Lecture 1 Wireless Channel Models

MIMO Communication Systems Lecture 1 Wireless Channel Models Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/3/2 Lecture 1: Wireless Channel Modeling 1

Outline Path Loss and Shadowing (Chapter 2 in Goldsmith s Book, Chapter 2 in Tse s Book) TX-RX Signal Models Free-Space Path Loss Ray Tracing Empirical Path-Loss Models Statistical Multipath Channels (Chapter 3 in Goldsmith s Book) Time-Varying Channel Impulse Response Narrowband Fading Models Wideband Fading Models 2017/3/2 Lecture 1: Wireless Channel Modeling 2

Path Loss and Shadowing Introduction Here we will characterize the variation in received signal power over distance due to path loss and shadowing. Path loss is caused by dissipation of the power radiated by the transmitter as well as effects of the propagation channel Shadowing is caused by obstacles between the transmitter and receiver that attenuate signal power through absorption, reflection, scattering, and diffraction Since variations due to path loss and shadowing occur over relatively large distances, they are sometimes referred to as largescale propagation effects Variations due to multipath occur over very short distances (on the order of the signal wavelength), so they are sometimes referred to as small-scale propagation effects Figure 2.1 illustrates the ratio of the received-to-transmit power in db versus log-distance for the combined effects of path loss, shadowing, and multipath. 2017/3/2 Lecture 1: Wireless Channel Modeling 3

Path Loss and Shadowing 2017/3/2 Lecture 1: Wireless Channel Modeling 4

Transmit and Receive Signal Models The transmitted signal can be modeled as (2.1) where u(t) =x(t)+jy(t) is a complex baseband signal with in-phase component, quadrature component.. The signal u(t) is called the complex envelope or complex low-pass equivalent signal of s(t). The received signal will have a similar form: (2.2) where the complex baseband signal v(t) will depend on the channel through which s(t) propagates. If s(t) is transmitted through a time-invariant channel then, where c(t) is the equivalent lowpass channel impulse response for the channel. 2017/3/2 Lecture 1: Wireless Channel Modeling 5

Doppler Shift The received signal may have a Doppler shift of, where θ is the arrival angle of the received signal relative to the direction of motion, v is the receiver velocity towards the transmitter in the direction of motion, and is the signal wavelength ( m/s is the speed of light). c =3 10 8 f D = v cos / The geometry associated with the Doppler shift is shown in Fig. 2.2. = c/f c The Doppler frequency is then obtained from the relationship between signal frequency and phase: (2.3) 2017/3/2 Lecture 1: Wireless Channel Modeling 6

Path Loss Model We will ignore the Doppler term in the free-space and ray tracing models since for typical vehicle speeds (75 Km/hr) and frequencies (around 1 GHz), it is on the order of 100 Hz Suppose s(t) of power P t is transmitted through a given channel, with corresponding received signal r(t) of power P r, where P r is averaged over any random variations due to shadowing. We define the path loss of the channel as the difference in db between the transmitted and received signal power: P L db = 10 log 10 P t P r (2.5) In general, the db path loss is a nonnegative number since the channel does not contain active elements, and thus can only attenuate the signal. The db path gain is defined as the negative of the db path loss: P G = P L =, which is generally a negative number. 10 log 10 (P r /P t ) db 2017/3/2 Lecture 1: Wireless Channel Modeling 7

Free-Space Path Loss Consider a signal transmitted through free space to a receiver located at distance d from the transmitter. A line-of-sight (LOS) channel: there are no obstructions between TX and RX and the signal propagates along a straight line between the two. The corresponding received signal is called the LOS signal or ray. The received signal that undergoes free-space path loss p Gl (2.6) where is the product of the transmit and receive antenna field radiation patterns in the LOS direction. The phase shift e j2 d/ is due to the distance d the wave travels. The power in the transmitted signal s(t) is, so the ratio of received to transmitted power The received signal power falls off inversely proportional to d 2 P r P t = applep Gl 4 d 2 (2.7) 2017/3/2 Lecture 1: Wireless Channel Modeling 8

