Questions and Answers of Principles Communications Systems

It is desired to transmit data ISI free at 10 kbps for which pulses with a raised-cosine spectrum are used. If the channel bandwidth is limited to 5 kHz, ideal low pass, what is the allowed roll-off
Two semi-infinite filaments on the z axis lie in the regions − ∞ < z < −a and a < z < ∞. Each carries a current I in the az direction.(a) Calculate H as a function of ρ and ϕ at
By appropriate solution of Laplace’s and Poisson’s equations, determine the absolute potential at the center of a sphere of radius a, containing uniform volume charge of density ρ0. Assume
The two conducting planes illustrated in Figure 6.14 are defined by 0.001 < Ï < 0.120 m, 0 < z < 0.1 m, Ï• = 0.179 and 0.188 rad. The medium surrounding the planes is
Coaxial conducting cylinders are located at ρ = 0.5 cm and ρ = 1.2 cm. The region between the cylinders is filled with a homogeneous perfect dielectric. If the inner cylinder is at 100 V and the
The derivation of Laplace’s and Poisson’s equations assumed constant permittivity, but there are cases of spatially varying permittivity in which the equations will still apply. Consider the
The functions V1(ρ, ϕ, z) and V2(ρ, ϕ, z) both satisfy Laplace’s equation in the region a < ρ < b, 0 ≤ v < 2π, −L < z < L; each is zero on the surfaces ρ = b for −L <
Show that in a homogeneous medium of conductivity σ, the potential field V satisfies Laplace’s equation if any volume charge density present does not vary with time.
Two conducting plates, each 3 × 6 cm, and three slabs of dielectric, each 1 × 3 × 6 cm, and having dielectric constants of 1, 2, and 3, are assembled into a capacitor with d = 3 cm. Determine the
A continuous data signal is quantized and transmitted using a PCM system. If each data sample at the receiving end of the system must be known to within ±0.20% of the peak-to-peak full-scale value,
The imperfect second-order PLL is defined as a PLL with the loop filterF(s) = s + α / s + λαin which λ is the offset of the pole from the origin relative to the zero location. In practical
In this problem we wish to develop a base band (low pass equivalent model) for a Costas PLL. We assume that the loop input is the complex envelope signal x̃(t) = Ac m(t) ejϕ(t)and that the VCO
A Costas PLL operates with a small phase error so that sin ψ ≈ ψ ≈ and cos ψ ≈ 1 Assuming that the low pass filter preceding the VCO is modeled as a/(s + α), where α is an arbitrary
Verify (4.120) by showing that Kte - Kttu(t) satisfies all properties of an impulse function in the limit as Kt → ∞.
Using a single PLL, design a system that has an output frequency equal to 7/3 f0, Where f0 is the input frequency. Describe fully, by sketching, the output of the VCO for your design. Draw the
By adjusting the values of R,L, and C in Figure 4.38, design a discriminator for a carrier frequency of 100 MHz, assuming that the peak frequency deviation is 4 MHz. What is the discriminator
Consider the FM discriminator shown in Figure 4.38. The envelope detector can be considered ideal with an infinite input impedance. Plot the magnitude of the transfer function E(f) / Xr (f).
A narrow band FM signal has a carrier frequency of 110 kHz and a deviation ratio of 0.05. The modulation bandwidth is 10 kHz. This signal is used to generate a wide band FM signal with a deviation
A sinusoidal message signal has a frequency of 250 Hz. This signal is the input to an FM modulator with an index of 8. Determine the bandwidth of the modulator output if a power ratio, Pr, of 0.8 is
An FM modulator is followed by an ideal band pass filter having a center frequency of 500 Hz and a bandwidth of 70 Hz. The gain of the filter is 1 in the pass band. The unmodulated carrier is given
Prove that Jn(Î²) can be expressed as and use this result to show thatJ-n(Î²) = (-1)n Jn(Î²) л cos(B sin x — пх)dx J„(B) = п
By making use of (4.30) and (4.39), show that ο0 Σ)-1 n=-00
An audio signal has a bandwidth of 15 kHz. The maximum value of |m(t)| is10 V. This signal frequency modulates a carrier. Estimate the peak deviation and the bandwidth of the modulator output,
An FM modulator has fc = 2000 Hz and fd = 20 Hz/V. The modulator has input m(t) = 5 cos[2π(10)t].(a) What is the modulation index?(b) Sketch, approximately to scale, the magnitude spectrum of
An FM modulator with fd= 10 Hz/V. Plot the frequency deviation in Hz and the phase deviation in radians for the three message signals shown in Figure 4.37.Figure 4.37 т(0) 4 3 1 3 4 т() т() 3 3 2
Repeat the preceding problem assuming that m(t) is the triangular pulse 4Î›[1/3(t - 6)].Data From Problem 11An FM modulator has outputwhere fd = 20 Hz/V. Assume that m(t) is the
An FM modulator has outputwhere fd = 20 Hz/V. Assume that m(t) is the rectangular pulse m(t) = 4II [1/8(t - 4)](a) Sketch the phase deviation in radians.(b) Sketch the frequency deviation
A transmitter uses a carrier frequency of 1000 Hz so that the unmodulated carrier is Ac cos(2πfct). Determine both the phase and frequency deviation for each of the following transmitter outputs:(a)
Determine and sketch the spectrum (amplitude and phase) of an angle-modulated signal assuming that the instantaneous phase deviation is ϕ(t) = β sin(2πfmt). Also assume β = 10, fm = 30 Hz, and fc
Given that J0(5) = -0.178 and that J1(5) = -0.328, determine J3(5) and J4(5).
