Principles Of Embedded Networked Systems Design 1st Edition Gregory J. Pottie ,William J. Kaiser - Solutions

Thresholds with costs Compute the threshold in Example 5.5 analytically.
Noise under orthogonal decomposition One property of the white noise n(t) is that the autocorrelation function E[n(t)n(tþ)] is equal to (N0/2()). When this noise is expanded as a set of orthonormal functions with weights wi, prove the variance of wi is also N0/2.
Cross-products for ML detection The ML decision is to choose the signal space point closest to the received vector of decision variables. This requires the calculation of the Euclidean distance. The complexity of this calculation can be very high in some situations. When the signal energy is the
Frequency shift keying (FSK)In an FSK scheme, transmitted signals are orthogonal to each other. For binary FSK (BFSK), the two equally likely signals are s1ðtÞ ¼ffiffiffiffi Ep ffiffiffiffiffiffiffiffiffi 2=T psin !1t; 0 t T;s2ðtÞ ¼ffiffiffiffi Ep ffiffiffiffiffiffiffiffiffi 2=T psin !2t;
Correlation vs. matched filter receiver In Section 5.2, it is shown that the correlation filter for signal x(t) can be replaced by a matched filter with impulse response x(T t). Show that the cross-product between the received signal x(t) and sk(t) can be implemented by a matched filter with
Matched filter and SNR Prove that the matched filter maximizes the output SNR and compute the maximum output SNR as a function of the energy of the signal s(t) and N0.
Vector and integral forms of likelihood function Show that the likelihood function lðkÞ ¼ N 2lnðpN0Þ 2 N0 XN i¼1ðxi skiÞ2 can be written in the form lkðxÞ ¼2 N0 Z T 0xðtÞskðtÞdt 1 N0 Z T 0s2 kðtÞdt:
Likelihood function for non-coherent detection Show that the likelihood functionk ¼ eEk N0 I0 2ffiffiffiffiffiffi Ek pN0 jLkj ;which is discussed in Section 5.2 for non-coherent detection, can be derived fromkðÞ ¼ ez;where z ¼ 2 N0 Z T 0xðtÞskðt; Þdt 1 N0 Z T 0s2 kðt; Þdt;and
On–off signaling Explain the reason why l0 is equal to 0 in Example 5.6.
Detection of two sinusoids Consider the problem of detecting two signals s1 (t) ¼ A1 sin (2p fctþ) and s2 (t) ¼ A2 sin (2p fctþ) over the interval (0,T), perturbed by AWGN with psd N0/2. The a priori probabilities of s1 and s2 are P(s1) and P(s2), respectively.What is the MAP detector,
Decision threshold for on–off signaling Show that in Example 5.6 the threshold is approximately E/4 at high SNR.
MR combining In MR combining, the received signals xk from each of the N sensors with channel coefficient kejk are cophased to provide coherent voltage addition and are individually weighted to provide optimal SNR. The noise power is N0/2.Show that after being cophased by multiplying by ejk the
MR combining with unequal SNRs Derive the weights forMRcombining when the noise variances are not equal and compute the resulting SNR.
Coverage area with cooperation In Example 5.12(d) it is shown that with the cooperation of two nodes, the coverage area per sensor is doubled. If the channel and the requirements are the same, compute the coverage radius and coverage area when four sensors arranged in a square participate in this
Mean square estimation Let X be a real-valued RV with a pdf of fX(x). Find an estimate ^x such that the mean square error of x by ^x is minimized when no observation is available. 10 d Ok d 2 3 Figure 5.24 Regular beamforming array.
Fitting functions to observations Consider RVs X and Y. Find a function g(.) of the observed X such that g(x)minimizes Y in the minimal mean square sense. (Note, you will need the results from Problem 5.21).
Orthogonality and MMSE filters In Section 5.3, it is shown that the input u is orthogonal to the estimation error e.Show that for the MMSE filter e and y are also orthogonal.
Three-tap equalizer In Example 5.16, an equalizer with two taps is considered. Now assume that three taps are available. Compute the tap coefficients and the overall response of the channel and the equalizer. Also compare this responsewith the response of two taps.