Free-Space Path Loss For other signal propagation models, the received signal power falls off more quickly relative to this distance The received signal power is also proportional to the square of the signal wavelength λ. (As the carrier frequency increases, the received power decreases) This dependence of received power on the signal wavelength λ is due to the effective area of the receive antenna Directional antennas can be designed so that receive power is an increasing function of frequency for highly directional links The received power can be expressed in dbm as Free-space path loss is defined as the path loss of the free-space model: The free-space path gain is thus (2.9) (2.8) (2.10) 2017/3/2 Lecture 1: Wireless Channel Modeling 9

Free-Space Path Loss Example 2.1: Consider an indoor wireless LAN with MHz, cells of radius 10 m, and nondirectional antennas. Under the free-space path loss model, what transmit power is required at the access point such that all terminals within the cell receive a minimum power of 10µW. How does this change if the system frequency is 5 GHz? Solution: We must find the transmit power such that the terminals at the cell boundary receive the minimum required power. We obtain a formula for the required transmit power as follows: apple 2 4 d P t = P r p Gl. 2017/3/2 Lecture 1: Wireless Channel Modeling 10

Ray Tracing In a typical urban or indoor environment, a radio signal transmitted from a fixed source will encounter multiple objects in the environment that produce reflected, diffracted, or scattered copies of the transmitted signal, as shown in Figure 2.3. These additional copies of the transmitted signal, called multipath signal components, can be attenuated in power, delayed in time, and shifted in phase and/or frequency from the LOS signal path at the receiver. The multipath and transmitted signal are summed together at the receiver, which often produces distortion in the received signal relative to the transmitted signal. 2017/3/2 Lecture 1: Wireless Channel Modeling 11

Two-Ray Model The two-ray model is used when a single ground reflection dominates the multipath effect, as illustrated in Figure 2.4. The received signal for the two-ray model is (2.11) where p is the p time delay of the ground reflection relative to the LOS ray,, Gr = p Gl = p G a G b G c G d, R is the ground reflection coefficient. 2017/3/2 Lecture 1: Wireless Channel Modeling 12

Two-Ray Model The delay spread of the two-ray model equals the delay between the LOS ray and the reflected ray: (x + x 0 l)/c If the transmitted signal is narrowband relative to the delay u ) then u(t) u(t ). With this approximation, the received power of the two-ray model for narrowband transmission is ( B 1 (2.12) where received signal components. We can show that is the phase difference between the two (2.13) where d denotes the horizontal separation of the antennas, transmitter height, and denotes the receiver height. h r h t denotes the 2017/3/2 Lecture 1: Wireless Channel Modeling 13

Two-Ray Model When d is very large compared to we can use a Taylor series to get (2.14) For asymptotically large d, Using these approximations, the received signal power is approximately or, in dbm, we have (2.17) (2.18) Thus, in the limit of asymptotically large d, the received power falls off inversely with the fourth power of d and is independent of the wavelength λ. 2017/3/2 Lecture 1: Wireless Channel Modeling 14

Two-Ray Model A plot of a function of distance is illustrated in Figure 2.5 for f = 900 MHz, and transmit power normalized so that the plot starts at 0 dbm. What can you learn from this plot? 2017/3/2 Lecture 1: Wireless Channel Modeling 15

Two-Ray Model Example 2.2: Determine the critical distance for the two-ray model in an urban microcell and an indoor microcell Solution: d c =4h t h r / = 800 meters for the urban microcell and 160 meters for the indoor system. A cell radius of 800 m in an urban microcell system is a bit large: urban microcells today are on the order of 100 m to maintain large capacity. However, if we used a cell size of 800 m under these system parameters, signal power would fall off as d 2 inside the cell, and interference from neighboring cells would fall off as d 4,and thus would be greatly reduced. Similarly, 160 m is quite large for the cell radius of an indoor system, as there would typically be many walls the signal would have to go through for an indoor cell radius of that size. So an indoor system would typically have a smaller cell radius, on the order of 10-20 m. 2017/3/2 Lecture 1: Wireless Channel Modeling 16

Ten-Ray Model This model assumes rectilinear streets with buildings along both sides of the street and transmitter and receiver antenna heights that are close to street level. Assume a narrowband model such that u(t) u(t i ) for all i, then the received power is (2.20) where R i denotes the path length of the ith reflected ray and i =2 (x i l)/. 2017/3/2 Lecture 1: Wireless Channel Modeling 17

