CHAPTER 6 Application to Control and Communications
6.6 What Have We Accomplished? What Is Next?
In this chapter we have illustrated the application of the Laplace and the Fourier analysis to the theories of control, communications, and filtering. As you can see, the Laplace transform is very appropriate for control problems where transients as well as steady-state responses are of interest.
On the other hand, in communications and filtering there is more interest in steady-state responses and frequency characterizations, which are more appropriately treated using the Fourier transform.
It is important to realize that stability can only be characterized in the Laplace domain, and that it is necessary when considering steady-state responses. The control examples show the importance of the transfer function and transient and steady-state computations. Block diagrams help to visualize the interconnection of the different systems. Different types of modulation systems are illustrated in the communication examples. Finally, this chapter provides an introduction to the design of analog filters. In all the examples, the application of MATLAB was illustrated.
Although the material in this chapter does not have sufficient depth, reserved for texts in control, communications, and filtering, it serves to connect the theory of continuous-time signals and systems with applications. In the next part of the book, we will consider how to process signals using comput- ers and how to apply the resulting theory again in control, communications, and signal processing problems.
PROBLEMS
6.1. Cascade implementation and loading
The transfer function of a filterH(s)=1/(s+1)2is to be implemented by cascading two first-order filters Hi(s)=1/(s+1),i=1, 2.
(a) ImplementHi(s)as a series RC circuit with inputvi(t)and outputvi+1(t),i=1, 2. Cascade two of these circuits and find the overall transfer functionV3(s)/V1(s). Carefully draw the circuit.
(b) Use a voltage follower to connect the two circuits when cascaded and find the overall transfer function V3(s)/V1(s). Carefully draw the circuit.
(c) Use the voltage follower circuit to implement a new transfer function
G(s)= 1
(s+1000)(s+1) Carefully draw your circuit.
6.2. Cascading LTI and LTV systems
The receiver of an AM system consists of a band-pass filter, a demodulator, and a low-pass filter. The received signal is
r(t)=m(t)cos(40000πt)+q(t)
where m(t) is a desired voice signal with bandwidth BW=5 KHz that modulates the carrier cos(40, 000πt)andq(t)is the rest of the signals available at the receiver. The low-pass filter is ideal with magnitude1and bandwidthBW. Assume the band-pass filter is also ideal and that the demodulator is cos(ct).
(a) What is the value ofcin the demodulator?
(b) Suppose we input the received signal into the band-pass filter cascaded with the demodulator and the low-pass filter. Determine the magnitude response of the band-pass filter that allows us to recover m(t). Draw the overall system and indicate which of the components are LTI and which are LTV.
(c) By mistake we input the received signal into the demodulator, and the resulting signal into the cascade of the band-pass and the low-pass filters. If you use the band-pass filter obtained above, determine the recovered signal (i.e., the output of the low-pass filter). Would you get the same result regardless of whatm(t)is? Explain.
6.3. Op-amps as feedback systems
An ideal operational amplifier circuit can be shown to be equivalent to a negative-feedback system. Con- sider the amplifier circuit in Figure 6.28 and its two-port network equivalent circuit to obtain a feedback system with inputVi(s)and outputV0(s). What is the effect ofA→ ∞on the above circuit?
−Av− Ro 0
+ + +
+
+ + R1
R2
R1
R2
v− v−
v+
v+ Ri
vi(t)
vi(t) vo(t)
vo(t)
→
→
−
−
− −
−
−
∞
FIGURE 6.28
6.4. RC circuit as feedback system
Consider a series RC circuit with input a voltage sourcevi(t)and output the voltage across the capacitor vo(t).
(a) Draw a negative-feedback system for the circuit using an integrator, a constant multiplier, and an adder.
(b) Let the input be a battery (i.e.,vi(t)=Au(t)). Find the steady-state errore(t)=vi(t)−vo(t).
6.5. RLC circuit as feedback system
A resistorR, a capacitorC, and an inductorLare connected in series with a sourcevi(t). Consider the output of the voltage across the capacitorvo(t). LetR=1,C=1F andL=1H.
(a) Use integrators and adders to implement the differential equation that relates the inputvi(t)and the outputvo(t)of the circuit.
(b) Obtain a negative-feedback system block diagram with inputVi(s)and outputV0(s). Determine the feedforward transfer functionG(s)and the feedback transfer functionH(s)of the feedback system.
(c) Find an equation for the errorE(s)=Vi(s)−V0(s)H(s)and determine its steady-state response when the input is a unit-step signal (i.e.,Vi(s)=1/s).
