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Molisch, Andreas F. Wireless communications / Andreas F. Molisch. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN Ove Edfors, Andreas F. Molisch, and Fredrik Tufvesson. WIRELESS textbook by A. F. Molisch . Satellite communications, wireless networking, cellular technology – .. caite.info Idea. Professor Andreas F. Molisch, renowned researcher and educator,has put together the Wireless Communications (eBook, PDF) - Molisch, Andreas F.
Figure Equations and figures without this prefix refer to the solution manual itself. Molisch combines intuitive explanations with mathematical descriptions and examples in this well-organized, comprehensive guide. The underlying mechanism is: This decrease is.
Here nk is the noise term with variance Es N0 , which is equal for all nk. The SNR at the output of the kth matched filter is. This result shows that the Rake functions as a maximum ratio combiner since the SNRs of the multipaths are added.
Let the mean SNR on both branches be. The cdf is therefore given by. Thus we have a 1. For twobranch MRC the gain would be 3 dB and for two-branch selection diversity the gain would be 1. This sum is for large mean SNRs dominated by the first term, and the average error rate approximately becomes. Note that this corresponds to a system with no diversity. Channel coding 1. If all these syndromes are unique a decoder can distinguish between them and we have proven the claim.
Yes, the entire code can be found from the given information.
By using the cyclic property and, e. The four codewords are in vector form: Codeword x6 x1 0 0 x2 1 x3 0 x4. These code words are linerarly independent and span the whole code space. Using the linear property, all other codewords can easily be calculated. To prove that the generator polynomial G x is the only codeword with degree N K we use two properties of cyclic codes: First we assume that there exist at least two codewords of degree N K.
If G x is a factor of all codewords, this leads to a contradiction. Hence, there is only one codeword with degree N K and that is the generator polynomial itself. Inspection shows that these are. This gives the trivial all-zero codeword last M essage. The corresponding all codewords,. Since the generator polynomial itself is the only codeword with highest degree 3, we know that X x above is the generator polynomial, i.
We can verify our H by calculating the product HGT and ensure that it becomes the all-zero matrix. We will use binary codes here, for simplicity, but the proof is easily extended to other bases. We start by listing the 2K codewords in the code in a table. Then we remove the dmin 1 first symbols from each codeword and get a new set of 2K words that are still unique the original codewords dier in at least dmin positions.
Since the new set of words contains N dmin 1 bits we also know that there cannot be more than 2N dmin 1 dierent ones. When performing syndrome decoding we have syndromes of length N K bits, meaning that we can at most have 2N K unique syndromes. On the other hand, to be able to correct t errors in code words of length N , we need our syndromes to uniquely identifyPall error patterns of Hamming weight up to and including t errors.
Combining these two facts we get the following requirement on syndrome decoding N K. Soft Viterbi decoding a Replacing ones and zeros with their antipodal signal constellation points in the trellis stage in Figure B We should, however, notice that in state B in the second to last trelllis stage there were two equal paths and one was eliminated using the toss of a fair coin. Had the coin given the opposite result, we would have obtained the equally valid: Neither of the alternatives gave exactly the same surviving paths as the hard decoding in Figure B Block codes on fading channels.
Speech coding 1. The two main drawbacks are i wasting resources, since the quality of the speech in that case becomes even better than would be necessary for good perception, and ii in many cases, the lossless coders result in a variable rate, which is dicult to match to the fixed rate provided by circuit-switched wireless systems.
The three main types are i waveform coders, which use the source model only implicitly to design an adaptive dynamical system which maps the original speech waveform on a processed waveform that can be transmitted with fewer bits over the given digital channel, ii model-based coders or vocoders, which rely on an explicit source model to represent the speech signal with a small set of parameters which the encoder estimates, quantizes, and transmits over the digital channel, and iii hybrid coders, which aim at an optimal mix of the two previous designs.
They start out with a model-based approach to extract speech signal parameters but still compute the modeling error explicitly on the waveform level. Speech shows a nearly periodic, discrete multi-tone DMT signal with a fundamental frequency f0 in the range of to Hz for male, to Hz for female speakers and to Hz for children.
Its spectrum slowly falls o towards higher frequencies and spans a frequency range of several Hz. Equalizers 1. The noise enhancement is a direct result of the filters construction. The main advantage of blind equalization is the improvement of spectral eciency as compared to conventional equalization that relies on training sequence.