Empirical Path Loss Models Most mobile communication systems operate in complex propagation environments that cannot be accurately modeled by free-space path loss or ray tracing. Many path loss models have been developed to predict path loss in typical wireless environments such as large urban macro-cells, urban microcells, and, more recently, inside buildings Analytical models characterize as a function of distance, so path loss is well defined. Empirical measurements of as a function of distance include the effects of path loss, shadowing, and multipath. In order to remove multipath effects, empirical measurements for path loss typically average their received power measurements and the corresponding path loss at a given distance over several wavelengths. This average path loss is called the local mean attenuation (LMA) at distance d, and generally decreases with d due to free space path loss and signal obstructions. 2017/3/2 Lecture 1: Wireless Channel Modeling 18

Empirical Path Loss Models The empirical path loss PL(d) for a given environment (e.g. a city, suburban area, or office building) is defined as the average of the LMA measurements at distance d, averaged over all available measurements in the given environment. Okumura model (for signal prediction in large urban macro cells): This model is applicable over distances of 1-100 Km and frequency ranges of 150-1500 MHz. The empirical path loss formula of Okumura at distance d parameterized by the carrier frequency is given by (2.28) where is free space path loss at distance d and carrier frequency is the median attenuation in addition to free space path loss across all environments, is the base station antenna height gain factor, is the mobile antenna height gain factor, and is the gain due to the type of environment. 2017/3/2 Lecture 1: Wireless Channel Modeling 19

Empirical Path Loss Models Hata Model : it is an empirical formulation of the graphical path loss data provided by Okumura and is valid over roughly the same range of frequencies, 150-1500 MHz. The Hata model simplifies calculation of path loss since it is a closedform formula and is not based on empirical curves for the different parameters. The standard formula for empirical path loss in urban areas under the Hata model is (2.31) The parameters in this model are the same as under the Okumura model, and is a correction factor for the mobile antenna height based on the size of the coverage area. Piecewise Linear (Multi-Slope) Model: it is anempirical method for approximating path loss in outdoor microcells and indoor channels. This approximation is illustrated in Figure 2.9 for db attenuation versus log-distance 2017/3/2 Lecture 1: Wireless Channel Modeling 20

Empirical Path Loss Models In the figure, the dots represent hypothetical empirical measurements and the piecewise linear model represents an approximation to these measurements. 2017/3/2 Lecture 1: Wireless Channel Modeling 21

Empirical Path Loss Models A special case of the piecewise model is the dual-slope model as shown in the following (2.35) d c is some critical distance, and are called path loss exponent. Indoor Attenuation Factors: Indoor environments differ widely in the materials used for walls and floors, the layout of rooms, hallways, windows, and open areas, the location and material in obstructing objects, and the size of each room and the number of floors. Indoor path loss models must accurately capture the effects of attenuation across floors due to partitions, as well as between floors. Partition materials and dielectric properties vary widely, and thus so do partition losses. 2017/3/2 Lecture 1: Wireless Channel Modeling 22

Empirical Path Loss Models Table 2.1 indicates a few examples of partition losses measured at 900-1300 MHz The partition loss obtained by different researchers for the same partition type at the same frequency often varies widely, making it difficult to make generalizations about partition loss from a specific data set. 2017/3/2 Lecture 1: Wireless Channel Modeling 23

Simplified Path Loss Model The following simplified model for path loss as a function of distance is commonly used for system design: (2.39) The db attenuation is thus (2.40) where K is a unitless constant which depends on the antenna characteristics and the average channel attenuation, d 0 is a reference distance for the antenna far-field, and is the path loss exponent. The values for K, d 0 and can be obtained to approximate either an analytical or empirical model. Due to scattering phenomena in the antenna near-field, the simplified model is generally only valid at transmission distances d>d 0, where is typically assumed to be 1-10 m indoors and 10-100 m outdoors. 2017/3/2 Lecture 1: Wireless Channel Modeling 24

Simplified Path Loss Model When the simplified model is used to approximate empirical measurements, the value of K<1is sometimes set to the free space path gain at distance assuming omnidirectional antennas: and this assumption is supported by empirical data for free-space path loss at a transmission distance of 100 m. depends on the propagation environment: for propagation that approximately follows a free-space or two-ray model is set to 2 or 4, respectively. K (db) = 20 log 10 4 d 0 2017/3/2 Lecture 1: Wireless Channel Modeling 25

