Lecture2
- 1. UNIT-1 Introduction to Wireless System and Channels Physical Modeling for Wireless Channels Input/Output Model of the Wireless Channel Time and Frequency Coherence Statistical Channel Models Specific Absorption Rate
- 3. The Wireless Channel • A defining characteristic of the mobile wireless channel is the variations of the channel strength over time and over frequency. • The variations can be roughly divided into two types: (1) large-scale fading: It is due to path loss of signal as a function of distance and shadowing by large objects such as buildings and hills. This occurs as the mobile moves through a distance of the order of the cell size, and is typically frequency independent. (2) Small-scale fading: It is due to the constructive and destructive interference of the multiple signal paths between the transmitter and receiver. This occurs at the spatial scale of the order of the carrier wavelength, and is frequency dependent. • Large-scale fading is more relevant to issues such as cell-site planning. • Small-scale multipath fading is more relevant to the design of reliable and efficient communication systems.
- 4. Physical Modeling for Wireless Channels • Wireless channels operate through electromagnetic radiation from the transmitter to the receiver. • Channel quality varies over multiple time scales. At a slow scale, channel varies due to large-scale fading effects. At a fast scale, channel varies due to multipath effects.
- 5. Physical Modeling for Wireless Channels • Cellular communication in the USA is limited by the Federal Communication Commission (FCC), and by similar authorities in other countries, to one of three frequency bands, one around 0.9 GHz, one around 1.9 GHz, and one around 5.8 GHz. • The wavelength Λ(f) of electromagnetic radiation at any given frequency f is given by Λ = c/f, where c = 3 × 108 m/s is the speed of light. • One of the important questions is where to choose to place the base stations, and what range of power levels are then necessary on the downlink and uplink channels. • Another major question is what types of modulation and detection techniques look promising. • To address this, we will construct stochastic models of the channel, assuming that different channel behaviors appear with different probabilities, and change over time (with specific stochastic properties).
- 6. Uplink and Downlink Uplink is the transmission path from the mobile station (cell phone) to a base station (cell site). Downlink is the transmission path from a cell site to the cell phone.
- 7. Free space, fixed transmitting and receive antennas • In the far field, the electric field and magnetic field at any given location are perpendicular both to each other and to the direction of propagation from the antenna. • They are also proportional to each other, so it is sufficient to know only one of them (just as in wired communication, where we view a signal as simply a voltage waveform or a current waveform). In response to a transmitted sinusoid cos 2πf t, we can express the electric far field at time t as: (1) • Here, (r, θ, ψ) represents the point u in space at which the electric field is being measured, where r is the distance from the transmitting antenna to u and where (θ, ψ) represents the vertical and horizontal angles from the antenna to u, respectively. The constant c is the speed of light, and αs(θ, ψ, f) is the radiation pattern of the sending antenna at frequency f in the direction (θ, ψ). phase of the field varies with fr/c.
- 8. • It is important to observe that, as the distance r increases, the electric field decreases as r−1 and thus the power per square meter in the free space wave decreases as r−2 . • We will see shortly that this r−2 reduction of power with distance is often not valid when there are obstructions to free space propagation. • Next, suppose there is a fixed receive antenna at the location u = (r, θ, ψ). The received waveform (in the absence of noise) in response to the above transmitted sinusoid is then: • (2) • where α(θ, ψ, f) is the product of the antenna patterns of transmitting and receive antennas in the given direction. Our approach to (2) is a bit odd since we started with the free space field at u in the absence of an antenna.
- 9. • Placing a receive antenna there changes the electric field in the vicinity of u, but this is taken into account by the antenna pattern of the receive antenna. Now suppose, for the given u, that we define • (3) • We then have • Thus, H(f) is the system function for an LTI (linear time-invariant) channel, and its inverse Fourier transform is the impulse response. , and its inverse Fourier transform is the impulse response.
- 10. Free space, moving antenna • up to that we deal with Fixed antenna and free space model : • Now consider receive antenna that is moving with speed v in the direction of increasing distance from the transmitting antenna. • That is, we assume that the receive antenna is at a moving location described as u(t) = (r(t), θ, ψ) with r(t) = r0 + vt. • Using equation (1) to describe the free space electric field at the moving point u(t) (for the moment with no receive antenna), we have (4)
- 11. • Note that we can rewrite f(t− r0 /c− vt/c) as f(1−v/c)t − f r0 /c. • Thus, the sinusoid at frequency f has been converted to a sinusoid of frequency f(1−v/c); there has been a Doppler shift of −fv/c due to the motion of the observation point. • If the antenna is now placed at u(t), and the change of field due to the antenna presence is again represented by the receive antenna pattern, the received waveform, in analogy to Eq (2), • (5) • we can represent the channel in 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.
- 12. Reflecting wall, fixed antenna • Consider Figure. below in which there is a fixed antenna transmitting the sinusoid cos 2πf t, 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.
- 13. • 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 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 Eq (2) for both the direct and the reflected wave, and assuming the same
- 14. • antenna gain α for both waves, we get • (6)
- 15. • The received signal is a superposition of two waves, both of frequency f. The phase difference between the two waves is • (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: where λ := c/f is the wavelength of the transmitted sinusoid.
- 16. • The constructive and destructive interference pattern also depends on the frequency f: for a fixed r, if f changes by • we move from a peak to a valley. The quantity • 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
- 17. • pattern changes significantly if the frequency changes by an amount of the order of 1/Td. This parameter is called the coherence bandwidth.
- 18. Reflecting wall, moving antenna • Suppose the receive antenna is now moving at a velocity v (Figure). • 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.
- 19. • 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 , we get: • For Moving antenna only(with velocity v)(There is no wall): We already calculated(from Eq 5): • Similarly for reflecting wall only(Fixed Antenna): • Now in this particular case, Moving antenna and reflecting wall
- 20. • For moving antenna frequency become f(1−v/c), secondly for reflecting wall distance become 2d-r, Taking r = r0 + vt hence will get • Hence conclude that: • 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 is called the Doppler spread. Ds := D2 − D1
- 21. • Example, if the mobile is moving at 60 km/h and f = 900 MHz, what is the Doppler spread.
- 22. Doppler spread =D2-D1 := +fv/c-(-fv/c) 900X106X60X103/3600X3X108-(-900X106X60X103/3600X3X108)=50-(- 50)=100 • It is 100 Hz.
- 23. • 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 can approximate the denominator of the second term by • r = r0 + vt. Then, combining the two sinusoids, we get • 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.
- 24. • When the difference in the length between two paths changes by a quarter wavelength, which causes a very significant change in the overall received amplitude. • Why it is so: • Since the carrier wavelength is very small (as the frequency is high) 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 denominator terms cause a significant change.
- 25. • 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 period
- 26. Reflection from a Ground Plane • Consider a transmitting and a receive antenna, both above a plane surface such as a road (see Figure). 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.
- 27. • In particular, the difference between the direct path length and the reflected path length goes to zero . • 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 place on roads
- 28. 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.
- 29. • 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.
- 30. • 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.
- 31. • 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 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.
- 32. 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 , the reflected field calculated in Eq 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.
- 33. • 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.
- 34. • Knowing how to find the amplitude of the reflected field from each type of reflector 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.
Editor's Notes
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