Tse and Viswanath:Fundamentals of Wireless Communication 30 on roads. 2.1.6 Power Decay with Distance and Shadowing The previous example with reflection from a ground plane suggests that the received power can decrease with distance faster than r- in the presence of disturbances to free space.In practice,there are several obstacles between the transmitter and the receiver and,further,the obstacles might also absorb some power while scattering the rest.Thus,one expects the power decay to be considerably faster than r-2.Indeed. e from expe erimental field studies suggests that while power decay rthe transmitter is like,at large distances the power decays wit ance. The ray tracing approach used so far provides a high degree of numerical accuracy in determi ing the electric field at the receiver,but requires a precise physical model including the location of the obstacles.But here,we are only looking for the order of decay of power with distance and can consider an alternative appronch.So we look for a model of the hys cal on t with the est numbe pa amet rs but one that still pro ide ful global info rm on a 00u the field properti A simpl probabilistic model with two parameters of the physical environment: e density o the obstacles and the nature of the obstacles (scatterer or absorber)is developed in Exercise 2.6.With each obstacle absorbing a positive fraction of the energy impinging on it,the model allows us to show that the power decays exponentially in distance at a rate that is proportional to the density of the obstacles. With a lim on the trar power (either r at the base station or at the mobile the largest di anc th base station an e at can reliably take place is called the coverage of the cell.For reliable communication a minimal received power level has to be met and thus the fast decay of power with distance constrains cell coverage.On the other hand,rapid signal attenuation with distance is also helpful:it reduces the interference between adjacent cells.As cellular systems become more opular,however,the major determinant of cell size is the of n th cell In in cell is said to be limited inste ead of overage e limited The size of cells has been adil decreasi and one talks of micro cells and pico cells as a response to this effect.With capacity limited cells,the inter-cell interference may be intolerably high.To alleviate the inter cell interference,neighboring cells use different parts of the frequency spectrum,and frequency is reused at cells that are far enough.Rapid signal attenuation with distance allows frequ encies to be reused at closer distances The de ity s betwee the transm and receive ennas depends v much on the physical environment. For example,outdoor plains have very little by Since the ground plane is modeled as a perfect scatterer (i.ethere isno ossof energy in scat tering),there are other receiver positions where the power decays slower thanr
Tse and Viswanath: Fundamentals of Wireless Communication 30 on roads4 . 2.1.6 Power Decay with Distance and Shadowing The previous example with reflection from a ground plane suggests that the received power can decrease with distance faster than r −2 in the presence of disturbances to free space. In practice, there are several obstacles between the transmitter and the receiver and, further, the obstacles might also absorb some power while scattering the rest. Thus, one expects the power decay to be considerably faster than r −2 . Indeed, empirical evidence from experimental field studies suggests that while power decay near the transmitter is like r −2 , at large distances the power decays exponentially with distance. The ray tracing approach used so far provides a high degree of numerical accuracy in determining the electric field at the receiver, but requires a precise physical model including the location of the obstacles. But here, we are only looking for the order of decay of power with distance and can consider an alternative approach. So we look for a model of the physical environment with the fewest number of parameters but one that still provides useful global information about the field properties. A simple probabilistic model with two parameters of the physical environment: the density of the obstacles and the nature of the obstacles (scatterer or absorber) is developed in Exercise 2.6. With each obstacle absorbing a positive fraction of the energy impinging on it, the model allows us to show that the power decays exponentially in distance at a rate that is proportional to the density of the obstacles. With a limit on the transmit power (either at the base station or at the mobile) the largest distance between the base station and a mobile at which communication can reliably take place is called the coverage of the cell. For reliable communication, a minimal received power level has to be met and thus the fast decay of power with distance constrains cell coverage. On the other hand, rapid signal attenuation with distance is also helpful; it reduces the interference between adjacent cells. As cellular systems become more popular, however, the major determinant of cell size is the number of mobiles in the cell. In engineering jargon, the cell is said to be capacity limited instead of coverage limited. The size of cells has been steadily decreasing, and one talks of micro cells and pico cells as a response to this effect. With capacity limited cells, the inter-cell interference may be intolerably high. To alleviate the intercell interference, neighboring cells use different parts of the frequency spectrum, and frequency is reused at cells that are far enough. Rapid signal attenuation with distance allows frequencies to be reused at closer distances. The density of obstacles between the transmit and receive antennas depends very much on the physical environment. For example, outdoor plains have very little by 4Since the ground plane is modeled as a perfect scatterer (i.e., there is no loss of energy in scattering), there are other receiver positions where the power decays slower than r −2
Tse and Viswanath:Fundamentals of Wireless Communicatior way of obstacles while indoor environments pose many obstacles.This randomness in the environment is captured by modeling the density of obstacles and their absorption behavior as random numbers:the overall phenomenon is called shadowing.The effect of shadow fading differs fr multipath fading in an important way.The duration of a shadow fade asts s for multip or minutes,and hence occurs at a much slower time-scale compared to multipath fading. 2.1.7 Moving Antenna,Multiple Reflectors Dealing with multiple reflectors,using the technique of ray tracing,is in principle simply a matter of modeling the received waveform as the sum of the responses from paths rather than just tw ths We hav enough exa oweve rsta that finding t and pha of these no imple large wall example in Figure 2.2,the reflected fiel calculated in(2.6)is valid only at distances from the wall that are small relative to the dimensions of the wall.At very large distances,the total power reflected from the wall is proportional to both d-2 and to the area of the cross section of the wall.The power reaching the receiver is proportional to (d())2.Thus,the p ver he lo dis case)is proporti al to (did_ rather than to (2d r() his shows ray tr cing must be used with some caution.Fortunately,however,linearity still holds in these more complex cases. Another type of reflection is known as scattering and can occur in the atmosphere or in reflections from very rough objects.Here there are a very large number of individual paths,and the received wavetorm is better modeled as an integral over paths with mall differences in ther lengths,rather than s to find th olitude of the ected field fro each ty e of re flector is helpful in determining the coverage of a base station (although,ultimately experimentation is necessary).This is an important topic if our objective is trying to determine where to place base stations.Studying this in more depth,however,would take us afield and too far into electromagnetic theory.In addition,we are primarily in- terested in questions of modulation,detection,multiple access.and network protocols rather than location of base stations.Thus,we turn our attention to understanding the nature of the aggregat eived waveform,given a repre ation for wave.This leads to modeling the input/output behavior of a channel rather than the detailed response on each path. 5This is called shadowing because it is similar to the effect of clouds partly blocking sunlight
Tse and Viswanath: Fundamentals of Wireless Communication 31 way of obstacles while indoor environments pose many obstacles. This randomness in the environment is captured by modeling the density of obstacles and their absorption behavior as random numbers; the overall phenomenon is called shadowing5 . The effect of shadow fading differs from multipath fading in an important way. The duration of a shadow fade lasts for multiple seconds or minutes, and hence occurs at a much slower time-scale compared to multipath fading. 