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
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
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
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
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
Linearity Linear systems Output is linear transformation of input Homogeneous and Additive 26   T x ( t ) y ( t ) ECE101 DR. NEMA SALEM FALL 2017
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
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
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
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
Example: Check for Linearity ) ( ) ( 2

#### 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

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern