1 / 29

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].

williampalmer
Download Presentation

Properties of LTI Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related