2.1.7 Moving Antenna, Multiple Reflectors Dealing with multiple reflectors, using the technique of ray tracing, is in principle simply a matter of modeling the received waveform as the sum of the responses from the different paths rather than just two paths. We have seen enough examples, however, to understand that finding the magnitude and phase of these responses is no simple task. Even for the very simple large wall example in Figure 2.2, the reflected field calculated in (2.6) is valid only at distances from the wall that are small relative to the dimensions of the wall. At very large distances, the total power reflected from the wall is proportional to both d −2 and to the area of the cross section of the wall. The power reaching the receiver is proportional to (d−r(t))−2 . Thus, the power attenuation from transmitter to receiver (for the large distance case) is proportional to (d(d − r(t)))−2 rather than to (2d − r(t))−2 . This shows that ray tracing must be used with some caution. Fortunately, however, linearity still holds in these more complex cases. Another type of reflection is known as scattering and can occur in the atmosphere or in reflections from very rough objects. Here there are a very large number of individual paths, and the received waveform is better modeled as an integral over paths with infinitesimally small differences in their lengths, rather than as a sum. Knowing how to find the amplitude of the reflected field from each type of re- flector is helpful in determining the coverage of a base station (although, ultimately experimentation is necessary). This is an important topic if our objective is trying to determine where to place base stations. Studying this in more depth, however, would take us afield and too far into electromagnetic theory. In addition, we are primarily interested in questions of modulation, detection, multiple access, and network protocols rather than location of base stations. Thus, we turn our attention to understanding the nature of the aggregate received waveform, given a representation for each reflected wave. This leads to modeling the input/output behavior of a channel rather than the detailed response on each path. 5This is called shadowing because it is similar to the effect of clouds partly blocking sunlight
Tse and Viswanath:Fundamentals of Wireless Communication 32 2.2 Input/Output Model of the Wireless Channel We derive an input/output model in this section.We first show that the multipath effectscan be time varying system.We then obtain a baseband representatio 10 this odel.The continud chan nnel is then sampled to obtair a discrete -time model.Finally we incorporate additive noise 2.2.1 The Wireless Channel as a Linear Time-Varying System In the previous section we focussed on the response to the sinusoidal input o(t)= cos 2 ft.The received signal can be written as ai(f.t)o(t-(f.t)),where ai(f,t) andft)are respectively the overall attenuation and agation delay at time t the t 41 on path i.The over att tion is ply the product of the attenuation factors due to the antenna pattern of the transmitter an the receiver,the nature of the reflector,as well as a factor that is a function of the distance from the transmitting antenna to the reflector and from the refector to the receive antenna.We have described the channel effect at a particular frequency f.If we further assume that the aft)'s and thef,t)'s do not depend on the frequency f then w se the principle of s ralize the above relation to an arbitrary input(t)with nonzero bandwidth: mnput-output ()=a()x(t-T(t)】 (2.14) In practice the attenuations and the propagation delays are usually slowly varying functions of frequency.These variations follow from the time-varying path lengths and also from frequency dependent antenna gains.However,we are primarily interested in transmitting over bands that are narrow relative to the carrier frequency,and over such ranges we can omit this frequency dependence.It should however be noted that although the individual atte ions and delays are mmed to be indep dent of the all char el response can still vary with frequency due to the fact erent paths have different delays. For the example of a perfectly reflecting wall in Figure 2.4,then, a a a(t)= (2.15) T。+t a2(t)= 2d-。-vt n()=。+t∠o n0=24-。-t (2.16) where the first expression is for the direct path and the second for the reflected path The term here is to account for possible phase changes at the transmitter,reflector and receiver.For the example here,there is a phase reversal at the reflector so we take Φ1=0 and o=T
