Questions and Answers of Principles Communications Systems

For a PN sequence of length 27 - 1 = 127 two possible feedback connections are x4 ⊕ x7 and x6 ⊕ x7.(a) Draw a block diagram of the feedback shift register for the
An m-sequence is generated by a continuously running feedback shift-resister with a clock rate of 10 kHz. Assume that the shift register has six stages and that the feedback connection is the
Consider the marker code C8 of Table 9.8. Find the Hamming distance between all possible shifts of it and the received sequence 10110 10110 00011 101011 (spaces for clarity). Is there a unique match
Find the difference in dB between (10.125) and (10.126). That is, find the ratio σ2ϵ, SL / σ2ϵ, AV expressed in dB.
Plot Ïƒ2Ï• versus z for the various cases given in Table 10.7. Assume 10% of the signal power is in the carrier for the PLL and all signal power is in the modulation for the
Draw the block diagram of an M -power law circuit for synchronizing a local carrier for 8-PSK. Assume that fc = 10 MHz and Ts = 0.1 ms. Care fully label all blocks, and give critical frequencies and
Derive the Fourier transform pair given by (10.120). To show:
Assume that a data stream d(t) consists of a random (coin-toss) sequence of + 1s and -1s each of which is T seconds in duration. The auto correlation function for such a sequence is (a) Find and
Note that from (10.116) and (10.122) that the 10% out-of-band power RF bandwidth for QPSK is where 1/2Tb is the first zero of sinc (2Tbf). From (10.117) we therefore conclude that for M-ary
Generalize the results for power-containment bandwidth for quadrature-modulation schemes given in Section 10.2 to M-ary PSK. (Is it any different than the result for QAM?) With appropriate
On the basis of 90% power-containment bandwidth, give the required transmission bandwidth to achieve a bit rate of 50 kbps for(a) BPSK(b) QPSK or OQPSK(c) MSK(d) 16-QAM
A channel of bandwidth 4 MHz is available through which it is desired to communicate at a data rate of 1 Mbps.(a) Find the largest value of M, which is a power of 2, for M-ary coherent FSK such that
A channel of bandwidth 1 MHz with N0 = 10-9 W/Hz is available through which it is desired to transmit data with an M-ary communication system at 4 Mbps.(a) What are candidate modulation schemes?(b)
Demonstrate that (10.89) is equivalent to (10.68).
Show that for M-ary QAMwhere Es is the symbol energy averaged over the constellation of M signals, which is (10.58). The summation formulaswill prove useful. ЗЕ, a 2 (M – 1) т (т+1) т
By substituting (10.60) through (10.62) into (10.59), collecting all like-argument terms in the Q-function, and neglecting squared Q-function terms, show that the symbol error probability for 16-QAM
Derive the three equations numbered (10.60) through (10.62) for M-QAM.
Two binary symmetrical channels are in cascade, as shown in Figure 12.45. Determine the capacity of each channel. The overall system with inputs x1and x2and out puts z1and z2can be represented as
A source has two outputs [A, B] with respective probabilities [5/8, 3/8]. Determine the entropy of the fourth order extension of this source using two different methods.
Calculate the entropy of the fourth-order extension of the source defined in Table 12.1. Determine LÌ…/n for n = 4, and add this result to those shown in Figure 12.9. Determine the
A source has seven equally likely output messages. Determine a Shannon-Fano code for the source, and determine the efficiency of the resulting code. Repeat for the Huffman code, and compare the
A source has five outputs denoted [m1, m2, m3, m4, m5] with respective probabilities [0.40, 0.20, 0.17, 0.13, 0.10]. Determine the code words to rep resent the source outputs using both the
A binary source has output probabilities [0.85, 0.15]. The channel can transmit 350 binary symbols per second at the capacity of 1 bit/symbol. Determine the maximum source symbol rate if transmission
A source output consists of eleven equally likely messages. Encode the source output using both binary Shannon--Fano and Huffman codes. Compute the efficiency of both of the resulting codes and
An analog source has an output described by the probability density functionThe output of the source is quantized into 10 messages using the eleven quantizing levelsxi = 0.1k,
A source output consists of five messages [m1, m2, m3, m4, m5] with respective probabilities [0.35, 0.25, 0.2, 0.15, 0.05]. Determine the binary code words for the second-order source extension using
It can be shown that a necessary and sufficient condition for the existence of an instantaneous binary code with word lengths Ii, 1 ‰¤ i ‰¤ N, is This is known as the
A continuous band pass channel can be modeled as illustrated in Figure 12.46. Assuming a signal power of 60 W and a noise power spectral density of 10-5W/Hz, plot the capacity of the channel as a
Consider again the band pass channel illustrated in Figure 12.46. The noise power spectral density is 10-5 and the bandwidth is 10 kHz. Plot the capacity of the channel as a function of signal power
Develop an analysis that shows that increasing n for a rate 1/n code always degrades system performance in an AWGN. In order to obtain specific results, assume PSK modulation.
