234 C H A P T E R 3: The Laplace Transform
FindY 2 (s)=L[y 2 (t)]using the convolution property of the Laplace transform and findy 2 (t).
(b) Consider then the result forN= 3 of these convolutions—that is,y 3 (t)=(x∗x∗x)(t)FindY 3 (s)=L[y 3 (t)]using the convolution property of the Laplace transform and findy 3 (t).
(c) The signalx(t)can be considered the impulse response of an averager that ”smooths” out a signal.
Lettingy 1 (t)=x(t), plot the three functionsyi(t)fori=1, 2, and 3. Compare these signals on their
smoothness and indicate their supports in time. (Fory 2 (t)andy 3 (t), how do their supports relate to
the supports of the signals convolved?)
3.29. Positive and negative feedback
There are two types of feedback, negative and positive. In this problem we explore their difference.
(a) Consider negative feedback. Suppose you have a system with transfer functionH(s)=Y(s)/E(s)
whereE(s)=C(s)−Y(s), andC(s)andY(s)are the transforms of the feedback system’s reference
c(t)and outputy(t). Find the transfer function of the overall systemG(s)=Y(s)/C(s).
(b) In positive feedback, the only equation that changes isE(s)=C(s)+Y(s); the other equations remain
the same. Find the overall feedback system transfer functionG(s)=Y(s)/C(s).
(c) Suppose thatC(s)= 1 /s,H(s)= 1 /(s+ 1 ). DetermineG(s)for both negative and positive feedback.
Findy(t)=L−^1 [Y(s)]for both types of feedback and comment on the difference in these signals.
3.30. Feedback stabilization
An unstable system can be stabilized by using negative feedback with a gainKin the feedback loop. For
instance, consider an unstable system with transfer functionH(s)=2
s− 1
which has a pole in the right-hands-plane, making the impulse response of the systemh(t)grow ast
increases. Use negative feedback with a gainK> 0 in the feedback loop, and putH(s)in the forward
loop. Draw a block diagram of the system. Obtain the transfer functionG(s)of the feedback system and
determine the value ofKthat makes the overall system BIBO stable (i.e., its poles in the open left-hand
s-plane).
3.31. All-pass stabilization
Another stabilization method consists in cascading an all-pass system with the unstable system to cancel
the poles in the right-hands-plane. Consider a system with a transfer functionH(s)=s+ 1
(s− 1 )(s^2 + 2 s+ 1 )
which has a pole in the right-hands-plane,s= 1 , so it is unstable.
(a) The poles and zeros of an all-pass filter are such that ifp 12 =−σ±j 0 are complex conjugate poles
of the filter, thenz 12 =σ±j 0 are the corresponding zeros, and for real polesp=−σthere is a
correspondingz=σ. The orders of the numerator and the denominator of the all-pass filter are equal.
Write the general transfer function of an all-pass filterHap(s)=KN(s)/D(s).
(b) Find an all-pass filterHap(s)so that when cascaded withH(s)the overall transfer functionG(s)=
H(s)Hap(s)has all its poles in the left-hands-plane.
(c) FindKof the all-pass filter so that whens= 0 the all-pass filter has a gain of unity. What is the relation
between the magnitude of the overall system|G(s)|and that of the unstable filter|H(s)|.
3.32. Half-wave rectifier—MATLAB
In the generation of DC from AC voltage, the ”half-wave” rectified signal is an important part. Suppose the
AC voltage isx(t)=sin( 2 πt)u(t).