Questions and Answers of Principles Communications Systems

Write a computer program to evaluate degradation due to flat Rayleigh fading at a specified error probability. Include PSK, FSK, DPSK, and non-coherent FSK.
Write a computer program to design equalizers for specified channel conditions for: (a) the zero-forcing criterion; (b) the MMSE criterion.
Redo the derivation of Section 9.1 for the case where the possible transmitted signals are either 0 or A for T seconds. Let the threshold be set at AT / 2.Express your result in terms of signal
As an approximation to the integrate-and-dump detector in Figure 9.3(a), we replace the integrator with a low pass RC filter with frequency response function where f3 is the 3-dB cut off
Write a computer simulation program to evaluate the bit error rate performance of a MMSE equalizer using LMS tap weight adjustment, thus verifying the results of Figure 9.33. 100 Unequalized; 400,000
Assume that the probabilities of sending the signals s1(t) and s2(t) are not equal, but are given by p and q = 1 - p, respectively. Derive an expression for PEthat replaces (9.32) that takes this
The general definition of a matched filter is a filter that maximizes peak signal-to-rms noise at some prechosen instant of time t0.(a) Assuming white noise at the input, use Schwarz€™s
Referring to Problem 9.11 for the general definition of a matched filter, find the following in relation to the two signals shown in Figure 9.34.(a) The causal matched-filter impulse responses.
(a) Find the optimum (matched) filter impulse response h0(t), as given by (9.45) for s1(t) and s2(t), shown in Figure 9.35.(b) Find Î¶2 as given (9.56).
Find the peak-signal-squared-to-mean-squared noise ratio for the output of a matched filter for each of the following signals in terms of A and T. Take the noise spectral density (single-sided) as
Given these signals:Assume that they are used in a binary digital data transmission system in the following combinations. Express B and C in terms of A so that their energies are the same. Sketch
Given the three signals(a) Sketch each one and show that each has energy of A2T.(b) Show that R12, = 0 for each of the combinations (A, B), (B, C), and (A, C). What is the optimum threshold for each
A channel of bandwidth 10 kHz with single-sided noise power spectral density 10-7 W/Hz is available. A data rate through it of at least 4 kbps is desired at a PE ≤ 10-6. Design at least two
Plot the results for PEgiven in Table 9.2 versus z = Eb / N0in decibels with PEplotted on a semilog axis. Estimate the additional Eb / N0at PE= 10-5in decibels over the case for no phase
Find z = Eb / N0 required to give PE = 10-5 for the following coherent digital modulation techniques:(a) binary ASK;(b) BPSK;(c) binary FSK;(d) BPSK with no carrier component but with a phase error
An NFSK receiver must be designed with additional bandwidth for the input filters to accommodate frequency uncertainty of the received signal (due to Doppler shift, for example). (a) Justify
A binary PSK-modulated signal with carrier component was written in Section 9.2.8 asSPSK (t) = A sin [Ï‰ct + cos-1 m d (t) + Î¸]where 0 ‰¤ m ‰¤ 1 is
Derive an expression for PEfor binary coherent FSK if the frequency separation of the two transmitted signals is chosen to give a minimum correlation coefficient between the two signals. That is,
Deferentially encode the following binary sequences. Arbitrarily choose a 1 as the reference bit to begin the encoding process. (a) 111 110 001 100 (b) 101 011 101 011(c) 111 111 111
(a) Consider the sequence 011 101 010 111. Deferentially encode it and assume that the deferentially encoded sequence is used to biphase modulate a sinusoidal carrier of arbitrary phase. Prove that
(a) In the analysis of the optimum detector for DPSK, show that the random variables n1, n2, n3, and n4 have zero means and variances N0T / 4.(b) Show that ω1,ω2,ω3, and ω4, have zero means and
A delay-and-multiply receiver for DPSK as shown in Figure 9.17 should have an input filter bandwidth of B = 2/T = 2R Hz to achieve the asymptotic probability of error given by (9.113). If the
Assume that the delay in a delay-and-multiply receiver for DPSK as shown in Fig. 9.17 is in error by |Î”T|. (a) Show that the asymptotic bit error probability becomes (Consider
A channel of bandwidth 100 kHz is available. Using null-to-null RF bandwidths, what data rates may be supported by:(a) BPSK;(b) coherent FSK (tone spacing = 1 ∕ 2T);(c) DPSK;(d) non-coherent
An M-ary communication system transmits at a rate of 2000 symbols per second. What is the equivalent bit rate in bits per second for M = 4? M = 8? M = 16? M = 32? M = 64? Show a plot of bit rate
Use MATLAB to plot out-of-band power for M-ary PSK, QPSK (or OQPSK), and MSK. Compare with Figure 10.16. Use trapz to do the required numerical integration.Figure 10.16 -10 BPSK -20 QPSK or OQPSK -30
A serial bit stream, proceeding at a rate of 10 kbps from a source, is given as110110 010111 011011 (spacing for clarity)Number the bits from left to right starting with 1 and going through 18 for
QPSK is used to transmit data through a channel that adds Gaussian noise with power spectral density N0 = 10-11 V2 / Hz. What are the values of the quadrature modulated carrier amplitudes required to
Use MATLAB to plot curves like those shown in Figure 10.25. Use fzero in MATLAB to find the processing gain required to give a desired probability of bit error for a given JSR and SNR. Note that your
Write a MATLAB simulation of GMSK that will simulate the modulated waveform. From this, compute and plot the power spectral density of the modulated waveform. Include the special case of ordinary MSK
A QPSK modulator produces a phase-imbalanced signal of the form(a) Show that the integrator outputs of Figure 10.2, instead of (10.5) and (10.7), are now given bywhere the ± signs depend on
Write a MATLAB simulation for an Alamouti type of MISO diversity system.
