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38 Ch. 1 / Signal and System Modeling Concepts (b) For the trapezoidal integration rule given in recursive form in Example 13. show that the
nonrecursive form up through time NT is 5W7) = 5(0) + [0(0) + a(NT) + 2 a(n Check this, using vaiues from Example 13.
13. Rework Example 1] if the acceleration proﬁle for the rocket is
a(r)=20mfsz. 05:5505 and zero otherwise. Assume the same parameter values as given in Example [1. Find and
plot the velocity proﬁle. 14. Rework Example 1! with the acceleration proﬁle shown. Sketch the resulting velocities. am, nuts2 20 0 IO 20 30 40
FIGURE P14 (a) Assume an initial velocity of zero.
(b) Assume an initial velocity of 1500 this. 15. In the satellite communications example. Example 14, let the transmitted signal bexit) = coswoz.
(a) Show that the received signal can be put into the form yo) = 1 + 2&3 €05 2000‘" + (0:13)2 cos(¢ouz — [anl 1 + (2,8 cos Zines! 0)) Plot the envelope of the cosine in the above equation versus (001' for at? = 0. 1. Note that
the received signal will “fade” as 'r varies due to satellite motion. Section 13 16. (a) A sampledata signal derived by taking samples of a continuoustime signal, m(r). is often
represented in terms of an inﬁnite sequence of rectangular pulse signals by multiplication.
That is, xsdﬂ) = mo) i 11(hfm°) nHon If n10) = exp(—n'10)u(:), r = 0.5 s, and to = 2 s, sketch lid“) for l S r s 10 s.
{b} Flattop sample representation of a signal is sometimes preferred over the above scheme. In
this case, the sampledata representation of a continuoustime signal is given by 3‘11“): 3} meta) n(" ‘7’“0). n=—m Sketch this sample—data representation for the same signal and parameters as given in
part (a). Discuss the major difference berveen the two representations. r Problems 39 17. Ideal sampling is represented in two ways depending on whether the signal is considered to be
continuous—time or discretetime.
(a) The continuoustime signal (305070 is to be represented in terms of samples by multiplying
by a train of impulses spaced by 0.1 seconds. Let the impulse train be written as i 50 — 0.1:!) NHoo and show that the ideal impulsesampled waveform is given by no 2 cos(0.27m)5(t ~— 0.1”)
by using appropriate properties of the unit impulse.
(b) Show that the discrete~time representation is the same, except that the continuoustime unit
impulse is replaced by a discretetime unit pulse function, 6[n], appropriately shifted. Sketch
for the signal cos(2 m) sampled each 0.1 seconds '18. Sketch the following signals:
(a) H(0.lr) (b) 1100:) (c) 110 — U2) (d) THU  2W] (e) “[(t — l)f2] + HO  l) 5119. What are the fundamental periods of the signals given below? (Assume units of seconds for the
rvariable.) 3,1,5 (a) sin 5011: (11) cos 607:": (c) cos 70111 ((1) sin 507:? + cos 60411 (e) sin 507:: + cos 70m 0. Given the two complex numbers A = 3 + j3 and B = 10 eprnB). (a) Put A into polar form and B into cartesian form. What is the magnitude of A ? The argument H _ ofA? The real part ofB? The imaginary part ofB? ' (b) Compute their sum. Show as a vector in the complex plane along with A and B. (c) Compute their difference. Show as a vector in the complex plane. (6) Compute their product in two ways: by multiplying both numbers in cartesian form and by
multiplying in polar form. Show that both answers are equivalent. (ﬁe) Compute the quotient ALB in two ways: by dividing with both numbers expressed in carte ;  ilsian form and by dividing with both numbers expressed in polar form. Show that both an ' swers are equivalent. d the periods and fundamental frequencies of the following signals:
' £0) = 2 cos(10m + m6)
Ibo) = 5 cos(l7m  711M)
') ICU) = 3 sin(197n — 118)
Ida) = xa(1) + xb(t)
x50) = _xa(r_) + x4!)
.90) = xb(r) + xr(t) p 40 Ch. 1 / Signal and System Modeling Concepts 112. 113. 114. 115. 116. 1‘17. 118. 1—19. (3) Write the signals of Problem lll as the real part of the sum of rotating phasors.
(b) Write the signals of Problem 1“ as the sum of counterrotating phasors. ((2) Plot the singlesided amplitude and phase spectra for these signals. (d) Plot the doublesided amplitude and phase spectra for these signals. (a) Write the signals given in Problem 19 as the real parts of rotating phasors.
