Download presentation

Presentation is loading. Please wait.

Published byDrake Daunt Modified over 3 years ago

1
Leo Lam © 2010-2012 Signals and Systems EE235

2
Leo Lam © 2010-2012 Pet Q: Has the biomedical imaging engineer done anything useful lately? A: No, he's mostly been working on PET projects.

3
Leo Lam © 2010-2012 Todays menu System properties examples –Invertibility –Stability –Time invariance –Linearity

4
Invertibility test Positive test: find the inverse For some systems, you need tools that well learn later in the quarter… Negative test: find an output that could be generated by two different inputs (note that these two different inputs might only differ at only one time value) Each input signal results in a unique output signal, and vice versa Invertible Leo Lam © 2010-2011

5
Invertibility Example Leo Lam © 2010-2011 1)y(t) = 4x(t) 2)y(t) = x(t –3) 3)y(t) = x 2 (t) 4)y(t) = x(3t) 5)y(t) = (t + 5)x(t) 6)y(t) = cos(x(t)) invertible: T i {y(t)}=y(t)/4 invertible: T i {y(t)}=y(t/3) invertible: T i {y(t)}=y(t+3) NOT invertible: dont know sign of x(t) NOT invertible: cant find x(-5) NOT invertible: x=0,2 π,4 π,… all give cos(x)=1

6
Stability test For positive proof: show analytically that –a bounded input signal gives a bounded output signal (BIBO stability) For negative proof: –Find one counter example, a bounded input signal that gives an unbounded output signal –Some good things to try: 1, u(t), cos(t), 0 Leo Lam © 2010-2012

7
Stability test Is it stable? Leo Lam © 2010-2012 Bounded input results in a bounded output STABLE!

8
Stability test How about this? Leo Lam © 2010-2012 Stable Let for all t

9
Stability test How about this, your turn? Leo Lam © 2010-2012 Not BIBO stable Counter example: x(t)=u(t) y(t)=5tu(t)=5r(t) Input u(t) is bounded. Output y(t) is a ramp, which is unbounded.

10
Stability test How about this, your turn? Leo Lam © 2010-2012 Stable NOT Stable Stable

11
System properties Leo Lam © 2010-2012 Time-invariance: A System is Time-Invariant if it meets this criterion System Response is the same no matter when you run the system.

12
Time invariance Leo Lam © 2010-2012 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 =

13
Time invariance example Leo Lam © 2010-2012 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!

14
Time invariance test Leo Lam © 2010-2012 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!

15
Time invariance example Leo Lam © 2010-2012 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 ).

16
Time invariance example Leo Lam © 2010-2012 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 ).

17
Time invariance example Leo Lam © 2010-2012 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 ).

18
Time invariance example Leo Lam © 2010-2012 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

19
Time invariance example Leo Lam © 2010-2012 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 0 3 4 shifted system output y(t-3) = x(5(t-3)) t 0 3 8 shifted system input x(t-3) 0.6 1.6 t system output for shifted system input T(x(t-3)) = x(5t-3)

20
Time invariance example Leo Lam © 2010-2012 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 ).

21
System properties Leo Lam © 2010-2011 Linearity: A System is Linear if it meets the following two criteria: Together…superposition Ifand Then If Then System Response to a linear combination of inputs is the linear combination of the outputs. Additivity Scaling

22
Linearity Leo Lam © 2010-2011 Order of addition and multiplication doesnt matter. = System T System T Linear combination System 1 st Combo 1 st Linear combination

23
Linearity Leo Lam © 2010-2011 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 doesnt hold

24
Linearity Proof Leo Lam © 2010-2011 Combo Proof Step 1: find y i (t) Step 2: find y_combo Step 3: find T{x_combo} Step 4: If y_combo = T{x_combo} Linear System T System T Linear combination System 1 st Combo 1 st Linear combination

25
Linearity Example Leo Lam © 2010-2011 Is T linear? T x(t)y(t)=cx(t) Equal Linear

26
Linearity Example Leo Lam © 2010-2011 Is T linear? Not equal non-linear T x(t)y(t)=(x(t)) 2

27
Linearity Example Leo Lam © 2010-2011 Is T linear? Not equal non-linear T x(t)y(t)=x(t)+5

28
Linearity Example Leo Lam © 2010-2011 Is T linear? =

29
Linearity unique case Leo Lam © 2010-2011 How about scaling with 0? If T{x(t)} is a linear system, then zero input must give a zero output A great negative test

30
Spotting non-linearity Leo Lam © 2010-2011 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. (although sometimes ok) NOT Formal Proofs!

Similar presentations

OK

Leo Lam © 2010-2013 Signals and Systems EE235. Transformers Leo Lam © 2010-2013 2.

Leo Lam © 2010-2013 Signals and Systems EE235. Transformers Leo Lam © 2010-2013 2.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on active listening skills Ppt on retail marketing strategy Ppt on earthquake in japan Ppt on business environment nature concept and significance definition Ppt on chromosomes and genes middle school Ppt on network traffic management Sources of energy for kids ppt on batteries Moral values for kids ppt on batteries Ppt on mass production of food Ppt on isobars and isotopes of elements