Simplified Path Loss Model Example 2.3: Consider the set of empirical measurements of P r /P t given in the table below for an indoor system at 900 MHz. Find the path loss exponent that minimizes the MSE between the simplified model and the empirical db power measurements, assuming that d 0 =1m and K is determined from the free space path gain formula at this. Find the received power at 150 m for the simplified path loss model with this path loss exponent and a transmit power of 1 mw (0 dbm). 2017/3/2 Lecture 1: Wireless Channel Modeling 26

Simplified Path Loss Model Solution: We first set up the MMSE error equation for the db power measurements as 2017/3/2 Lecture 1: Wireless Channel Modeling 27

Shadow Fading A signal transmitted through a wireless channel will typically experience random variation (shadow fading) due to blockage from objects in the signal path, giving rise to random variations of the received power at a given distance. Such variations are also caused by changes in reflecting surfaces and scattering objects. The most common model for characterizing this fading attenuation is log-normal shadowing. In the log-normal shadowing model, the ratio of transmit-to-receive power is assumed random with a log-normal distribution given by (2.43) where, is the mean of in db and is the standard deviation of, also in db. 2017/3/2 Lecture 1: Wireless Channel Modeling 28

Shadow Fading The mean of (the linear average path gain) can be obtained by (2.44) The conversion from the linear mean (in db) to the log mean (in db) is derived as Performance in log-normal shadowing is typically parameterized by the log mean, which is referred to as the average db path loss and is in units of db. The distribution of the db value of is Gaussian and given by (2.45) (2.46) 2017/3/2 Lecture 1: Wireless Channel Modeling 29

Shadow Fading Example 2.4: In Example 2.3 we found that the exponent for the simplified path loss model that best fits the measurements in Table 2.3 was. Assuming the simplified path loss model with this exponent and the same K = 31.54 db, find, the variance of lognormal shadowing about the mean path loss based on these empirical measurements. Solution The sample variance relative to the simplified path loss model with =3.71 is where and is the path loss measurement in Table 2.3 at distance. Thus 2017/3/2 Lecture 1: Wireless Channel Modeling 30

Combined Path Loss and Shadowing Models for path loss and shadowing can be superimposed to capture power falloff versus distance along with the random attenuation about this path loss from shadowing. For the combined model of path loss and shadowing, the ratio of received to transmitted power in db is given by: (2.51) where is a Gauss-distributed random variable with mean zero and variance. As shown in Figure 2.1, the path loss decreases linearly relative to with a slope of 10 db/decade, where is the path loss exponent. Examples 2.3 and 2.4 illustrate the combined model for path loss and log-normal shadowing based on Table 2.3, where path loss obeys the simplified path loss model with K = 31.54 db and path loss exponent =3.71 and shadowing obeys the log normal model with mean given by the path loss model and standard deviation 2017/3/2 Lecture 1: Wireless Channel Modeling 31

Outage Probability under Path Loss and Shadowing In wireless systems there is typically a target minimum received power level below which performance becomes unacceptable (e.g. the voice quality in a cellular system is too poor to understand). The outage probability under path loss and shadowing is defined as the probability that the received power at a given distance, falls below For the combined path loss and shadowing model, the outage probability is (2.52) where the Q function is defined as the probability that a Gaussian random variable x with mean zero and variance one is bigger than z: or 2017/3/2 Lecture 1: Wireless Channel Modeling 32

Outage Probability under Path Loss and Shadowing Example 2.5: Find the outage probability at 150 m for a channel based on the combined path loss and shadowing models of Examples 2.3 and 2.4, assuming a transmit power of P t = 10 mw and minimum power requirement P min = 110.5 dbm. Solution: We have An outage probability of 1% is a typical target in wireless system designs. 2017/3/2 Lecture 1: Wireless Channel Modeling 33

Cell Coverage Area The cell coverage area in a cellular system is defined as the expected percentage of area within a cell that has received power above a given minimum. Consider a base station inside a circular cell of a given radius R. All mobiles within the cell require some minimum received SNR (Signalto-Noise Ratio) for acceptable performance. The transmit power at the base station is designed for an average received power at the cell boundary of, averaged over the shadowing variations. Shadowing will cause some locations within the cell to have received power below, and others will have received power exceeding. Figure 2.10 shows contours of constant received power based on a fixed transmit power at the base station for path loss and average shadowing and for path loss and random shadowing. 2017/3/2 Lecture 1: Wireless Channel Modeling 34