6.6. Ideal and lossy integrators
An ideal integrator has a transfer function1/s, while a lossy integrator has a transfer function1/(s+K).
(a) Determine the feedforward transfer function G(s) and the feedback transfer function H(s) of a negative-feedback system that implements the overall transfer function
Y(s) X(s) = K
K+s
whereX(s)andY(s)are the Laplace transforms of the inputx(t)and the outputy(t)of the feedback system. Sketch the magnitude response of this system and determine the type of filter it is.
Problems 411
(b) If we letG(s)=sin the previous feedback system, determine the overall transfer functionY(s)/X(s) whereX(s)andY(s)are the Laplace transforms of the inputx(t)and the outputy(t)of this new feed- back system. Sketch the magnitude response of the overall system and determine the type of filter it is.
6.7. Feedback implementation of an all-pass system
Suppose you would like to obtain a feedback implementation of an all-pass filter T(s)= s2√
2s+1 s2√
2s+1
(a) Determine if theT(s)is the transfer function corresponding to an all-pass filter by means of the poles and zeros ofT(s).
(b) Determine the feedforward transfer function G(s) and the feedback transfer function H(s) of a negative-feedback system that hasT(s)as its overall transfer function.
(c) Would it be possible to implement T(s) using a positive-feedback system? If so, indicate its feedforward transfer functionG(s)and the feedback transfer functionH(s).
6.8. Filter stabilization
The transfer function of a designed filter is
G(s)= 1
(s+1)(s−1)
which is unstable given that one of its poles is in the right-hands-plane.
(a) Consider stabilizingG(s)by means of negative feedback with a gainK>0in the feedback. Determine the range of values ofKthat would make the stabilization possible.
(b) Use the cascading of an all-pass filterHa(s)with the givenG(s)to stabilize it. GiveHa(s). Would it be possible for the resulting filter to have the same magnitude response asG(s)?
6.9. Error and feedforward transfer function
Suppose the feedforward transfer function of a negative-feedback system isG(s)=N(s)/D(s), and the feedback transfer function is unity.
(a) Given that the Laplace transform of the error is
E(s)=X(s)[1−H(s)]
whereH(s)=G(s)/(1+G(s))is the overall transfer function of the feedback system, find an expres- sion for the error in terms ofX(s),N(s), andD(s). Use this equation to determine the conditions under which the steady-state error is zero forx(t)=u(t).
(b) If the input isx(t)=u(t), the denominatorD(s)=(s+1)(s+2), and the numeratorN(s)=1, find an expression forE(s)and from it determine the initial valuee(0)and the final valuelimt→∞e(t)of the error.
6.10. Product of polynomials ins—MATLAB Given a transfer function
Y(s) X(s) = N(s)
D(s)
whereY(s)andX(s)are the Laplace transforms of the outputy(t)and of the inputx(t)of an LTI system, andN(s)andD(s)are polynomials ins, to find the output
Y(s)=X(s)N(s) D(s)
we need to multiply polynomials to getY(s)before we perform partial fraction expansion to gety(t). (a) Find out about the MATLAB functionconvand how it relates to the multiplication of polynomials.
LetP(s)=1+s+s2andQ(s)=2+3s+s2+s3. Obtain analytically the productZ(s)=P(s)Q(s)and then useconvto compute the coefficients ofZ(s).
(b) Suppose thatX(s)=1/s2, and we haveN(s)=s+1,D(s)=(s+1)((s+4)2+9). Useconvto find the numerator and the denominator polynomials ofY(s)=N1(s)/D1(s). Use MATLAB to findy(t), and to plot it.
(c) Create a function that takes as input the values of the coefficients of the numerators and denominators ofX(s)and of the transfer functionH(s)of the system and provides the response of the system. Show your function, and demonstrate its use with theX(s)andH(s)given above.
6.11. Feedback error—MATLAB
Control systems attempt to follow the reference signal at the input, but in many cases they cannot follow particular types of inputs. Let the system we are trying to control have a transfer functionG(s), and the feedback transfer function beH(s). IfX(s)is the Laplace transform of the reference input signal, andY(s) the Laplace transform of the output, then the close-loop transfer function is
Y(s)
X(s) = G(s) 1+G(s)H(s) The Laplace transform of the error signal isE(s)=X(s)−Y(s)H(s),
G(s)= 1
s(s+1)(s+2) and H(s)=1
(a) Find an expression forE(s)in terms ofX(s),G(s), andH(s).
(b) Letx(t)=u(t)and the Laplace transform of the corresponding error beE1(s). Use the final value property of the Laplace transform to obtain the steady-state errore1ss.
