Problems 567(a) Use the given Z-transform to find a difference equation for which the outputy[n]is a discrete-time
cosineAcos(ω 0 n)and the input isx[n]=δ[n]. What should you use as initial conditions?
(b) Verify your algorithm by generating a signaly[n]=2 cos(πn/ 2 )u[n]by implementing your algorithm
in MATLAB. Plot the input and the output signalsx[n]andy[n].
(c) Indicate how to change your previous algorithm to generate a sine functiony[n]=2 sin(πn/ 2 )u[n].
Use MATLAB to findy[n], and to plot it.9.14. Inverse Z-transform and poles and zeros
When finding the inverse Z-transform of functions withz−^1 terms in the numerator, the fact thatz−^1 can
be thought of as a delay operator can be used to simplify the computation. Consider
X(z)=
1 −z−^10
1 −z−^1(a) Use the Z-transform ofu[n]and the properties of the Z-transform to findx[n].
(b) If we considerX(z)a polynomial in negative powers ofz, what would be its degree and the values of
its coefficients?
(c) Find the poles and the zeros ofX(z)and plot them on thez-plane. Is there a pole or zero atz= 1?
Explain.9.15. Initial conditions and steady state
Consider a second-order system represented by the difference equation
y[n]=0.25y[n−2]+x[n]wherex[n]is the input andy[n]is the output.
(a) For the zero-input case (i.e., whenx[n]= 0 ), find the initial conditionsy[−1]andy[−2]so thaty[n]=
0.5nu[n].
(b) Suppose the input isx[n]=u[n]. Without solving the difference equation can you find the correspond-
ing steady stateyss[n]? Explain how and give the steady-state output. Verify by inverse Z-transform
that the steady-state responseyss[n]is the one obtained.9.16. Initial conditions and impulse response
A second-order system has the difference equation
y[n]=0.25y[n−2]+x[n]wherex[n]is the input andy[n]is the output.
(a) Find the inputx[n]so that for zero initial conditions, the output is given asy[n]=0.5nu[n].
(b) Ifx[n]=δ[n]+0.5δ[n−1]is the input to the above difference equation, find the impulse response of
the system.9.17. Convolution sum and product of polynomials
The convolution sum is a fast way to find the coefficients of the polynomial resulting from the multiplication
of two polynomials.
(a) Supposex[n]=u[n]−u[n−3]. Find its Z-transformX(z), a second-order polynomial inz−^1.
(b) MultiplyX(z)by itself to get a new polynomialY(z)=X(z)X(z)=X^2 (z). FindY(z).
(c) Graphically show the convolution ofx[n]with itself and verify that the result coincides with the
coefficients ofY(z).