ECE 461 Fall 2006Wireless CommunicationChannel Model for Mobile CommunicationsPSfrag replacementss(t)h(ξ)y(t)w(t)r(t)Figure 1: Complex baseband point-to-point communications channelSo far we studied the complex baseband model for point-to-point communications shown in Figure 1.Our goal now is to modify this channel model to incorporate the effects of the mobility. We will focuson terrestrial mobile communications channels – satellite channels are more “well-behaved”. Thefollowing are points worth noting in making the transition to the mobile communication channelmodel.◦ The additive noise term w(t) is always present whether the channel is point-to-point or mobile,and usually w(t) is modelled as proper complex WGN.◦ For point-to-point communications the channel response is generally well modelled by a lineartime invariant (LTI) system (h(ξ) may or may not be known at the receiver). For mobile com-munications, the channel response is time-varying, and we will see that it is well modelled as alinear time-varying (LTV) system.To study the mobile communications channel, consider the situation where the mobile station (MS)is at location (x, y) or (d, ϕ) in a coordinate system with the base station (BS) at the origin asshown in Figure 2. A 3-d model may be more appropriate in some situations, but for simplicity wewill consider a 2-d model. Also, we restrict our attention now to the channel connecting one pairof transmit (Tx) and receive (Rx) antennas.If the mobile is fixed at location (d, ϕ), the channel that it sees is time-invariant. The response ofthis time-invariant channel is a function of the location, and is determined by all paths connectingthe BS and the MS. Thus we have the system shown in Figure 3, where hd,ϕ(ξ) is the impulseresponse of a causal LTI system, which is a function of the multipath profile between the BS andMS.Referring to Figure 2, suppose the n-th path connecting the BS and MS has amplitude gain βn(d, ϕ)and delay τn(d, ϕ). The delay of τn(d, ϕ) introduces a carrier phase shift ofφn(d, ϕ) = −2πfcτn(d, ϕ) + constantwhere the constant depends on the reflectivity of the surface(s) that reflect the path. Then we cancV.V. Veeravalli, 2006 1PSfrag replacementsXYBSMSϕdFigure 2: Multipath channel seen at location (d, ϕ) for one Tx&Rx antenna pairPSfrag replacementss(t) y(t)hd,ϕ(ξ)Figure 3: Causal LTI system representing multipath profile at location (d, ϕ)write the output y(t) in terms of the input s(t) asy(t) =Xnβn(d, ϕ) ejφn(d,ϕ)s(t − τn(d, ϕ))which implies that the impulse response ishd,ϕ(ξ) =Xnβn(d, ϕ) ejφn(d,ϕ)δ(ξ − τn(d, ϕ))Scales of VariationAs the MS moves, (d, ϕ) change with time and the linear system associated with the channelbecomes LTV. There are two scales of variation:◦ The first is a small-scale variation due to rapid changes in the phase φnas the mobile moves overdistances of the order of a wavelength of the carrier λc= c/fc, where c is the velocity of light.This is because movements in space of the order of a wavelength cause changes in τnof the orderof 1/fc, which in turn cause changes in φnof the order of 2π. (Note that for a 900 MHz carrier,λc≈ 1/3 m.)Modeling the phases φnas independent Uniform[0, 2π] random variables, we can see that theaverage power gain in the vicinity of (d, ϕ) is given byPnβ2n(d, ϕ). We denote this averagepower gain by G(d, ϕ).cV.V. Veeravalli, 2006 2◦ The second is a large-scale variation due to changes in {βn(d, ϕ)} and {τn(d, ϕ)} – both in thenumber of paths and their strengths. These changes happen on the scale of the distance betweenobjects in the environment.To study these two scales of variation separately, we redraw Figure 3 in terms of two componentsas shown in Figure 4.xPSfrag replacementss(t) y(t)pG(d, ϕ)cd,ϕ(ξ)Figure 4: Small-scale and large-scale variation components of channelHere cd,ϕis normalized so that the average power gain introduced by cd,ϕis 1, i.e.cd,ϕ(ξ) =Xnβn(d, ϕ)ejφn(d,ϕ)δ(ξ − τn(d, ϕ))where {βn(d, ϕ)} is normalized so thatPnβ2n(d, ϕ) = 1. The large-scale variations in (average)amplitude gain are then lumped into the multiplicative termpG(d, ϕ).The goal of wireless channel modeling is to find useful analytical models for the variations in thechannel. Models for the large scale variations are useful in cellular capacity-coverage optimizationand analysis, and in radio resource management (handoff, admission control, and power control).Models for the small scale variations are more useful in the design of digital modulation anddemodulation schemes (that are robust to these variations). We hence focus on the small scalevariations in this class.Small-scale Variations in GainPSfrag replacementss(t) y(t)√Gcd,ϕ(ξ)Figure 5: Small-scale variations in the channel (with large-scale variations treated as constant).Recall that the small scale variations in the channel are captured in a linear system with responsecd,ϕ(ξ) =Xnβn(d, ϕ)ejφn(d,ϕ)δ(ξ − τn(d, ϕ)) ,cV.V. Veeravalli, 2006 3where the {βn(d, ϕ)} are normalized so thatPnβ2n(d, ϕ) = 1. As (d, ϕ) changes with t, the channelcorresponding to the small-scale variations becomes time-varying and we get:c(t; ξ) := cd(t),ϕ(t)(ξ) =Xnβn(t) ejφn(t)δ(ξ − τn(t)) .Treating the large scale variationspG(d, ϕ) as roughly constant (see Figure 5), we obtain:y(t) =√GZ∞0c(t; ξ)s(t − ξ)dξ .Finally, we may absorb the scaling factor√G into the signal s(t), with the understanding that thepower of s(t) is the received signal power after passage through the channel. Theny(t) =Z∞0c(t; ξ)s(t − ξ)dξ .Doppler shifts in phaseFor movements of the order of a few wavelengths, {βn(t)} and {τn(t)} are roughly constant, andthe time variations in c(t; ξ) are mainly due to changes in {φn(t)}, i.e.,c(t; ξ) ≈Xnβnejφn(t)δ(ξ − τn) .From this equation it is clear that the magnitude of the impulse response |c(t; ξ)| is roughly inde-pendent of t. A typical plot of |c(t; ξ)| is shown in Figure 6. The width of the delay profile (delayspread) is of the order of tens of microseconds for outdoor channels, and of the order of hundredsof nanoseconds for indoor channels. Note that the paths typically appear in clusters in the delayprofile (why?).To study the phase variations φn(t) in more detail, consider a mobile that is traveling with velocityv and suppose that the n-th path has an angle of arrival θn(t) with respect to the velocity