Signals and Systems - Electrical Engineering

Problems 565

(d) From the above results, indicate the region in thes-plane to which the wholez-plane is mapped into.

Sinceω=ω+ 2 π, is this mapping unique? Explain.

9.3. Z-transform and ROCs
Consider the noncausal sequence
s[n]=s 1 [n]+s 2 [n]
wheres 1 [n]=u[n]is causal ands 2 [n]=−u[−n]is anti-causal. This signal is thesignum, or sign function,
that extracts the sign of a real-valued signal—that is,

s[n]=sgn(x[n])=







− 1 x[n]< 0

0 x[n]= 0

1 x[n]> 0

(a) Find the Z-transforms ofs 1 [n]ands 2 [n], indicating the corresponding ROC.

(b) Determine the Z-transformS(z).

9.4. Z-transform and ROC
Given the anti-causal signal
x[n]=−αnu[−n]

(a) Determine the Z-transformX(z), and carefully plot the ROC whenα=0.5andα= 2. For which of the

two values ofαdoesX(ejω)exist?

(b) Find the signal that corresponds to the derivativedX(z)/dz. Express it in terms ofα.

9.5. Significance of ROC
Consider a causal signalx 1 [n]=u[n]and an anti-causal signalx 2 [n]=−u[−n−1].
(a) Find the Z-transformsX 1 (z)andX 2 (z)and carefully plot their ROCs. If the ROCs are not included with
the Z-transforms, would you be able to tell which is the correct inverse? Explain.
(b) Determine if it is possible to find the Z-transform ofx 1 [n]+x 2 [n].

9.6. Fibonacci sequence generation—MATLAB
Consider the Fibonacci sequence generated by the difference equation

f[n]=f[n−1]+f[n−2] n≥ 0

with initial conditionsf[−1]= 1 ,f[−2]=− 1.

(a) Find the Z-transform off[n], orF(z).

(b) Find the polesφ 1 andφ 2 and the zeros ofF(z)and plot them. How are the poles connected? How are

they related to the “golden ratio”?

(c) The Fibonacci difference equation has zero input, but its response is a sequence of ever-increasing

integers. Obtain a partial fraction expansion ofF(z)and findf[n]in terms of the polesφ 1 andφ 2 , and

show that the result is always integer. Use MATLAB to implement the inverse in term of the poles.

9.7. Laplace and Z-transforms of sampled signals
An analog pulsex(t)=u(t)−u(t− 1 )is sampled using a sampling periodTs=0.1.
(a) Obtain the discrete-time signalx(nTs)=x(t)|t=nTsand plot it as a function ofnTs.
(b) If the sampled signal is represented as an analog signal as

xs(t)=

N∑− 1

n= 0

x(nTs)δ(t−nTs)

determine the value ofNin the above equation.