The main drawback is the computational complexity and reliability of blind equalization techniques. Three established blind techniques are i Constant modulus algorithm, ii Blind maximum likelihood, iii Algorithms based on the cyclostationarity of the received signal.
The autocorrelation can be computed as follows: In matrix form, we have. However, since the length or memory of the channel transfer function is longer than that of the equalizer, residual ISI values exist just outside of the enforced region. The presence of the mixed term 1. From a standard calculus textbook, the anti-clockwise rotation angle of the axes is given by cot 2. Substituting into the MSE equation. Using the above equation, the elliptical contours can be drawn in the rotated e1 , e 2 coordinate system see Fig.
The contour lines must then be rotated anticlockwise by 45 to the desired MSE contours in e1 , e2 coordinate system. The gradient is known exactly and no longer stochastic as in the case of LMS algorithm. The gradient has been provided as. The MSE for each of the three cases is given in Fig. Since R and p are known exactly and fixed, the convergence curve is smooth and monotonic as opposed to the case in Fig. Indeed, Eq. Convergence paths for dierent step sizes.
The MSE surface contours obtained from Exercise Multiple access and the cellular principle 1. During a busy hour each subscriber generates one call of two minutes on average. The system can be modeled as Erlang B. To find the maximum number of subscribers per cell we first need the cluster size, which can be found using Tab.
A blocking probability of 0. From Tab. By using the figure on slide 18, a maximum blocking probability of 0.
Erlang B the most commonly used trac model. Erlang B is used to work out how many lines are required if the trac figure during the busiest hour is known.
A total of 24 channels is needed. Erlang C this model assumes that all blocked calls are queued in the system until they can be handled. Call centers can use this calculation to determine how many call agents to sta, based on the number of calls per hour, the average duration of class and the amount of time calls are left in the queue. A total of 27 channels is needed.
Erlang C systems require more resources than Erlang B. For one operator the waiting time is 0. TDMA requires a temporal guard interval. For the downlink we have the approximation. This is a pessimistic approximation, as at least some BSs have a larger distance. The approximation holds the better the larger the reuse distance is. The worst-case scenario is when MS-0 is on the boundary of its cell, i.
The received power of the user of interest, S, and received power from the interfering user, I, is equally strong at the receiver. Hence, a full overlap of one packet is admissible. The throughout without lost packets is p Tp , where p is the average transmission rate in packets per second. The eective throughput is the percentage of time during which the channel is used in a meaningful way, i. Spread spectrum systems 1. For the following simulations, a generator for PN sequences is required.
L-1 ]; end Here a long sequence with an 8-state shift register is used and two users use dierent shifted sub-sequences. After Despreading, User 1 20 15 10 5 0 -5 After Despreading, User 2 20 15 10 5 0 -5 After Despreading, User 1 50 0 After Despreading, User 1 8 6 4 2 0 -2 -4 -6 The results for a delay-dispersive channel acting on both signals are found in Figs.
The results for delay-dispersive channels that are dierent for the two users can be found in Fig. For a longer spreading sequence, this eect is smaller. Hadamard sequences have better cross-correlation properties, therefore is less vulnerable to ISI, which can be seen from d.
After Despreading, User 1 30 20 10 0 An exhaustive computer search shows that no such sequences exist. If no time shifts are applied, the following sequences have zero or one collision: For this interpretation, we have to actually consider periodically extended sequences, so that the first number "follows" the fourth one.
We assume a block fading channel model, i. For a slowly varying channel, i. For WSS channels, the correlation function depends on t1 and t2 only through the dierence t2 t1 , i. In SS parlance, Rs is the periodic time autocorrelation function of the baseband spread signature sequence.
This implies that. Together with the slowly varying channel assumption, this gives Rh t2 t1 ; Rh 0;. The function 0; is known as the power delay profile or multipath intensity profile.
By use of With maximum-ratio combining, the diversity orders of the dierent branches add up.