Beamforming for interference suppression Assume M sensors are deployed to implement beamforming, as depicted in Figure 5.24. The distance from the reference sensor to the ith sensor is denoted by di. There are M sources. The first source is the desired source and the others are the interferers. The
LMS algorithm Show that the gradient G0 in the LMS algorithm is equal to E e0u0.
Cramer–Rao bound Let X be the sample mean from n independent Gaussian RVs X1 ,X2 , . . . , Xn with distribution N(, 2). Assume 2 is known. First, derive the Cramer–Rao bound. Then, show that X is the most efficient unbiased estimate for (i.e., it attains the RHS of the Cramer–Rao
ML estimates of the mean and variance of Gaussian random variables Consider n independent random samples from N(, 2). Let ¼(, ). That is,1¼and 2¼. Find the ML estimates of and .
Spectral density of non-zero mean sequences In Section 6.1 the baseband-equivalent psd is given for a zero mean sequence.Derive the baseband-equivalent psd of linearly modulated signals if the mean of the sequence is not zero. Assume the information symbols {In} in the sequence are real and
Power spectral density of PAM The low-pass equivalent representation of a PAM signal isuðtÞ ¼X n Ingðt nTÞ:Suppose g(t) is a rectangular pulse and In ¼ an an3;where {an} is a sequence of uncorrelated binary-valued (1, 1) random variables that occur with equal probability.(a) Determine the
Impulse response of raised-cosine filter Determine the inverse Fourier transform of the frequency response P(f) of the raised-cosine filter.
QAM transmission with raised-cosine filtering The bandwidth BT of the raised-cosine filter is defined by 2W f1 ¼ W(1þ). A computer outputs binary data at the rate of 56 kb/s. The computer output is transmitted using a baseband 16-QAM system that is designed to have a raisedcosine spectrum.
Gray code The property of the Gray code is that the code words of adjacent symbols only differ in one bit. For example, the code words of 8-PAM symbols are as illustrated in Figure 6.21. This results in a minimum expected number of bit errors per symbol error, in conditions of low symbol error
Error probability for FSK Using the approximation of symbol error probability PðeÞ NdminQ dffimffiffiffiiffinffiffiffiffi 2N0 p ;prove the symbol error probability of M-FSK is PðeÞ ðM 1ÞQ ffiffiffiffiffiffi EN0r : 0(1) 8(t) LPF Kd V2(t) F(f)=F(f) o(t) VCO Kv F(f)= j2f Figure 6.22
Hard-decisions vs. soft-decisions In Example 6.5, if hard-decisions are used instead of soft-decisions:(a) How many errors can the code detect and correct, respectively?(b) Compute the error probability, i.e., the probability that the decoder cannot make the correct decision.(c) Compare the error
Minimum HD decoding A given code consists of the codewords: 0000000, 0011110, 0101101, 0111000, 1001100, 1011001, 1101010, 1110100. If 1001110 is received, what is the decoded codeword based on the minimum HD?
A rate-1/3 convolutional encoder Consider a convolutional encoder for a K¼4 shift register with v¼3 modulo-2 adders. At each clock time the outputs of the v adders are sampled by a commutator.Thus v output symbols (v1,v2 ,v3) are generated for each input symbol, giving a code of rate 1/v. The
Linearized PLL The linear model of the analog PLL is depicted in Figure 6.22.(a) Derive the closed-loop transfer function o (f )/i(f ).(b) If F(s)¼1/(s¼ffiffiffi2 p), using the Laplace transform and the final value theorem, find an expression for the steady-state phase error.Hints: The Laplace
PLL transfer function Assume that the loop filter of a PLL is a low-pass filter, as shown in Figure 6.23.Evaluate the closed-loop transfer functionH(f ) for a linearized PLL, and compute the 3 dB frequency.
