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Lecture 11 Linear Systems
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Past History Matters the purpose of this lecture is to give you tools that provide a way of relating what happened in the past to what’s happening today
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Example: no past history needed Flame with time-varying heat h(t) Thermometer measuring temperature (t) Flame instantaneously heats the thermometer Thermometer retains no heat (t) h(t)
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Example: past history needed Flame with time-varying heat h(t) Thermometer measuring temperature (t) Heats takes time to seep through plate Plate retains heat (t=t’) history of h(t) for time t<t’ Steel plate
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Linear System (t) history of h(t) for all times, past and future linear in the sense that doubling h(t) doubles (t) if h 1 (t) causes 1 (t) then 2h 1 (t) causes 2 1 (t) and also in the sense of additivity if h 1 (t) causes 1 (t) and if h 2 (t) causes 2 (t) then h 1 (t)+h 2 (t) causes 1 (t)+ 2 (t)
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Special Case: Causal Linear System (t) history of h(t=t’) for all past times, t’<t When the variable “t” actually represents “time”, then most linear systems are causal But when the independent variable represents something else (e.g. spatial position) then the linear system probably obeys a “non-causal” rule, (x) h(x=x’) for all positions x’ both to the left & right of x
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Special Case: Time-Shift Invariant Linear System (t) history of h(t’), but only time difference (t-t’) matter The temperature depends only on the time elapsed since the flame was turned on, and not on the whether I performed it on Monday or Wednesday. Same shape (Might not be true if combustion depended on barometric pressure that changed from day-today) (t)
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How to write a Linear System (t) history of h(t’) for all times, past and future (t 1 ) = … + g 1,-2 h(t -2 ) + g 1,-1 h(t -1 ) + g 1,0 h(t 0 ) + g 1,1 h(t 1 ) + g 1,2 h(t 2 ) + … (t 2 ) = … + g 2,-2 h(t -2 ) + g 2,-1 h(t -1 ) + g 2,0 h(t 0 ) + g 2,1 h(t 1 ) + g 2,2 h(t 2 ) + … Doesn’t quite do it, since time is a continuous variable …
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How to write a Linear System (t) history of h(t’) for all times, past and future (t) = - + g(t,t’) h(t’) dt’ (t 1 ) = … + g 1,-2 h(t -2 ) + g 1,-1 h(t -1 ) + g 1,0 h(t 0 ) + g 1,1 h(t 1 ) + g 1,2 h(t 2 ) + …
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Special cases Causal (t) = - t g(t,t’) h(t’) dt’ Time-shift invariant (t) = - + g(t-t’) h(t’) dt’ Causal and Time-shift invariant (t) = - t g(t-t’) h(t’) dt’
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Different ways to write the Time-shift invariant case (t) = - t g(t-t’) h(t’) dt’ but rename t’ to (t) = - t g(t- ) h( ) d alternatively let =t-t’ so d =-dt’, (t’=- )= , (t’=t)=0 (t) = 0 g( ) h(t- ) d Note symmetry of integrand
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Interpretation of g( ): Impulse Response Suppose h(t) is a impulse (spike) at time 0 t 0 h(t) Then the resulting (t) is called the impulse response and denoted g(t): t 0 (t)=g(t)
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Suppose h(t) is two impulses of different amplitude at two different times t t1t1 h(t) Then by additivity, (t) is the sum of two time-shifted impulse responses of correspondingly scaled amplitudes t2t2 t (t) t1t1 t2t2 t t1t1 t2t2 A g(t-t 1 ) B g(t-t 2 ) A g(t-t 1 )+B g(t-t 2 ) amplitude A amplitude B
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An arbitrary function can be viewed as a sum of many impulses of different amplitudes occurring at a sequence of times H(t) Then by additivity, (t) is the sum of all the corresponding responses: (t) t’<t g(t-t’) h(t’) tt’ amplitude h(t’) tt’ (t) = t g(t-t’) h(t’) dt’ Same formula as before