(a) A BPSK system and a QPSK system are designed to transmit at equal rates; that is, two bits are transmitted with the BPSK system for each symbol (phase) in the QPSK system. Compare their symbol
Given the serial data sequence101011 010010 100110 110011associate every other bit with the upper and lower data streams of the block diagrams of Figures 10.2 and 10.4. Draw on the same time scale
Sketch excess phase tree and phase trellis diagrams for each of the cases of Problem 10.7. Show as a heavy line the actual path through the tree and trellis diagrams represented by the data sequence
Derive Equation (10.25) for the spectrum of an MSK signal by multiplying |G(f)|2, given by (10.23), times SBPSK (f), given by (10.24). That is, show that serial modulation of MSK works from the
An MSK system has a carrier frequency of 10 MHz and transmits data at a rate of 50 kbps.(a) For the data sequence 1010101010 …, what is the instantaneous frequency?(b) For the data sequence
Show that the scalar-product definitions given by (11.44) and (11.45) satisfy the properties listed in the subsection entitled €˜€˜Scalar Product€™€™ in
Determine the quantizing levels, in terms of σ, so that the entropy at the output of a quantizer is maximized. Assume that there are six quantizing levels and that the input to the quantizer is a
Repeat the preceding problem assuming that the input to the quantizer has the Rayleigh probability density function Data From Problem 12.11The input to a quantizer is a random signal having an
The input to a quantizer is a random signal having an amplitude probability density function The signal is to be quantized using four quantizing levels xi as shown in Figure 12.44. Determine the
From the entropy definitions given in (12.25) through (12.29), derive (12.30) and (12.31).
Determine the capacity of the channel described by the channel matrix shown below. Sketch your result as a function of p and give an intuitive argument that supports your sketch. (Note: q = 1 - p.).
A binary symmetric channel with error probability p1 is followed by an erasure channel with erasure probability p2. Describe the channel matrix that results from this cascade combination of
In Computer Example 12.7, a runstream of W and B symbols was generated. Develop a MATLAB program to develop the run-length code.
A channel has two inputs, (0, 1), and three outputs, (0, e,1), where e indicates an erasure; that is, there is no output for the corresponding input. The channel matrix is Compute the channel
Repeat Computer Example 12.6 for y = 0.1 and y = 0.3 and y = 0.4 What do you conclude from these results combined with the results of Computer Example 12.6, which were generated for y = 0.2?
A binary symmetric channel has an error probability of 0.007. How many of these channels can be cascaded before the overall error probability exceeds 0.02?
Develop a MATLAB program for generating the tree diagram illustrated in Figure 12.25. 0000 (000) (000) 0001 (111) (000) 0010 (101) (111) 0011 (010) (000) 0100 (100) (101) 0101 (011) (111) 0110 (001)
In implementing the Tomen technique for comparing codes on the basis of information bit error probability, the MATLAB function nchoosek was used. Using this function for large values of n and k can
A channel has the following transition matrix:(a) Sketch the channel diagram showing all transition probabilities.(b) Determine the channel output probabilities assuming that the input
Table 12.5 gives a short list of rate 1/2 and rate 3/4 BCH codes. Using the Torrieri bound and an appropriate MATLAB program, plot together on a single set of axes the bit error probability for the
A source consists of six outputs denoted [A, B, C, D, E, F] with respective probabilities [0.30, 0.25, 0.20, 0.1, 0.1, 0.05]. Determine the entropy of the source.
