Tse and Viswanath:Fundamentals of Wireless Communication 25 terms of a system function followed by translating the frequency f by the Doppler shift -fu/c.It is important to observe that the amount of shift depends on the frequency f.We will come back to discussing the importance of this Doppler shift and of the time varying attenuation after the next exampe. The a ve analysis do depend c whether it is s the (or both)th ansmi tter or the receive are moving.So long asr(t)is interpreted as the distanc etween th antennas (and the relative orientations of the antennas are constant),(2.4)and(2.5) are valid. 2.1.3 Reflecting wall,fixed antenna Consider fig re22 below in which there sinusd cos2x.ft. e ante and a ingle perfe cting large ked wall.We assume that in the absence of the recei e antenna,the elec ctromagnetic field at the point where the receive antenna will be placed is the sum of the free space field coming from the transmit antenna plus a reflected wave coming from the wall.As before,in the presence of the receive antenna.the perturbation of the field due to the antenna is represented by the antenna pattern.An additional assumption here is that the pre of the re ceive antenna does on the wall. appreciably affectthe In essence,what we have done here is to approximate the solution o Transmit d Wall receive antenna Figure 2.2:Illustration of a direct path and a reflected path Maxwell's equations by a method called ray tracing.The assumption here is that the received waveform can be approximated by the sum of the free space wave from the sending transmitter plus the reflected free space waves from each of the reflecting obstacles. In the pre sent situation,if we a sume that the wall is very large, the reflected wa at a given point is the same (except sign change)as the free space wave tha t would exist on the opposite side of the wall if the wall were not present (see Figure 2.3).This means that the reflected wave from the wall has the intensity of a free space wave at a distance equal to the distance to the wall and then back to the receive antenna,i.e. 2d-r.Using (2.2)for both the direct and the reflected wave,and assuming the same
Tse and Viswanath: Fundamentals of Wireless Communication 25 terms of a system function followed by translating the frequency f by the Doppler shift −fv/c. It is important to observe that the amount of shift depends on the frequency f. We will come back to discussing the importance of this Doppler shift and of the time varying attenuation after considering the next example. The above analysis does not depend on whether it is the transmitter or the receiver (or both) that are moving. So long as r(t) is interpreted as the distance between the antennas (and the relative orientations of the antennas are constant), (2.4) and (2.5) are valid. 2.1.3 Reflecting wall, fixed antenna Consider Figure 2.2 below in which there is a fixed antenna transmitting the sinusoid cos 2πft, a fixed receive antenna, and a single perfectly reflecting large fixed wall. We assume that in the absence of the receive antenna, the electromagnetic field at the point where the receive antenna will be placed is the sum of the free space field coming from the transmit antenna plus a reflected wave coming from the wall. As before, in the presence of the receive antenna, the perturbation of the field due to the antenna is represented by the antenna pattern. An additional assumption here is that the presence of the receive antenna does not appreciably affect the plane wave impinging on the wall. In essence, what we have done here is to approximate the solution of ☛ ✡ ✟ ✄ ✠ ✂ ✁ ✄ ✂ ✁ ✲ ✛ ✲ Wall Transmit Antenna receive antenna r d Figure 2.2: Illustration of a direct path and a reflected path. Maxwell’s equations by a method called ray tracing. The assumption here is that the received waveform can be approximated by the sum of the free space wave from the sending transmitter plus the reflected free space waves from each of the reflecting obstacles. In the present situation, if we assume that the wall is very large, the reflected wave at a given point is the same (except for a sign change) as the free space wave that would exist on the opposite side of the wall if the wall were not present (see Figure 2.3). This means that the reflected wave from the wall has the intensity of a free space wave at a distance equal to the distance to the wall and then back to the receive antenna, i.e., 2d − r. Using (2.2) for both the direct and the reflected wave, and assuming the same
Tse and Viswanath:Fundamentals of Wireless Communicatior 26 Figure 2.3:Relation of reflected wave to wave without wall. antenna gain a for both waves,we get E,,=0os2mf-且-a0os2afe-)】 (2.6 2d-r osition of two waves,both of frequency f.The phase △9=(2f2-+n）-(2r）=d-r+元 (2.7) When the phase difference is an integer multiple of 2,the two waves add constructively and the received signal is strong.When the phase difference is an odd integer multiple of m,the two waves add destructively,and the received signal is weak.As a function of r,this translates into a a spatial patte of structiv nd destr ctive interference of the waves.The distance from a peak toavalley is called the coherence distance A= (2.8) where A:=c/f is the wavelength of the transmitted sinusoid The constructive and destructive interference pattern also depends on the frequency f:for a fixed r,if f changes by (2.9) we move from a peak to a valley.The quantity T4= (2.10) is called the delay spread of the channel:it is the difference between the propagation delays along the two signal paths.Thus,the constructive and destructive interference
