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"Professor Andreas F. Molisch, renowned researcher and educator, has put together the comprehensive book, Wireless Communications. The second edition . PDF | In the last two decades, wireless communications has become a vast field of application and research consisting of subdomains, such as. "Professor Andreas F. Molisch, renowned researcher and educator, has put together cover image of Wireless Communications Adobe PDF eBook MB.


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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 – .. homeranking.info Idea. Professor Andreas F. Molisch, renowned researcher and educator,has put together the Wireless Communications (eBook, PDF) - Molisch, Andreas F.

Endfire direction: Sajal Mittal. The Okumura—Hata Model 7. Molisch ieee. Consequently the model is valid.

The received power varies inversely as the fourth power of the separation distance between TX and RX. The condition is 4hTX hRX 4 10 1. Received power at the BS and MS as a function of separation distance.

The parameters given are: The scenario is illustrated in Figure 4. The received power at the MS and the reference antenna with the BS as a transmitter is. MS BS Figure 4. Hence, for a separation distance between the BS and the MS of d 92 m, the radiation from a mobile unit at a distance of 3 m will be greater than the radiation from the BS note, however, again the breakpoint.

In many real-life situations the distance from a mobile to the base station is much greater than that, and the distance to people using mobile phones less than 3 meters.

This means that exposure to radiation from surrounding base stations is less than the exposure to radiation from mobiles used by surrounding people. Remember, though, that the studied case is based on very simple models.

First we redraw the geometry according to Figure 4. The height h and the distance from the transmitter dT X are given by simple trigonometry as. Redrawn geometry for Problem While the diraction angle becomes the same as in a , the Fresnel parameter and Fresnel integral now become s s 2dT X dRX 2 We have still the same diraction angle as in a , while the new Fresnel parameter and Fresnel integral become s r 2dT X dRX 2 Statistical description of the wireless channel 1.

Using the complex baseband notation: The power per unit area is P 0. I0 x is the modified Bessel function of the first kind, zero order. Thus, the outage probability can be obtained as the tail probability of the normal distribution,. We assume that only the mobile moves in a stationary environment , and that the maximal Doppler spread is twice the maximal Doppler shift, i. We are to design a system communicating at both MHz and MHz. For the Doppler shift we have the relation,.

As can be seen in Eq. Expressed in statistical terms, the outage probability is given by Pout. Yes, increasing the transmit power will increase the average received power. The probability of receiving a power level below the sensitivity level decreases and therefore so does the outage probability.

However, it might not be a possible solution due to spectrum regulations. Further, increased power may lead to unacceptable interference in the system. No, assuming that we want to communicate over a certain distance the only way to reduce the deterministic path loss is to change the environment removing obstacles such as houses and hills , which is in most cases not an option.

Yes, increasing the gain of the antennas will increase the average received power and decrease the outage probability. However, increasing the antenna gain will also increase the directivity so be sure to point the antennas in the right directions.

No, the F is determined by the environment and cannot be changed. Yes, a better receiver can tolerate a weaker received signal and therefore the outage probability can be decreased. However, a better receiver may be more complex and expensive, and may also use more power.

The Rayleigh CDF is 2 rmin. With the instantaneous received power, C, the receiver sensitivity level minimum received power for the we have system to work properly , Cmin , and the average received power, C,. Figure 5. Approximation error as a function of Pout. Using the derived expressions we have. It can be seen that the error increases with Pout. Since the mean value is larger than the median value, the fading margin referring to the mean value has to be larger by those 1.

We have for rmin the level crossing rate fading dips per second r 2 m in 2 rmin r2 0 ,. The link budget diagram is shown in Figure 5. Link Budget. Note that the small-scale fading margin has been given relative to the median path loss, since the propagation path loss model gives a median loss.

The total transmit power required is PTX dB. Tx-B interferes with the reception of Rx-A. The scenario is shown in Fig. We have. Wideband and directional channel characterization 1. Both the spreading and scattering functions describe the channels delay and Doppler characteristics. However, the respective underlying models for the channel are dierent. In the case of the spreading function, the channel is viewed as a deterministic function. In contrast, in deriving the scattering function we have assumed a statistical model for the channel.

