Questions and Answers of Principles Communications Systems

(a) Given II(t) ↔ sinc(f), find the Fourier transforms of the following signals using the frequency-translation followed by the time-delay theorem. (i) x1(t) = II (t - 1) exp [j4π(t -
By applying appropriate theorems and using the signals defined in Problem 2.28, find Fourier transforms of the following signals:(a) xa (t) = 1/2 x1 (t) + 1/2 x1 (-t)(b) xb (t) = 1/2
Use the superposition, scale-change, and time-delay theorems along with the transform pairs II(t) †” sinc(f), sinc(t) †” II(f), ˆ§ (t), †” sinc2(f), and
Use the Poisson sum formula to obtain the Fourier series of the signal οο 4m ΣπΠ x (t) = П m -ο
Without actually computing them, but using appropriate sketches, tell if the Fourier transforms of the signals given below are real, imaginary, or neither; even, odd, or neither. Give your reasoning
Find and plot the energy spectral densities of the following signals. Dimension your plots fully. Use appropriate Fourier-transforms pairs and theorems.(a) x1(t) = 10e-5tu (t)(b) x2(t) = 10 sine
Evaluate the following integrals using Rayleigh€™s energy theorem (Parseval€™s theorem for Fourier transforms).(a)(b)(c)(d) I = L 00 a2+(2xf )² -∞ I, = [ sinc 2(7 f)df 00 %3D
Obtain and sketch the convolutions of the following signals.(a) y1(t) = e-atu(t) ∗ Il(t - τ), a and τ positive constants(b) y2(t) = [II(t/2) + II(t)] ∗ Il(t)(c) y3(t) = e-α|t| ∗ II(t), a
Find the signals corresponding to the following spectra. Make use of appropriate Fourier-transform theorems.(a) X1 (f) = 2 cos (2πf) II (f) exp (-j4πf) (b) X2 (f) = Λ (f/2) exp
Given the following signals, suppose that all energy spectral components outside the bandwidth |f| ≤ W are removed by an ideal filter, while all energy spectral components within this
(a) Find the Fourier transform of the cosine pulse x(t) = AII (2t / T0) cos (Ï‰0t), where Ï‰0= 2Ï€ / T0Express your answer in terms of a sum of sinc functions. Provide
Provide plots of the following functions of time and find their Fourier transforms. Tell which ones should be real and even functions of f and which ones should be imaginary and odd functions of f.
(a) Obtain the time-average auto correlation function of x(t) = 3 + 6 cos (20πt) + 3 sin (20πt). (Combine the cosine and sine terms into a single cosine with a phase angle.)(b) Obtain the power
Find the power spectral densities and average powers of the following signals.(a) x1 (t) = 2 cos (20πt + π/3)(b) x2 (t) = 3 sin (30πt)(c) x3 (t) = 5 sin (10πt - π/6)(d) x4 (t) = 3 sin (30πt) +
By applying the properties of the auto correlation function, determine whether the following are acceptablefor auto correlation functions. In each case, tell why or why not.(a) R1 (τ) = 2
A filter has amplitude response and phase shift shown in Figure 2.39. Find the output for each of the inputs given below. For which cases is the transmission distortionless? Tell what type of
Find the auto correlation functions of the signals having the following power spectral densities. Also give their average powers.(a) S1 (f) = 4δ (f - 15) + 4δ (f + 15)(b) S2 (f) = 9δ (f - 20) +
Find the auto correlation functions corresponding to the following signals.(a) x1 (t) = 2 cos(10πt + π/3)(b) x2 (t) = 2 sin(10πt + π/3)(c) x3 (t) = Re [3 exp (j10πt) + 4j exp (j10πt)](d) x4 (t)
A system is governed by the differential equation (a, b, and c are non-negative constants) (a) Find H(f) (b) Find and plot |H (f)| and (c) Find and plot |H (f)| and dy (t) dx (t)
For each of the following transfer functions, determine the unit impulse response of the system.(a) (b) (c)(d) H,(f) = 7+ j2nf j2nf 7+ j2¤f Н.(Г) %3
A filter has frequency response function H(f) = II (f / 2B) and input x(t) = 2W sinc (2W t).(a) Find the output y (t) for W < B.(b) Find the output y (t) for W > B.(c) In which case does the
A second-order active band pass filter (BPF), known as a band pass Sallen-Key circuit, is shown in Figure 2.37.(a) Show that the frequency response function of this filter is given
For the two circuits shown in Figure 2.38, determine H(f) and h(f) Sketch accurately the amplitude and phase responses. Plot the amplitude response in decibels. Use a logarithmic frequency axis. R1
Using the Paley-Wiener criterion, show that|H(f)| = exp (-βf2)is not a suitable amplitude response for a causal, linear time-invariant filter.
