Unformatted text preview: ts excitation. Classify the system as to
homogeneity, additivity, linearity, timeinvariance, 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, timeinvariance 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 zeroexcitation 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 timeinvariance, 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 327 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 complexsinusoidal 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 328 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.
 Spring '09
 ATOUSA

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