Computer Example 12.2 did not contain the MATLAB program used to generate Figure 12.18. Develop a MATLAB program for generating Figure 12.18, and use your program to verify the correctness
A source has five outputs denoted [m1, m2, m3, m4, m5] with respective probabilities [0.30, 0.25, 0.20, 0.15, 0.10]. Determine the entropy of the source. What is the maximum entropy of a source with
Develop a MATLAB program that generates the Huffman source code for an input random binary bit stream of arbitrary length.
Assume that you have a standard deck of 52 cards (jokers have been removed).(a) What is the information associated with the drawing of a single card from the deck?(b) What is the information
Develop a computer program that allows you to plot the entropy of a source with variable output probabilities. We wish to observe that the maximum source entropy does indeed occur when the source
A binary three-hop communication system has transition probabilities αi, βi, and γi (i = 1,2,3,4). Determine the transition probabilities for the overall three-hop system. Solve this problem two
Given that the impulse response of an ideal integrator over T seconds is h(t) = 1/T [u (t) - u (t - T)] where u(T) is the unit-step function, show that its equivalent noise bandwidth is BN,idealint =
Assume a biphase modulated signal in white Gaussian noise of the form y(t) = ˆš2P sin (Ï‰ct ± cos-1 m + Î¸) + n(t), 0 ‰¤ t ‰¤
Generalize the estimation of a sample of a PAM signal, expressed by (11.207), to the case where the sample value m0is a zero-mean Gaussian random variable with variance Ïƒ2m. 1 No -° (E
Given K independent measurements (Z1, Z2,.....,ZK) of a noise voltage Z(t) at the RF filter output of a receiver: (a) If Z(t) is Gaussian with mean zero and var {σ2n}, what is the ML estimate
For which of the cost functions and posterior pdfs shown in Figure 11.14 will the conditional mean be the Bayes estimate? Tell why or why not in each case. х 3DА-ӑ Cх) fAlZ х (a) C(x) fAlZ
Let an observed random variable Z depend on a parameter Î» according to the conditional pdf The Î± priori pdf of Î» iswhere Î² and m are
Investigate the use of diversity to improve the performance of binary non-coherent FSK signaling over the flat-fading Rayleigh channel. Assume that the signal energy Esis divided equally among N sub
Generalize the fading problem of binary non-coherent FSK signaling to the M-ary case. Let the ith hypothesis be of the formi = 1, 2,......., M; 0 ‰¤ t ‰¤ Tswhere Gi is
This problem develops the simplex signaling set.13Consider M orthogonal signals, si(t), i = 0, 1, 2,...., M - 1 each with energy Es. Compute the average of the signals and define a new signal
Go through the steps in deriving (11.144). LI SR, (r:|H1,) dr SR, (ri ,) dr |P (E\H¡) = 1 1+; (20²E/No) 8. I|
(a) Referring to the signal set defined by (11.98), show that the minimum possible Δf = Δω / 2π such that (si, sj) = 0 is Δf = 1/(2Ts).(b) Using the result of part (a), show that for a given
Consider vertices-of-a-hyper-cube signaling, for which the in which the coefficients Î±ik are permuted through the values +1 and -1, Es is the signal energy, and the
Consider (11.128) for M = 2. Express PE as a single Q-function. Show that the result is identical to binary, coherent FSK. VE,+n n;/No e-n;/No Pei - dp, VANO dn; νTNο -00 -00 M-1 E,/No+y dy
For M-ary PSK, the transmitted signal is of the form si (t) = A cos(2πt + iπ / 2), i = 0, 1, 2, 3, for 0 ≤ t ≤ 1si (t) = A cos[4πt + (i - 4)π / 2], i = 4, 5, 6, 7, for 0 ≤ t ≤ 1(a)
Rework Example 11.6 for half-cosine pulses given by – kt Pr(t) = II [t – kt (t – kt cos |T т k = 0,±1,±2,±K
Use the Gram--Schmidt procedure to find an orthonormal basis for the signal set given below. Express each signal in terms of the orthonormal basis set found.s1(t) = 1, 0 ≤ t ≤ 2s2 (t) = cos
(a) Find a set of orthonormal basis functions for the signals given below which are defined on the interval -1 ≤ t ≤ 1:x1 (t) = t x2 (t) = t2 x3 (t) = t3 x4 (t) = t4 (b)
(a) Using the Gram--Schmidt procedure, find an orthonormal basis set corresponding to the signalsx1 (t) = exp (-t) u(t)x2 (t) = exp (-2t) u (t)x3 (t) = exp (-3t) u (t)(b) See if you can find a