Noise in a PLL Consider the noise characteristic of a PLL. The internal phase noise of the VCO is modeled by the input n(t) as shown in Figure 6.24.(a) Find an expression for the closed-loop transfer function o ( f ) / n( f ), wherei(t)¼0.(b) If F1 ( f ) is a low-pass filter given in the last
Costas loop The Costas loop uses both in-phase and quadrature phase detectors to keep a VCO centered at the suppressed-carrier frequency, as shown in Figure 6.25. Let the input be b(t)cos (!ctþi), where b(t)¼1. Label the signals for each path in the block diagram and show that the Costas loop
Differential PSK (DPSK)In DPSK, the information is encoded using the differences between bits in two successive bit intervals. The output digits after the arbitrary starting digit are determined by the rule that there is no change in the output state if a 1 is present.There is a change in output
Sinc pulse trains The overall pulse shape p(t) of a baseband binary PAM system is defined by pðtÞ ¼ sinc tTb ;where Tb is the bit duration of the input binary data. The amplitude levels at the pulse modulator output are þ1 or 1, depending on whether the binary symbol at the input is 1 or 0,
Early–late gate In the early–late gate synchronizer, the outputs of two correlators can be viewed as the following two samples although their values are not equal. The correlator can be replaced by a matched filter and the output sampled at t¼T and t¼Tþ.If the input of the multipliers is
Multipath interference In the presence of multipath interference, replicas of the transmitted signal arrive at the receiver with various attenuations and delays. In this problem, assume that there is only one predominant source of multipath. If the transmitted signal in such a system is x(t), then
ZF LEQ Find the tap coefficients required for a ZF transversal equalizing filter for the case of a three-tap filter. For a single flat-topped pulse into the transmit filter the responses of the channel at the sampling times are wc(2Ts)¼0.08, wc(Ts)¼0.25, wc(0)¼1.0, wc(Ts)¼0.3, and
Five-tap ZF LEQ Find the tap coefficients if the three-tap filter in Problem 6.18 is replaced by a fivetap filter. In the desired output sequence, there are two samples before and after the peak value. Compare filter coefficients and the output sequences of the threetap and the five-tap filters.
Equalization of a non-return-to-zero (NRZ) signal A2-kb/s unipolarNRZsignal is sent over a band-limited channel. The channel has the impulse response hc(t)¼e2000tu(t). Design a five-tap ZF transversal filter to equalize the channel response. How will the coefficients change if more taps are used?
DPSK on a fading channel Given the bit error rate of DPSK in the AWGN channel, compute the bit error rate of DPSK with a Rayleigh fading channel and plot the bit error rate for each as a function of Eb /N0 over a range 0 –20 dB.
PN sequence properties A PN sequence is generated using a feedback shift register of length m¼5. The chip rate is 106 chips per second. Find the following parameters:(a) the PN sequence length;(b) the chip duration of the PN sequence;(c) the PN sequence period. Clock Flip-flop Modulo-2 adder
PN sequence generation Figure 6.26 shows a four-stage feedback shift register. The initial state of the register is 1000. Find the output sequence of the shift register.
PN sequence statistical properties For the feedback shift register given in Problem 6.23, two properties can be demonstrated.(a) The balance property states the number by which the number of 1s exceeds the number of 0s. Determine this number for Problem6.23.(b) A subsequence of identical symbols
Processing gain The processing gain of a DS-SS system may be expressed as the ratio of the spread bandwidth of the transmitted signal to the despread bandwidth of the received signal. Justify this statement for the DS/BPSK system.
Selection diversity It is assumed that there are M diversity branches and each branch has the same average SNR given by .(a) Derive the cdf of the SNR after selection diversity processing.(b) Derive the average SNR improvement offered by selection diversity.
MR combining for diversity In MR combining, the optimal weight is Gi¼ri/N0, where ri is the received signal amplitude at the ith branch. Derive the average SNR improvement.
Five-branch diversity Assume five-branch diversity is used, where each branch receives an independent Rayleigh fading signal. If the average SNR is 20 dB, determine the probability that the SNR will drop below 10 dB. Compare this with the case of a single receiver without diversity.
