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EG1110 SIGNALS AND SYSTEMS • Split this integral between time before a reference point,0− , and a time after: y(t) = Input/Output analysis of systems Z 0 − −∞ g(τ )u(t − τ )dτ + Z ∞ 0− g(τ )u(t − τ )dτ • Define Free and Forced response of systems y0(t) = u(t) y(t) G Z 0 − −∞ g(τ )u(t − τ )dτ This is the response of the system due to all inputs up until time 0 −. • Thus we get • Let G have impulse response g(t). General expression for y(t) is y(t) = y0(t) + y(t) = Z ∞ −∞ Z ∞ 0− g(τ )u(t − τ )dτ i.e. the system is a function of its output at time t = 0− plus an additional term. g(τ )u(t − τ )dτ 1 2 • Assume g(t) is causal: • Thus for a general causal linear time invariant system, output is given by ⇒ y(t) = 0 for all τ > t y(t) = y0(t) + (Output at time t is not dependent on anything which happens in the future) i.e. Z t 0− g(τ )u(t − τ )dτ • We denote – yF REE (t) = y0(t) response of system if no input were applied after time t = 0. y(t) = y0(t) + = y0(t) + Z t 0− Z t 0− g(τ )u(t − τ )dτ + g(τ )u(t − τ )dτ Z ∞ |t g(τ )u(t − τ )dτ {z =0 } (1) i.e. the response due to INITIAL CONDITIONS – yF ORCED (t) = Rt 0− g(τ )u(t − τ )dτ response of system due to inputs applied after time t = 0, assuming inputs zero for all τ < t. • Thus total response of system depends on – “Output” at time t = 0 due to inputs applied before hand - initial conditions – Convolution of its impulse response and input from time t = 0 to present time (t). – Often we may have initial condition values which are easier to use than input values i.e. y 0(t) is a fairly simple function. 3 4 Electrical example • Let y(t) = e(t), then integrating the above equation we get: e(t) C q charge e voltage across capacitor e(t) = e(t = 0) + 1 C Z t 0 i(τ )dτ • Now take C capacitance i – y(t) = e(t) (output) current – u(t) = i(t) (input) Let i(t) i(t) = • Consider a capacitor which stores charge in the standard way: 1 0 < t1 < t < t2 < ∞ 0 otherwise q = Ce q̇ = i = C ė 5 • Thus we have 6 Mechanical example x(t) y(t) = = = = 1 Zt e(t = 0) + i(τ )dτ C 0 Z 1 t2 i(τ )dτ e(t = 0) + C t1 1 Z t2 1dτ e(t = 0) + C t1 1 e(t = 0) + [t2 − t1] C i.e. the voltage across the capacitor (the system’s output) depends on: F(t) • Consider a cart rolling on wheels (assume no friction in bearings etc.) Input – The voltage across it at time t = 0 M • u(t) Force applied F (t) Ouput y(t) velocity of ball – The length of time for which further current is being applied (the input is being applied) • Define input as u(t) = 7 1 0 < t1 < t < t2 < ∞ 0 otherwise 8 Then as F = mẍ (x displacement) More on free response • Thus velocity of cart depends on y = ẋ 1 Zt F (τ )dτ + ẋ(0) = m 0 1 Zt F (τ )dτ = y(0) + m 0 1 Z t2 1dτ = y(0) + m t1 1 = y(0) + [t2 − t1] m – Its initial velocity y(0). • Systems we are interested in often described by ordinary differential equations (ODE’s). • Electrical example. Consider discharge of capacitor through resistor – The time interval over which the external force is applied. C • Obviously if y(0) = 0 the velocity of the ball after time t is different to if it was R some constant y(0) = c. No forcing voltage. 9 • We know that iC = C ė and e = iR R, or IR = e/R 10 • Integrate on l.h.s from e(0) to e(t) and on r.h.s from 0 to t: • Using Kirchoff’s current law Z e(t) e(0) e R 1 e ⇒ ė = − CR 1 de = − e dt CR de 1 ⇒ = dt e CR C ė = − [by separating the variables] 1 de = e ⇒ ln[e(t)] − ln[e(0)] = e(t) ⇒ ln = e(0) e(t) ⇒ = e(0) ⇒ e(t) = 1 dt CR 1 t − CR 1 − t CR 1 exp[− t] CR 1 e0 exp[− t] CR Z t 0 • Hence voltage will be an exponentially decaying function of time determined by – e(0) - initial voltage across capacitor/resistor – CR - time constant of system • Note no external input so no forced response! 11 12 (2) (3) (4) (5) More on forced response • If “initial conditions” are zero, it turns out that yF REE (t) = Z 0 −∞ g(τ )u(t − τ )dτ = 0 for all t > 0. • Hence, can disregard this in computing response system after t = 0 i.e. y(t) = yF ORCED (t) = Z t 0 g(τ )u(t − τ )dτ • This is equivalent to considering – Motion of mechanical systems “at rest” – Electrical systems with no initial voltages across/currents through circuit elements. 13