Consider the set of signalswhere N is an integer and i = 0,1,2,3,4,5,6,7.(a) Find an orthonormal basis set for the space spanned by this set of signals.(b) Draw a set of coordinate axes, and plot the
Use the Gram--Schmidt procedure to find a set of orthonormal basis vectors corresponding the the vector space spanned by the vectors x1 = 3î + 2ĵ – k, x, = -2î + 5f +k, x3 = 6î – 2ĵ + Tk,
(a) Use the Gram--Schmidt procedure to find a set of orthonormal basis functions corresponding to the signals given in Figure 11.12.(b) Express s1, s2, and s3 in terms of the orthonormal basis set
Verify Schwarz’s inequality for the 3-space vectors of Problem 11.6.Data From Problem 11.6For the following vectors in 3-space with x, y, z components as given, evaluate their magnitudes and the
Verify Schwarz€™s inequality forwhere the Ï•n(t)s are orthonormal and the ans and bns are constants. N x1 (t) = Ea,ø, (t) and x, (1) = b,4, (1) n=1 n=1
Evaluate||x1||,||x2||,||x3||, (x2, x1), and (x3, x1) for the signals in Figure 11.11. Use these numbers to construct a vector diagram and graphically verify that x3= x1+ x2.Figure 11.11 X1 Хз X2 -1
Let x1(t) and x2(t) be two real-valued signals. Show that the square of the norm of the signal x1(t) + x2(t) is the sum of the square of the norm of x1(t) and the square of the norm of x2(t), if and
Using the appropriate definition, (11.44) or (11.45), calculate (x1,x2) for each of the following pairs of signals:(a) e-ItI, 2e-3tu(t)(b) e-(4+j3)t u(t), 2e-(3+j5)t u(t)(e) cos 2πt, cos 4πt(d) cos
For the following vectors in 3-space with x, y, z components as given, evaluate their magnitudes and the cosine of the angle between them (î, ĵ, and k̂ are the orthogonal unit vectors along the
Show that ordinary three-dimensional vector space satisfies the properties listed in the subsection entitled ‘‘Structure of Signal Space’’ in Section 11.2, where x(t) and y (t) are
Referring to Problem 10.3, find general expressions for the probabilities of false alarm and detection for each case. Assume that c12 = 1 in all cases. Numerically evaluate them for the cases where
Write a computer simulation of the PLL estimation problem. Do this by generating two independent Gaussian random variables to form Z1 and Z2 given by (11.196). Thus, for a given (11.201).
Assume that data of the form Z = S + N are observed where S and N are independent, Gaussian random variables representing signal and noise, respectively, with zero means and variances
Consider the hypothesesH1 : Z = N and H2 : Z = S + Nwhere S and N are independent random variables with the pdfsfs(x) = 2e-2x u(x) and fN(x) = 10e-10x u(x)(a) Show thatfz(z|H2) =
Write a computer program to make plots of σ2p versus K, the number of observations, for fixed ratios of σ2A /σ2n, thus verifying the conclusions drawn at the end of Example 10.8.
In practical communications systems and radar systems we desire that the system operate with a probability of detection that is nearly one and a probability of false alarm that is only slightly
Consider a two-hypothesis decision problem where(a) Find the likelihood ratio Î›(Z).(b) Letting the threshold n be arbitrary, find the decision regions R1 and R2 illustrated in Figure
Show that the variance of Ngas given by (10.145) is N0Tb. -TB 2n(t)c(t) cos(@,t + 0)dt (10.145) Ng
Show that the variance of N1 as given by (10.150) is approximated by the result given by (10.151). [You will have to make use of the fact that T-1c Î›(Ï„ /Tc) is
Compute the number of users that can be supported at a maximum bit error probability of 10-3 in a multi user DSSS system with a code length of n = 255. [Take the limit as Eb / N0 → ∞ in
Consider a DSSS system employing BPSK data modulation. PE = 10-5 is desired with Eb / N0 → ∞. For the following JSRs tell what processing gain, Gp, will give the desired PE. If none, so state.(a)
A DSSS system employing BPSK data modulation operates with a data rate of 10 kbps. A processing gain of 1000 (30 dB) is desired.(a) Find the required chip rate.(b) What is the RF transmission
The a periodic auto correlation function of a binary code is of interest in some synchronization applications. In computing it, the code is not assumed to periodically repeat itself, but (10.130) is

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