SIR in a cellular network with a reuse factor of 7 A cellular network with a frequency reuse factor of 7 is depicted in Figure 7.12, where cells with the same letter share the same set of frequencies. In this system, mobiles communicate with base stations through two 25-kHz simplex channels, and a
Linear prediction Consider a zero-mean stationary random process x(n) that has the following correlation function:RXðnÞ ¼1 n ¼ 0 0:8 n ¼ 1 0:5 n ¼ 2 0:3 n ¼ 3 0 any other n:8>>>>>>>>:Linear predictors with two and three delay taps are shown in Figure 7.13.(a)
MIMO equalization A discrete time two by two MIMO system is shown in Figure 7.14, where s1(n), s2(n) are source sequences, and y1 and y2 are measurement sequences. The channel transfer functions are:h11ðzÞ ¼ 1 þ az1; h22ðzÞ ¼ 1 þ bz1; h12ðzÞ ¼ cz1; h21ðzÞ ¼ dz1:Besides the
MR combining Consider the system shown in Figure 7.15, where the signal s is coherently sensed by two receivers. The measurements y1 and y2 can be expressed as follows:y1 y2 ¼h1 h2 s þ1 1 ;where the channel gains are given by h1 ¼ aej1 ; and h2 ¼ a2ej2 . Suppose the noise 1 and 2 are
Array processing A uniform linear array consists of ten sensors in the pattern depicted in Figure 7.16.The array is steered to 0¼p/3. The sensor separation is d¼l/4, where l is the wavelength of interest.(a) Compute the electrical angle for the signal coming at the designated direction of 0.(b)
Source–sensor separation Consider the problem of deploying sensors to monitor a certain area, in this case a square of unit area. Within the square, there is a uniformly distributed point source that we would like to measure. In other words, the source appears anywhere in the square with equal
Source–sensor separation Consider deploying n sensors in a disk of unit area. A point source appears in the disk according to a uniform distribution. Supposing the sensors are placed in the disk based on the same distribution, determine the mean distance between the point source and the closest
MAC schemes Consider the following cases in conjunction with MAC schemes. Which MAC scheme appropriately describes each situation?(a) Cars are waiting at a crossing for the traffic lights to turn green.(b) You raise your hand before asking a question in the classroom.(c) Many radio stations are
Maximal length sequence Consider the feedback shift register of length 4 depicted in Figure 7.18. It consists of four flip-flops and a modulo-2 adder.(a) Assuming the initial state of the shift register is r1 r2 r3 r4¼1000, determine the output sequence y(n). What is the period of y(n)?(b) Replace
DS-SS The information bit duration of a DS-SS communication system is Tb¼8.191 ms. The PN sequence has a chip duration of Tc¼1ms. What is the processing gain of this spread spectrum system?What is the length of the shift register that is used to generate the PN sequence?
Ground reflection model In a two-ray ground reflection model, the transmitter and receiver are placed ht and hr above the ground, as shown in Figure 7.19. Assume the wave obeys the free-space propagation law, and is given byin which E0 is the field at the reference distance d0. Note that the phase
Little’s Theorem A packet arrives at the server every K seconds. Each packet requires 2K/3 seconds for transmission and K/2 seconds for processing. What is the average number of packets in the server? (Hint: Little’s Theorem states that the average number ofpackets in the system N, the average
Aloha In a MAC system, users transmit equal length packets through a shared channel.The transmission time for each packet is T seconds.Acollision occurs when two or more packet transmissions overlap in time, and a transmission is considered successful if there is no collision during transmission.
Connectivity Connectivity is usually used to describe the reliability of the network. In a network, if there is a path connecting nodes i and j after removing (k1) other nodes and their associated links, then i and j are said to be k-connected. Anetwork is said to be k-connected if any pair of
Diameter The distance between a pair of nodes in the network is the length of the shortest path connecting the two nodes. In the case of disconnected nodes, the distance is set to infinity. The diameter of a network is then defined as the maximum distance between any pair of nodes in the network.