This function can be useful mainly for two applications: This applies to both time division multiplex and frequency division multiplex systems since RH t, f depends on both the time and frequency separation. In general, it is desirable that the correlation between the channels in the uplink and the downlink is high, so that channel estimates from one can be used for the adaptive transmission in the other.

More details about this aspect can also be found in Sec. Details about diversity systems can be found in Chapter 13, but the basic principle is that the signal is transmitted on independent paths, so that there is a low probability that all signal paths are in a fading dip simultaneously. In this case, it is desirable that the correlation coecient is as low as possible. The information about the required spacing in time or frequency can be found from PH t, f.

Correcting for a misprint in the problem formulation, we write the power delay profile as. With a coherence bandwidth of kHz, the channel is regarded as approximately frequency-flat. From Figure 6. Thus, the coherence time is therefore approximately 25 ms. As discussed in Sec. Therefore the number of delay bins becomes the ratio between the excess delay and the chip duration.

We obtain five delay bins in the case of 1. The system can be considered wideband in the former case, but not in the latter. Channel models 1. Assume that the transmitted signal is reflected at several reflection points before arriving at the receiver.

From the Central Limit Theorem sum of equally distributed random variables , h dB is Gaussian distributed. Hence the linear, h, is log-normal distributed. A rule of thumb is that the sum should consist of at least 10 random variables, hence that requires 10 reflection points. After 10 reflections there is not much power left. For this reason, this process for the justification of a lognormal distribution is still under discussion in the literature. We are considering propagation in a medium-sized city where we have got the following parameters.

Parameters hb hm fc d. The resulting Okumura path-loss is: Note that this value is only correct for a base station height of m and a mobile station height of 3 m. For other antenna heights the. For the given BS height we get a correction factor of: Finally, b Table 7.

Hence, the total propagation loss can be written as: That they agree well is not too surprising, since the latter calculation Okumura-Hata is based on a parameterization of the first Okumura. Consequently the model is valid. Since only hb has changed, L0 and Lrts are the same. Figure 7. With a BS below rooftops the attenuation due to multiscreen loss Lmsd is higher than for a BS above rooftops.

This is reasonable since high surrounding buildings shadow the transmissions from a low BS resulting in higher path loss. We are considering the COST wideband model. The result is displayed in Figure 7. The result is shown in Figure 7. RA - The rural area does not contain many reflection points in terms of buildings and other obstacles, which results in a quite short delay spread echos of the signal.

TU - In the urban city area there are a large number of scattering points, resulting in a larger delay spread compared to RA. BU - In the urban area there might also be some high rise builidings, resulting in strong reflections delayed a few s relative to the first peak and occurring at a delay such that the first peak has not fully decayed. HT - In hilly terrain, the direct path first arrives with a little delay spread.

Hence, it is only a measure of the spread of the channel. Note, however, that the RA has a Rician-fading first component. Thus, while the rms delay spread can be mathematically defined and computed as above, great care must be taken when interpreting the result and applying it, e. The rms delay spread S and the coherence bandwidth Bcoh are obviously related: Based on this insight, an uncertainty relationship could be derived Bcoh.

The correlation between element 1 and 2, r, is then " N N X X. For another distribution of the APS the correlation will change. Simulation results for the correlation in Exercise 7. Results in a correlation between 0 and 1. From the definition of the cross-correlation. Figure 8. Auto correlation function for the code waveform is periodic. Consequently, the impulse response of a linear system such as the wireless propagation channel can be evaluated using white noise and some form of correlation processing.

In practice it is not possible to generate true white noise so channel sounding systems use deterministic waveforms with noise like properties.

The most widely used example of such waveforms are the pseudo-random binary maximallength sequences m-sequences, PN-sequences. This sequence is periodic with period 7. This spreading waveform has the autocorrelation function. Note, however, that better dynamic range can be achieved with nonlinear evaluation methods, e.

