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- Question : 1 - (a) Derive an expression for the distance traveled by the rocket of Example 1-1 valid up to the time of bumout. (b) Suppose a voltage analog to distance is to be provided by another operational amplifier inte
- Question : 2 - (a) For the rectangular integration rule given in recursive form in Example 1-3, show that the nonrecursive form up through time NT is
- Question : 3 - (b) For the trapezoidal integration rule given in recursive form in Example 1-3, show that the nonrecursive form up through time NT is
- Question : 4 - Rework Example 1-1 if the acceleration profile for the rocket is and zero otherwise. Assume the same parameter values as given in Example 1-1. Find and plot the velocity profile.
- Question : 5 - Rework Example 1-1 with the acceleration profile shown. Sketch the resulting velocities.(a) Assume an initial velocity of zero. ' S (b) Assume an initial velocity of 1500 m/s.
- Question : 6 - In the satellite communications example, Example 1-4, let the transmitted signal be x(t) = cos<oQt. (a) Show that the received signal can be put into the form (b) Plot the envelope of the cosine in the above equation versus a>or for w/3 = 0.1. Note that the received signal will
- Question : 7 - (a) A sample-data signal derived by taking samples of a continuous-time signal, m(t), is often represented in terms of an infinite sequence of rectangular pulse signals by multiplication.
- Question : 8 - That is,(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 Sketch this sample-data representation for the same signal and parameters as given in part (a). Discuss the major difference between the two representations.
- Question : 9 - 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 cos(2rrZ) 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 and show that the ideal impulse-sampled waveform is given byby 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, <S[n J, appropriately shifted. Sketch for the signal cos(2tt/) sampled each 0.1 seconds
- Question : 10 - ketch the following signals: (a) 11(0.1/) (b) 11(10/) (c) ri(z - 1 /2 ) (d) n((r - 2)/5] (e) n i(/ - l)/2] + II(/ - 1)
- Question : 11 - What are the fundamental periods of the signals given below? (Assume units of seconds for /-variable.) (a) sin 5O7rt (b) cos 60/7/ (c) cos 70/rt (d) sin 50/7/ + cos 60 77/ (e) sin 50?// + cos 70/rt
- Question : 12 - Given the two complex numbers A
- Question : 13 - Find the periods and fundamental frequencies of the following signals: (a) xa(t) = 2 cos(10t7/ + tt/6) (b) xb(t) = 5 cos(17tt/ - tt/4) (c) xc(t) = 3 sin(197r/
- Question : 14 - (e) xt (r) = xa(t) + xc(t)
- Question : 15 - (0 x^t) = Xb(t) + xc(t)
- Question : 16 - (a) Write the signals of Problem l-l 1 as the real part of the sum of rotating phasors. (b) Write the signals of Problem 1-11 as the sum of counterrotating phasors. (c) Plot the single-sided amplitude and phase spectra for these signals. (d) Plot the double-sided amplitude and phase spectra for these signals.
- Question : 17 - (a) Write the signals given in Problem 1-9 as the real parts of rotating phasors. (b) Write each of the signals given in Problem 1-9 as one-half the sum of a rotating phasor and its complex conjugate. (c) 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 1-9.
- Question : 18 - (a) Express the signal given below in terms of step functions. Sketch it first. (b) Express the derivative of the signal given above in terms of unit impulses.
- Question : 19 - Suppose that instead of writing a sinusoid as the real part of a rotating phasor, we agree to usd the convention (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? (b) What change, if any, will there be to the two-sided phase spectrum of a signal from the case where the real-part convention is used?
- Question : 20 - Sketch the following signals: (a) u[(r - 2)/4] (d) n ( - 3 r + 1) (b) r[(t + l)/3] (e) FI[(r - 3)/2]
- Question : 21 - (c) r (-2 t + 3)
- Question : 22 - Derive expressions for singularity functions u/t) for i =
- Question : 23 - Plot accurately the following signals defined in terms of singularity functions:
- Question : 24 - (a) xa(t) = r(t)u(2 - t)
- Question : 25 - (b) x
- Question : 26 - (a) Sketch the signal y(t) = u(t
- Question : 27 - Express the signals shown in terms of singularity functions (they are all zero for t < 0):
- Question : 28 - Represent the signals shown in terms of singularity functions.
- Question : 29 - Write the signals shown in Figure Pl-22 in terms of singularity functions.
- Question : 30 - (a) Show that has the properties of a delta function in the limit as e -> 0 (b) Show that exp[
- Question : 31 - (a) By plotting the derivative of the function given in Problem 1 -23(b) for a = 0.2 and a = 0.05, deduce what a unit doublet must
- Question : 32 - In taking derivatives of product functions, one of which is a singularity function, one must exer
- Question : 33 - Evaluate the following integrals.f to js (b) ( cos 2irt 8(t
- Question : 34 - Evaluate the following integrals (dots over a symbol denote time derivative).