Flooding Flooding is often used in a local network to update path state information. To limit the number of transmissions, two rules are observed during flooding: the node does not relay the packet back to the sender; the node only relays the packet to its neighbor once. Consider the network
Shortest-path routing: Bellman–Ford algorithm In this problem, we use the network topology in Figure 8.29 to illustrate the Bellman–Ford algorithm for finding the shortest route to a node. In the figure, the number beside a node serves as the label for the node, and the number near an arc
Shortest-path routing: Dijkstra algorithm We will use Figure 8.29 to illustrate the Dijkstra algorithm for finding the shortest route to a destination node. The length of the direct arc connecting nodes i and j is defined to be dij. For a detailed description of the figure and the definition of
DSR For the network topology in Figure 8.30, assuming the same transmission delay on all the links, use the DSR algorithm to determine the routes from node 1 to node 12, from node 3 to node 12, and from node 6 to node 8. 2 + 7 10 3 Figure 8.30 DSR. 5 2 1 12 2 4 P+ 3 (a) Figure 8.31 Spanning tree.
Spanning tree Find all possible spanning trees for the two graphs in Figure 8.31 subject to the constraint that node 1 must be the root. Determine the number of nodes N and arcs A in each of these spanning trees. Can you see a relation between N and A?
Directed diffusion Consider the situation in Figure 8.32, which is similar to Example 8.6. The solid lines represent transmission links between nodes, and dashed lines indicate boundaries of tiers. Here node A wants to transmit to node D. Suppose the transmission takes the branches within the same
M/M/1 queues Consider the infinite length M/M/1 queue discussed in Example 8.7.(a) Given that the probability that there are n customers in the queue is P(n)¼(1)n, where ¼l / , show that the average number of customers in the queue is N ¼ EðnÞ ¼X1 n¼0 nPðnÞ ¼1 :(b) Plot N as a
ARQ Consider a simple ARQ scheme through a single transmission link of data rate R.The ARQ scheme works as follows. The sender transmits a data packet across the link. Once the receiver receives the whole packet, it checks whether data have been corrupted. If there is no error, a packet is sent to
Source localization Consider source localization using clusters of sensors. Shown in Figure 8.33 are the source and two clusters of sensors located at coordinates (0,0) and (1,0). Using the beamforming technique discussed in Chapter 7, the two clusters of sensors are able to determine that the
Channel capacity – Gaussian channel The channel capacity of a continuous Gaussian channel is given as C ¼ B log2 1 þS NB in which S is the signal power, N is the noise power density, and B is the channel bandwidth. Find the C when B!1.
Slepian–Wolf coding Consider two discrete random variables X and Y. Their joint distribution is given in Table 8.5.(a) Compute H(X), H(Y), H(X|Y), H(Y|X), and H(X,Y).(b) Supposing X and Y are observed and coded by two sensors, compare the source rates with independent and Slepian–Wolf coding
Network capacity Use R1 and R2 as two coordinates. Plot the rate region that is achievable in a twouser Gaussian multiple access channel assuming P/N¼1.
Rate distortion Consider rate distortion coding with a single helper. Suppose 2 X ¼ 1, DX¼0.1.Plot the lower bound of RX as a function of R1 for ¼0, 0.4, 0.7, and 1. Comment on how this bound changes as R1 and change.
Timing offset and GPS GPS uses a constellation of 24 satellites and their ground stations as reference points to calculate positions accurate to a matter of meters. Suppose we find our distance measurements from three satellites to be 18 000, 19 000, and 20 000 km respectively. Collectively this
Linearizing GPS equations In order to find position using the GPS system, we need to know the location of at least three satellites and the distance to each of those satellites. Assume that the three satellites are located respectively at (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), and that the
Averaging to reduce error in TOA TOA is based upon the measurement of the arrival time of a signal transmitted from the to-be-located object to several reference nodes. For radio signals, the distance is ct, where c is the velocity of light and t is time of travel from the object to the reference
TOA with low-cost clocks In order to make accurate range measurements in a GPS system, the receiver and satellite both need clocks that can be synchronized down to a nanosecond, which potentially could require atomic clocks not only on all the satellites, but also in the receivers. However, atomic
TDOA in a two-dimensional space TOA requires that all the reference nodes and the receiver have precise synchronized clocks and the transmitted signals be labeled with time-stamps. TDOA measurements remove the requirement of an accurate clock at the receiver.Assume that five reference nodes have
TDOA in a three-dimensional space Assume that five reference nodes are known to be at (0,3,0), (6,0,0), (3,4,0),(4,3,0), and (0,0,8) respectively. Also, t12¼0 s, t13¼1 s, t14¼0.7 s, t15¼0.7 s, and t16¼1.7 s. The velocity of propagation is n.(a) Use (9.10) to find the unknown location (xt,
Position estimation without the range error covariance matrix Let u denote the node with unknown position, and let the position of reference node j be (xj, yj). Denote by rj,u the range measurement. It has been shown that the position estimate of the unknown can be updated according to
An alternative approach for position estimation Assume a¼(1,1), b¼(1,1), c¼(1,1), and d¼(1,1). Locate node d using the noisy range estimates la¼0.7, lb¼1.5, lc¼1.6, ld¼2.2.(a) Calculate the centroid of the three known locations.(b) Obtain R and y according to (9.25) and (9.26)
Weighted centroid computation Three beacons are located at a¼(1,1), b¼(1,1), and c¼(1,1). The received powers from nodesa, b, and c are 1.2, 1.5, and 1.7 respectively. Calculate the unknown position of the receiver through a weighted centroid computation.
