Signals and Systems EE235. Leo Lam

Transcription

1 Signals and Systems EE35 Leo Lam

2 oday s menu Good weekend? System properties ime invariance (cont ) Linearity Superposition!

3 System properties ime-invariance: A System is ime-invariant if it meets this criterion System Response is the same no matter when you run the system.

4 ime invariance he system behaves the same no matter when you use it If { x( t)} y( t) System st { x( t t )} y( t t ) then 0 0 Input is delayed Systemby t 0 seconds, Delayoutput y(t-t is the 0 ) same but delayed t 0 seconds x(t) Delay st Delay t 0 y(t) x(t-t 0 ) t 0 System = [x(t-t 0 )]

5 ime invariance example Still you (x(t)) = x(5t). y(t) = x(5t). y(t 3) = x(5(t-3)) = x(5t 5) 3. (x(t-3)) = x(5t- 3) 4. Oops Not time invariant! Does it make sense? KEY: In step you replace the t by t-t 0. In step 3 you replace the x(t) by x(t-t 0 ). Shift then scale

6 ime invariance example Graphically: (x(t)) = x(5t). y(t) = x(5t). y(t 3) = x(5(t-3)) = x(5t 5) 3. (x(t-3)) = x(5t- 3) system input x(t) 0 system output y(t) = x(5t) shifted system output y(t-3) = x(5(t-3)) t 0 5 shifted system input x(t-3) t system output for shifted system input (x(t-3)) = x(5t-3) t t t

7 ime invariance example Integral [ x( t)] x( ) d. First:. Second: 3. hird: 4. Lastly: t ime invariant! t y( t) x( ) d tt y( t t0) x( ) d 0 t KEY: In step you replace the t by t-t 0. In step 3 you replace the x(t) by x(t-t 0 ). [ x( t t )] x( t ) d x( v) dv v t tt 0 0 tt tt x( v) dv x( ) d 0

8 System properties Linearity: A System is Linear if it meets the following two criteria: If { x ( t)} y ( t) and { x ( t)} y ( t) hen { x ( t) x ( t)} { x ( t)} { x ( t)} If { x( t)} y( t) hen { ax( t)} a{ x( t)} System Response to a linear combination of inputs is the linear combination of the outputs. ogether superposition Additivity Scaling { ax ( t) bx ( t)} ay ( t) by ( t)

9 Linearity Order of addition and multiplication doesn t matter. System st System y ( t), y ( t) Linear combination ay ( t) by ( t) = x ( t), x ( t) Linear combination ax ( t) bx ( t) System { ax ( t) bx ( t)} Combo st

10 Linearity Positive proof Prove both scaling & additivity separately Prove them together with combined formula Negative proof Show either scaling OR additivity fail (mathematically, or with a counter example) Show combined formula doesn t hold

11 Linearity Proof Combo Proof Step : find y i (t) Step : find y_combo System st System Step 3: find {x_combo} Step 4: If y_combo = {x_combo} Linear y ( t), y ( t) Linear combination ay ( t) by ( t) x ( t), x ( t) Linear combination ax ( t) bx ( t) System { ax ( t) bx ( t)} Combo st

12 Linearity Example Is linear? x(t) y(t)=cx(t) y ( t) cx ( t); y ( t) cx ( t) ay ( t) by ( t) acx ( t) bcx ( t) c( ax ( t) bx ( t)) { ax ( t) bx ( t)} c( ax ( t) bx ( t)) Equal Linear

13 Linearity Example Is linear? x(t) y(t)=(x(t)) y( t) ( x( t)) ay( t) a( x( t)) { ax( t)} ( ax( t)) a ( x( t)) Not equal non-linear

14 Linearity Example Is linear? x(t) y(t)=x(t)+5 y( t) x( t) 5 ay( t) a( x( t) 5) ax( t) 5a { ax( t)} ax( t) 5 Not equal non-linear

15 Linearity Example Is linear? y ( t) x ( t ) d y( t) x( t ) d y ( t) x ( t ) d ay ( t) by ( t) a x ( t ) d b x ( t ) d ( ax ( t ) bx ( t )) d { ax ( t) bx ( t)} ( ax ( t ) bx ( t )) d =

16 Linearity unique case How about scaling with 0? y( t) { x( t)} ay( t) a{ x( t)} 0 if a 0 { ax( t)} ay( t) 0 if linear If {x(t)} is a linear system, then zero input must give a zero output A great negative test

17 Linearity Rules of thumbs multiplying x(t) by another x() y(t)=g[x(t)] where g() is nonlinear piecewise definition of y(t) in terms of values of x, e.g. x( t) x( t) 0 y( t) x( t) x( t) x( t) 0 (although sometimes ok)

18 Superposition Superposition is If x ( t) y ( t) k k x( t) ak xk ( t) y( t) ak yk ( t) k Weighted sum of inputs weighted sum of outputs Divide & conquer k

19 Superposition example Graphically x (t) y (t) x (t) y (t) 3 x ( t) x ( t) y ( t) y ( t) -? y (t) -y (t) 9

20 Superposition example Slightly aside (same system) x (t) y (t) x (t) y (t) 3 Is it time-invariant? No idea: not enough information Single input-output pair cannot test positively 0

21 Superposition example Unique case can be used negatively x (t) x (t) y (t) NO ime Invariant: Shift by shift by y (t) x (t)=u(t) S y (t)=tu(t) NO Stable: Bounded input gives unbounded output

22 Summary: System properties Causal: output does not depend on future input times Invertible: can uniquely find system input for any output Stable: bounded input gives bounded output ime-invariant: ime-shifted input gives a time-shifted output Linear: response to linear combo of inputs is the linear combo of corresponding outputs