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(t) = - t g(t- ) h( ) d Or equivalently (t) = 0 g( ) h(t- ) d The h( ) is “forward in time” The g( ) is “forward in time” This form of integral is called a “convolution”
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approximation of functions as time series h k =h[k t] with k= … -2, -1, 0, 1, 2, … hkhk t tktk h(t) tt (t) = - t g(t- ) h( ) d k = t p=- k g k-p h p or equivalently (t) = 0 g( ) h(t- ) d k = t p=o g p h k-p
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Matrix formulation (t) = - t g(t- ) h( ) d k = t p=- k g k-p h p k=0: 0 = t p=- 0 g 0-p h p = t{ … + g 0 h 0 } k=1: 1 = t p=- 1 g 1-p h p = t{ … + g 1 h 0 + g 0 h 1 } k=2: 2 = t p=- 2 g 2-p h p = t{… + g 2 h 0 + g 1 h 1 + g 0 h 2 } 01…N01…N h0h1…hNh0h1…hN g 0 0 0 0 0 0 g 1 g 0 0 0 0 0 … g N … g 3 g 2 g 1 g 0 = t Note problem with parts of the equation being “off the ends” of the matrix
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Might be especially useful when we know g T k = t p=- k g k-p H p 01…N01…N h0h1…hNh0h1…hN g 0 0 0 0 0 0 g 1 g 0 0 0 0 0 … g N … g 3 g 2 g 1 g 0 = t = G h
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Alternate Matrix formulation (t) = 0 g( ) h(t- ) d k = t p=0 g p h k-p k=0: 0 = t p=0 g p h 0-p = t{ g 0 h 0 + …} k=1: 1 = t p=0 g p h 1-p = t{ g 0 h 1 + g 1 h 0 + …} k=2: 2 = t p=0 g p h 2-p = t{ g 0 h 2 + g 1 h 1 + g 2 h 0 + … } 01…N01…N g0g1…gNg0g1…gN h 0 0 0 0 0 0 h 1 h 0 0 0 0 0 … h N … h 3 h 2 h 1 h 0 = t Note again the problem with parts of the equation being “off the ends” of the matrix
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k = t p=0 g p h k-p 01…N01…N g0g1…gNg0g1…gN h 0 0 0 0 0 0 h 1 h 0 0 0 0 0 … h N … h 3 h 2 h 1 h 0 = t = H g Might be especially useful when we know h
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Returning to the flame problem Approximation: very conductive plate that heats very rapidly, but slowly cools down due to warming up the surrounding air d /dt = -c + h(t) Heat input from flame heat loss to air faster when plate hotter change in temperature of plate
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Impulse response let h(t) = (t-t’) d /dt = -c + (t-t’) solution (take my word for it) unit impulse at time t’ (t) = g(t,t’) = if t<t’ exp{-c(t-t’)} if t t’
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impulse response for c=1 and t’=2 t g(t,t’=2)
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MatLab Code implements =Gh N = 51; tmin = 0; tmax = 10; dt = (tmax-tmin)/(N-1); t = tmin+dt*[0:N-1]'; c=1; g = exp(-c*t); G=zeros(N,N); for p = [1:N] G(p:N,p) = dt*g(1:N-p+1); end f=0.2; h = 2+sin(2*pi*f*t); theta = G*h; define g(t) create matrix G setup time define h(t) compute (t)
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But suppose and G are known =Gh So h = G -1
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g(t) h true (t) true (t) obs (t)= true (t)+noise h est (t) and h true (t)
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another example in this case t and h(t) are known and g(t) is unknown
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Thermometer measuring plate temperature (t) Goal: infer “physics” of plate, as embodied in its impulse response function plateThermometer measuring flame heat h(t)
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=Hg So g = H -1
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g(t) h true (t) true (t) Set up of problem
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obs (t)= true (t)+noise h obs (t)=h true (t)+noise Simulate noisy data
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Results g true (t) and g est (t) … yuck!
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Fix-up try for shorter g(t) and use damped least-squares Damping: 2 =10
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Another Fix-up try for shorter g(t) and use 2 nd derivative damping Damping: 2 =100
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