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Signals and SystemsSpring 2004

Lecture #2(2/5/04)

“Figures and images used in these lecture notes by permission,copyright 1997 by Alan V. Oppenheim and Alan S. Willsky”

1) Several examples of systems

2) System properties and examples

a) Causality

b) Linearity

c) Time invariance

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SYSTEM EXAMPLESIn 6.003, systems are described from input/output perspective,

that is, input to the system x causes the output y

x(t) y(t)CT System DT Systemx[n] y[n]

Ex. #1 RLC circuit –– an electrical system

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i(t) = C dv(t)dt

capacitance1 2 3

+v(t)R

resistance{

+1L

v(τ )dτ−∞

t∫inductance

1 2 4 4 3 4 4 .

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Ex. #2 A shock absorber – a mechanical system

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f (t) = M dv(t)dt

inertial force1 2 4 3 4

+ Bv(t)friction1 2 3 + K v(τ )dτ

−∞

t∫spring force

1 2 4 3 4 .

Force Balance:

This equation looks quite familiar, we just saw it earlier.

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Observation: different systems could bedescribed by the same input/output relations ––they are the same as far as 6.003 is concerned

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⇔

Largely for this reason, 6.003 is the most general among allthe EECS subjects, and perhaps the most important.

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Example #3 –– Human vocal system

Med. school description 6.003 version

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Example #4: A robot car – a close-loop system

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Ex. #5 A thermal system

Cooling Fin in Steady State

Coolant

Coolant

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Ex. #5 (Continued)

Observations

• Independent variable can be something other than time,such as space.

• Such systems may, more naturally, have boundaryconditions, rather than “initial” conditions.

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Ex. #6 Financial system

Observation: Even if the independent variable is time, there areinteresting and important systems which have boundaryconditions.

Fluctuations in the price of zero-coupon bondst = 0 Time of purchase at price y0

t = T Time of maturity at value yT

y(t) = Values of bond at time tx(t) = Influence of external factors on fluctuations in bond price

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Observations1) A very rich class of systems (but by no means all systems of

interest to us) are described by differential and differenceequations.

2) Such an equation, by itself, does not completely describe theinput-output behavior of a system: we need auxiliaryconditions (initial conditions, boundary conditions).

3) In some cases the system of interest has time as the naturalindependent variable and is causal. However, that is notalways the case.

4) Very different physical systems may have very similarmathematical descriptions.

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SYSTEM PROPERTIES(Causality, Linearity, Time-invariance, etc.)

Why bother with such general properties?

• Important practical/physical implications. We can makemany important predictions of the system behaviorswithout having to do any mathematical derivations.

• They allow us to develop powerful tools(transformations, more on this later) for analysis anddesign.

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CAUSALITY

• A system is causal if the output does not anticipate futurevalues of the input, i.e., if the output at any time dependsonly on values of the input up to that time.

• All real-time physical systems are causal, because time onlymoves forward. Effect occurs after cause. (Imagine if youown a noncausal system whose output depends ontomorrow’s stock price.)

• Causality does not apply to spatially varying signals. (Wecan move both left and right, up and down.)

• Causality does not apply to systems processing recordedsignals, e.g. taped sports games vs. live broadcast.

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CAUSALITY (continued)

If x1(t) → y1(t) x2(t) → y2(t)

and x1(t) = x2(t) for all t ≤ to

Then y1(t) = y2(t) for all t ≤ to

• More explicit mathematical definition of causality later.

• Mathematically (in CT): A system x(t) → y(t) is causal

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CAUSAL OR NONCAUSAL

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LINEAR AND NONLINEARSYSTEMS

• Many systems are nonlinear.Ex. Economic system.Input: fiscal and monetary policies, labor, resources, etc.→ Output: GDP, inflation, etc.System behavior is very unpredictable because it is highlynonlinear.

• In 6.003, we will deal with only linear systems, which aregood approximations of nonlinear systems in certain ranges.Ex. Small-signal conductance of a nonlinear diode.

• Linear systems can be analyzed accurately.

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LINEARITY

A (CT) system is linear if it has the superposition property:

If x1(t) → y1(t) and x2(t) → y2(t)

then ax1(t) + bx2(t) → ay1(t) + by2(t)

• For linear systems, zero input → zero output

"Proof" 0 = 0 ⋅ x[n]→ 0 ⋅ y[n] = 0

Question: Is the system y = A + x linear?

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PROPERTIES OF LINEAR SYSTEMS

• Superposition

If xk[n] → yk[n]

Then akk∑ xk [n] → ak

k∑ yk[n]

This property seems to be almost trivial now, but it isone of the most important ones used in 6.003.

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Properties of Linear Systems (Continued)

• A linear system is causal if and only if it satisfies the conditions of initial rest: (both necessary and sufficient)

“Proof”

x(t) = 0 for t ≤ to → y(t) = 0 for t ≤ to (*).

a) Suppose system is causal. Show that (*) holds.

b) Suppose (*) holds. Show that the system is causal.

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TIME-INVARIANCE (TI)

Informally, a system is time-invariant (TI) if its behavior does notdepend on the choice of t = 0. Then two identical experimentswill yield the same results, regardless the starting time.

• Mathematically (in DT): A system x[n] → y[n] is TI if forany input x[n] and any time shift n0,

If x[n] → y[n]

then x[n - n0] → y[n - n0] .

• Similarly for CT time-invariant system,

If x(t) → y(t)

then x(t - to) → y(t - to) .

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TIME-INVARIANT OR TIME-VARYING ?

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NOW WE CAN DEDUCE SOMETHING!

Fact: If the input to a TI system is periodic, then the outputis also periodic with the same period.

“Proof”: Suppose x(t + T) = x(t)and x(t) → y(t)

Then by TI x(t + T) → y(t + T)

↑ ↑

But these arethe same input!

So these must bethe same output,i.e., y(t) = y(t+T)

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LINEAR TIME-INVARIANT (LTI) SYSTEMS

• Focus of most of this course

– Practical importance (Eg. #1-2 earlier this lecture are both LTI systems.)

– The powerful analysis tools (transformation) associated with LTI systems

• A basic fact: If we know the response of an LTI systemto some inputs, we actually know the response to manyinputs

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Example: DT LTI System