Tse and Viswanath: Fundamentals of Wireless Communication 26 Sending Antenna Wall Figure 2.3: Relation of reflected wave to wave without wall. antenna gain α for both waves, we get Er(f, t) = α cos 2πf t − r c r − α cos 2πf t − 2d−r c 2d − r . (2.6) The received signal is a superposition of two waves, both of frequency f. The phase difference between the two waves is ∆θ = 2πf(2d − r) c + π − 2πfr c = 4πf c (d − r) + π. (2.7) When the phase difference is an integer multiple of 2π, the two waves add constructively, and the received signal is strong. When the phase difference is an odd integer multiple of π, the two waves add destructively, and the received signal is weak. As a function of r, this translates into a spatial pattern of constructive and destructive interference of the waves. The distance from a peak to a valley is called the coherence distance: ∆xc := λ 4 (2.8) where λ := c/f is the wavelength of the transmitted sinusoid. The constructive and destructive interference pattern also depends on the frequency f: for a fixed r, if f changes by 1 2 2d − r c − r c −1 , (2.9) we move from a peak to a valley. The quantity Td := 2d − r c − r c (2.10) is called the delay spread of the channel: it is the difference between the propagation delays along the two signal paths. Thus, the constructive and destructive interference
Tse and Viswanath:Fundamentals of Wireless Communication 27 pattern changes significantly if the frequency changes by an amount of the order of 1/Ta.This parameter is called the coherence bandwidth. 2.1.4 Reflecting wall,moving antenna Suppose the receive antenna is now moving at a velocity v(Figure 2.4).As it moves through the pattern of constructive and destructive interference created by the two waves,the strength of the received signal increases and decreases.This is the phe- nomenon of multipath fading.The time taken to travel from a peak to a valley is c/(4fv):this is the time-scale at which the fading occurs,and it is called the coher- ence time of the channel. Sending Antenna d Wall oO U Figure 2.4:Illustration of a direct path and a reflected path An equivalent way of seeing this is in terms of the Doppler shifts of the direct and the reflected waves.Suppose the receive antenna is at location ro at time 0.Taking r=ro vt in (2.6),we get B(ft)=acos2f(1)acos2mf (+t+ T。+vt (2.11) 2d-r。-vt The first term,the direct wave,is a sinusoid of slowly decreasing magnitude at frequency f(1-v/c),experiencing a Doppler shift D:=-fu/c.The second is a sinusoid of smaller but increasing magnitude at frequency f(1+v/c),with a Doppler shift D2:=+fv/c.The parameter D。=D2-D1 (2.12) is called the Doppler spread.For example,if the mobile is moving at 60 km/h and read can be visualized to e transmi antenna.In this case the attenuations are roughly the same for both paths,and we
Tse and Viswanath: Fundamentals of Wireless Communication 27 pattern changes significantly if the frequency changes by an amount of the order of 1/Td. This parameter is called the coherence bandwidth. 2.1.4 Reflecting wall, moving antenna Suppose the receive antenna is now moving at a velocity v (Figure 2.4). As it moves through the pattern of constructive and destructive interference created by the two waves, the strength of the received signal increases and decreases. This is the phenomenon of multipath fading. The time taken to travel from a peak to a valley is c/(4fv): this is the time-scale at which the fading occurs, and it is called the coherence time of the channel. ☛ ✡ ✟ ✄ ✠ ✂ ✁ ✄ ✂ ✁ ✲ ✛ ✲ Wall Sending Antenna v ✲ r(t) d Figure 2.4: Illustration of a direct path and a reflected path. An equivalent way of seeing this is in terms of the Doppler shifts of the direct and the reflected waves. Suppose the receive antenna is at location r0 at time 0. Taking r = r0 + vt in (2.6), we get Er(f, t) = α cos 2πf[(1 − v c )t − r0 c ] r0 + vt − α cos 2πf h (1 + v c )t + r0−2d c i 2d − r0 − vt . (2.11) The first term, the direct wave, is a sinusoid of slowly decreasing magnitude at frequency f(1 − v/c), experiencing a Doppler shift D1 := −fv/c. The second is a sinusoid of smaller but increasing magnitude at frequency f(1 + v/c), with a Doppler shift D2 := +fv/c . The parameter Ds := D2 − D1 (2.12) is called the Doppler spread. For example, if the mobile is moving at 60 km/h and f = 900 MHz, the Doppler spread is 100 Hz. The role of the Doppler spread can be visualized most easily when the mobile is much closer to the wall than to the transmit antenna. In this case the attenuations are roughly the same for both paths, and we
Tse and Viswanath:Fundamentals of Wireless Communication 28 E,(t) Figure 2.5:The received waveform oscillating at frequency f with a slowly varying envelope at frequency D,/2. can approximate the denominator of the second term byr=ro+ut.Then,combining the two sinusoids.we get 2asin2f sin2f E(,t)≈ (2.13) r。+Ut This is the product of two sinusoids,one at the input frequency f,which is typically on the order of GHz,and the other one at fu/c=D,/2,which might be on the order of 50Hz Thus the to a sinusoid at f is another sinusoid at f with a time-varying en nvelope,with pe 1 s going to zeros arou very 5 ms (Figure 2.5).The envelope is at its widest when the mobile is at a peak of the interference pattern and at its narrowest when the mobile is at a valley.Thus,the Doppler spread determines the rate of traversal across the interference pattern and is inversely proportional to the coherence time of the channel. red the denominator terms in(2.11)and (2.13).Whe nce in the length between two paths change wavelength the phase differene between the a quarte s changes by /2,which causes a very significant change in the overall received amplitude.Since the carrier wavelength is very small relative to the path lengths,the time over which this phase effect causes a significant change is far smaller than the time over which the