Cell Coverage Area For path loss and average shadowing, constant power contours form a circle around the base station since combined path loss and average shadowing is the same at a uniform distance from the base station. For path loss and random shadowing the contours form an amoeba-like shape due to the random shadowing variations about the average. 2017/3/2 Lecture 1: Wireless Channel Modeling 35

Cell Coverage Area Let be the received power in da from combined path loss and shadowing. The total area within the cell where the minimum power requirement is exceeded is obtained by integrating over all incremental areas where this minimum is exceeded: where 1[ ] denotes the indicator function. (2.55) Define Then we can have The outage probability of the cell is defined as the percentage of area within the cell that does not meet its minimum power requirement (2.56) 2017/3/2 Lecture 1: Wireless Channel Modeling 36

Cell Coverage Area Given the log-normal distribution for the shadowing, Using this outage probability in calculating C, we get (2.57) (2.58) where (2.59) and is the received power at the cell boundary (distance R from the base station) due to path loss alone. A closed-form solution for C in terms of a and b is given by (2.60) 2017/3/2 Lecture 1: Wireless Channel Modeling 37

Cell Coverage Area If the target minimum received power equals the average power at the cell boundary: then a = 0 and the coverage area simplifies to (2.61) (Note that with this simplification C depends only on the ratio ) Example 2.6: Find the coverage area for a cell with the combined path loss and shadowing models of Examples 2.3 and 2.4, a cell radius of 600 m, a base station transmit power of, and a minimum received power requirement of dbm and of dbm. Solution: We first consider and check if a = 0 to determine whether to use the full formula (2.60) or the simplified formula (2.61). We have so we use (2.60). Evaluating a and b from (2.59) yields 2017/3/2 Lecture 1: Wireless Channel Modeling 38

Substituting these into C yields Cell Coverage Area which would be a very low coverage value for an operational cellular system (lots of unhappy customers). Now considering the less stringent received power requirement dbm yields a = ( 120+114.9) / 3.65 = 1.479 and the same b = 4.41. Substituting these values into C in (2.60) yields C =.988, a much more acceptable value for coverage area. (End of Chapter 2 in Goldsimth s Book) 2017/3/2 Lecture 1: Wireless Channel Modeling 39

Statistical Multipath Channel Models Here, we will examine fading models for the constructive and destructive addition of different multipath components introduced by the channel. We model the multipath channel by a random time-varying impulse response. If a single pulse is transmitted over a multipath channel, the received signal will appear as a pulse train, with each pulse in the train corresponding to the LOS component or a distinct multipath component associated with a distinct scatter or cluster of scatters. An important characteristic of a multipath channel is the time delay spread it causes to the received signal. This delay spread equals the time delay between the arrival of the first received signal component (LOS or multipath) and the last received signal component associated with a single transmitted pulse. If the delay spread is small compared to the inverse of the signal bandwidth, then there is little time spreading in the received signal. However, when the delay spread is relatively large, there is significant time spreading of the received signal which can lead to substantial signal distortion. 2017/3/2 Lecture 1: Wireless Channel Modeling 40

Time-Varying Channel Impulse Response Recall the transmitted signal The corresponding received signal is the sum of the line-of-sight (LOS) path and all resolvable multipath components: (3.1) (3.2) where n = 0 corresponds to the LOS path. The unknowns in this expression are the number of resolvable multipath components N(t), and for the LOS path and each multipath component, its path length and corresponding delay, Doppler phase shift and amplitude. 2017/3/2 Lecture 1: Wireless Channel Modeling 41

Time-Varying Channel Impulse Response We say that two multipath components with delay and are resolvable if their delay difference significantly exceeds the inverse signal bandwidth: Multipath components that do not satisfy this resolvability criteria cannot be separated out at the receiver since and thus these components are nonresolvable. 2017/3/2 Lecture 1: Wireless Channel Modeling 42

Time-Varying Channel Impulse Response We can simplify r(t) by letting Then the received signal can be rewritten as (3.4) Since is a function of path loss and shadowing while depends on delay and Doppler, we typically assume that these two random processes are independent. The received signal r(t) is obtained by convolving the baseband input signal u(t) with the equivalent lowpass time-varying channel impulse response c(τ, t) of the channel and then upconverting to the carrier frequency (3.5) 2017/3/2 Lecture 1: Wireless Channel Modeling 43