Write the parity-check matrix and the generator matrix for a (15, 11) single error-correcting code. Assume that the code is systematic. Calculate the code word corresponding to the all-ones
A parity-check code has the parity-check matrixDetermine the generator matrix and find all possible code words. 1 1 1 1 0 0 1 0 1 1 0 1 0 [H] = 1 1 0 1 0 0 1
For the code described in the preceding problem, find the code words [T1] and [T2] corresponding to the information sequences Using these two code words, illustrate the group property. [A]] =
Determine the relationship between n and k for a Hamming code. Use this result to show that the code rate approaches 1 for large n.
Determine the generator matrix for the coder illustrated in Figure 12.15. Use the generator matrix to generate the complete set of code words and use the results to check the code words shown in
Use the result of the preceding problem to determine the parity-check matrix for the coder shown in Figure 12.15. Use the parity-check matrix to decode the received sequences 1101001 and 1101011.
Consider the coded system examined in Computer Example 12.1. Show that the probability of three symbol errors in a code word is negligible compared to the probability of two symbol errors in a code
Consider the convolutional coder shown in Figure 12.24. The shift register contents are S1S2S3where S1represents the most recent input. Compute the output sequence v1v2v3for S1= 0 and for S1= 1. Show
What is the constraint span of the convolutional coder shown in Figure 12.47? Draw the state diagram, giving the output for each state transition.Figure 12.47 Input S4 S1 S2 S3 Vị V2 Output
Repeat the preceding problem for the convolutional coder illustrated in Figure 12.47. For the coder shown in Figure 12.47, the shift register contents are S1S2S3S4, where S1 represents the most
Determine the state diagram for the convolutional coder shown in Figure 12.48. Draw the trellis diagram through the first set of steady-state transitions. On a second trellis diagram, show the
Derive (12.149), the expression for the expected number of transmissions for the feedback channel.
Develop the appropriate analysis to verify the correctness of (12.149).
Repeat the preceding problem assuming a (23,12) Golay code.Data From Problem 12.38A source produces binary symbols at a rate of 5000 symbols per second. The channel is subjected to error bursts
A source produces binary symbols at a rate of 5000 symbols per second. The channel is subjected to error bursts lasting 0.2 s Devise an encoding scheme using an interleaved (n,k) Hamming code, which
Compare FM with pre-emphasis to an optimal modulation system for β = 1, 5, and 10. Consider only operation above threshold, and assume 20 dB as the value of PT / N0W at threshold.
Develop a computer simulation of an integrate-and dump detector for antipodal base band signaling based on (9.1). Generate AT or -AT randomly by drawing a uniform random number in [0, 1] and
Carry out the integrations leading to (9.206)[use (9.205) as a pattern], (9.207), and (9.208) given that the signal-to-noise ratio pdf is given by fz(z) = =e=z/Z, z > 0.
(a) Show that the distance from a hexagon cell center to a vertex is 1 / √3 if adjacent cell centers are 1 unit apart.(b) Show that the distance between cochannel cell centers is Dco = √N in a
Fading margin can be defined as the incremental Eb / N0, in decibels, required to provide a certain desired error probability in a fading channel as could be achieved with the same modulation
Find the probability of error for non-coherent ASK, with signal setwhere Î¸ is a uniformly distributed random variable in [0, 2Ï€). White Gaussian noise of two-sided power
Rework Example 10.11 with everything the same, except assume SIRdB,min = 10 dB.
