Leo Lam © Signals and Systems EE235

Leo Lam © Pet Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's mostly been working on PET projects.

Leo Lam © Today’s menu Homework due now! Tomorrow: The Hossein Lecture Friday: Lecture will be online for download System properties examples –Stability –Time invariance –Linearity

System properties Leo Lam © Time-invariance: A System is Time-Invariant if it meets this criterion “System Response is the same no matter when you run the system.”

Time invariance Leo Lam © The system behaves the same no matter when you use it Input is delayed by t 0 seconds, output is the same but delayed t 0 seconds If then System T Delay t 0 System T Delay t 0 x(t) x(t-t 0 ) y(t) y(t-t 0 ) T[x(t-t 0 )] System 1 st Delay 1 st =

Time invariance example Leo Lam © T{x(t)}=2x(t) x(t) y(t)= 2x(t) y(t-t 0 ) T Delay x(t-t 0 ) 2x(t-t 0 ) Delay T Identical  time invariant!

Time invariance test Leo Lam © Test steps: 1.Find y(t) 2.Find y(t-t 0 ) 3.Find T{x(t-t 0 )} 4.Compare! IIf y(t-t 0 ) = T{x(t-t 0 )} Time invariant!

Time invariance example Leo Lam © T(x(t)) = x 2 (t) 1.y(t) = x 2 (t) 2.y(t-t 0 ) =x 2 (t-t 0 ) 3.T(x(t-t 0 )) = x 2 (t-t 0 ) 4.y(t-t 0 ) = T(x(t-t 0 )) Time invariant! KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ).

Time invariance example Leo Lam © Your turn! T{(x(t)} = t x(t) 1.y(t) = t*x(t) 2.y(t-t 0 ) =(t-t 0 ) x(t-t 0 ) 3.T(x(t-t 0 )) = t x(t-t 0 ) 4.y(t-t 0 )) != T(x(t-t 0 )) Not time invariant! KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ).

Time invariance example Leo Lam © Still you… T(x(t)) = 3x(t - 5) 1.y(t) = 3x(t-5) 2.y(t – t 0 ) = 3x(t-t 0 -5) 3.T(x(t – t 0 )) = 3x(t-t 0 -5) 4.y(t-t 0 )) = T(x(t-t 0 )) Time invariant! KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ).

Time invariance example Leo Lam © Still you… T(x(t)) = x(5t) 1.y(t) = x(5t) 2.y(t – 3) = x(5(t-3)) = x(5t – 15) 3.T(x(t-3)) = x(5t- 3) 4.Oops… Not time invariant! Does it make sense? KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ). Shift then scale

Time invariance example Leo Lam © Graphically: T(x(t)) = x(5t) 1.y(t) = x(5t) 2.y(t – 3) = x(5(t-3)) = x(5t – 15) 3.T(x(t-3)) = x(5t- 3) t 0 system input x(t) 5 t 0 system output y(t) = x(5t) 1 t shifted system output y(t-3) = x(5(t-3)) t shifted system input x(t-3) t system output for shifted system input T(x(t-3)) = x(5t-3)

Time invariance example Leo Lam © Integral 1.First: 2.Second: 3.Third: 4.Lastly: Time invariant! KEY: In step 2 you replace t by t-t 0. In step 3 you replace x(t) by x(t-t 0 ).