Time-Varying Channel Impulse Response We know that c(,t) must be given by (3.6) c(,t) where represents the equivalent lowpass response of the channel at time t to an impulse at time t-τ. So it follows that 2017/3/2 Lecture 1: Wireless Channel Modeling 44

Time-Varying Channel Impulse Response Some channel models assume a continuum of multipath delays, in which case c(,t) becomes Figure 3.2 shows each multipath component corresponds to a single reflector. (3.7) 2017/3/2 Lecture 1: Wireless Channel Modeling 45

Time-Varying Channel Impulse Response 2017/3/2 Lecture 1: Wireless Channel Modeling 46

Time-Varying Channel Impulse Response Example 3.1: Consider a wireless LAN operating in a factory near a conveyor belt. The transmitter and receiver have a LOS path between them with gain, phase and delay. Every seconds a metal item comes down the conveyor belt, creating an additional reflected signal path in addition to the LOS path with gain, phase and delay. Find the time-varying impulse response c(τ,t) of this channel. Solution: For the channel impulse response corresponds to just the LOS path. For the channel impulse response has both the LOS and reflected paths. Thus, c(τ,t) is given by Note that for typical carrier frequencies, the nth multipath component will have. For example, with GHz and ns (a typical value for an indoor system),. 2017/3/2 Lecture 1: Wireless Channel Modeling 47

Time-Varying Channel Impulse Response If f c n (t) 1 then a small change in the path delay can lead to a very large phase change in the nth multipath component with phase Rapid phase changes in each multipath component gives rise to constructive and destructive addition of the multipath components, which causes rapid variation in the received signal strength. This phenomenon is called fading. The impact of multipath on the received signal depends on whether the spread of time delays associated with the LOS and different multipath components is large or small relative to the inverse signal bandwidth. If the channel delay spread is small then the LOS and all multipath components are typically nonresolvable, leading to the narrowband fading model. If the delay spread is large then the LOS and all multipath components are typically resolvable into some number of discrete components, leading to the wideband fading model. 2017/3/2 Lecture 1: Wireless Channel Modeling 48

Narrowband Fading Models Suppose the delay spread of a channel is small relative to the inverse signal bandwidth B of the transmitted signal, i.e.. Under most delay spread characterizations implies that the delay associated with the ith multipath component so. Thus, In order to characterize the random scale factor caused by the multipath we choose s(t) to be an unmodulated carrier with random phase offset which is narrowband for any. With this assumption the received signal becomes (3.13) 2017/3/2 Lecture 1: Wireless Channel Modeling 49

Narrowband Fading Models where the in-phase and quadrature components are given by and where 2017/3/2 Lecture 1: Wireless Channel Modeling 50

Narrowband Fading Models Envelope and Power Distributions: For any two Gaussian random variables X and Y, both with mean zero and equal variance, it can be shown that is Rayleigh-distributed and is exponentially distributed. 2017/3/2 Lecture 1: Wireless Channel Modeling 51

Envelope and Power Distributions For uniformly distributed, and are both zero-mean Gaussian random variables. If we assume a variance of for both in-phase and quadrature components then the signal envelope r I is Rayleigh-distributed with distribution (3.32) where is the average received signal power of the signal, i.e. the received power based on path loss and shadowing alone. We can obtain the received signal power is exponentially distributed with mean : (3.33) Example 3.2: Consider a channel with Rayleigh fading and average received power dbm. Find the probability that the received power is below 10 dbm. 2017/3/2 Lecture 1: Wireless Channel Modeling 52 r Q

Envelope and Power Distributions Solution: We have that. Thus. We want to find the probability If the channel has a fixed LOS component, and are not zero-mean. In this case, the received signal equals the superposition of a complex Gaussian component and a LOS component, and the signal enveloep can be shown to have a Rician distribution, given by (3.34) where is the average power in the non-los multipath components and is the power in the LOS component. The function is the modified Bessel function of 0th order. 2017/3/2 Lecture 1: Wireless Channel Modeling 53