Therefore, the total diversity order is 8. We have furthermore seen in Chapter 12 that the moment-generating function of Nakagami fading is. SNR curve is plotted in Fig. Since the MS operates in a rich multipath environment, the available frequency diversity allows the Rake receiver to eliminate the small-scale fading. Since the MS is in soft handover, it has links to two BSs, whose shadow fading is independent. The Rake receiver at the MS is capable of adding up the contributions from the two MSs note that for the uplink, typically the base station controller would only select the stronger of the two available components.
Therefore, we need to find the probability that the sum of two independent, lognormally distributed variables falls below a certain threshold value. It is well known that the sum of two lognormally distributed variables can be approximated as another lognormally distributed variable. Matching the first and second moment of the approximation and of the sum of the two constituent random variables is the well-known Fenton-Wilkinson method.
BER vs. This results in a lognormally distributed variable with a mean of 12 dB and a of 4 dB. Thus, the outage probability is the probability that a Gaussian variable has a value that is 2 standard deviations below its mean, which is 2.
Assume the position of BS1 is 0, 0 , and that of BS2 is , 0. For the ease of simulations, we assume the cells have circular shapes. One MS is randomly located in Cell 1, and the average received interference power at BS2 is calculated by simulation. The transmit power of the MS is calculated based on the power control criteria 90 dBm and the path loss: The provided channel gains without power control represent a scenario with serious near-far eect, so the detections of users 1 and 2 are incorrect, because of the strong interference from user 3.
With power control the received signal of all the three users are the same and equal to the smallest one , the performance can be greatly enhanced and the data of all the users can be correctly detected.
The results are plotted in Figs. Signal received with zero-forcing multiuser detector, Exercise The results obtained with that program are plotted in Figs. For any modulation signal with independent input sequences, the power spectral density can be written generally as.
The spectrum of a Signal received with serial interference cancellation multiuser detector, Exercise Rewriting Eq. Ts 2 With. We find after some calculations. Replacing S f in Eq. Tns S e. In another word, there is no intercarrier interference due to frequency selectivity.
At the baseband, the receiver discards the first 3 received samples. As mentioned in Sec.
The distribution of the total amplitude is thus Rayleigh. Let the block Xk be the i-th block of the transmission, with k indexing the N complex modulation symbols.
Throughout the paper we assume perfect synchronization of carriers and blocks, and further, if not otherwise stated, absence of noise. We now write the Discrete Fourier Transform of Eq. Yk and Yi,k are again the parts of Yk that originate from the own symbol i and the previous symbol i 1, respectively. If we use Note that if X i1 was detected successfully and the channel is completely known, as we always assume , Y i,i1 can be computed and subtracted from Y i.
A simulation plot is shown in Fig. For the evaluation of this problem, we apply Eq. Inserting this, and Eq. The result is shown in Fig. See Exercise The capacity for waterfilling is given by Eq. Thus, the third channel is not selected by the system. The SNRs in the two used channels are then The key to this exercise is to recognize that the dierent carriers are fading independently, and thus the behavior of the coded system is the same as a properly interleaved block-coded single-carrier system in a flat-fading channel.
Thus, Eq. Spacing the carriers at B then ensures complete orthogonality. It thus seems that OFDM shows the better spectral eciency. For this comparison, we disregard the possible introduction of a cyclic prefix: Multiantenna systems 1. The advantages of using a smart antenna system include Increase of coverage area Increase of user capacity. Improvement of user localization 2. Consider now the reduction of the received desired signal power due to angular spreading.
The array factor is given as i h. The three dierent purposes are: The underlying mechanism is: More explicitly, the MIMO channel matrix H contains more of one significant singular values, each of which corresponds to the complex gain of the new single channels.
For the case of no channel information at the TX, equal transmit power is assigned to all the TX antennas,. To perform waterfilling, let the eigenvalues of the matrix HH be 1 , 2 , 3. Relative capacity gain for dierent SNRs. It appears in Fig. The fact that feedback gain reduces at higher SNR levels can be intuitively explained by the following fact.
Knowledge of the transmit channel mainly provides transmit array gain. In contrast, gains such as diversity gain and multiplexing gain do not require this knowledge as these gains can be captured by blind transmit schemes such as STCs and V-BLAST. Since the relative importance of transmit array gain in boosting average SNR decreases in the high SNR region, the benefit of channel state information at the transmitter is also reduced. Each transmission traces a path from a TX antenna to one scatterer, and then to the RX antenna.
