Chap3StudentSolutions

23k1e 523 t 076k 2e 076 t t 0 523k1 076k 2 and h0

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Unformatted text preview: the unit sequence with the impulse response to get the overall system response, n +1 n +1 n +1 1 − − 0.6 1 − 0.6 1 − (−0.8) u[ n ] n] = 0.6455 0.6455 y[ + − 1.7746 0.2254 1.8 ( ) 19. Find the impulse responses of these systems: (a) y′ ( t) + 5 y( t) = x( t) Follow the example in the text. (b) y′′ ( t) + 6 y′ ( t) + 4 y( t) = x( t) h′′ ( t ) + 6 h′ ( t ) + 4 h ( t ) = δ ( t ) For t < 0, h( t) = 0 . For t > 0, h h ( t) = K1e −5.23 t + K 2e −0.76 t Solutions 3-16 ( ) M. J. Roberts - 8/16/04 Since the highest derivative of “ x ” is two less than the highest derivative of “y”, the general solution is of the form, h( t) = (K1e −5.23 t + K 2e −0.76 t ) u( t) (See the discussion in the text of what the solution form must be for different derivatives of x and y.) Integrating the differential equation once from t = 0 − to t = 0 + , [ h′ (0 ) − h′ (0 ) + 6 h(0 ) − h(0 + − + 0+ − 0+ 0− 0− )] + 4 ∫ h(t)dt = ∫ δ (t)dt = 1 We know that the impulse response cannot contain an impulse because its second derivative would be a triplet and there is no triplet excitation. We also know that the impulse response cannot be discontinuous at time, t = 0, because if it were the second derivative would be a doublet and there is no doublet excitation. Therefore, h′ (0 + ) − h′ (0 − ) = 1 ⇒ h′ (0 + ) = 1 This requirement, along with the requirement that the solution be continuous at time, t = 0, leads to the two equations, [ h′ (0 + ) = 1 = −5.23K1e −5.23 t − 0.76K 2e 0.76 t t = 0+ = −5.23K1 − 0.76K 2 and h(0 + ) = 0 = K1 + K 2 . (This second equation can also be found by integrating the differential equation twice from from t = 0 − to t = 0 + .) Solving, K1 = −0.2237 and K 2 = 0.2237 Then the total impulse response is h( t) = 0.2237(e −0.76 t − e −5.23 t ) u( t) . (c) 2 y′ ( t) + 3 y( t) = x′ ( t) (d) 4 y′ ( t) + 9 y( t) = 2 x( t) + x′ ( t) The homogeneous solution is y h ( t) = K h e h( t) = K h e 3 −t 2 9 −t 4 . The impulse response is of the form, u( t) + K iδ ( t) . Solutions 3-17 M. J. Roberts - 8/16/04 The solution is h( t) = − 9 1 −4t 1 e u( t) + δ ( t) 16 4 20. Sketch g( t) . (a) g ( t ) = rect ( t ) ∗ rect ( t ) = 1 2 ∞ ∫ rect (τ ) rect (t − τ ) dτ = ∫ rect (t − τ ) dτ −∞ − 1 2 Probably the easiest way to find this solution is graphically through the “flipping and shifting” process. When the second rectangle is flipped, it looks exactly the same because it is an even function. This is the “zero shift” position, the t = 0 position. At this position the two rectangles coincide and the area under the product is one. If t is increased from this position the two rectangles no longer coincide and the area under the product is reduced linearly until at t = 1 the area goes to zero. Exactly the same thing happens for decreases in t until it gets to -1. The convolution is obviously a unit triangle function. This fact is the reason the unit triangle fu...
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