Envelope and Power Distributions The average received power in the Rician fading is given by (3.35) The Rician distribution is often described in terms of a fading parameter K, defined by K is the ratio of the power in the LOS component to the power in the other (non-los) multipath components. For K = 0 we have Rayleigh fading, and for K = we have no fading, i.e. a channel with no multipath and only a LOS component. The fading parameter K is therefore a measure of the severity of the fading: a small K implies severe fading, a large K implies more mild fading. 2017/3/2 Lecture 1: Wireless Channel Modeling 54

Envelope and Power Distributions Nakagami fading distribution was developed based on a variety of empirical measurements. This distribution is given by (3.38) where is the average received power and Γ( ) is the Gamma function. For m = 1, the Nakagami distribution reduces to Rayleigh fading. For the, it is approximately Rician fading with K. For m =, there is no fading: is a constant. The power distribution for Nakagami fading, obtained by a change of variables, is given by (3.39) 2017/3/2 Lecture 1: Wireless Channel Modeling 55

Wideband Fading Models When the signal is not narrowband we get another form of distortion due to the multipath delay spread. In this case, a short transmitted pulse of duration T will result in a received signal that is of duration T + T m, where T m is the multipath delay spread. Thus, the duration of the received signal may be significantly increased. This is illustrated in Figure 3.11. 2017/3/2 Lecture 1: Wireless Channel Modeling 56

Wideband Fading Models If the multipath delay spread T m T then the multipath components are received roughly on top of one another, as shown on the upper right of the figure. The resulting constructive and destructive interference causes narrowband fading of the pulse, but there is little time-spreading of the pulse and therefore little interference with a subsequently transmitted pulse. If the multipath delay spread T m T, then each of the different multipath components can be resolved, as shown in the lower right of the figure. However, these multipath components interfere with subsequently transmitted pulses. This effect is called intersymbol interference (ISI). ISI mitigation is not necessary if T m T, but this can place significant constraints on data rate. 2017/3/2 Lecture 1: Wireless Channel Modeling 57

Wideband Fading Models The difference between wideband and narrowband fading models is that as the transmit signal bandwidth B increases so that, the approximation is no longer valid. Thus, the received signal is a sum of copies of the original signal, where each copy is delayed in time by and shifted in phase by. The received signal is a sum of copies of the original signal, where each copy is delayed in time by and shifted in phase by. Although the received signal in narrowband fading no longer applies when the signal bandwidth is large relative to the inverse of the multipath delay spread, if the number of multipath components is large and the phase of each component is uniformly distributed then the received signal will still be a zero-mean complex Gaussian process with a Rayleigh-distributed envelope. However, wideband fading differs from narrowband fading in terms of the resolution of the different multipath components. 2017/3/2 Lecture 1: Wireless Channel Modeling 58

Wideband Fading Models The starting point for characterizing wideband channels is the equivalent lowpass time-varying channel impulse response c(,t). We can take the Fourier transform of c(,t) with respect to t as (3.49) We call the deterministic scattering function of the lowpass equivalent channel impulse response c(,t). The statistical characterization of c(,t) is thus determined by its autocorrelation function, defined as (3.50) Most channels in practice are wide-sense stationary (WSS), such that the joint statistics of a channel measured at two different times t and t + t depends only on the time difference t. 2017/3/2 Lecture 1: Wireless Channel Modeling 59

Wideband Fading Models We will assume that our channel model is WSS, in which case the autocorrelation becomes independent of t: Uncorrelated scattering (US): If a channel has a US, then the channel response associated with a given multipath component of delay is uncorrelated with the response associated with a multipath component at a different delay. Incorporating the US property into yields (3.51) (3.52) where gives the average output power associated with the channel as a function of the multipath delay and the difference Δt in observation time. 2017/3/2 Lecture 1: Wireless Channel Modeling 60

Wideband Fading Models The scattering function for random channels is defined as the Fourier transform of A c ( ; t) with respect to the Δt parameter: (3.53) 2017/3/2 Lecture 1: Wireless Channel Modeling 61

Coherent Bandwidth The time-varying multipath channel in the frequency domain can be characterized by taking the Fourier transform of c( ; t) with respect to τ. Specifically, define the random process (3.57) Since c( ; t) is WSS, its integral C(f; t) is as well. Thus, the autocorrelation of C(f; t) is given by (3.58) (3.59) 2017/3/2 Lecture 1: Wireless Channel Modeling 62