Modify the simulation program given in Computer Example 4.4 by replacing the trapezoidal integrator by a rectangular integrator. Show that for sufficiently high sampling frequencies the two PLLs give
The power of an un-modulated carrier signal is 50 W and the carrier frequency is fc = 40 Hz. A sinusoidal message signal is used to FM modulate it with index β = 10. The sinusoidal message signal
Compute the single-sided amplitude and phase spectra of xc3 (t) = A sin[2Ï€fct + Î² sin(2Ï€fmt)]and xc4 (t) = Ac sin[2Ï€fct + Î²
We previously computed the spectrum of the FM signal defined by xc1 (t) = Ac cos[2πfct + β sin(2πfmt)]Now assume that the modulated signal is given byxc2 (t) = Ac cos[2πfct + β
Redraw Figure 4.4 assuming m(t) = A sin (2Ï€fmt + Ï€/6)Figure 4.4 R(t) fm fm fm Ф() R(t) Re Re Ac Ac fm (a) if fc +fm fe - fm fc fe-fm fe fe+ fm (b) fe +fm .f fe-fm fe fe-fm fe
Repeat the preceding problem for kp-1/2Ï? and 3/8Ï?. Data From problem 1 Let the input to a phase modulator be m(t) = u(t - t0), as shown in Figure 4.1(a). Assume that the unmodulated carrier is
The purpose of this exercise is to demonstrate the properties of SSB modulation. Develop a computer program to generate both upper-side band and lower-side band SSB signals and display both the
Using the same message signal and value for fm used in the preceding computer exercise, show that carrier reinsertion can be used to demodulate an SSB signal. Illustrate the effect of using a
Assume that a DSB signal xc (t) = Acm(t) cos (2πfct + ϕ0) is demodulated using the demodulation carrier 2 cos [2πfct + θ(t)]. Determine, in general, the demodulated output yD (t). Let
A message signal is given byand the carrier is given by c(t) = 100 cos(200Ï€t)Write the transmitted signal as a Fourier series and determine the transmitted power. 5 5 m(t) = > 10
Design an envelope detector that uses a full-wave rectifier rather than the half-wave rectifier shown in Figure 3.3. Sketch the resulting wave forms, as was done in for a half-wave rectifier. What
In this computer exercise we investigate the properties of VSB modulation. Develop a computer program (using MATLAB) to generate and plot a VSB signal and the corresponding amplitude spectrum. Using
Three message signals are periodic with period T, as shown in Figure 3.32. Each of the three message signals is applied to an AM modulator. For each message signal, determine the modulation
Using MATLAB simulate delta modulation. Generate a signal, using a sum of sinusoids, so that the bandwidth is known. Sample at an appropriate sampling frequency (no slope overload). Show the stair
In Example 3.1 we determined the minimum value of m(t) using MATLAB. Write a MATLAB program that provides a complete solution for Example 3.1. Use the FFT for finding the amplitude and phase spectra
The positive portion of the envelope of the output of an AM modulator is shown in Figure 3.33. The message signal is a waveform having zero DC value. Determine the modulation index, the carrier
Sketch Figure 3.20 for the case where fLO= fc- fIF. Desired signal Local oscillator fi+ f2=f LO Signal at mixer ini output 2=fF 2f +f2 Image signal fi + 2f2 = fe+ f IF Image signal at mixer output
Using a sum of sinusoids as the sampling frequency, sample and generate a PAM signal. Experiment with various values of τfs. Show that the message signal is recovered by low pass filtering. A
A message signal is a square wave with maximum and minimum values of 8 and -8 V, respectively. The modulation index α = 0.7 and the carrier amplitude Ac = 100 V. Determine the power in the side
In this problem we examine the efficiency of AM for the case in which the message signal does not have symmetrical maximum and minimum values. Two message signals are shown in Figure 3.34. Each is
An AM modulator operates with the message signalm(t) = 9 cos (20πt) - 8 cos (60πt)The unmodulated carrier is given by 110 cos(200πt) and the system operates with an index of 0.8.(a) Write the
Rework Problem 3.8 for the message signalm(t) = 9 cos(20πt) + 8 cos(60πt)Data from Problem 3.8(a) Write the equation for mn(t), the normalized signal with a minimum value of -1.(b) Determine
An AM modulator has outputxc (t) = 40 cos [20π(200)t] + 5 cos[2π(180)t] + 5 cos[2π(220)t]Determine the modulation index and the efficiency.
