Problems 413

(d) Consider the situation wheref 0 =2 KHz, but the parameters of the paths are random, trying to

simulate real situations where these parameters are unpredictable, although somewhat related. Let

r(t)=α 0 cos( 2 π(f 0 −ν)(t−L 0 /c))+α 1 cos( 2 π(f 0 −ν)(t−L 1 /c))

whereν= 50 ηHz,L 0 =1,000η,L 1 =10,000η,α 0 = 1 −η,α 1 =α 0 / 10 , andηis a random number

between 0 and 1 with equal probability of being any of these values (this can be realized by using the

randMATLAB function). Generate the received signal for 10 different events, useFs=10,000 Hzas

the sampling rate, and plot them together to observe the effects of the multipath and Doppler.

6.13. RLC implementation of low-pass Butterworth filters
Consider the RLC circuit shown in Figure 6.29 whereR= 1 .
(a) Determine the values of the inductor and the capacitor so that the transfer function of the circuit when
the output is the voltage across the capacitor is
Vo(s)
Vi(s)

1

s^2 +

√

2 s+ 1

That is, it is a second-order Butterworth filter.

(b) Find the transfer function of the circuit, with the values obtained in (a) for the capacitor and the induc-

tor, when the output is the voltage across the resistor. Carefully sketch the corresponding frequency

response and determine the type of filter it is.

FIGURE 6.29

L

R= 1 Ω C

vi(t)

+

−

6.14. Design of low-pass Butterworth/Chebyshev filters
The specifications for a low-pass filter are:
n p=1500 rad/sec,αmax=0.5 dBs
n s=3500 rad/sec,αmin=30 dBs
(a) Determine the minimum order of the low-pass Butteworth filter and compare it to the minimum
order of the Chebyshev filter that satisfy the specifications. Which is the smaller of the two?
(b)Determine the half-power frequencies of the designed Butterworth and Chebyshev low-pass
filters by lettingα(p)=αmax. Use the minimum orders obtained above.
(c) For the Butterworth and the Chebyshev designed filters, find the loss function values atpand
s. How are these values related to theαmaxandαminspecifications? Explain.
(d)If new specifications for the passband and stopband frequencies arep=750 rad/secands=
1750 rad/sec, respectively, are the minimum orders of the Butterworth and the Chebyshev filters
changed? Explain.

6.15. Low-pass Butterworth filters
The loss at a frequency=2000 rad/secisα( 2000 )=19.4 dBsfor a fifth-order low-pass Butterworth
filter. If we letα(p)=αmax=0.35 dBs, determine