Determine whether or not the filters with impulse responses given below are BIBO stable. α and f0 are positive constants.(a) h1 (t) = exp(-α |t|) cos (2πf0t)(b) h2 (t) = cos (2πf0t) u(t)(c) h3
Given a filter with frequency response functionH (f) = 5 / 4 + j (2πf)and input x(t) = e-3tu(t), obtain and plot accurately the energy spectral densities of the input and output.
A filter with frequency response functionH(f) = 3II (f / 62)has as an input a half-rectified cosine waveform of fundamental frequency 10 Hz. Determine an analytical expression for the output of the
Another definition of bandwidth for a signal is the 90% energy containment bandwidth. For a signal with energy spectral density G(f) = |X(f)|2, it is given by B90in the relation Obtain
An ideal quadrature phase shifter has frequency response function Find the outputs for the following inputs:(a) x1 (t) = exp (j100Ï€t)(b) x2 (t) = cos (100Ï€t)(c) x3 (t) =
Determine and accurately plot, on the same set of axes, the group delay and the phase delay for the systems with unit impulse responses:(a) h1 (t) = 3e-5t u(t)(b) h2 (t) = 5e-3tu (t) - 2e-5tu
A system has the frequency response functionDetermine and accurately plot the following: (a) The amplitude response(b) The phase response(c) The phase delay(d) The group delay J2лf (8+
The nonlinear system defined by y(t) = x(t) + 0.1x2 (t) has an input signal with the band pass spectrumSketch the spectrum of the output, labeling all important frequencies and amplitudes.
Find the impulse response of an ideal high pass filter with the frequency response function -2л fto Ннp() %3D Но | 1-П 2W
Given the band pass signal spectrum shown in Figure 2.42, sketch spectra for the following sampling rates fsand indicate which ones are suitable.(a) 2B (b) 2.5B (c) 3B (d) 4B (e) 5B (f) 6B X(f) Д -f
(a) Show that the frequency response function of a second-order Butter worth filter iswhere f3 is the 3-dB frequency in hertz.(b) Find an expression for the group delay of this filter. Plot the group
Using appropriate Fourier-transform theorems and pairs, express the spectrum Y(f) of y(t) = x(t) cos (ω0t) + x̂(t) sin (ω0t)in terms of the spectrum X(f) of x(t), where X(f) is low pass with
Given a filter with frequency response functionDetermine and accurately plot the following: (a) The amplitude response; (b) The phase response; (c) The phase delay; (d) The group
Given a nonlinear, zero-memory device with transfer characteristic y(t) = x3(t),Find its output due to the inputx(t) = cos (2πt) + cos (6πt)List all frequency components and tell whether they
Verify the pulse width-bandwidth relationship of Equation (2.234) for the following signals. Sketch each signal and its spectrum.(a) x(t) = A exp (-t2 / 2τ2) (Gaussian pulse) (b) x(t) = A exp
A sinusoidal signal of frequency 1 Hz is to be sampled periodically.(a) Find the maximum allowable time interval between samples.(b) Samples are taken at 1/3-S intervals (i.e., at a rate of fs =
A flat-top sampler can be represented as the block diagram of Figure 2.40.(a) Assuming Ï„ << Ts, sketch the output for a typical x(t).(b) Find the spectrum of the output, Y(f), in
Figure 2.41 illustrates so-called zero-order-hold reconstruction.(a) Sketch y(t) for a typical x(t). Under what conditions is y(t) a good approximation to x(t)?(b) Find the spectrum of y(t) in terms
Determine the range of permissible cutoff frequencies for the ideal low pass filter used to reconstruct the signal x(t) = 10 cos2(600πt) cos(2400πt)Which is sampled at 4500 samples per
