Problems 291(b) Suppose that the system is represented by a first-order differential equation,y′(t)+ 5 y(t)=x(t)wherey(t)is the output of the system and the givenx(t)is the input of the system. Find the steady-
state responsey(t)due tox(t)using the eigenfunction property.4.5. Steady state of LTI systems
The transfer function of an LTI system is
H(s)=
Y(s)
X(s)=
s+ 1
s^2 + 3 s+ 2
If the input to this system isx(t)= 1 +cos(t+π/ 4 ),−∞<t<∞, what is the outputy(t)in the steady
state?
4.6. Eigenfunction property of LTI systems and Laplace
The transfer function of an LTI system is given by
H(s)=
Y(s)
X(s)=
1
s^2 + 3 s+ 2
and its input is
x(t)= 4 u(t)(a) Use the eigenfunction property of LTI systems to find the steady-state responsey(t)of this system.
(b) Verify your result in (a) by means of the Laplace transform.4.7. Different ways to compute the Fourier coefficients—MATLAB
We would like to find the Fourier series of a sawtooth periodic signalx(t)of periodT 0 = 1. The period of
x(t)is
x 1 (t)=r(t)[u(t)−u(t− 1 )]
(a) Carefully plotx(t)and compute the Fourier coefficientsXkusing the integral definition.
(b) An easier way to do this is to use the Laplace transform ofx 1 (t). FindXkthis way.
(c) Use MATLAB to plot the signalx(t)and its magnitude and phase line spectra.
(d) Obtain a trigonometric Fourier seriesˆx(t)consisting of the DC term and 40 harmonics to approximate
x(t). Use MATLAB to find the values ofˆx(t)fort= 0 to 10 in steps of0.001. How does it compare with
x(t)?4.8. Addition of periodic signals—MATLAB
Consider a sawtooth signalx(t)with periodT 0 = 2 and period
x 1 (t)={
t 0 ≤t< 1
0 otherwise(a) Find the Fourier coefficientsXkusing the Laplace transform. Consider the cases whenkis odd and
even (k6= 0 ). You need to computeX 0 directly from the signal.
(b) Lety(t)=x(−t). Find the Fourier coefficientsYk.
(c) The sumz(t)=x(t)+y(t)is a triangular function. Find the Fourier coefficientsZkand compare them
toXk+Yk.
(d) Use MATLAB to plotx(t),y(t), andz(t)and their corresponding magnitude line spectra. Find an
approximate of z(t) using the dc value and10 harmonics and plot it.