- Question : 35 - (a) I e3
- Question : 36 - ?J
- Question : 37 - f10
- Question : 38 - (b) I cos(2-7rt)8(t
- Question : 39 - Jo
- Question : 40 - (c) I [e*3' + cos(2rrt)]S(t) dt
- Question : 41 - Find the unspecified constants, denoted as C,, C2, . . . . in the following expressions. (a) 10S(r) + C,8(t) + (2 + C2)8(t) = (3 + C3)5(r) + 58(t) + 68(t) (b) (3 + C}) 9 ^ t) + C28(t) + C35(t) = C48<3>(t) + C58(t)
- Question : 42 - (a) Sketch the following signals: (1) x,(t) = r(t + 2) - 2 d 0 + r(t - 2) (2) x2(t) =
- Question : 43 - For the signal shown, write an equation in terms of singularity functions.
- Question : 44 - Evaluate the integrals given below.
- Question : 45 - (a) J tJ8(t
- Question : 46 - (b) I (3z + cos 2irt)8(t - 5)d J -00
- Question : 47 - (c) f (1 + t 2)S(t - 1.5) dt
- Question : 48 - ?' -oo
- Question : 49 - Write the signals in Figure P l-32 in terms of singularity functions.
- Question : 50 - Sketch the following signals and calculate their energies.
- Question : 51 - (a) e_ |0 'u(0 (b) u(t) - u(t - 15) (c) cos 10-rrt u(t)u(2
- Question : 52 - (d) rtf) - 2r(r - 1) + rtf - 2)
- Question : 53 - Obtain the energies of the signals in Problem 1-22.
- Question : 54 - Which of the signals given in Problem 1-18 are energy signals? Justify your answers.
- Question : 55 - Obtain the average powers of the signals given in Problem 1-9.
- Question : 56 - Obtain the average powers of the signals given in Problem 1-11.
- Question : 57 - Which of the following signals are power signals and which are energy signals? Which arc nei
- Question : 58 - (c) e ~ 5'utf)
- Question : 59 - (d) (e~ * + l)u(r)
- Question : 60 - (e) (1
- Question : 61 - (g) r(t) - rtf - 1) (h) r V4
- Question : 62 - Given the signal
- Question : 63 - xtf) = 2 cos(6rrt - tt/3) + 4 sinflOrrt)
- Question : 64 - (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. (e) Show that it is a power signal.
- Question : 65 - Which of the following signals are energy signals ? Find the energies of those that are. Sketch each signal.
- Question : 66 - (a) u(t) - u ( t - l )
- Question : 67 - 0>) rtf) - rtf - 1) - r(t - 2) + rtf - 3) (c) texp(-2r)u(t)
- Question : 68 - (d) r(r) ~ r ( t - 2)
- Question : 69 - (e) utf) - - 10)
- Question : 70 - Given the following signals:
- Question : 71 - (1) cos 5irt + sin 6-rrt
- Question : 72 - (2) sin 2t + cos nt
- Question : 73 - (3) e-'M O (4) e*u(f)
- Question : 74 - (a) Which are periodic? Give their periods. (b) Which are power signals? Compute their average powers. (c) Which are energy signals? Compute their energies.
- Question : 75 - Prove Equation (1-84) by starting with (1-76).
- Question : 76 - Given the signal (a) 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.
- Question : 77 - Plot the power spectral density of the signal given in Problem 1-43.
- Question : 78 - Given the signal
- Question : 79 - x(t) = 16 cos(207rt + rr/4) + 6 cos(307rt + tt/6) + 4 cos(40-m + ir/3) (a) Find and plot its power spectral density. (b) Compute the power contained in the frequency interval 12 Hz to 22 Hz.
- Question : 80 - Use the functions given in Section 1-6 for the step and ramp signals to plot the signal shown in Problem 1-30 using Matlab.
- Question : 81 - Use the functions given in Section 1-6 for the step and ramp signals to plot the signals shown in Problem 1-32 using Matlab.
- Question : 82 - Use the elementary function programs given in Section 1-6 to compute and plot the following. (a) A step of height 3 starting at t - 3 and going backwards to t = (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 t < 0, jumps to a value of 1 at r = 0, a value of 2 at t = 1, a value of 3 at t = 2, a value of 4 at t = 3, and stays at 4 thereafter; (d) A ramp starting at t = 2 and going downward with a slope of -3 .
- Question : 83 - (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
- Question : 84 - a) Write a Matlab function to generate a sequence of impulses spaced by an arbitrary amount t_ re p , lasting for t_ w id th and centered on t = 0. Call it cmb_f n (t . t_ r e p . t_ w id th . d e lta ) (b) Generate an impulse comb with spacing between impulses of 1.25 seconds and containing 5 impulses starting at z = 0.
- Question : 85 - (a) Write a Matlab function to generate a unit parabola singularity function. (b) Write a Matlab function to generate a unit cubic singularity function. (c) Use them to generate a plots of the signals shown in Problem 1-21.

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