Tse and Viswanath: Fundamentals of Wireless Communication 28 t Er(t) Figure 2.5: The received waveform oscillating at frequency f with a slowly varying envelope at frequency Ds/2. can approximate the denominator of the second term by r = r0 + vt. Then, combining the two sinusoids, we get Er(f, t) ≈ 2α sin 2πf h v c t + (r 0−d) c i sin 2πf[t − d c ] r0 + vt . (2.13) This is the product of two sinusoids, one at the input frequency f, which is typically on the order of GHz, and the other one at fv/c = Ds/2, which might be on the order of 50Hz. Thus, the response to a sinusoid at f is another sinusoid at f with a time-varying envelope, with peaks going to zeros around every 5 ms (Figure 2.5). The envelope is at its widest when the mobile is at a peak of the interference pattern and at its narrowest when the mobile is at a valley. Thus, the Doppler spread determines the rate of traversal across the interference pattern and is inversely proportional to the coherence time of the channel. We now see why we have partially ignored the denominator terms in (2.11) and (2.13). When the difference in the length between two paths changes by a quarter wavelength, the phase difference between the responses on the two paths changes by π/2, which causes a very significant change in the overall received amplitude. Since the carrier wavelength is very small relative to the path lengths, the time over which this phase effect causes a significant change is far smaller than the time over which the
Tse and Viswanath:Fundamentals of Wireless Communication 29 Sending Antenna n r2 Receive antenna Ground Plane hr Figure 2.6:Illustration of a direct path and a reflected path off a ground plane. denominator terms cause a significant change.The effect of the phase changes is on the order of milliseconds,whereas the effect of changes in the denominator are of the order of seconds or minutes.In terms of modulation and detection,the time scales of interest are in the range of milliseconds and less,and the denominators are effectively constant over these pe The night notice that we are constantly making in trying to understand wireless communication,much more so than for wired communication.This is partly because wired channels are typically time-invariant over a very long time-scale while wireless channels are typically time varying,and appropriate models depend very much on the time scales of interest.For wireless systems,the most important issue is what approximations to make.Solving and manipulating equations is far less important.Thus it is important to understand these modeling i ssues thoroughly 2.1.5 Reflection from a Ground Plane Consider a transmitting and a receive antenna,both above a plane surface such as a road(see Figure 2.6).When the horizontal distance r between the antennas becomes very large relative to their vertical displacements from the ground plane(i.e.,height) a very surprising thing happens.In particular,the differ ace between the direct path length and the reflected path length zero as r-1 with ir easing r(see Exer 2.5).When r is large enough, ngths becomes small relative to the wavelength c/f.Since the sign of the electric field is reversed on the reflected path,these two waves start to cancel each other out.The electric wave at the receiver is then attenuated as r-2,and the received power decreases as r-4.This situation is particularly important in rural areas where base stations tend to be placed
Tse and Viswanath: Fundamentals of Wireless Communication 29 ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ☎ ✠ ✡ ✠ ☛ ✠ ☞✌ ✠ ✟ ☎ ✍ ✠ ☎ ☎ ✟ ✎ ✠ ☎ ✆ ☞☎ ✏ ✑ ☎ ✍ ✠ ☎ ☎ ✟ ✒ ✒ ✓ ✔ ✕ ✔ ✖ ✒ ✗ Figure 2.6: Illustration of a direct path and a reflected path off a ground plane. denominator terms cause a significant change. The effect of the phase changes is on the order of milliseconds, whereas the effect of changes in the denominator are of the order of seconds or minutes. In terms of modulation and detection, the time scales of interest are in the range of milliseconds and less, and the denominators are effectively constant over these periods. The reader might notice that we are constantly making approximations in trying to understand wireless communication, much more so than for wired communication. This is partly because wired channels are typically time-invariant over a very long time-scale, while wireless channels are typically time varying, and appropriate models depend very much on the time scales of interest. For wireless systems, the most important issue is what approximations to make. Solving and manipulating equations is far less important. Thus, it is important to understand these modeling issues thoroughly. 2.1.5 Reflection from a Ground Plane Consider a transmitting and a receive antenna, both above a plane surface such as a road (see Figure 2.6). When the horizontal distance r between the antennas becomes very large relative to their vertical displacements from the ground plane (i.e., height), a very surprising thing happens. In particular, the difference between the direct path length and the reflected path length goes to zero as r −1 with increasing r (see Exercise 2.5). When r is large enough, this difference between the path lengths becomes small relative to the wavelength c/f. Since the sign of the electric field is reversed on the reflected path, these two waves start to cancel each other out. The electric wave at the receiver is then attenuated as r −2 , and the received power decreases as r −4 . This situation is particularly important in rural areas where base stations tend to be placed