The scatterer positions are placed according to random realizations of DoD and DoA of signal paths, denoted by t and r , respectively. Each signal path has a complex gain drawn from a Rayleigh distribution.
The ideal Rayleigh case is also included for comparison. The impact of having fewer scatterers than Nt or Nr on the capacity distribution is obvious. It is also noted that the capacity gain.
In fact, a saturation point is eventually obtained when NS is increased further. Also note that in this case, the capacity distribution tends towards the Rayleigh case for a large NS. Ntest H: RRX is similarly defined. From the figure, it is clear that the outage capacity suers at lower array spacing, and particularly for lower angular spread.
This is because when antennas are brought closer together, there is less fading diversity available for the received signals. Note, however, that here the eect of mutual coupling is neglected. In reality, mutual coupling can significantly aect the capacity performance at smaller array spacings. By definition, hnm is a circularly symmetric complex Gaussian random variable with i. The channel capacity for each channel realization is given by.
In Figure The cdf for the NLOS case is given as the dotted line. The increase of transmit power by trial and error brings the expected capacity back to that of the LOS case is shown as the dot-dashed line. Table A loss of LOS path can cause a large drop in capacity, which can only be compensated for by a large increase in TX power. Here we assume equal power at the TX antennas, i. As mentioned in the text, when the TX has no channel knowledge, then there is little point in having more transmit than receive antennas: Of course, we can transmit the same data stream from multiple transmit antennas, but that does not increase the SNR for that stream at the receiver without channel knowledge at the TX, the streams add up incoherently at the receiver.
As we have seen in Part II, the pathloss is significantly lower at lower frequencies; this is partly related to the lower free-space pathloss, and also to the diraction around obstacles. More quantitative results about the frequency dependence can be found from the Okumura-Hata models described in Chapter 7. For all these reasons, the carrier frequencies around MHz are mostly suitable for covering larger areas.
On the other hand, the lower propagation loss implies that the interference to neighboring cells can be higher, so that the reuse distance might have to be increased compared to GSM Furthermore, the available absolute bandwidth is three times larger at MHz, so that it is typically cheaper to buy spectrum in this frequency range.
Thus, the MHz cells are better suited for high-capacity areas. Summarizing, it is preferable to use GSM in urban areas where a high capacity is desired, while GSM should be applied in rural areas where a large coverage area can be achieved.
In many cases, an operator can only buy a nationwide spectral license. In that case, GSM can be gainfully used in urban areas to establish "umbrella cells". Such cells have a larger coverage area, and can accomodate users that move fast without having an excessive numbers of handovers. Alternatively, GSM can also be established to provide better indoor coverage.
From Eq. The t between the beginning of a burst and its end is after bit durations. Thus, the time between the training sequence and the end of the burst is about 74 3.
The the maximum Doppler frequency is fc 1. Between the beginning and the end of the burst, the argument of the Bessel function is doubled, so that the correlation coecient becomes J02 2 0. Inserting those values into the above equation, we get i 0.
We thus see that for cases ii and iii , there is complete decorrelation. The MS informs the BS about the strength and quality of the signal received from the serving, as well as from the neighboring, BSs.
This information need not be transmitted very often, and therefore the channel is called slow ; however, the information needs to be transmitted continuously. This information has to be transmitted fast, but only rarely. The signalling bits just before and after the midamble in a burst signify whether the contained information is payload data or FACCH data.
The block FEC of the voice data is only an error-detecting code, to determine whether the class 1a voice bits were received correctly after convolutional decoding. For this reason, it only contains 3 parity check bits for the 50 class 1a data bits. None of the other voice data are encoded with a block encoder. The signalling bits, on the other hand, are encoded with a , Fire code. Fire codes are block codes which are capable of correcting burst errors.
The stronger protection is used because transmission errors in the control data can have catastrophic results for a link. All midambles have an autocorrelation function that has a peak of amplitude 26 for delay 0 , and value 0 for at least 4 sample values to the left and right of this peak.
Several sequences of length 26 fulfill this requirement. Dierent BSs use the dierent midambles. The reason is that this allows the MSs to distinguish signals from dierent BSs, and thus to better track the received power from those BSs. This provides important information for the handover procedure. Subscriber A pays for international call charges Denmark to Finland , as well as for roaming charges being outside the coverage area of the home network , and possibly a fee for "conference call".