Normalized beam forming spectrum versus DOA for suciently separated angles. If we increase the m-sequence period by the factor 63 31 2, the maximum Doppler shift which the receiver can handle is reduced by half.

Thus the maximum permissible mobile velocity will be reduced by half, e. An increase in Mc by a factor 2 will increase the maximum measurable delay of the multipath components by the same factor. The angles at which peaks occur in the spectrum are the estimates for the angles of arrival.

Molisch wireless pdf communications

The results are shown in Fig. The matlab code is provided below. Normalized beam forming spectrum versus DOA for angle separation less than angle resolution of beamformer.

Reading the angles coresponding to the spectrum peaks gives 43 and 86 as the respective estimates. However, the spectrum as shown in Fig. The explanation comes from the true DOAs in this example which are 60 and These DOAs correspond to an electrical angle separation of 2.

Hence some super resolution algorithm has to be used to estimate both angles. Following the steps outlined in App. A, the DOAs are estimated as The matlab implementation is provided below: Inserting Y and N into Eq.

Antennas 1. When we place the handset to the head the total attenuation is increased by 6 dB, which corresponds to a factor 4 on a linear scale. As a first step, we need to compute the size of a patch antenna. Placement along the top of the PDA is preferable. One of the solutions found in commercial products is to have a WiFi card partly protruding from the top of the PDA, thus minimizing the risk of having it covered by the hand.

Placement within the casing is often dicult, as the casing can be essentially a Faraday cage. Additional solutions are the solutions to: The directivity is 2.

For the gain of the whole array, we would also have to take the gain of the antenna elements into account. Structure of a wireless communication link 1. The structure of the directional coupler is sketched in Fig. As we have seen in Chapter 3, Eq. Let us now prove that M is the correct criterion for placing an amplifier first. The statement to be proved is that the following should be fulfilled: This demonstrates that the output signal has a DC component that is proportional to the squared amplitude of the signal, and thus to the power.

For the intercept point, i. In order for the relative variance to stay below a treshold of K dB, we thus require that 1 2 1 The required number of bits thus doubles. Modulation formats 1. The advantage of a smoother filter is that the spectral eciency becomes higher. The drawback is that the signal does not always reach the ideal MSK signal points. Therefore the smoother filter implies a higher sensitivity to noise and other disturbances.

The second term is the integral of the carrier frequency, which is not of interest. Solve for the first term. For non-coherent demodulation, the minimum separation is T1. The minimum value of the envelope is, e. Since the mean envelope is unity, the requested ratio becomes 0. Therefore, there are no advantages of such a modulation compared to "normal" 8-PSK. In App. A, we found that p. In order to ensure unit energy,.

We get the following expansions: Note, however, that this function cannot be represented completely as a linear combination of the two expansion functions.

Due to the symmetry of the problem, it is sucient to consider only one quadrant in the signal-space diagram. The average signal energy is thus Es. A transmit power of 20 W means 43 dBm, so we need an attenuation of 93 dB for the out-of-band emissions. Demodulation 1. N0 min. Then the minimum received power at the antenna output is Cmin. Figure QPSK signal constellation.

Here, Es is the symbol energy, Ts the symbol length and fc the carrier frequency. Note that the total symbol energy is twice the bit energy since two bits are transmitted per symbol. There is no interference between the signals Therefore, r! The data rate can thus be doubled without sacrificing BER performance or increasing transmission bandwidth.

In the case of BPSK, the symbol error probability is the same as the bit error probability. For QPSK, the probability of correct decision of one symbol i. Thus, the symbol error probability is ps,QPSK. N0 4 N0. Thus, s.

Recall that the original requirement on maximum BER was , which means that the BER has been degraded by a factor of However, coherent detection of the QPSK symbols is no longer needed. The full union bound is obtained by using s. With the specified amount of dierence we have!

