of LTI System u[n] s[n] h[n] The step response of a discrete-time LTI system is the convolution of the unit step with the impulse response:- s[n]=u[n]*h[n]. Via commutative property of convolution, s[n]=h[n]*u[n]. That means s[n] is the response to the input h[n] of a discrete-time LTI system with unit impulse response u[n]. h[n] s[n] u[n]

  5. Using the convolution sum:-

  6. Unit Step Response of Continuous-time LTI System Similarly, unit step response is the running integral of its impulse response. The unit impulse response is the first derivative of the unit step response:-

  7. Causal LTI Systems Described By Differential & Difference Equations

  8. Example 2.14

  9. Example 2.14 with impulse input(Problem 2.56 (a) Pg 158-159 OWN)

  10. General Higher N-order DE

  11. Linear Constant-Coefficient Difference equations

  12. Linear Constant-Coefficient Difference equations

  13. Recursive case when N > or = 1.

  14. Block Diagram Representations of First- Order Systems. • Provides a pictorial representation which can add to our understanding of the behavior and properties of these systems. • Simulation or implementation of the systems. • Basis for analog computer simulation of systems described by DE. • Digital simulation & Digital Hardware implementations

  15. First-Order Recursive Discrete-time System. y[n]+ay[n-1]=bx[n] y[n]=-ay[n-1]+bx[n] y[n] + b x[n] D -a y[n-1]

  16. First-Order Continuous-time System Described By Differential Equation y(t) b/a + x(t) Difficult to implement, sensitive to errors and noise. D -1/a

  17. First-Order Continuous-time System Described By Differential Equation Alternative Block Diagram. y(t) b + x(t) -a

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