Consider the inputx(t) = II(t/τ) cos [2π(f0 + Δf)t], Δf << f0to a filter with impulse responseh(t) = αe-αt cos (2πf0t) u(t)
Following Example 2.30, considerx(t) = 2 cos (52πt)Find x̂ (t), xp (t), x̃ (t), xR (t), and xI (t) for the following cases: (a) f0 = 25 Hz; (b) f0 = 27 Hz; (c) f0 = 10
Assume that the Fourier transform of x(t) is real and has the shape shown in Figure 2.43. Determine and plot the spectrum of each of the following signals:(a) (b)(c) (d) x, (t) =
Show that x(t) and x̂ (t) are orthogonal for the following signals: (a) x1(t) = sin (ω0t)(b) x2(t) = 2cos (ω0t) + sin (ω0t) cos (2 ω0t)(c) x3(t) = A exp (jω0t)
A rectangular conducting plate lies in the xy plane, occupying the region 0 < x < a, 0 < y < b. An identical conducting plate is positioned directly above and parallel to the first, at z
Let V = 10(ρ + 1)z2 cos ϕ V in free space.(a) Let the equipotential surface V = 20 V define a conductor surface. Find the equation of the conductor surface.(b) Find ρ and E at that point on the
A coaxial transmission line has inner and outer conductor radii a and b. Between conductors (a < ρ < b) lies a conductive medium whose conductivity is σ(ρ) = σ0/ρ, where σ0 is a constant.
Given the potential field V = 100xz/(x2 + 4) V in free space:(a) Find D at the surface z = 0.(b) Show that the z = 0 surface is an equipotential surface.(c) Assume that the z = 0 surface is a
Two parallel circular plates of radius a are located at z = 0 and z = d. The top plate (z = d) is raised to potential V0; the bottom plate is grounded. Between the plates is a conducting material
Let V = 20x2yz − 10z2 V in free space.(a) Determine the equations of the equipotential surfaces on which V = 0 and 60 V.(b) Assume these are conducting surfaces and find the surface charge density
Two point charges of −100πμC are located at (2, −1, 0) and (2, 1, 0). The surface x = 0 is a conducting plane.(a) Determine the surface charge density at the origin.(b) Determine ρS at P(0, h,
Let the surface y = 0 be a perfect conductor in free space. Two uniform infinite line charges of 30 nC/m each are located at x = 0, y = 1, and x = 0, y = 2.(a) Let V = 0 at the plane y = 0, and find
A dipole with p = 0.1az μC · m is located at A(1, 0, 0) in free space, and the x = 0 plane is perfectly conducting.(a) Find V at P(2, 0, 1).(b) Find the equation of the 200 V equipotential surface
At a certain temperature, the electron and hole mobilities in intrinsic germanium are given as 0.43 and 0.21 m2/V · s, respectively. If the electron and hole concentrations are both 2.3 × 1019
Electron and hole concentrations increase with temperature. For pure silicon, suitable expressions are ρh = −ρe = 6200T1.5e−7000/T C/m3. The functional dependence of the mobilities on
A semiconductor sample has a rectangular cross section 1.5 by 2.0 mm, and a length of 11.0 mm. The material has electron and hole densities of 1.8 × 1018 and 3.0 × 1015 m−3, respectively. If μe
Two equal but opposite-sign point charges of 3 μC are held x meters apart by a spring that provides a repulsive force given by Fsp = 12(0.5 − x) N. Without any force of attraction, the spring
Two perfect dielectrics have relative permittivities ∈r1 = 2 and ∈r2 = 8. The planar interface between them is the surface x − y + 2z = 5. The origin lies in region 1. If E1 = 100ax + 200ay −
Region 1 (x ≥ 0) is a dielectric with ∈r1 = 2, while region 2(x < 0) has ∈r2 = 5. Let E1 = 20ax − 10ay + 50az V/m.(a) Find D2.(b) Find the energy density in both regions.