Subscriber C pays for a call forwarding from France to England. Subscribers B and D do not pay any fees. As a first step, we analyze the worst-case phase deviation of GMSK. We know that for regular MSK, the phase deviation in each bit is 0. We furthermore observe that 4. The symbol error probability for 8-PSK can be computed from Eq.
Assuming that each symbol error leads to one bit error, this results in a BER of 0. The spreading factors are independent of the rate set; it is the code rate that is dierent. For the downlink, the data rate after encoding is For the uplink, the data rate is IS can adjust the power by 1 dB in each 1.
In the following, we derive a simplified estimate. We find that the envelope correlation between two signals with 1 dB dierence is 0. Inserting the equation for the maximum Doppler frequency, we obtain 2. As the capacity is limited by interference, the capacity per cell is not aected by this increase assuming that the pilot does not act as a significant interferer, which can be achieved, e.
On the other hand, the increase of the power allows an increase of the cell radius, by a factor 1. Note that an increase in the cell size leads to a corresponding decrease in the area spectral eciency. From this we can conclude that a significant decrease in pilot power is not beneficial for capacity, and also does not lead to a significant increase in range. On the other hand, weak reception of the pilot tone leads to a host of other problems. In IS, three dierent code rates are used: In IS, dierent channels are assigned dierent channelization codes either Walsh codes or spreading codes, for downlink or uplink.
The receiver can separate the channels by correlating with the dierent channelization codes. In GSM, dierent channels are assigned dierent timeslots and frequencies, where bits can also have dierent meanings depending on their position within a superframe.
In the uplink, transmission always occurs at full power. If the data rate is lower than the maximum considered for a rateset, then bits are repeated. A gating algorithm described in Sec. In the downlink, the channels with a lower data rate are transmitted with lower power.
In the uplink, the 3x mode uses a higher chip rate. Thus, each symbol is inherently spread over the whole transmission bandwidth, and assuming that a Rake receiver is used , the frequency diversity For the downlink, the information is first encoded, and then multiplexed onto the three used carriers.
The drawback of this method is that for a low coding rate, the distribution onto the carriers may not be as eective for obtaining diversity as spreading by using shorter chip durations. The service classes in UMTS are: The delays for this type of service should be on the order of ms or less; larger values are experienced as unpleasant interruptions by users. BERs should be on the order of or less; ii streaming: Larger delays in excess of ms can be tolerated, as the receiver typically buers several seconds of streaming material.
The BERs are typically smaller, as noise in the audio music signal is often considered to be more irksome than in a voice telephone conversation; iii interactive: The most important category is web-browsing, but database retrievals and interactive computer games also fall into this category. Also for this category, there are upper limits to the tolerable delay - the time between choosing a certain website and its actual appearance on the screen should not exceed a few seconds.
BERs have to be lower; typically or less; iv background class: We assume that the sensitivity level of the MS is dBm; since our considered MS is operating at 3 dB above sensitivity, the received power is dBm.
The receiver is still functional if the interfering power is equal to the noise power, assumed to be dBm. We furthermore see from Fig. Thus, we require a pathloss of more than dB. At a carrier frequency of MHz, this means that the attenuation from 1 m to the desired distance is 80 dB. Also note that the interfering signal is not necessarily suppressed: For this reason, some coordination between operators is desirable.
In IS, the pilot is transmitted on a separate channel, i. Furthermore, it is noteworthy that the transmission of the pilot channel is independent of the channels. In UMTS, there is a common pilot channel that is transmitted with a specific spreading code, and thus bears resemblance to the pilot channel in IS However, in addition, there is also a pilot transmitted as part of each dedicated channel.
The pilot is part of the dedicated control channel that is mapped onto the quadrature-phase component of the transmit signal. Within that control channel, the position of the symbols within each frame determines whether the bits carry pilot information or other types of control information. That rate is achieved the following way: Each of those three codes has a spreading factor of 4.
Thus, the datarate for each of the channels is 3. Note, however, that this is the maximum possible throughput. It assumes that the desired user can take up most of the capacity in the cell, does not suer from significant interference, and is close enough. Also note that ineciencies of higher layers are not considered for this number. Feedback information is transmitted once in every timslot, which has a duration of 0.