(1)} Write each of the signals given in Problem 19 as onehalf the sum of a rotating phasor and its complex conjugate.
(1:) Sketch the singlesided amplitude and phase spectra of the signals given in Problem 19.
(d) Sketch the doublesided amplitude and phase spectra of the signals given in Problem [9. (a) Express the signal given below in terms of step functions. Sketch it ﬁrst.
ran) = l'l[(t ~ 3)/6] + l'l[(t  4)/2] (11) Express the derivative of the signal given above in terms of unit impulses. Suppose that instead of writing a sinusoid as the real part of a rotating phasor. we agree to use
the convention sinuour + 6) = Im expmwot + 6H}
or
sincwor + 9) = explﬂwof + one; — explﬂwo: + one; (a) What change, if any. will there be to the twosided amplitude spectrum of a signal from the
case where the realpart convention is used? (13) What change, if any. will there be to the twowsided phase spectrum of a signal from the case
where the realpan convention is used? Sketch the following signals: (a) u[(£ — 2)l4] (d) H(~3t + l)
(b) :10 + W3] (6} “[(t F 3V2]
(c) r(2: + 3) Derive expressions for singularity functions aim fort = —4 and i = —5. Generalize to arbitrary
negative values of i. Plot accurately the following signals defined in terms of singularity functions: (3) xxx) = NILd2 — I) (blxb(f}=r(0—r(f"1)*F(f—2)+F(f—3) (c) £0) = 2:0 x80 — 2») (Plot for 0 E r E 8, and use three dots to indicate its semi~infinite
extent.) (d) xd(t) = 2;“ 151::  3n) (Plot for 0 s t s 9 and use three dots to indicate its semiinfinite
extent.) (a) Sketch the signal y(t) = :50 u(t — 2n)u(1 + 2n — r).
(b) Is it periodic? If so, what is its period? If not, why not? (c) Repeat parts (a) and (b) for the signal y(t) = 23”“ at: — 2n)a(l + 2n — t). Problems 41
. 120. Express the signals shown in terms of singularity functions (they are all zero for t < 0):
1,0) _ xv (r) I
2 _ 0 5
(a) _ ('3)
xi, (0 “:2 xdlr)
j/ \E _ [
.v r
0 2.5 5 7.5 0 3 6
(c) ((1) FIGURE P1 20 121. Represent the signals shown in terms of singularity functions. FIGURE P121 Igril'e the signais shown in Figure Pl22 in terms of singularity functions.
Show that et/e 64:) = 6 no) the properties of a delta function in the limit as e —> 0. P 42 Ch. 1 1 Signal and System Modeling Concepts It“) x30) nu} FIGURE P1 22 (b) Show that BXp[‘—12}20'2]IV27T0'2 has the properties of a unit impulse function as a —) 0.
(Hint: Look up the integral of exp( —at3) in a table of deﬁnite integrals.) 124. (a) By plotting the derivative of the function given in Problem l23(b) for or = 0.2 and a = 0.05.
deduce what a unit doublet must “look like.”
(1)) By plotting the second derivative of the function given in Problem 123(b) for the same val
ues of or as given in part (a). deduce what a unit triplet must “look like." 125. In taking derivatives of product functions, one of which is a singularity function, one must exer
cise care. Consider the second derivative of Mr) = e’muﬂ) Blindly carrying out the derivative twice yields
— = ale—“MD — Zoe—“’t‘Xt) + e_“’3(t) Use the fact that
cre‘"'5(r) = ore—“(’80) = 06(1‘) to obtain the correct result. 4. "_ Problems 43
126. Evaluate the following integrals. 10 (a) J cos 2m 30 — 2) d1
5 .
5 (b) I cos 2m so A 2) d:
0
S . (c) I cos 217: so — 0.5m
l] (d) I” (r — 2)26(z — 2) d: (e) r 1250 — 2) dz . Evaluate the following integrals (dots over a symbol denote time derivative). (3) r 8'50 — 2) d:
10 II (mj cos(2m)5(t — 0.5) d:
0 (c) f [e3‘ + cos(2m)]é(:) d: . Find the unspeciﬁed constants. generth as C I, C2. . . . , in the follOWing expressions.