Let the cylindrical surfaces ρ = 4 cm and ρ = 9 cmenclose two wedges of perfect dielectrics, ∈r1 = 2 for 0 < ϕ < π/2 and ∈r2 = 5 for π/2 < ϕ < 2π. If E1 = (2000/ρ)aρ V/m,
Given the current density J = −104[sin(2x)e−2yax + cos(2x)e−2yay] kA/m2(a) Find the total current crossing the plane y = 1 in the ay direction in the region 0 < x < 1, 0 < z < 2.(b)
Given J = −10−4(yax + xay)A/m2, find the current crossing the y = 0 plane in the −ay direction between z = 0 and 1, and x = 0 and 2.
Let J = 400 sin θ/(r2 + 4) ar A/m2.(a) Find the total current flowing through that portion of the spherical surface r = 0.8, bounded by 0.1π < θ < 0.3π, 0 < ϕ < 2π.(b) Find the
If volume charge density is given as ρv = (cos ωt)/r2 C/m2 in spherical coordinates, find J. It is reasonable to assume that J is not a function of θ or ϕ.
Let J = 25/ρaρ − 20/(ρ2 + 0.01) az A/m2.(a) Find the total current crossing the plane z = 0.2 in the az direction forρ < 0.4.(b) Calculate ∂ρν/∂t.(c) Find the outward current crossing
In spherical coordinates, a current density J = −k/(r sin θ) aθ A/m2 exists in a conducting medium, where k is a constant. Determine the total current in the az direction that crosses a circular
Assuming that there is no transformation of mass to energy or vice versa, it is possible to write a continuity equation for mass.(a) If we use the continuity equation for charge as our model, what
A truncated cone has a height of 16 cm. The circular faces on the top and bottom have radii of 2 mm and 0.1 mm, respectively. If the material from which this solid cone is constructed has a
(a) Using data tabulated in Appendix C, calculate the required diameter for a 2-m-long nichrome wire that will dissipate an average power of 450 W when 120 V rms at 60 Hz is applied to it.(b)
A large brass washer has a 2-cm inside diameter, a 5-cm outside diameter, and is 0.5 cm thick. Its conductivity is σ = 1.5 × 107 S/m. The washer is cut in half along a diameter, and a voltage is
Two perfectly conducting cylindrical surfaces of length are located at ρ = 3 and ρ = 5 cm. The total current passing radially outward through the medium between the cylinders is 3 A dc.(a)
Two identical conducting plates, each having area A, are located at z = 0 and z = d. The region between plates is filled with a material having z-dependent conductivity, σ(z) = σ0e−z/d, where σ0
A hollow cylindrical tube with a rectangular cross section has external dimensions of 0.5 in. by 1 in. and a wall thickness of 0.05 in. Assume that the material is brass, for which σ = 1.5 × 107
A positive point charge of magnitude q1 lies at the origin. Derive an expression for the incremental work done in moving a second point charge q2 through a distance dx from the starting position (x,
If E = 120aρ V/m, find the incremental amount of work done in moving a 50-μC charge a distance of 2 mm from(a) P(1, 2, 3) toward Q(2, 1, 4);(b) Q(2, 1, 4) toward P(1, 2, 3).
An electric field in free space is given by E = xax + yay + zaz V/m. Find the work done in moving a 1-μC charge through this field(a) From (1, 1, 1) to (0, 0, 0);(b) From (ρ = 2, ϕ = 0) to (ρ =
Compute the value of ʃPA G· dL for G = 2yax with A(1,−1, 2) and P(2, 1, 2) using the path(a) Straight-line segments A(1,−1, 2) to B(1, 1, 2) to P(2, 1, 2);(b) Straight-line segments A(1,−1,
An electric field in free space is given as E = x âx + 4z ây + 4y âz. Given V(1, 1, 1) = 10 V, determine V(3, 3, 3).