A transceiver can now determine the channel state from the pilot tones during one timeslot, and transmit it in the subsequent timeslot. The other station can make use of that information in yet another timeslot. Thus, the time between the observation of the CSI and its use is two timeslots, 1.
If the angular power spectrum is isotropic and similarly the antenna pattern , then the Doppler spectrum is uniformly distributed between max and max. The temporal autocorrelation function is the Fourier transform of the Doppler spectrum, i. The requirement for the velocity in that case is 0. It is stated there that since feedback occurs in every slot 0.
However, this neglects the eects of the delay between measurement and useage of the information, and also assumes that full information can be transmitted by the feedback bits in one slot.
Wireless Local Area Networks 1. The loss in spectral eciency is thus 0.
The total loss in spectral eciency due to all of the above eects is 1 [ 1 0. The use of OFDM allows better use of the assigned spectral band, since OFDM can more easily achieve a sharp rollo of the spectrum than can single-carrier modulation.
Thus, the required guard bands are smaller. Furthermore, Note, however, that such a high-order modulation format can only be employed for good SINR. There is a number of N -bit long codewords, and - depending on the data symbol to be transmitted one out of those possible code vectors is transmitted.
Since the codewords can be complex, we have in principle 4N codewords available. If actually all of those 4N codewords are valid codewords, then the transmission scheme becomes just standard QPSK, using a 1: The point of CCK is to define a subset of codewords with "good" properties as valid codewords, while other codewords are not valid.
In the Out of those, only 64 codewords are actually admissible ones, representing 6 bits. Thus, a total of 8 bits is transmitted in one codeword. In either case, each symbol combination of K input bits is associated with a transmit symbol of length N. The association can be done via a lookup table as in the case of MOK or block codes , or by some algebraic rules the more common form for block codes.
Thus, the transmission lasts s. Solution manual of Wireless communications by Andreas. F Molisch. Flag for inappropriate content. Related titles. Woods] Probability and Rando BookFi. Wireless Communications 1st Ed. Solution Manual. Jump to Page. Search inside document.
Solution Manual for the exercises in the textbook Wireless Communications by A. Chapter 1 Applications and requirements of wireless services 1. Chapter 2 Technical challenges of wireless communications 1. Chapter 3 Noise- and interference-limited systems 1. Thus, Gpar 0.
The link budget. Hence, 4. Using the derived expressions we have which is shown in Figure 5. We have M dB 5. The average mean delay is R Tm P d 6. The rms delay spread is S sR 2 d Pm 2 Tm 6. Let the PDP that is terminated after 20 s be P 0 , i. Inter- 8. Endfire direction: S14 Directional coupler for Exercise G1 Substituting this, we obtain x21 cos2 t 2 N0 4 N0 Thus, to calculate the nearest neighbor union bound we must evaluate s M X X d2ij 1 nij , pb Q Q 6 N0 b When Gray coding is used, only one bit changes between adjacent constellation points.
Es N0 However, it should be noted that shadow fading has not been considered. Nr We now need to find numerically the smallest Nr for which! N0 The combiner adds the outputs of the matched filters to obtain qtot Nr X qk Inspection shows that these are 4 Combining these two facts we get the following requirement on syndrome decoding N K t X N. In matrix form, we have e2 1.
One trellis stage. Trellis for the equalization performed by the Viterbi equalizer. MSE contours for the 2-tap equalizer. MSE curves for dierent step sizes.
Ncluster 19 Tcell 0. Acell For the downlink we have the approximation C 1 R. After Despreading, User 1 8 6 4 2 0 -2 -4 -6 -8 1. After Despreading, User 1 20 15 10 5 0 -5 1. After Despreading, User 1 30 20 10 0 1. After Despreading, User 1 50 0 1. Received signal before despreading with Hadamard spreading. Received signal after despreading with Hadamard spreading. This implies that Rs t1 Ts Rs t2 Ts 0 , t2 t1 2. Received signal before despreading - no power control. Setup for Problem Molisch Author Andreas F.
He received his M. We want your feedback! Click here. Molisch ebook. Subjects Technology Nonfiction. Technology Nonfiction. Publication Details Publisher: Andreas F. More about Andreas F. Wireless Communications Embed. Media Wireless Communications.