An AM modulator has outputxc(t) = A cos[2π(200)t] + B cos[2π(180)t] +B cos[2π(220)t]The carrier power is P0 and the efficiency is Eff. Derive an expression for Eff in terms of P0, A, and B.
An AM modulator has outputxc(t) = 25 cos[2π(150)t] + 5 cos[2π(160)t] + 5 cos[2π(140)t]Determine the modulation index and the efficiency.
An AM modulator is operating with an index of 0.8.The modulating signal ism(t) = 2 cos(2πfmt) + cos(4πfmt) + 2 cos(10πfmt)(a) Sketch the spectrum of the modulator output showing the weights of all
Consider the system shown in Figure 3.35. Assume that the average value of m(t) is zero and that the maximum value of |m(t)| is M. Also assume that the square-law device is defined by y(t) = 4x (t) +
Assume that a message signal is given bym(t) = 4 cos(2πfmt) + cos(4πfmt)Calculate an expression forxc(t) = 1/2 Acm(t) cos(2πfmt) ± 1/2 Acm̂ (t) sin (2πfct)for Ac = 10. Show, by sketching the
Redraw Figure 3.10 to illustrate the generation of upper-side band SSB. Give the equation defining the upper side band filter. Complete the analysis by deriving the expression for the output of an
Squaring a DSB or AM signal generates a frequency component at twice the carrier frequency. Is this also true for SSB signals? Show that it is or is not.
Prove analytically that carrier reinsertion with envelope detection can be used for demodulation of VSB.
Figure 3.36 shows the spectrum of a VSB signal. The amplitude and phase characteristics are the same as described in Example 3.3. Show that upon coherent demodulation, the output of the demodulator
A mixer is used in a short-wave superheterodyne receiver. The receiver is designed to receive transmitted signals between 10 and 30 MHz. High-side tuning is to be used. Determine an acceptable IF
A superheterodyne receiver uses an IF frequency of 455 kHz. The receiver is tuned to a transmitter having a carrier frequency of 1100 kHz. Give two permissible frequencies of the local oscillator and
A DSB signal is squared to generate a carrier component that may then be used for demodulation. (A technique for doing this, namely the phase-locked loop, will be studied in the next chapter.) Derive
A continuous-time signal is sampled and input to a holding circuit. The product of the holding time and the sampling frequency is τ fs. Plot the amplitude response of the required equalizer as a
A continuous data signal is quantized and transmitted using a PCM system. If each data sample at the receiving end of the system must be known to within ±0.25% of the peak-to-peak full-scale value,
Five messages band limited to W, W, 2W, 5W and 7W, Hz, respectively, are to be time-division multiplexed. Devise a sampling scheme requiring the minimum sampling frequency.
A delta modulator has the message signalm (t) = 3 sin 2π(10)t + 4 sin 2π(20)tDetermine the minimum sampling frequency required to prevent slope overload, assuming that the impulse weights δ0 are
Five messages band limited to W, W, 2W, 4W, and 4W Hz,are to be time-division multiplexed. Devise a commutator configuration such that each signal is periodically sampled at its own minimum rate and
Repeat the preceding problem assuming that the commutator is run at twice the minimum rate. What are the advantages and disadvantages of doing this?Repeat the preceding problemFive messages band
In an FDM communication system, the transmitted base band signal isx(t) = m1(t) cos(2πf1t) + m2(t) cos(2πf2t)This system has a second-order non-linearity between transmitter output and receiver
Write a computer program to evaluate the coefficients of the complex exponential Fourier series of a signal by using the fast Fourier transform (FFT). Check it by evaluating the Fourier series
Generalize the computer program of Computer Example 2.1 to evaluate the coefficients of the complex exponential Fourier series of several signals. Include a plot of the amplitude and phase spectrum
Sketch the single-sided and double-sided amplitude and phase spectra of the following signals:(a) x1(t) = 10 cos(4πt + π/8) + 6 sin(8πt + 3π/4)(b) x2(t) = 8 cos(2πt + π/3) + 4 cos(6πt +
Writea computer program to sum the Fourier series for the signals given in Table 2.1. The number of terms in the Fourier sum should be adjustable so that one may study the convergence of each Fourier
