Time Invariant System System is TI if a time shift in the input signal results

Time invariant system system is ti if a time shift in

This preview shows page 21 - 32 out of 37 pages.

Time-Invariant System System is TI if a time shift in the input signal results in an identical time shift in the output signal. X[n] ---------------- y[n] X[n-n 0 ] ------------ y[n-n 0 ] X(t) ----------------- y(t) X(t-t 0 ) -------------- y(t-t 0 ) 21 x ( t ) y ( t ) x ( t t 0 ) y shifted ( t ) ECE101 DR. NEMA SALEM FALL 2017
Image of page 21
Example: is this system TI? ) ( sin ) ( t x t y 22 TI is system t t y t y t t x t t y t t x t x t y t x t y let ) ( ) ( ) ( sin ) ( ) ( sin ) ( sin ) ( ) ( sin ) ( : 0 1 2 0 1 0 1 0 1 2 2 1 1 Solution ECE101 DR. NEMA SALEM FALL 2017
Image of page 22
Example: is this system TI? TV n n x n n n n y n n nx n nx n y n nx n y ) 2 ( ) 1 ( ) 2 ( ] [ ] [ ) 1 ( ] [ ] [ ] [ ] [ ] [ 0 1 0 0 1 0 1 2 2 1 1 23 ] [ ] [ n nx n y Solution ECE101 DR. NEMA SALEM FALL 2017
Image of page 23
Example: is this system TI? TV t t x t t y t t x t x t y t x t y ) 2 ( ) 1 ( ) 2 ( )) ( 2 ( ) ( ) 1 ( ) 2 ( ) 2 ( ) ( ) 2 ( ) ( 0 1 0 1 0 1 2 2 1 1 24 ) 2 ( ) ( t x t y Solution ECE101 DR. NEMA SALEM FALL 2017
Image of page 24
ECE101 DR. NEMA SALEM FALL 2017 25 ) 2 ( ) ( t x t y Is not time-invariant. Shift t by 2 shift Shift x(t) Y(t)=scale x(t) Y(t)=scale x(t-2) Shift y(t) by 2
Image of page 25
Linearity Linear systems Output is linear transformation of input Homogeneous and Additive 26   T x ( t ) y ( t ) ECE101 DR. NEMA SALEM FALL 2017
Image of page 26
Homogeneity If we scale the input by a scalar constant K and the output is scaled by the same amount, K, then the system is homogeneous. 27 k x(t) k y(t) ECE101 DR. NEMA SALEM FALL 2017
Image of page 27
Additive If we add two input signals, and the output is the sum of their respective outputs, then the system is linear. 28 x 1 (t) + x 2 (t) y 1 (t) + y 2 (t) ECE101 DR. NEMA SALEM FALL 2017
Image of page 28
Linearity The two properties can be combined in one equation: ] [ ] [ ] [ ] [ 2 1 2 1 n y b n y a n x b n x a 29 ) ( ) ( ) ( ) ( 2 1 2 1 t y b t y a t x b t x a ECE101 DR. NEMA SALEM FALL 2017
Image of page 29
Example: Check for Linearity linear is system t by t ay t btx t atx t y t bx t ax t t y t bx t ax t x t x t t y t x t x t t y t x ) ( ) ( ) ( ) ( ) ( )) ( ) ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 3 2 1 3 2 1 3 2 2 2 1 1 1 30 ) ( ) ( t tx t y Solution ECE101 DR. NEMA SALEM FALL 2017
Image of page 30
Example: Check for Linearity ) ( ) ( 2
Image of page 31
Image of page 32

You've reached the end of your free preview.

Want to read all 37 pages?

  • Fall '17
  • -
  • LTI system theory, Memoryless Systems, NEMA SALEM FALL, DR. NEMA SALEM

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes