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physics
principles communications systems

Principles of Communications Systems, Modulation and Noise 7th edition Rodger E. Ziemer, William H. Tranter - Solutions

In tossing a single six-sided fair die, event A = {1 or 3}, event B = {2 or 3 or 4}, and event C = {4 or 5 or 6}. Find the following probabilities:(a) P(A);(b) P(B);(c) P(C);(d) P (A ∪ B);(e) P (A ∪ C);(f) P (B ∪ C);(g) P (A ∩ B);(h) P (A ∩ C);(i) P (B ∩ C);(j) P (A ∩ (B ∩ C));(k) P
In tossing a six-sided fair die, we define event A = {2 or 4 or 6} and event B = {1 or 3 or 5 or 6}.Using equal likelihood and the axioms of probability, find the following:(a) P(A);(b) P(B);(c) P(A ∪ B);(d) P (A ∩ B);(e) P (A | B);(f) P (B |A);
A fair coin and a fair die (six sides) are tossed simultaneously with neither affecting the outcome of the other. Give probabilities for the following events using the principle of equal likelihood:(a) A head and a six;(b) A tail and a one or a two;(c) A tail or a head and a
Give advantages and disadvantages of the carrier modulation methods illustrated in Figure 5.19. Figure 5.19 Data ASK PSK FSK
Judging from the results of Figures 5.16. and 5.18, which method for generating a spectral component at the data clock frequency generates a higher-power one: the squarer or the delay-and-multiply circuit? Figure 5.16 Figure 5.18 (a) 100 200 400 500 600 300 1.5 1 (b) 0.5 100 400 600 200 300 500
Choose the correct adjective: A narrower bandwidth channel implies (more) (less) amplitude jitter.
Choose the correct adjective: A wider bandwidth channel implies (more) (less) timing jitter.
How many total samples of the incoming pulse are required to force the following number of zeros on either side of the middle sample for a zero-forcing equalizer?(a) 1;(b) 3;(c) 4;(d) 7;(e) 8;(f) 10.
For the data sequence of Problem 5.3 provide waveform sketches for:(a) Unipolar RZ;(b) Polar RZ;(c) Bipolar RZ.Data From Problem 5.3(a) Given the random binary data sequence 0 1 1 0 0 0 1 0 1 1. Provide waveform sketches for:(i) NRZ change;(ii) Split phase.(b) Demonstrate
Which of the following pulse spectra have inverse Fourier transforms with the zero-ISI property?(a) P1 (f) = II (Tf) where T is the pulse duration;(b) P2 (f) = Λ (Tf / 2);(c) P3 (f) = II(2T);(d) P4 (f) = II(Tf) + II (2Tf).
What is meant by a pulse having the zero-ISI property? What must be true of the pulse spectrum in order that it have this property?
(a) Given the random binary data sequence 0 1 1 0 0 0 1 0 1 1. Provide waveform sketches for:(i) NRZ change;(ii) Split phase.(b) Demonstrate satisfactorily that the split-phase waveform can be obtained from the NRZ waveform by multiplying the NRZ waveform by a ±1-valued clock signal of
Explain what happens to a line-coded data sequence when passed through a severely band limited channel.
Tell which binary data format(s) shown in Figure 5.2 satisfy the following properties, assuming random (fair coin toss) data: Figure 5.2 (a)?Zero DC level;(b)?A zero crossing for each data bit;(c) Binary 0 data bits represented by 0 voltage level for transmission and the waveform has nonzero DC
For the ? 1- amplitude wave forms of Figure 5.2, show that the average powers are: (a) NRZ change--- Pave = 1 W;(b) NRZ mark--- Pave = 1 W; (c) Unipolar RZ--- Pave = 1/4 W; (d) Polar RZ--- Pave = 1/2 W; (e) Bipolar RZ--- Pave = 1/4 W; (f) Split phase--- Pave = 1 W; Figure 5.2 2 4 10 12 14 16 18
Which data formats, for a random (coin toss) data stream, have (a) zero dc level; (b) built in redundancy that could be used for error checking; (c) discrete spectral lines present in their power spectra; (d) nulls in their spectra at zero frequency; (e) the most compact power spectra (measured to
Evaluate the time-average Poynting vector, = (12) Re {EsÃ— Hˆ—s} for the Hertzian dipole, assuming the general case that involves the field components as given by Eqs. (10), (13a), and (13b). Compare your result to the far-zone case, Eq. (26).Eqs. (10)Eqs. (13a)Eqs. (13b)Eqs.
Evaluate the time-average Poynting vector, = (1/2)Re EsÃ— Hˆ—sfor the magnetic dipole antenna in the far zone, in which all terms of order 1/r2and 1/r4are neglected in Eqs. (48), (49), and (50). Compare your result to the far-zone power density of the Hertzian dipole, Eq.
A 50-Ω load is attached to a 50-m section of the transmission line of Problem 10.1, and a 100-W signal is fed to the input end of the line.(a) Evaluate the distributed line loss in dB/m.(b) Evaluate the reflection coefficient at the load.(c) Evaluate the power that is dissipated by the load
A coaxial cable has conductor radii a and b, where a < b. Material of permeability μr ≠ 1 exists in the region a < ρ < c, whereas the region c < ρ < b is air filled. Find an expression for the inductance per unit length.
Revisit Problem 14.21, but with the current phase allowed to vary with frequency (this will automatically occur if the phase difference is established by a simple time delay between the feed currents). Now, the current phase difference will be ξ' = ξ f/ f0, where f0 is the original (design)
A two-element dipole array is configured to provide zero radiation in the broadside (ϕ = ± 90◦) and end-fire (ϕ = 0, 180◦) directions, but with maxima occurring at angles in between. Consider such a set-up with ψ = π at ϕ = 0 and ψ = −3π at ϕ = π, with both values determined in the
Repeat Problem 14.17, but with a full-wave antenna (2ℓ = λ). Numerical integrals may be necessary.Consider a lossless half-wave dipole in free space, with radiation resistance, Rrad = 73 ohms, and maximum directivity Dmax = 1.64. If the antenna carries a 1-A current amplitude,(a) How much total
Suppose that Ï• in Figure 12.17 is Brewster€™s angle, and that Î¸1is the critical angle. Find n0in terms of n1and n2. пz По пy П2
A uniform plane wave is normally incident onto a slab of glass (n = 1.45) whose back surface is in contact with a perfect conductor. Determine the reflective phase shift at the front surface of the glass if the glass thickness is(a) λ/2;(b) λ/4;(c) λ/8.
Sulfur hexafluoride (SF6) is a high-density gas that has refractive index, ns = 1.8 at a specified pressure, temperature, and wavelength. Consider the retro-reflecting prism shown in Fig. 12.16, that is immersed in SF6. Light enters through a quarter-wave antireflective coating and then totally
Suppose that the length of the medium of Problem 11.31 is made to be twice that determined in the problem. Describe the polarization of the output wave in this case.In Problem 11.31A linearly polarized uniform plane wave, propagating in the forward z direction, is input to a lossless anisotropic
(a) Most microwave ovens operate at 2.45 GHz. Assume that σ = 1.2 × 106 S/m and μr = 500 for the stainless steel interior, and find the depth of penetration.(b) Let Es = 50 0◦ V/m at the surface of the conductor, and plot a curve of the amplitude of Es versus the angle of Es as the field
Perfectly conducting cylinders with radii of 8 mm and 20 mm are coaxial. The region between the cylinders is filled with a perfect dielectric for which ∈ = 10−9/4π F/m and μr = 1. If E in this region is (500/ρ) cos(ωt − 4z)aρ V/m, find(a) ω, with the help of Maxwell’s equations in
Describe how the attenuation coefficient of a liquid medium, assumed to be a good conductor, could be determined through measurement of wavelength in the liquid at a known frequency. What restrictions apply? Could this method be used to find the conductivity as well?
A lossless line having an air dielectric has a characteristic impedance of 400Ω. The line is operating at 200 MHz and Zin = 200 − j200 Ω. Use analytic methods or the Smith chart (or both) to find(a) s;(b) ZL, if the line is 1 m long;(c) The distance from the load to the nearest voltage maximum.
The incident voltage wave on a certain lossless transmission line for which Z0 = 50Ω and νp = 2 × 108 m/s is V+(z, t) = 200 cos(ωt − πz) V.(a) Find ω.(b) Find I+(z, t). The section of line for which z > 0 is replaced by a load ZL = 50 + j30 at z = 0. Find:(c) L;(d) Vs− (z);(e) Vs at
In Section 9.1, Faraday’s law was used to show that the field E = −1/2 kB0ektρaϕ results from the changing magnetic field B = B0ektaz.(a) Show that these fields do not satisfy Maxwell’s other curl equation.(b) If we let B0 = 1 T and k = 106 s−1, we are establishing a fairly large magnetic
Repeat the previous computer exercise with 20 values of θ uniformly distributed in the range - π/4 ≤ θ < π/4.Data From Problem 7.1In this computer exercise we reexamine Example 7.1. A random process is defined byX(t) = A cos (2πf0t + θ)
Show that the variance of?âML?(Z) given by (11.178) is the result given by (11.179). (11.178) (11.179) K âM, (Z) K k=1
Consider the reception of a BPSK signal in noise with unknown phase, Î¸, to be estimated. The two hypotheses may be expressed asH1 : y(t) = A cos(Ï‰ct + Î¸) + n(t), 0 ‰¤ t ‰¤ TsH2 : y(t) = -A cos(Ï‰ct + Î¸) + n(t), 0
Write a computer program to evaluate various digital modulation techniques:(a) For a specified data rate and error probability, find the required bandwidth and Eb / N0 in decibels. Corresponding to the data rate and required Eb / N0, find the required received-signal power for N0 = 1 W/Hz.(b) For a
Consider the random process of Problem 7.3.(a) Find the time-average mean and the auto correlation function.(b) Find the ensemble-average mean and the auto correlation function.(c) Is this process wide-sense stationary? Why or why not?Data From Problem 7.3A random process is composed of
A random variable X has the probability-density functionwhere Î± is an arbitrary positive constant.(a) Determine the characteristic function Mx(jv).(b) Use the characteristic function to determine E[X] and E[X2].(c) Compute Ïƒ2x. -ах {3 аеах, х > 0 Sx (х) %— 0, х
An algorithm for generating a Gaussian random variable from two independent uniform random variables is easily derived.(a) Let U and V be two statistically independent random numbers uniformly distributed in [0, 1]. Show that the following transformation generates two statistically independent
In this exercise we examine a useful technique for generating a set of samples having a given pdf.(a) First, prove the following theorem: If X is a con tinuous random variable with cdf FX(x), the random variableY = Fx(X)is a uniformly distributed random variable in the interval (0, 1).(b) Using
(a) Write the signals of Figure 2.37 as the linear combination of two delayed triangular functions. That is, write xa(t) = a1 Î› ((t - t1) / T1) + a2Î›((t - t2) / T2) by finding appropriate values for a1, a2, t1, t2, T1, and T2. Do similar expressions for all four signals
Write a computer program to find the time duration of a low pass energy signal that contains a certain specified percentage of its total energy, for example, 95%. In other words, write a program to find
How would you use the same approach as in Computer Exercise 2.3 to evaluate the Fourier transform of a pulse-type signal. How do the two outputs differ? Compute an approximation to the Fourier transform of a square pulse signal 1 unit wide and compare with the theoretical result.
A toroidal core has a square cross section, 2.5 cm < ρ < 3.5 cm, −0.5 cm < z < 0.5 cm. The upper half of the toroid, 0 < z < 0.5 cm, is constructed of a linear material for which μr = 10, while the lower half, −0.5 cm < z < 0, has μr = 20. An mmf of 150 A· t
A sinusoidal voltage source drives the series combination of an impedance, Zg = 50 − j50Ω, and a lossless transmission line of length L, shorted at the load end. The line characteristic impedance is 50Ω, and wavelength λ is measured on the line.(a) Determine, in terms of wavelength, the
In Figure 10.17, let ZL= 250, Z0= 50, find the shortest attachment distance d and the shortest length d1of a short-circuited stub line that will provide a perfect match on the main line to the left of the stub. Express all answers in wavelengths. Z.
In Figure 12.5, let regions 2 and 3 both be of quarter-wave thickness. Region 4 is glass, having refractive index, n4= 1.45; region 1 is air.
A filamentary conductor is formed into an equilateral triangle with sides of length ℓ carrying current I. Find the magnetic field intensity at the center of the triangle.
Let V(x, y) = 4e2x + f (x) − 3y2 in a region of free space where ρν = 0. It is known that both Ex and V are zero at the origin. Find f (x) and V(x, y).
Given the spherically symmetric potential field in free space, V = V0e−r/a, find.(a) ρν at r = a;(b) The electric field at r = a;(c) The total charge.
Let V = 2xy2z3 and ∈ = ∈0. Given point P(1, 2,−1), find.(a) V at P;(b) E at P;(c) ρν at P;(d) The equation of the equipotential surface passing through P;(e) The equation of the streamline passing through P.(f) Does V satisfy Laplace’s equation?
A two-wire transmission line consists of two parallel perfectly conducting cylinders, each having a radius of 0.2 mm, separated by a center-to-center distance of 2 mm. The medium surrounding the wires has r = 3 and σ = 1.5 mS/m. A 100-V battery is connected between the wires.(a) Calculate the
A potential field in free space is given in spherical coordinates aswhere Ï0 and a are constants.(a) Use Poisson€™s equation to find the volume charge density everywhere.(b) Find the total charge present. [Po/(6e0)][3a? – r²] (r a)
In this computer exercise we reexamine Example 7.1. A random process is defined byX(t) = A cos (2πf0t + θ)Using a random number generator program generate 20 values of θ uniformly distributed in the range 0 ≤ θ < 2π. Using these 20 values of θ generate 20 sample functions of the process
Check the correlation between the random variables X and Y generated by the random number generator of Computer Exercise 6.2 by computing the sample correlation coefficient of 1000 pairs according to the definition whereand N Х (х, - йх) (Ү, — Ay) P(X,Y) = (N – 1) ô¡ô2 n=1 Âx : :
Write a MATLAB program to plot the Ricean pdf. Use the form (7.150) and plot for K = 1, 10, 100 on the same axes. Use r / Ïƒ as the independent variable and plot Ïƒ2f (r). а-p-к}(v).rz0 exp (7.150) [202 r20
Let the sample functions of a random process be given byX(t) = A cos 2Ï€f0twhere Ï‰0 is fixed and A has the pdf This random process is passed through an ideal integrator to give a random process Y(t).(a) Find an expression for the sample functions of the output process
In discussing thermal noise at the beginning of this chapter, we stated that at standard temperature (290 K) the white-noise assumption is valid to bandwidths exceeding 1000 GHz. If the temperature is reduced to 5 K, the variance of the noise is reduced, but the bandwidth over which the white-noise
The received signal in a digital base band system is either +A or -A, equally likely, for T-second contiguous intervals. However, the timing is off at the receiver so that the integration starts Î”T seconds late (positive) or early (negative). Assume that the timing error is less than
(a) Consider the transmission of digital data at a rate of R = 50 kbps and at an error probability of PE = 10-6. Using the bandwidth of the main lobe as a bandwidth measure, give an estimate of the required transmission bandwidth and Eb / N0 in decibels required for the following coherent
Use MATLAB to plot curves of Pbversus Eb/ N0, M = 2, 4, 8, 16, and 32, for(a) M-ary coherent FSK (use the upper-bound expression as an approximation to the actual error probability)(b) M-ary non-coherent FSK Compare your results with Figures 10.15(a) and (b). M-ary CFSK M-ary NCFSK 100 100 10- 10-
Show that the noise components N1and N1for QPSK, given by Equations (10.6) and (10.8), are uncorrelated; that is, show that E[N1N2] = 0. (Explain why N1and N12are zero mean.) (10.6) n(t) cos(27f t) dt N1 N = N2 = n(t) sin(27f.t) di (10.8)
Show that (10.26) and (10.27) are Fourier transform pairs. In 2 (f (10.26) н (f)— еxp 2л? в 2л -B exp In 2 h(t) = (10.27) In 2
(a) Sketch the signal space with decision regions for 16-ary PSK [see (10.47)].(b) Use the bound (10.50) to write down and plot the symbol error probability versus Eb / N0.(c) On the same axes, compute and plot the bit error probability assuming that Gray encoding is used. 2л (і — 1) 2E,
(a) Using (10.93) and appropriate bounds for PE,symbol, obtain Eb / N0 required for achieving PE,bit = 10-4 for M-ary PSK with M = 8, 16, 32.(b) Repeat for QAM for the same M values using (10.63).
Repeat Example 10.6 with everything the same except for a propagation delay uncertainty of ± 1.5 ms and a false-alarm penalty of Tfa = 100Ti.
(a) Consider a multi path channel with a delay spread of 5 microseconds through which it is desired to transmit data at 500 kbps. Design an MCM system that has a symbol period at least a factor of ten greater than the delay spread if the modulation to be used on each sub carrier is QPSK.(b) If an
Rework Examples 10.11 and 10.12 for an attenuation exponent of α = 4.
(a) Using (10.95), (10.96), and (10.67), obtain Eb / N0 required for achieving PE,bit = 10-3 for M-ary coherent FSK for M = 2,4,8, 16,32. Program your calculator to do an iterative solution or use MATLAB.(b) Using (10.95), (10.96), and (10.68), repeat for non-coherent M-ary FSK for M = 2,4,8, 16,32.
Given the following parameter values from the IEEE802.16 standard:Modulation Coding
Consider the system shown in Figure 8.21, in which an RC high pass filter is followed by an ideal low pass filter having bandwidth W. Assume that the input to the system is A cos(2Ï€fct), where fc< W, plus white noise with double-sided power spectral density 1/2 N0. Determine the SNR
The waveform at the input of a base band system has signal power PT and white noise with single-sided power spectral density N0. The signal bandwidth is W. In order to pass the signal without significant distortion, we assume that the input waveform is band limited to a bandwidth B = 2W using
Execute the computer program used to generate the FM discriminator performance characteristics illustrated in Figure 8.14. Add to the performance curves for Î² = 1, 5, 10, and 20 the curve for Î² = 0.1. Is thethreshold effect more or less pronounced? Why?Figure 8.14 ß =20 ß
A signal is given byx(t) = 5 cos [2Ï€(5)t]and the noise PSD is given byDetermine the largest permissible value of N0 that ensures that the SNR is ‰¥ 30 dB. If|< 8 S„(f) = No.
The value of the input SNR at threshold is often defined as the value of PT / N0W at which the denominator of (8.172) is equal to 2. Note that this value yields a post detection SNR, (SNR)D, that is 3 dB below the value of (SNR)D predicted by the above threshold (linear) analysis. Using this
Derive the equation for yD(t) for an SSB system assuming that the noise is expanded about the frequency fx = fc ±1/2W. Derive the detection gain and (SNR)D. Determine and plot Snc (f) and Sns (f).
In analyzing the performance of an FM discriminator, operating in the presence of noise, the post detection SNR, (SNR)D, is often determined using the approximation that the effect of modulation on (SNR)Dis negligible. In other words, |Î´Ì… fÌ…| is set equal to
In Section 8.1.3 we expanded the noise component about fc. We observed, however, that the noise components for SSB could be expanded about fc ± 1/2 W, depending on the choice of side bands. Plot the power spectral density for each of these two cases and, for each case, write the expressions
The preceding computer exercise problem examined the behavior of a PLL in the acquisition mode. We now consider the performance in the tracking mode. Develop a computer simulation in which the PLL is tracking an unmodulated sinusoid plus noise. Let the predetection SNR be sufficiently high to
Develop a computer program to verify the performance curves shown in Figure 8.17. Compare the performance of the non-coherent FSK system to the performance of both coherent FSK and coherent PSK with a modulation index of 1. We will show in the following chapter that the bit-error probability for
Assume that an AM system operates with an index of 0.5 and that the message signal is 10 cos(8πt). Compute the efficiency, the detection gain in dB, and the output SNR in decibels relative to the base band performance PT / N0W. Determine the improvement (in decibels) in the output SNR that
In Section 8.2 we described a technique for estimating the gain, delay, and the SNR at a point in a system given a reference signal. What is the main source of error in applying this technique? How can this error source be reduced, and what is the associated cost? Develop a test signal and sampling
An AM system has a message signal that has a zero mean Gaussian amplitude distribution. The peak value of m(t) is taken as that value that |m(t)| exceeds 1.0% of the time. If the index is 0.8, what is the detection gain?
Assume a three-bit ADC (eight quantizing levels). We desire to design a compounding system consisting of both a compressor and ex-pander. Assuming that the input signal is a sinusoid, design the compressor such that the sinusoid falls into each quantizing level with equal probability. Implement the
The threshold level for an envelope detector is sometimes defined as that value of (SNR)T for which Ac > rn with probability 0.99. Assuming that a2m̅2n ≅ 1, derive the SNR at threshold, expressed in decibels.
A compressor is often modeled asxout (t) = A tanh[axin(t)]We assume that that the input to the compressor is an audio signal having frequency content in the range 20 ≤ f ≤ 15,000 where frequency is measured in Hz. Select α so that the compressor gives a 6-dB amplitude attenuation at 20 Hz.
Verify the correctness of (8.59). = E {Ir?t) + n?t»F} – E²in?u) + n?t»)l = 4c; (8.59) ?+n?
An envelope detector operates above threshold. The modulating signal is a sinusoid. Plot (SNR)D in decibels as a function of PT/N0W for the modulation index equal to 0.3, 0.5, 0.6, and 0.8.
A square-law demodulate for AM is illustrated in Figure 8.20. Assuming that xc(t) = Ac[1 + amn (t)] cos(2Ï€fct) and m(t) = cos(2Ï€fmt) + cos(4Ï€fmt), sketch the spectrum of each term that appears in yD(t). Do not neglect the noise that is assumed to be band
Assume that a zero-mean message signal m(t) has a Gaussian pdf and that in normalizing the message signal to form mn(t), the maximum value of m(t) is assumed to be kσm, where k is a parameter and σm is the standard deviation of the message signal. Plot (SNR)D as a function of PT/N0W with α = 0.5
Compute (SNR)D as a function of PT / N0W for a linear envelope detector assuming a high predetection SNR and a modulation index of unity. Compare this result to that for a square-law detector, and show that the square-law detector is inferior by approximately 1.8 dB. If necessary, you may assume
The input to a communications receiver isr(t) = 5 sin(20πt + 7 /4 π) + ¡(t) + n(t)wherei(t) = 0.2 cos (60πt)and n(t) is noise having standard deviation σn = 0.1. The transmitted signal is 10 cos(20πt). Determine the SNR at the receiver input and the delay from the transmitted signal to the
An SSB system is to be operated with a normalized mean-square error of 0.06 or less. By making a plot of output SNR versus demodulation phase-error variance for the case in which normalized mean-square error is 0.4%, show the region of satisfactory system performance. Repeat for a DSB system. Plot
Repeat the preceding problem for a normalized mean-square error of 0.1 or less.Data From Problem 8.16An SSB system is to be operated with a normalized mean-square error of 0.06 or less. By making a plot of output SNR versus demodulation phase-error variance for the case in which normalized
Draw a phasor diagram for an angle-modulated signal for (SNR)T >> 1 illustrating the relationship between R(t), Ac, and rn(t). Show on this phasor diagram the relationship between ψ(t), ϕ(t) and ϕn(t). Using the phasor diagram, justify that for (SNR)T >> 1, the approximationψ(t) ≈
Develop a set of performance curves, similar to those shown in Figure 8.8, that illustrate the performance of a coherent demodulator as a function of the phase-error variance. Let the SNR be a parameter and express the SNR in decibels. As in Figure 8.8, assume a QDSB system. Repeat this exercise
The process of stereophonic broadcasting was illustrated in Chapter 4. By comparing the noise power in the I(t) - r(t) channel to the noise power in the I(t) + r(t) channel, explain why stereophonic broadcasting is more sensitive to noise than non-stereophonic broadcasting.
An FDM communication system uses DSB modulation to form the base band and FM modulation for transmission of the base band. Assume that there are eight channels and that all eight message signals have equal power P0 and equal bandwidth W. One channel does not use sub carrier modulation. The other
Using (8.146), derive an expression for the ratio of the noise power in yD(t) with de-emphasis to the noise power in yD(t) without de-emphasis. Plot this ratio as a function of W/f3. Evaluate the ratio for the standard values of f3 = 2.1 kHz and W = 15 kHz, and use the result to determine the

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