Collaborative multilateration Execute the second iteration of Example 9.5 using (9.5)–(9.8). That is, calculate u2 using A, C, and v1, and then calculate v2 using B, D, and u1.
Linearization of AOA location determination The intersection of the angles from two or more sites may be used to provide an unknown position in the plane. For this triangulation problem, denote the position of the two known nodes as ri¼[xi yi]T, i¼1, 2, and the unknown node’s position as r¼[x
Energy supply In Example 10.1, a node uses a zinc–air battery (at 1 V) instead of a lithium-ion battery. How large must the battery be for reliable operation of the node for 15 days? Further, the weather is not friendly and provides only 8 h of daylight. How large must the photovoltaic cell be?
Cost of storing energy and service life(a) An installation requires 60 cells in series to produce 120 V. Depending on the application, this output is converted to different ac voltages. The cost of a 450 A–h, twelve-cell battery is $3000. In order to get the longest possible life, the batteries
Dynamic power consumption in CMOS.(a) Discuss the different ways of reducing power consumption in CMOS.(b) What is the net effect on the power consumption in CMOS if the transition frequency is doubled and the voltage swing is halved?(c) What is the effect of transistor size on power consumption
Parallelism, power, and delay How does parallelism affect power consumption and delay in circuits? Consider the power consumption and delay equations as in Example 10.3.
Scheduling based on deadlines Consider an application of a sensor network that demands a distributed processor platform. We have three processors (P1, P2, P3), each having four modes of operation – boot-up mode (power consumption of 3 W), execution mode (power consumption of 5 W), idle mode, and
Direct transmission vs. multihop.Consider the scenario depicted in Figure 10.5. The Nþ1 nodes are placed in a straight line. All adjacent nodes are spatially separated by distance r. The radios on the nodes consume E J per bit when turned on, in either transmit or receive mode. The radio front-end
Dynamic modulation scaling revisited Redo Example 10.12 taking the energy consumed by the receiving radio, CR, into account. Assume CR¼12 nJ.
Network lifetime The lifetime of any network depends on the power dissipated. Using relaying rather than direct transmission extends the lifetime, as well as giving frequency reuse benefits. Let ES be the energy needed to sense a bit, ER that to receive a bit and ET that to transmit a bit over some
Best-energy route Find the minimum-energy path in Figure 10.7 from source S to destination D.The diagram shows the connectivity with associated energy cost.
Traveling salesman problem The energy mule example is an instance of the well-known traveling salesman problem. It is not always possible to use brute force to solve the equations. Two algorithms found useful for large networks are:(i) Nearest neighbor algorithm Begin at the starting node. Whenever
ACID properties Which ACID properties are illustrated by the following transactions?(a) While a new row is being added to a table by one process, another process reads the entire table but the new row is not included.(b) While a new row is being added to a table by one process, the process is

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Principles Of Embedded Networked Systems Design 1st Edition Gregory J. Pottie ,William J. Kaiser - Solutions

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