A signal has the double-sided amplitude and phase spectra shown in Figure 2.33. Write a time-domain expression for the signal. |Phase |Amplitude 4 -4 -2 4 4
The sum of two or more sinusoids may or may not be periodic depending on the relationship of their separate frequencies. For the sum of two sinusoids, let the frequencies of the individual terms be
Sketch the single-sided and double-sided amplitude and phase spectra of (a) x1 (t) = 5 cos (12πt - π/6)(b) x2 (t) = 3 sin(12πt) + 4 cos (16πt) (c) x3 (t) = 4 cos (8πt) cos
(a) Show that the function Î´Îµ(t) sketched in Figure 2.4(b) has unity area.Figure 2.4 (b)(b) Show thatÎ´Îµ(t) = e-1e-t/Îµu(t)has unity
Write a computer program to find the bandwidth of a lowpass energy signal that contains a certain specified percentage of its total energy, for example, 95%. In other words, write a program to find
Use the properties of the unit impulse function given after (2.14) to evaluate the following relations.(a)(b) 10+ means just to the right of 10; -10- means just to the left of
Which of the following signals are periodic and which are aperiodic? Find the periods of those that are periodic. Sketch all signals.(a) xa (t) = cos (5Ï€t) + sin
Write the signal x(t) = cos(6πt) + 2 sin(10πt) as(a) The real part of a sum of rotating phasors.(b) A sum of rotating phasors plus their complex conjugates.(c) From your results in parts (a) and
Find the normalized power for each signal below that is a power signal and the normalized energy for each signal that is an energy signal. If a signal is neither a power signal nor an energy signal,
Classify each of the following signals as an energy signal or as a power signal by calculating E, the energy, or P, the power (A, B, θ, ω, and τ are positive constants).(a) x1(t) = A| sin (ωt +
Find the powers of the following periodic signals. In each case provide a sketch of the signal and give its period.(a) x1 (t) = 2 cos (4Ï€t - Ï€/3)(b) (c) (d)
For each of the following signals, determine both the normalized energy and power. Tell which are power signals, which are energy signals, and which are neither.(Note: 0 and ∞ are possible
Show that the following are energy signals. Sketch each signal.(a) x1(t) = II(t/12) cos (6Ï€t)(b) x2(t) = e-|t|/3(c) x3(t) = 2u(t) - 2u(t - 8)(d) r1-10 u (2) da u(1) dà + S x4(1) =
Find the energies and powers of the following signals (note that 0 and ˆž are possible answers). Tell which are energy signals and which are power signals.(a) x1(t) = cos(10Ï€t)
Using the uniqueness property of the Fourier series, find exponential Fourier series for the following signals (f0 is an arbitrary frequency):(a) x1(t) = sin2(2πf0t)(b) x2(t) = cos(2πf0t) +
Expand the signal x(t) = 2t2 in a complex exponential Fourier series over the interval |t| ≤ 2. Sketch the signal to which the Fourier series converges for all t.
Ifare the Fourier coefficients of a real signal, x(t), fill in all the steps to show that:(a) (b) Xn is a real, even function of n for x(t) even.(c) Xn is imaginary and an odd function of
Obtain the complex exponential Fourier series coefficients for the (a) pulse train, (b) half-rectified sinewave, (c) full-rectified sinewave, and (d) triangular waveform as given in Table 2.1. Table
Find the ratio of the power contained in a rectangular pulse train for |n f0| ≤ τ -1 to the total power for each of the following cases:(a) τ/T0 = 1/2(b) τ/T0 = 1/5(c) τ/T0 = 1/10(d) τ/T0 =
(a) If x(t) has the Fourier and y(t) = x(t - t0), show that Yn = Xne-j2Ï€nf0t0 where the Yn's are the Fourier coefficient for y(t).(b) Verify the theorem proved in part (a) by
Use the Fourier series expansions of periodic square wave and triangular wave signals to find the sum of the following series: (a) 1 - 1/3 + 1/5 - 1/7 + ....(b) 1 + 1/9 + 1/25 + 1/49 + ....Write
Using the results given in Table 2.1 for the Fourier coefficients of a pulse train, plot the double-sided amplitude and phase spectra for the wave forms shown in Figure 2.34.(Note that xb(t) = -xa(t)
Sketch each signal given below and find its Fourier transform. Plot the amplitude and phase spectra of each signal (A and τ are positive constants).(a) x1(t) = A exp (-t/τ) u(t) (b)
(a) Use the Fourier transform of x(t) = exp (-Î±t) u (t) - exp (Î±t) u (-t) where Î± > 0 (b) Use the result above and the relation u(t) = 1/2[sgn (t) + 1] to find Fourier transform of

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