ECE_308_problem_copy_1

# ECE_308_problem_copy_1 - P-38 Ch 1 Signal and System...

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Unformatted text preview: P -38 Ch. 1 / Signal and System Modeling Concepts (b) For the trapezoidal integration rule given in recursive form in Example 1-3. 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 1-3. 1-3. 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. 1-4. Rework Example 1-! with the acceleration proﬁle shown. Sketch the resulting velocities. am, nuts2 20 0 IO 20 30 40 FIGURE P1-4 (a) Assume an initial velocity of zero. (b) Assume an initial velocity of 1500 this. 1-5. In the satellite communications example. Example 1-4, 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 — [an-l 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 1-3 1-6. (a) A sample-data signal derived by taking samples of a continuous-time 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°) nH-on If n10) = exp(—n'10)u(:), r = 0.5 s, and to = 2 s, sketch lid“) for -l S r s 10 s. {b} Flat-top sample representation of a signal is sometimes preferred over the above scheme. In this case, the sample-data representation of a continuous-time 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 1-7. Ideal sampling is represented in two ways depending on whether the signal is considered to be continuous—time or discrete-time. (a) The continuous-time 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:!) NH-oo and show that the ideal impulse-sampled 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 continuous-time unit impulse is replaced by a discrete-time unit pulse function, 6[n], appropriately shifted. Sketch for the signal cos(2 m) sampled each 0.1 seconds '1-8. 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 r-variable.) 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 eprn-B). (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 — 11-8) Ida) = xa(1) + xb(t) x50) = _xa(r_) + x4!) .90) = xb(r) + xr(t) p 40 Ch. 1 / Signal and System Modeling Concepts 1-12. 1-13. 1-14. 1-15. 1-16. 1‘17. 1-18. 1—19. (3) Write the signals of Problem l-ll 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 single-sided amplitude and phase spectra for these signals. (d) Plot the double-sided amplitude and phase spectra for these signals. (a) Write the signals given in Problem 1-9 as the real parts of rotating phasors. (1)} Write each of the signals given in Problem 1-9 as one-half the sum of a rotating phasor and its complex conjugate. (1:) Sketch the single-sided amplitude and phase spectra of the signals given in Problem 1-9. (d) Sketch the double-sided 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; — expl-ﬂwo: + one; (a) What change, if any. will there be to the two-sided amplitude spectrum of a signal from the case where the real-part convention is used? (13) What change, if any. will there be to the twowsided phase spectrum of a signal from the case where the real-pan 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) = NIL-d2 — 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 semi-infinite 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 . 1-20. 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 1-21. Represent the signals shown in terms of singularity functions. FIGURE P1-21 Igril'e the signais shown in Figure Pl-22 in terms of singularity functions. Show that e-t/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.) 1-24. (a) By plotting the derivative of the function given in Problem l-23(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 1-23(b) for the same val- ues of or as given in part (a). deduce what a unit triplet must “look like." 1-25. 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 1-26. 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 [e-3‘ + 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 P1-29 l' l . 44 Ch. 1 / Signal and System Modeling Concepts 1-30. For the signal shown, write an equation in terms of singularity functions. x0) FIGURE P1-30 1-31. Evaluate the integrals given below. (a) F 1360 — 3) d: (b) I (3: + cos 211030 — 5)d (c) r (1 + than — 1.5) d: 1-32. Write the signals in Figure Pl-32 in terms of singularity functions. Section 1—4 1-33. 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) 1-34. Obtain the energies of the signals in Problem 1-22. 1-35. Which of the signals given in Problem [—18 are energy signals? Justify your anSWers. 1-36. Obtain the average powers of the signals given in Problem 1-9. 1-37. Obtain the average powers of the signals given in Problem 1- l 1. 1-33. 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 single-sided 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(0-r(r— l)-rf!*2)+r(r-3) (c) texp(—2r)u(:) -. (d) r(1)- If! - 2) (e) um — gnu — IO) 46 Ch. 1 ‘/ Signal and System Modeling Concepts 1-41. 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. 1-42. Prove Equation (l-84) by starting with (1-76). 1-43. Given the signal x0) = sinZG’m — m'ﬁ) + cos(3m - 1713) (3) Sketch its single-sided amplitude and phase spectra. (b) Sketch its double-sided amplitude and phase spectra after writing it as the sum of complex conjugate rotating phasors. Section 1-5 1-44. Plot the power spectral density of the signal given in Problem 1-43. 1-45. 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 l-6 for the step and ramp signals to plot the signal shown in Problem 1-30 using MATLAB. 1-2. Use the functions given in Section [-6 for the step and ramp signals to plot the signals shown in Problem 1-32 using MATLAB. 1-3. 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. 1-4. (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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