Wireless Communications

From the symmetry of the constellation shown in Figure B Since the signal alternatives are equally probable, the average symbol energy is M. If the sum over j, i. Note that this is not a true bound, but rather an approximation of the BER. Thus, to calculate the nearest neighbor union bound we must evaluate. In this case, taking the number of bit errors that occur into account, we have. The nearest. Thus, the nearest neighbor union bound is given by!

In this case we have GGray. In the first case, where BPSK is analyzed, Gaussian noise samples are generated and a threshold is used for determining if a noise sample would have generated a symbol error. In the second case, where QPSK is analyzed, complex noise samples are generated and added to the transmitted signal constellation point. Then, the angles in the complex plane are used to determine which of the symbols are in error.

Solution manual- Wireless Communications Molisch | Signal To Noise Ratio | Decibel

Finally, in the case of 8-PSK, complex noise samples are generated and added to the transmitted symbol just as in the QPSK case above, but Euclidean distances are then calculated to all signal alternatives and used for detecting symbol errors. It can be seen in Figure It is however hard to determine in which region the union bound appears to be tight for 8-PSK. We start by calculating nearest neighbor union bound on BER for the QAM modulation scheme, expressed as a function of minimum distance dmin.

Since the constellation is Gray coded only one bit error occurs when a nearest neighbor constellation point is selected. Due to the symmetry of the QAM constellation, there are three groups of constellation points, where the points in each group have the same distances to the origin and the same numbers of nearest neighbor constellation points see Figure Next, replace dmin with the average bit energy.

The average symbol energy assuming equally probable symbols is s E. Upper bounding this expression using the pairwise probabilities for binary orthogonal signaling, we have ps. Thus, the transmit power must be increased by a factor of 10 log10 Using the Okumura-Hata path loss model and the specified carrier frequency and base station and mobile station heights, the maximum distance can be calculated.

Finally, with B. The probability that one or more of the bits are in error i. Thus, pb,max. The result in a is 2 3 1. Figure B Outage will occur whenever the SNR goes below a specified value. The probability for such an event can be read out directly from the figure since it is the cdf of the normalized SNR that is shown in the figure.

It is evident that the diversity gain increases when the specified values of BER and outage probability are small. Let the average SNR be. This decrease is. Equation B However, the total energy in the power delay profile is fixed; in other words, increasing Nr does not change the mean SNR. Thus we must replace in Eq. We now need to find numerically the smallest Nr for which. Thus, four resolvable multipaths reduce the fluctuations of the combiner, or Rake, output suciently to achieve the BER requirement.

The scheme in this problem is called the Alamouti scheme see also Chapter The SNR then becomes. For the two transmit antenna case, the SNR becomes 2.

Let the instantaneous SNR at the kth antenna, or branch, be given by the random variable k , with mean value k. The output of a maximal ratio combiner consists of a sum of the individual branch SNRs, i.

To derive the pdf of the combiner output M RC , we can perform an Nr -fold convolution of the branch pdfs. Another approach is to utilize a transform. Let the moment generating function MGF of a random variable X be defined as. The inverse of M R C s is then found by table look-up. We can therefore again utilize The kth matched filter correlates the received signal with a complex-conjugate replica of the wideband transmitted signal. The replica is delayed k , and before combining the output of the kth matched filter is weighed with the complex-conjugate of the fading gain of the kth multipath, hk.

We now use the assumption that the wideband transmitted signal has perfect autocorrelation properties. This is equivalent to. 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.

Wireless Communications, 2nd Edition

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.

Molisch pdf communications wireless

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.

Pdf wireless communications molisch

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 Fully updated to incorporate the latest research and developments, Wireless Communications, Second Edition provides an authoritative overview of the principles and applications of mobile communication technology. The author provides an in-depth analysis of current treatment of the area, addressing both the traditional elements, such as Rayleigh fading, BER in flat fading channels, and equalisation, and more recently emerging topics such as multi-user detection in CDMA systems, MIMO systems, and cognitive radio.

The dominant wireless standards; including cellular, cordless and wireless LANs; are discussed. Topics featured include: Combines mathematical descriptions with intuitive explanations of the physical facts, enabling readers to acquire a deep understanding of the subject.

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