Signals and Systems - Electrical Engineering

566 C H A P T E R 9: The Z-Transform

(c) Compute the Laplace transform of the sampled signal (i.e.,Xs(s)=L[xs(t]).

(d) Determine the Z-transform ofx(nTs), orX(z).

(e) Indicate how to transformXs(s)intoX(z)

9.8. Computation of Z-transform—MATLAB

Consider a discrete-time pulsex[n]=u[n]−u[n−10].

(a) Plotx[n]as a function ofnand use the definition of the Z-transform to findX(z).

(b) Use the Z-transform ofu[n]and properties of the Z-transform to findX(z). Verify that the expressions

obtained above forX(z)are identical.

(c) Find the poles and the zeros ofX(z)and plot them in thez-plane. Use MATLAB to plot the poles and

zeros.

9.9. Computation of Z-transform

A causal exponentialx(t)= 2 e−^2 tu(t)is sampled using a sampling periodTs= 1. The corresponding

discrete-time signal isx[n]= 2 e−^2 nu[n].

(a) Express the discrete-time signal asx[n]= 2 αnu[n]and give the value ofα.

(b) Find the Z-transformX(z)ofx[n]and plot its poles and zeros in thez-plane.

9.10. Computation of Z-transform

Consider the signalx[n]=0.5( 1 +[−1]n)u[n].

(a) Plotx[n]and use the definition of the Z-transform to obtain its Z-transform,X(z).

(b) Use the linearity property and the Z-transforms ofu[n]and[−1]nu[n]to find the Z-transformX(z)=

Z[x[n]].

(c) Determine and plot the poles and the zeros ofX(z).

9.11. Solution of difference equations with Z-transform

Consider a system represented by the first-order difference equation

y[n]=x[n]−0.5y[n−1]

wherey[n]is the output andx[n]is the input.

(a) Find the Z-transformY(z)in terms ofX(z)and the initial conditiony[−1].

(b) Find an inputx[n]6= 0 and an initial conditiony[−1]6= 0 so that the output isy[n]= 0 forn≥ 0. Verify

you get this result by solving the difference equation recursively.

(c) For zero initial conditions, find the inputx[n]so thaty[n]=δ[n]+0.5δ[n−1].

9.12. Transfer function, stability, and impulse response—MATLAB

Consider a second-order discrete-time system represented by the difference equation

y[n]− 2 rcos(ω 0 )y[n−1]+r^2 y[n−2]=x[n] n≥ 0

wherer> 0 and 0 ≤ω 0 ≤ 2 π,y[n]is the output, andx[n]is the input.

(a) Find the transfer functionH(z)of this system.

(b) Find the value ofω 0 and determine the values ofrthat would make the system stable. Use the

MATLAB functionzplaneto plot the poles and the zeros forr=0.5andω 0 =π/ 2 radians.

(c) Letω 0 =π/ 2. Find the corresponding impulse responseh[n]of the system. What other value ofω 0

would get the same impulse response?

9.13. Generation of discrete-time sinusoid—MATLAB

Given that the Z-transform of a discrete-time cosineAcos(ω 0 n)u[n]is

A( 1 −cos(ω 0 )z−^1 )

1 −2 cos(ω 0 )z−^1 +z−^2