Coherent Bandwidth where f = f 2 f 1 and the third equality follows from the WSS and US properties of c( ; t). The autocorrelation of C(f; t) in frequency depends only on the frequency difference f. (3.60) The frequency B c where A c ( f) 0 for all f>b c is called the coherence bandwidth of the channel. By the Fourier transform relationship between, if the then for. Thus, the minimum frequency separation for which the channel response is roughly independent is, where T is typically taken to be the rms delay spread. 2017/3/2 Lecture 1: Wireless Channel Modeling 63

Coherent Bandwidth In general, if we are transmitting a narrowband signal with bandwidth B B c, then fading across the entire signal bandwidth is highly correlated, i.e. the fading is roughly equal across the entire signal bandwidth. This is usually referred to as flat fading. If the signal bandwidth B B c, then the channel amplitude values at frequencies separated by more than the coherence bandwidth are roughly independent. Thus, the channel amplitude varies widely across the signal bandwidth. In this case, the channel is called frequencyselective. T s Tm Note that in linear modulation the signal bandwidth B is inversely proportional to the symbol time, so flat fading corresponds to T s 1/B 1/B c =, i.e. the case where the channel experiences negligible ISI. Frequency-selective fading corresponds to T s 1/B 1/B c = Tm, i.e. the case where the linearly modulated signal experiences significant ISI. 2017/3/2 Lecture 1: Wireless Channel Modeling 64

Coherent Bandwidth The power delay profile A c ( ) and its Fourier transform A C ( f) in Figure 3.13. 2017/3/2 Lecture 1: Wireless Channel Modeling 65

Coherent Bandwidth Example 3.6: In indoor channels ns whereas in outdoor microcells sec. Find the maximum symbol rate R s =1/T s for these environments such that a linearly-modulated signal transmitted through these environments experiences negligible ISI. Solution: We assume that negligible ISI requires T s Tm, i.e. T s 10 Tm. This translates to a symbol rate R s =1/T s apple.1/ Tm. For ns this yields Mbps and for T m 30µ sec this yields R s apple 3.33 Kbps. Note that indoor systems currently support up to 50 Mbps and outdoor systems up to 200 Kbps. To maintain these data rates for a linearly-modulated signal without severe performance degradation due to ISI, some form of ISI mitigation is needed. Moreover, ISI is less severe in indoor systems than in outdoor systems due to their lower delay spread values, which is why indoor systems tend to have higher data rates than outdoor systems. 2017/3/2 Lecture 1: Wireless Channel Modeling 66

Doppler Power Spectrum and Channel Coherence Time A Doppler shift in the received signal is caused by the time variations of the channel which arise from transmitter or receiver motion. It can be characterized by taking the Fourier transform of relative to Δt: A C ( f; t) (3.61) In order to characterize Doppler at a single frequency, we set Δf to zero and define S C ( ), S C (0; ) which is given by (3.62) where A C ( t), A C ( f = 0; t). Note that A C ( t) is an autocorrelation function defining how the channel impulse response decorrelates over time. A C ( t = T )=0 indicates that observations of the channel impulse response at times separated by T are uncorrelated and therefore independent since the channel is a Gaussian random process. 2017/3/2 Lecture 1: Wireless Channel Modeling 67

Doppler Power Spectrum and Channel Coherence Time The channel coherence time T c to be the range of values over which A C ( t) is approximately nonzero. Thus, the time-varying channel decorrelates after approximately seconds. The function is called the Doppler power spectrum of the channel: as the Fourier transform of an autocorrelation it gives the PSD of the received signal as a function of Doppler ρ. The maximum ρ value for which is greater than zero is called the Doppler spread of the channel, and is denoted by. ( ) If the transmitter and reflectors are all stationary and the receiver is moving with velocity v, then T c Recall that in the narrowband fading model samples became independent at time, so in general where k depends on the shape of. 2017/3/2 Lecture 1: Wireless Channel Modeling 68

Doppler Power Spectrum and Channel Coherence Time Figure 3.14 illustrates the Doppler power spectrum S C ( ) and its inverse Fourier transform A C ( t). Example 3.7: For a channel with Doppler spread B d = 80 Hz, what time separation is required in samples of the received signal such that the samples are approximately independent. Solution: The coherence time of the channel is T c 1/B d =1/80, so samples spaced 12.5 ms apart are approximately uncorrelated and thus these samples are approximately independent. 2017/3/2 Lecture 1: Wireless Channel Modeling 69