Chap3StudentSolutions

38 graph the magnitude and phase of the complex

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Unformatted text preview: ts excitation. Classify the system as to homogeneity, additivity, linearity, time-invariance, stability, causality, memory, and invertibility. y( t) = x 3 ( t) Invertibility: 1 3 Solve y( t) = x ( t) for x( t) . x( t) = y ( t) . The cube root operation is multiple valued. Therefore the system is not invertible, unless we assume that the excitation must be realvalued. In that case, the response does determine the excitation because for any real y there is only one real cube root. 3 34. A CT system is described by the differential equation, t y′ ( t) − 8 y( t) = x( t) . Classify the system as to linearity, time-invariance and stability. Stability: The homogeneous solution to the differential equation is of the form, t y′ ( t) = 8 y( t) To satisfy this equation the derivative of “ y ” times “ t ” must be of the same functional form as “y” itself. This is satisfied by a homogeneous solution of the form, y( t) = Kt 8 If there is no excitation, but the zero-excitation response is not zero, the response will increase without bound as time increases. Unstable 35. A CT system is described by the equation, y( t) = t 3 ∫ x(λ )dλ . −∞ Classify the system as to time-invariance, stability and invertibility. Time Invariance: Let x1 ( t) = g( t) . Then y1 ( t) = Let x 2 ( t) = g( t − t0 ) . t 3 ∫ g(λ )dλ . −∞ Solutions 3-27 M. J. Roberts - 8/16/04 Then y 2 ( t) = t 3 t − t0 3 t −t 30 ∫ g(λ − t )dλ = ∫ g(u)du ≠ y (t − t ) = ∫ g(λ )dλ . 0 −∞ 1 0 −∞ −∞ Time Variant Stability: t 3 −∞ If x( t) is a constant, K, then y( t) = t 3 −∞ ∫ Kdλ = K ∫ dλ and, as t → ∞ , y( t) increases without bound. Unstable Invertibility: Differentiate both sides of y( t) = t 3 ∫ x(λ )dλ −∞ that x( t) = y′ ( 3t) . Invertible. t w.r.t. t yielding y′ ( t) = x . Then it follows 3 36. A CT system is described by the equation, y( t) = t +3 ∫ x(λ )dλ . −∞ Classify the system as to linearity, causality and invertibility. 37. Show that the system described by y( t) = Re( x( t)) is additive but not homogeneous. (Remember, if the excitation is multiplied by any complex constant and the system is homogeneous, the response must be multiplied by that same complex constant.) 38. Graph the magnitude and phase of the complex-sinusoidal response of the system described by y′ ( t) + 2 y( t) = e − j 2πft as a function of cyclic frequency, f. Similar to Exercise 30. 39. A DT system is described by y[n] = n +1 ∑ x[m] . m = −∞ Classify this system as to time invariance, BIBO stability and invertibility. Time Invariance: Let x1[ n ] = g[ n ] . Then y1[ n ] = n +1 ∑ g[m] . m =−∞ Let x 2 [ n ] = g[ n − n 0 ] . Then y 2 [ n ] = n +1 ∑ g[m − n ] . m =−∞ 0 Solutions 3-28 M. J. Roberts - 8/16/04 The first equation can be rewritten as y1[ n − n 0 ] = n − n 0 +1 n +1 m =−∞ q =−∞ ∑ g[m] = ∑ g[q − n ] = y [n] 0 2 Time invariant Invertibility: Inverting the functional relationship, y[ n ] = n +1...
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This note was uploaded on 06/19/2013 for the course ENSC 380 taught by Professor Atousa during the Spring '09 term at Simon Fraser.

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