(3) 1060) + C180) + (?.+ C2)8({) = (3 + C980) + 580') + 650)
(b) (3 + C,)8l“l(t) + C280} + C380) = Cﬁmo) + C560) . (3') Sketch the following signals:
(1) 110) = r0 + 2) — 2:10 + r(! — 2)
(2) xztr) = u(r)u(10 — x)
{3) x3(:) = 2M!) + 6(r — 2)
(4) x40) = 2u(t)6(r ~ 2) _
(b) For the signal shown, write an equation in terms of singularity functions. —4 ' ' o 5
FIGURE P129 l' l
. 44 Ch. 1 / Signal and System Modeling Concepts 130. For the signal shown, write an equation in terms of singularity functions. x0) FIGURE P130 131. Evaluate the integrals given below. (a) F 1360 — 3) d:
(b) I (3: + cos 211030 — 5)d (c) r (1 + than — 1.5) d: 132. Write the signals in Figure Pl32 in terms of singularity functions. Section 1—4 133. Sketch the following signals and calculate their energies.
is) e’m‘uﬂ)
(b) no) — u(t 15)
(c) cos 10m u(r)u(2 ‘ t)
(d) r0) # Zd: — l) + r(r — 2) 134. Obtain the energies of the signals in Problem 122. 135. Which of the signals given in Problem [—18 are energy signals? Justify your anSWers.
136. Obtain the average powers of the signals given in Problem 19. 137. Obtain the average powers of the signals given in Problem 1 l 1. 133. Which of the following signals are power signals and which are energy signals? Which are nei
ther? Justify your answers.
(a) u(t) + 5t¢(t — l)  2u(r  2)
(b) u(t) + 5u(t . 1) — 6tt(! — 2)
(c) e ‘ 5’ u(t)
(d) (e— 5' + l)u(t)
(e) (l  8‘ 50W)
(0 "(0
(3) fl!) * Ii! “ 1)
(h) rmttu  3) r Problems d5 x.(r) (b) 1:0.) FIGURE P1 32 72.1939. Given the signal
1:0} = 2 cos(6m  m’3) + 4 sinUOn'I) (a) Is it periodic? if so, ﬁnd its period. (b) Sketch its singlesided amplitude and phase spectra. (c) Write it as the sum of rotating phasors plus their complex conjugates.
(d) Sketch its two~sided amplitude and phase spectra. (B) Show that it is a power signal. . Which of the following signals are energy signals? Find the energies of those that are. Sketch
each signal.
(a) “(1’) ' “(I _ 1)
(b) r(0r(r— l)rf!*2)+r(r3)
(c) texp(—2r)u(:)
. (d) r(1) If!  2)
(e) um — gnu — IO) 46 Ch. 1 ‘/ Signal and System Modeling Concepts 141. Given the following signals:
(1) cos 5m + sin 6m
{2) sin ‘2: + cos m
(3) e ’m'uU)
(4) eZ‘uU)
(a) Which are periodic? Give their periods.
(b) Which are power signals? Compute their average powers.
to) Which are energy signals? Compute their energies. 142. Prove Equation (l84) by starting with (176). 143. Given the signal
x0) = sinZG’m — m'ﬁ) + cos(3m  1713) (3) Sketch its singlesided amplitude and phase spectra.
(b) Sketch its doublesided amplitude and phase spectra after writing it as the sum of complex
conjugate rotating phasors. Section 15
144. Plot the power spectral density of the signal given in Problem 143. 145. Given the signal
x0) = 16 cos(20n'r + 'm‘4) + 6 cos(30m + 17/6) + 4 cos(40m + 17/3) (3) Find and plot its power spectral density.
(b) Compute the power contained in the frequency interval 12 Hz to 22 Hz. Computer Exercises [1. Use the functions given in Section l6 for the step and ramp signals to plot the signal shown in
Problem 130 using MATLAB. 12. Use the functions given in Section [6 for the step and ramp signals to plot the signals shown in
Problem 132 using MATLAB. 13. Lisa the elementary function programs given in Section 1—6 to compute and plot the following:
(a) A step of height 3 starting at r = 3 and going backwards tot = —00;
(b) A signal that starts at t = 1. increases linearly to a value of 2 at t = 2, and is constant thereafter;
(c) A stairstep signal that is 0 for: < 0,jumps to a value of l at t = 0. a value of2 at r = 1,21 value
of 3 at: = 2, a value of4 at: = 3, and stays at 4 thereafter;
(d) A ramp starting at t = .2 and going downward with a slope of —3. 14. (a) Generate a cosine burst of frequency 2 Hz, lasting for 5 seconds; (b) generate a sine burst of
frequency 2 Hz and lasting for ﬁve seconds; (c) combine the results of (a) and (b) to produce a si _ nusoidal burst of frequency 2 Hz, 5 seconds long. and with starting phase at: = 0 of 1714 radians. '
[Hintt recall the trigonometric identity sin(x + y) = 0.5(sin it cos y — cos x sin 3;). Set y = 1144.} ...
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