Let G = 3xy2ax + 2zay Given an initial point P(2, 1, 1) and a final point Q(4, 3, 1), find ʃG· dL using the path(a) Straight line: y = x − 1, z = 1;(b) Parabola: 6y = x2 + 2, z = 1.
Given E = −xax + yay,(a) Find the work involved in moving a unit positive charge on a circular arc, the circle centered at the origin, from x = a to x = y = a/√2;(b) Verify that the work done in
A uniform surface charge density of 20 nC/m2 is present on the spherical surface r = 0.6 cm in free space.(a) Find the absolute potential at P(r = 1 cm, θ = 25◦, ϕ = 50◦).(b) Find VAB, given
A sphere of radius a carries a surface charge density of ρs0 C/m2.(a) Find the absolute potential at the sphere surface.(b) A grounded conducting shell of radius b where b > a is now positioned
Let a uniform surface charge density of 5 nC/m2 be present at the z = 0 plane, a uniform line charge density of 8 nC/m be located at x = 0, z = 4, and a point charge of 2μC be present at P(2, 0, 0).
In spherical coordinates, E = 2r/(r2 + a2)2ar V/m. Find the potential at any point, using the reference(a) V = 0 at infinity;(b) V = 0 at r = 0;(c) V = 100 V at r = a.
Three identical point charges of 4 pC each are located at the corners of an equilateral triangle 0.5 mm on a side in free space. How much work must be done to move one charge to a point equidistant
Given the electric field E = (y + 1)ax + (x − 1)ay + 2az find the potential difference between the points(a) (2,−2,−1) and (0, 0, 0);(b) (3, 2,−1) and (−2,−3, 4).
Two uniform line charges, 8 nC/m each, are located at x = 1, z = 2, and at x = −1, y = 2 in free space. If the potential at the origin is 100 V, find V at P(4, 1, 3).
A spherically symmetric charge distribution in free space (with 0 < r < ∞) is known to have a potential function V(r ) = V0a2/r2, where V0 and a are constants.(a) Find the electric field
Uniform surface charge densities of 6 and 2 nC/m2 are present at ρ = 2 and 6 cm, respectively, in free space. Assume V = 0 at ρ = 4 cm, and calculate V at(a) ρ = 5 cm;(b) ρ = 7 cm.
Find the potential at the origin produced by a line charge ρL = kx/(x2 + a2) extending along the x axis from x = a to + ∞, where a > 0. Assume a zero reference at infinity.
The annular surface 1 cm < ρ < 3 cm, z = 0, carries the nonuniform surface charge density ρs = 5ρ nC/m2. Find V at P(0, 0, 2 cm) if V = 0 at infinity.
In a certain medium, the electric potential is given bywhere Ï0 and a are constants.(a) Find the electric field intensity, E.(b) Find the potential difference between the points x = d and
Let V = 2xy2z3 + 3 ln(x2 + 2y2 + 3z2) V in free space. Evaluate each of the following quantities at P(3, 2,−1)(a) V;(b) |V|;(c) E;(d) |E|;(e) aN;(f) D.
A line charge of infinite length lies along the z axis and carries a uniform linear charge density of Ï„“C/m. A perfectly conducting cylindrical shell, whose axis is the z axis,
It is known that the potential is given as V = 80ρ0.6 V. Assuming free space conditions, find.(a) E;(b) the volume charge density at ρ = 0.5 m;(c) The total charge lying within the closed surface
A certain spherically symmetric charge configuration in free space produces an electric field given in spherical coordinates bywhere Ï0 is a constant.(a) Find the charge density as a
Within the cylinder ρ = 2, 0 < z < 1, the potential is given by V = 100 + 50ρ + 150ρ sin ϕV.(a) Find V, E,D, and ρν at P(1, 60◦, 0.5) in free space.(b) How much charge lies within the

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