Integrate (9.121) by recasting the integrand into the form of a Ricean pdf and therefore use the fact that it integrates to 1. You will have to redefine some parameters and multiply and divide by
Gray encoding of decimal numbers ensures that only one bit changes when the decimal number changes by one unit. Let b1b2b3 ... bn represent an ordinary bi nary representation of a decimal number,
Consider a base band antipodal PAM system with channel bandwidth of 5 kHz and a desired data rate of 20 kbps. (a) What is the required value for M?(b) What value of Ed / N0 in dB will give a bit
Recompute the entries in Table 9.5 for a bit error probability of 10-4and an RF bandwidth of 100 kHz. Table 9.5 Comparison of Binary Modulation Schemes at Pp = 10-6 Required E,N, for P, = (dB) R
A channel of 100 kHz is available. What RF modulation scheme can be used to communicate through it at a data rate of:(a) 50 kbps;(b) 100 kbps;(c) 150 kbps;(d) 200
Show that (9.162) is the average energy in terms of Î” for binary antipodal ASK.(9.162) Hc(f) = 1, |f| < %3D
Rework Example 9.9 for Eb / N0 = 20 and β = 0.1. That is, recompute the matrices [Ryy] and [Ryd]. Find the equalizer coefficients and the MMSE. Comment on the differences from Example 9.9.
Consider (9.246) with all elements of both [Ryy] and [Ryd] divided by 10. (a) Do the weights remain the same? (b) What is an acceptable range for μ for an adaptive MMSE weight
For the numerical auto- and cross-correlation matrices of Example 9.9, find explicit expressions (write out an equation for each weight) for the steepest-descent tap weight adjustment algorithm
(a) Consider the design of an MMSE equalizer for a multi path channel whose output is of the formy(t) = A d(t) + b A d(t - Tm) + n(t)where the second term is a multi path component and the third term
Given the following pulse-response samples:pc (-4T) = 0; pc (-3T) = -1/9;pc (-2T) = 1/10; pc (-T) = -1;pc (0) = 1/2; pc (T) = -1;pc (2T) = 1/2; pc (3T) = -1/9;pc (4T) = 0(a) Find the tap coefficients
Given the following pulse-response samples:pc (-4T) = 0; pc (-3T) = -1/9;pc (-2T) = 1/2; pc (-T) = -1;pc (0) = 1/2; pc (T) = -1;pc (2T) = 1/2; pc (3T) = -1/9;pc (4T) = 0(a) Find the tap coefficients
Show the details in making the substitution σ2w = 1 / 2(1 + 1 / Z̅) in (9.203) so that it gives (9.206) after integration.
Redraw Figure 9.29 for PE= 10-5. Write a MATLAB program using the find function to obtain the degradation for various values of Î´ and Ï„m/ T.Figure 9.29 Degradation, D dB
Plot PE from Equation (9.183) versus z0 for δ = 0.5 and τm / T = 0.2, 0.6, and 1.0. Develop a MATLAB program to plot the curves.
Data are to be transmitted through a band limited channel at a rate R = 1 / T = 9600 bps. The channel filter has frequency response function HC (f) = 1 / 1 + j(f / 4800)The noise is white with
(a) Sketch the trapezoidal spectrum (b) By appropriate sketches, show that it satisfies Nyquist€™s pulse-shaping criterion. P(f) =A(f /b) – A(F/a), b > a > 0, for a =1 and b = 2. %|
Assume a raised-cosine pulse with Î² = 0.2 additive noise with power spectral densityand a channel filter with transfer-function-squared magnitude given by Find and plot the optimum
A symmetric dielectric slab waveguide has a slab thickness d = 10μm, with n1 = 1.48 and n2 = 1.45. If the operating wavelength is λ = 1.3μm, what modes will propagate?
A right-circularly polarized plane wave in air is incident at Brewster’s angle onto a semi-infinite slab of plexiglas (∈'r = 3.45, ∈"r = 0).(a) Determine the fractions of the incident power
A uniform plane wave in air, E+x1 = E+x10 cos(1010t − βz) V/m, is normally incident on a copper surface at z = 0. What percentage of the incident power density is transmitted into the copper?
A 2 GHz uniform plane wave has an amplitude Ey0 = 1.4 kV/m at (0, 0, 0, t = 0) and is propagating in the az direction in a medium where ∈" = 1.6 × 10−11 F/m, ∈' = 3.0 × 10−11 F/m, and μ
A certain lossless material has μr = 4 and ∈r = 9. A 10-MHz uniform plane wave is propagating in the ay direction with Ex0 = 400 V/m and Ey0 = Ez0 = 0 at P(0.6, 0.6, 0.6) at t = 60 ns. Find(a) β,
An electric field in free space is given in spherical coordinates as Es (r) = E0(r)e−jkr aθ V/m.(a) Find Hs (r) assuming uniform plane wave behavior.(b) Find < S >.(c) Express the average
The phasor magnetic field intensity for a 400 MHz uniform plane wave propagating in a certain lossless material is (2ay − j5az)e−j25x A/m. Knowing that the maximum amplitude of E is 1500 V/m,
A uniform plane wave has electric field Es = (Ey0 ay − Ez0 az) e−αx e−jβx V/m. The intrinsic impedance of the medium is given as η = |η| ejϕ, where ϕ is a constant phase.(a) Describe the
A 150 MHz uniform plane wave in free space is described by Hs = (4 + j10) (2ax + jay)e−jβz A/m.(a) Find numerical values for ω, λ, and β.(b) Find H(z, t) at t = 1.5 ns, z = 20 cm.(c) What is
Small antennas have low efficiencies (as will be seen in Chapter 14), and the efficiency increases with size up to the point at which a critical dimension of the antenna is an appreciable fraction of
An H field in free space is given as H(x, t) = 10 cos(108t − βx)ay A/m. Find(a) β;(b) λ;(c) E(x, t) at P(0.1, 0.2, 0.3) at t = 1 ns.