(c) Letx(t)=tu(t) (i.e., a ramp signal) andE2(s)be the Laplace transform of the corresponding error signal. Use the final value property of the Laplace transform to obtain the steady-state errore2ss. Is this error value larger than the one above? Which of the two inputsu(t)andr(t)is easier to follow?
(d) Use MATLAB to find the partial fraction expansions ofE1(s)andE2(s)and use them to finde1(t)and e2(t)and then plot them.
6.12. Wireless transmission—MATLAB
Consider the transmission of a sinusoidx(t)=cos(2πf0t)through a channel affected by multipath and Doppler. Let there be two paths, and assume the sinusoid is being sent from a moving transmitter so that a Doppler frequency shift occurs. Let the received signal be
r(t)=α0cos(2π(f0−ν)(t−L0/c))+α1cos(2π(f0−ν)(t−L1/c))
where0≤αi≤1are attenuations,Liare the distances from the transmitter to the receiver that the signal travels in theith path i=1, 2,c=3×108 m/sec, and the frequency shiftν is caused by the Doppler effect.
(a) Letf0=2 KHz,ν=50 Hz,α0=1,α1=0.9, andL0=10,000 meters. What would beL1if the two sinusoids have a phase difference ofπ/2?
(b) Is the received signalr(t), with the parameters given above butL1=10,000, periodic? If so, what would be its period and how much does it differ from the period of the original sinusoid? Ifx(t)is the input andr(t)the output of the transmission channel, considered a system, is it linear and time invariant? Explain.
(c) Sample the signalsx(t)andr(t)using a sampling frequencyFs=10 KHz. Plot the sampled sentx(nTs) and receivedr(nTs)signals forn=0to2000.
Problems 413
(d) Consider the situation where f0=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)=α0cos(2π(f0−ν)(t−L0/c))+α1cos(2π(f0−ν)(t−L1/c))
whereν=50ηHz,L0=1,000η,L1=10,000η,α0=1−η,α1=α0/10, andηis a random number between0and1with 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 s2+√
2s+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 C R=1Ω
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 atp and
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
n The half-power frequencyhpof the filter.
n The passband frequencypof the filter.
6.16. Design of low-pass Butterworth/Chebyshev filters The specifications for a low-pass filter are:
n α(0)=20 dBs
n p=1500 rad/sec,α1=20.5 dBs
n s=3500 rad/sec,α2=50 dBs
(a) Determine the minimum order of the low-pass Butterworth and Chebyshev filters, and determine which is smaller.
(b) Give the transfer function of the designed low-pass Butterworth and Chebyshev filters (make sure the dc loss is as specified).
(c) Determine the half-power frequency of the designed filters by lettingα(p)=αmax.
(d) Find the loss function values provided by the designed filters at p ands. How are these val- ues related to theαmax andαmin specifications? Explain. Which of the two filters provides more attenuation in the stopband?
(e) If new specifications for the passband and stopband frequencies are p=750 rad/sec ands= 1750 rad/sec, respectively, are the minimum orders of the filter changed? Explain.
6.17. Butterworth, Chebyshev, and Elliptic filters—MATLAB
Design an analog low-pass filter satisfying the following magnitude specifications:
n αmax=0.5 dB;αmin=20 dB
n p=1000 rad/sec;s=2000 rad/sec
(a) Use the Butterworth method. Plot the poles and zeros and the magnitude and phase of the designed filter. Verify that the specifications are satisfied by plotting the loss function.
(b) Use the Chebyshev methodcheby1. Plot the poles and zeros and the magnitude and phase of the designed filter. Verify that the specifications are satisfied by plotting the loss function.
(c) Use the elliptic method. Plot the poles and zeros and the magnitude and phase of the designed filter. Verify that the specifications are satisfied by plotting the loss function.
(d) Compare the three filters and comment on their differences.
6.18. Chebyshev filter design—MATLAB
Consider the following low-pass filter specifications:
n αmax=0.1 dB;αmin=60 dB
n p=1000 rad/sec;s=2000 rad/sec
(a) Use MATLAB to design a Chebyshev low-pass filter that satisfies the above specifications. Plot the poles and zeros and the magnitude and phase of the designed filter. Verify that the specifications are satisfied by plotting the loss function.
(b) Compute the half-power frequency of the designed filter.
6.19. Getting rid of 60-Hz hum with different filters—MATLAB A desirable signal
x(t)=cos(100πt)−2 cos(50πt)
is recorded asy(t)=x(t)+cos(120πt)—that is, as the desired signal but with a60-Hzhum. We would like to get rid of the hum and recover the desired signal. Use symbolic MATLAB to plotx(t)andy(t). Consider the following three different alternatives (use symbolic MATLAB to implement the filtering and use any method to design the filters):
(a) Design a band-eliminating filter to get rid of the 60-Hzhum in the signal. Plot the output of the band-eliminating filter.
