1. 1 Linear Time-Invariant ( “LTI” ) Systems Montek Singh Thurs., Feb. 7, 2002 3:30-4:45 pm, SN115

2. 2 What we will learn  How to represent a circuit as an input-output system (“black box”)  What are LTI systems?  How is their behavior described?

3. 3 Why treat circuits as I/O systems? A system representation … is not bound to a particular input is not bound to a particular input allows us to distill the essence of an arbitrarily complex circuit into a concise description allows us to distill the essence of an arbitrarily complex circuit into a concise description  e.g., Thevenin and Norton equivalents can incorporate other (non-electrical) technologies can incorporate other (non-electrical) technologies  e.g., acoustic, optical, magnetic etc. input output input output f

4. 4 What are LTI systems? LTI systems are linear and time-invariant: Linearity: Linearity:  output for a sum of inputs = sum of individual outputs  i.e., Time-Invariance: Time-Invariance:  inherent system properties do not change with time  delaying the input by time  simply delays the output by   i.e.,

5. 5Examples LTI systems: Most physical systems when operated at small amplitudes: Most physical systems when operated at small amplitudes:  an LCR electrical network  a mechanical spring, a glass prism, a loudspeaker … Non-linear systems: Most physical systems when “stretched to the limit”: Most physical systems when “stretched to the limit”:  a blaring loudspeaker Some systems that are intentionally operated in that mode: Some systems that are intentionally operated in that mode:  diodes, transistors, logic gates, digital systems … Time-variant systems: Systems whose properties change with time: Systems whose properties change with time:  a resistor getting hotter  the human eye

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7. 7 An LTI system’s behavior System’s behavior = mapping from input to output How to represent? Describe the underlying physical phenomena Describe the underlying physical phenomena  goes back to circuit theory Enumerate all (interesting) input-output pairs Enumerate all (interesting) input-output pairs  unwieldy description Describe output for a select set of inputs Describe output for a select set of inputs  choose some special input  compute output behavior for that input  infer behavior for arbitrary inputs

8. 8 Choosing that special input … Unit impulse function:  ( t ) Unit impulse = a pulse of: infinitesimal duration infinitesimal duration infinite amplitude infinite amplitude unit area unit area Also known as: Dirac delta function tF(t)1 1 t F  (t)  1/  t  (t) 1

9. 9 Unit impulse: properties Examples: t 2  (t) 2 t  (t- a ) 1 at -½  (t+ 1 ) ½

10. 10 Unit impulse: used as a sampler Multiplying a signal by  (t-a) and integrating has the effect of sampling it at t = a. tx(t) t  (t- a ) 1 a t x(t)  (t- a ) 1 a Sampling Theorem:

11. 11 Reconstituting a signal from samples (1) Swap the roles of t and a : Sampling Theorem: x(t) can be regarded as an infinite sum of infinitesimal samples, i.e., sample x(a) summed over all a.

12. 12 Reconstituting a signal from samples (2) tx(t)  (t-a) a t x(t) a da 1/da t x(t) a x(a)  (t-a)da t x(t) a

13. 13 Unit impulse: system’s response Output of a system when input =  (t) is called the “unit impulse response” Denoted by h(t): Example: human eye th(t) latency persistence peak

14. 14 Generalization: Arbitrary input Given: unit impulse response h(t), i.e., Find: system response y(t) to an arbitrary input x(t) Method: express input x(t) as an infinite sum of weighted impulses express input x(t) as an infinite sum of weighted impulses compute response to each individual impulse compute response to each individual impulse weight and add up all the individual responses weight and add up all the individual responses

15. 15Convolution Definition: y(t) is the “convolution of” x(t) and h(t) if: Notation:Properties: 1. commutativity: 2. associativity: 3. distributivity: 4. scalability: 5. derivatives:

16. 16 Convolution: example

17. 17 http://www.jhu.edu/~signals/convolve/index.html Check it out!

18. 18 Homework: Due 2/19 1. The output of a particular system S is the time derivative of its input. a) Prove that system S is linear time-invariant (LTI). b) What is the unit impulse response of this system? 2. Prove Property 5. That is, prove that, for an arbitrary LTI system, for a given input waveform x(t), the time derivative of its output is identical to the output of that system when subjected to the time derivative of its input. In other words, differentiation on the input and output sides are equivalent.