Tse and Viswanath: Fundamentals of Wireless Communication 32 2.2 Input/Output Model of the Wireless Channel We derive an input/output model in this section. We first show that the multipath effects can be modeled as a linear time varying system. We then obtain a baseband representation of this model. The continuous-time channel is then sampled to obtain a discrete-time model. Finally we incorporate additive noise. 2.2.1 The Wireless Channel as a Linear Time-Varying System In the previous section we focussed on the response to the sinusoidal input φ(t) = cos 2πft. The received signal can be written as P i ai(f, t)φ(t − τi(f, t)), where ai(f, t) and τi(f, t) are respectively the overall attenuation and propagation delay at time t from the transmitter to the receiver on path i. The overall attenuation is simply the product of the attenuation factors due to the antenna pattern of the transmitter and the receiver, the nature of the reflector, as well as a factor that is a function of the distance from the transmitting antenna to the reflector and from the reflector to the receive antenna. We have described the channel effect at a particular frequency f. If we further assume that the ai(f, t)’s and the τi(f, t)’s do not depend on the frequency f, then we can use the principle of superposition to generalize the above input-output relation to an arbitrary input x(t) with nonzero bandwidth: y(t) = X i ai(t)x(t − τi(t)). (2.14) In practice the attenuations and the propagation delays are usually slowly varying functions of frequency. These variations follow from the time-varying path lengths and also from frequency dependent antenna gains. However, we are primarily interested in transmitting over bands that are narrow relative to the carrier frequency, and over such ranges we can omit this frequency dependence. It should however be noted that although the individual attenuations and delays are assumed to be independent of the frequency, the overall channel response can still vary with frequency due to the fact that different paths have different delays. For the example of a perfectly reflecting wall in Figure 2.4, then, a1(t) = |α| r0 + vt , a2(t) = |α| 2d − r0 − vt , (2.15) τ1(t) = r0 + vt c − ∠φ1 2πf , τ2(t) = 2d − r0 − vt c − ∠φ2 2πf , (2.16) where the first expression is for the direct path and the second for the reflected path. The term ∠φj here is to account for possible phase changes at the transmitter, reflector, and receiver. For the example here, there is a phase reversal at the reflector so we take φ1 = 0 and φ2 = π
Tse and Viswanath:Fundamentals of Wireless Communicatior 33 Since the channel (2.14)is linear,it can be described by the response h(r,t)at time t to an impulse transmitted at time t-r.In terms of h(,t),the input-output relationship is given by ut)=/h(r,t)z(t-r)dr. (2.17) Comparing(2.17)and(2.14),we see that the impulse response for the fading multipath channel is h(r,t)=>a(t)6(r-(t) (2.18) This expression is really quite nice.It says that the effect of mobile users,arbitrarily moving reflectors and absorbers tions, ally reduce and all of the complexities of solving Maxwell's equa utput r relation between transmit and I receiv e antenn RA3四Bou四BI0 asuodsol asmduy a se poquasod61dstp四 filter The effect of the Doppler shift is not immediately evident in this representation. From (2.16)for the single reflecting wall example,(t)=vi/c where v;is the velocity with which the ith path length is increasing.Thus,the Doppler shift on the ithpath is -f). In the special case when the transmitter,receiver and the environ nt are all stationary,the attenuations()'s and propagation delays('s do not depend or time t,and we have the usual linear time-invariant channel with an impulse response h(r)=>ad(r-). (2.19) For the time-varying impulse response h(,t),we can define a time-varying fre quency response IU0=Mr,rh-∑a0en (2.20) In the special case when the channel is time-invariant,this reduces to the usual fre- quency response.One way of interpreting H(f:t)is to think of the system as a slowly varying function of t with a frequency response H(f;t)at each fixed time t.Corre- sponding,h(r,t)can be thought of as the impulse response of the system at a fixed time t.This is a legitimate and useful way of thinking about multipath fading chan scale at which the chan varies is typi ch longer than the dely pread of the impulse response at a fiedl time.In the rellecting wall example ir Section 2.1.4,the time taken for the channel to change significantly is of the order of milliseconds while the delay spread is of the order of microseconds.Fading channels which have this characteristic are sometimes called underspread channels
Tse and Viswanath: Fundamentals of Wireless Communication 33 Since the channel (2.14) is linear, it can be described by the response h(τ, t) at time t to an impulse transmitted at time t − τ . In terms of h(τ, t), the input-output relationship is given by y(t) = Z ∞ −∞ h(τ, t)x(t − τ )dτ. (2.17) Comparing (2.17) and (2.14), we see that the impulse response for the fading multipath channel is h(τ, t) = X i ai(t)δ(τ − τi(t)). (2.18) This expression is really quite nice. It says that the effect of mobile users, arbitrarily moving reflectors and absorbers, and all of the complexities of solving Maxwell’s equations, finally reduce to an input/output relation between transmit and receive antennas which is simply represented as the impulse response of a linear time-varying channel filter. The effect of the Doppler shift is not immediately evident in this representation. From (2.16) for the single reflecting wall example, τ 0 i (t) = vi/c where vi is the velocity with which the i th path length is increasing. Thus, the Doppler shift on the i th path is −fτ 0 i (t). In the special case when the transmitter, receiver and the environment are all stationary, the attenuations ai(t)’s and propagation delays τi(t)’s do not depend on time t, and we have the usual linear time-invariant channel with an impulse response h(τ ) = X i aiδ(τ − τi). (2.19) For the time-varying impulse response h(τ, t), we can define a time-varying frequency response H(f;t) := Z ∞ −∞ h(τ, t)e −j2πfτdτ = X i ai(t)e −j2πfτi(t) . (2.20) In the special case when the channel is time-invariant, this reduces to the usual frequency response. One way of interpreting H(f;t) is to think of the system as a slowly varying function of t with a frequency response H(f;t) at each fixed time t. Corresponding, h(τ, t) can be thought of as the impulse response of the system at a fixed time t. This is a legitimate and useful way of thinking about multipath fading channels, as the time-scale at which the channel varies is typically much longer than the delay spread of the impulse response at a fixed time. In the reflecting wall example in Section 2.1.4, the time taken for the channel to change significantly is of the order of milliseconds while the delay spread is of the order of microseconds. Fading channels which have this characteristic are sometimes called underspread channels
Tse and Viswanath:Fundamentals of Wireless Communication 34 +S(f) --孚-j+妥 e-孚人+号 +S(f) Figure 2.7:Illustration of the relationship between a passband spectrum S(f)and its baseband equivalent S(f) 2.2.2 Baseband Equivalent Model In typical wireless applications,communication occurs in a passband [fef of bandwidth W around a center frequency fe,the spectrum having been specified ost of the proce sing,such as oding/de oding synch onization,et ctually don t the baseb od At the transmitter,the last stage of the operation isto onvert"the signal to the carrier frequency and transmit it via the antenna.Similarly,the first step at the receiver is to "down-convert"the RF(radio-frequency)signal to the baseband before further processing.Therefore from a communication system design point of view,it is most useful to have a baseband equivalent representation of the system.We first start with defining the baseband equivalent representation of signals. with Fourier transformf),bandlimited inf <2fe.Define its compler baseband equivalent su(t)as the signal having Fourier transform: 50={v2SU+月f+6>0 f+fe≤0 (2.21) Since s(t)is real,its Fourier transform is Hermitian around f=0,which means that s(t)contains exactly the same information as s(t).The factor of v2 is quite arbitrary
Tse and Viswanath: Fundamentals of Wireless Communication 34 fc − W − 2 fc − W 2 − W 2 −fc + W 2 fc + W 2 W 2 √ 2 1 Sb(f) S(f) f f Figure 2.7: Illustration of the relationship between a passband spectrum S(f) and its baseband equivalent Sb(f). 2.2.2 Baseband Equivalent Model In typical wireless applications, communication occurs in a passband [fc − W 2 , fc + W 2 ] of bandwidth W around a center frequency fc, the spectrum having been specified by regulatory authorities. However, most of the processing, such as coding/decoding, modulation/demodulation, synchronization, etc, is actually done at the baseband. At the transmitter, the last stage of the operation is to “up-convert” the signal to the carrier frequency and transmit it via the antenna. Similarly, the first step at the receiver is to “down-convert” the RF (radio-frequency) signal to the baseband before further processing. Therefore from a communication system design point of view, it is most useful to have a baseband equivalent representation of the system. We first start with defining the baseband equivalent representation of signals. Consider a real signal s(t) with Fourier transform S(f), bandlimited in [fc − W/2, fc + W/2] with W < 2fc. Define its complex baseband equivalent sb(t) as the signal having Fourier transform: Sb(f) = √ 2S(f + fc) f + fc > 0 0 f + fc ≤ 0 . (2.21) Since s(t) is real, its Fourier transform is Hermitian around f = 0, which means that sb(t) contains exactly the same information as s(t). The factor of √ 2 is quite arbitrary