(b) Design a high-pass filter to get the hum signal and then subtract it fromy(t). Plot the output of the high-pass filter.
Problems 415
(c) Design a band-pass filter to get rid of the hum. Plot the output of the band-pass filter.
(d) Is any of these alternatives better than the others? Explain.
6.20. Demodulation of AM—MATLAB The signal at the input of an AM receiver is
u(t)=m1(t)cos(20t)+m2(t)cos(100t)
where the messagesmi(t),i=1, 2are the outputs of a low-pass Butterworth filter with inputs
x1(t)=r(t)−2r(t−1)+r(t−2) x2(t)=u(t)−u(t−2)
respectively. Suppose we are interested in recovering the messagem1(t).
(a) Design a10th-order low-pass Butterworth filter with half-power10 rad/sec. Implement this filter using MATLAB and find the two messagesmi(t), i=1, 2using the indicated inputsxi(t),i=1, 2, and plot them.
(b) To recover the desired messagem1(t), first use a band-pass filter to obtain the desired signal con- tainingm1(t)and to suppress the other. Design a band-pass Butterworth filter with a bandwidth of 10 rad/sec, centered at 20 rad/sec and order 10 that will pass the signalm1(t)cos(20t)and reject the other signal.
(c) Multiply the output of the band-pass filter by a sinusoidcos(20t)(exactly the carrier in the transmitter), and low-pass filter the output of the mixer (the system that multiplies by the carrier frequency cosine).
Design a low-pass Butterworth filter of bandwidth10 rad/sec, and order 10 to filter the output of the mixer.
(d) Use MATLAB to display the different spectra. Compute and plot the spectrum ofm1(t),u(t),the output of the band-pass filter, the output of the mixer, and the output of the low-pass filter. Write numeric functions to compute the analog Fourier transform and its inverse.
6.21. Quadrature AM—MATLAB
Suppose we would like to send the two messagesmi(t),i=1, 2, created in Problem 6.20 using the same bandwidth and to recover them separately. To implement this, consider the QAM approach where the transmitted signal is
s(t)=m1(t)cos(50t)+m2(t)sin(50t)
Suppose that at the receiver we receives(t)and that we only need to demodulate it to obtainmi(t),i=1, 2.
Design a low-pass Butterworth filter of order 10 and a half-power frequency10 rad/sec(the bandwidth of the messages).
(a) Use MATLAB to plots(t)and its magnitude spectrum|S()|. Write numeric functions to compute the analog Fourier transform and its inverse.
(b) Multiplys(t)bycos(50t), and filter the result using the low-pass filter designed before. Use MATLAB to plot the result and to find and plot its magnitude spectrum.
(c) Multiplys(t)bysin(50t), and filter the result using the low-pass filter designed before. Use MATLAB to plot the result and to find and plot its magnitude spectrum.
(d) Comment on your results.
3 P A R T
Theory and Application of Discrete-Time Signals and Systems
C H A P T E R 7
S a m p l i n g T h e o r y
The pure and simple truth is rarely pure and never simple.
Oscar Wilde (1854–1900) Irish writer and poet
7.1 INTRODUCTION
Since many of the signals found in applications such as communications and control are analog, if we wish to process these signals with a computer it is necessary to sample, quantize, and code them to obtain digital signals. Once the analog signal is sampled in time, the amplitude of the obtained discrete-time signal is quantized and coded to give a binary sequence that can be either stored or processed with a computer.
The main issues considered in this chapter are:
n How to sample—As we will see, it is the inverse relation between time and frequency that provides the solution to the problem of preserving the information of an analog signal when it is sampled.
When sampling an analog signal one could choose an extremely small value for the sampling period so that there is no significant difference between the analog and the discrete signals—
visually as well as from the information content point of view. Such a representation would, however, give redundant values that could be spared without losing the information provided by the analog signal. If, on the other hand, we choose a large value for the sampling period, we achieve data compression but at the risk of losing some of the information provided by the analog signal. So how do we choose an appropriate value for the sampling period? The answer is not clear in the time domain. It does become clear when considering the effects of sampling in the frequency domain: The sampling period depends on the maximum frequency present in the analog signal. Furthermore, when using the correct sampling period the information in the analog signal will remain in the discrete signal after sampling, thus allowing the reconstruction of the original signal from the samples. These results, introduced by Nyquist and Shannon, constitute
Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00011-9
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