A 10 GHz uniform plane wave propagates in a lossless medium for which ∈r = 8 and μr = 2. Find(a) νp;(b) β;(c) λ;(d) Es;(e) Hs;(f) S.
Show that Exs= Aej(k0z+Ï•)is a solution of the vector Helmholtz equation, Eq. (30), for k0= Ï‰ˆšÎ¼0ˆˆ0and any Ï• and A. d²Exs = -k,Es.
The characteristic admittance (Y0 = 1/Z0) of a lossless transmission line is 20 mS. The line is terminated in a load YL = 40 − j20 mS. Use the Smith chart to find(a) s;(b) Yin if l = 0.15λ;(c) the
Two lossless transmission lines having different characteristic impedances are to be joined end to end. The impedances are Z01 = 100 and Z03 = 25 Ω. The operating frequency is 1 GHz.(a) Find the
The characteristic impedance of a certain lossless transmission line is 72. If L = 0.5μH/m, find(a) C;(b) νp;(c) β if f = 80 MHz.(d) The line is terminated with a load of 60 Ω. Find Γ and s.
A sinusoidal wave on a transmission line is specified by voltage and current in phasor form:Vs (z) = V0 eαz ejβz and Is (z) = I0 eαz ejβz ejφwhere V0 and I0 are both real.(a) In which direction
The rails in Figure 9.6 each have a resistance of 2.2/m. The bar moves to the right at a constant speed of 9 m/s in a uniform magnetic field of 0.8 T. Find I (t), 0 < t < 1 s, if the bar is at
Conducting planes in air at z = 0 and z = d carry surface currents of ±K0ax A/m.(a) Find the energy stored in the magnetic field per unit length (0 < x < 1) in a width w(0 < y < w).(b)
White noise with two-sided power spectral density 1/2 N0 is added to a signal having the power spectral density shown in Figure 8.22. The sum (signal plus noise) is filtered with an ideal low
Consider the system shown in Figure 8.23 The signal x(t) is defined by x(t) = A cos (2Ï€fct)The low pass filter has unity gain in the pass band and bandwidth W, where fc < W.
Consider the system shown in Figure 8.24. The noise is white with two-sided power spectral density 1/2 N0. The power spectral density of the signal isThe parameter f3 is the 3-dB bandwidth of the
Derive an expression, similar to (8.172), that gives the output SNR of an FM discriminator output for the case in which the message signal is random with a Gaussian amplitude pdf. Assume that the
Assume that a PPM system uses Nyquist rate sampling and that the minimum channel bandwidth is used for a given pulse duration. Show that the post detection SNR can be written as and evaluate K.
A planar transmission line consists of two conducting planes of width b separated d m in air, carrying equal and opposite currents of I A. If b >> d, find the force of repulsion per meter of
Show that the differential work in moving a current element IdL through a distance dl in a magetic field B is the negative of that done in moving the element Idl through a distance dL in the same
A rectangular loop of wire in free space joins point A(1, 0, 1) to point B(3, 0, 1) to point C(3, 0, 4) to point D(1, 0, 4) to point A. The wire carries a current of 6 mA, flowing in the az direction
Show that a charged particle in a uniform magnetic field describes a circular orbit with an orbital period that is independent of the radius. Find the relationship between the angular velocity and
A point charge for which Q = 2 × 10−16 C and m = 5 × 10−26 kg is moving in the combined fields E = 100ax − 200ay + 300az V/m and B =−3ax + 2ay − az mT. If the charge velocity at t = 0 is
Compare the magnitudes of the electric and magnetic forces on an electron that has attained a velocity of 107 m/s. Assume an electric field intensity of 105 V/m, and a magnetic flux density

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