1 Wireless Channel Models 8-1
2 Why do we need models? Help analyze the performance of large systems Ask fundamental questions about a system E.g. what is the best I can do with a given system? Allows comparison of two or more systems Caveats: Less accurate compared to experimentation and simulation Every model makes assumptions Results obtained using a model must be validated against reality
3 Wireless Channel Models: Outline We want to understand how wireless channel parameters such as Carrier frequency Bandwidth Delay spread Doppler spread Mobile speed affect how a channel behaves from a communication system point of view Strategy: start with physical models and end with statistical models Slides used in this class is based on material from Chapter 2 of Tse & Viswanath and from presentations by David Tse
4 Wireless Channel Wireless channel varies over two spatial scales: olarge scale variations osmall scale variations
5 Free Space, fixed Tx Antenna Consider a fixed Tx antenna in free space Electric far field at time t: E r f, t,( r,, ) s (,, f )cos 2f ( t r r / c) (r,,) is the point u in space where field is measured f: carrier frequency, c: speed of light r: distance between point u and antenna s (,,f) radiation pattern of sending antenna
6 Free space, fixed Tx antenna E r f, t,( r,, ) s (,, f )cos 2 f r ( t r / c) r P t Electric field decreases as 1/r Power density decreases as 1/r 2 Note surface area increases as r 2 Total power radiated through the sphere is constant
7 Free Space, fixed Tx and Rx Antennas Rx antennas at point u Electric far field at time t: (,, f )cos 2 f ( t r / c) E r f, t,( r,, ) r (,,f) product of antenna patterns of sending and receiving antennas Varies depending on omni vs directional antennas Accounts for antenna losses Receive field linear in input
8 Free Space, Moving Antenna Consider now a receiver Rx is moving away from Tx with velocity v Initially at r 0 ; r(t) = r 0 + vt Electric field at Rx at time t: E r f, t,( r 0 vt,, ) (,, (,, f )cos 2 f ( t r vt r / c vt / c) f )cos 2 f ((1 v / c) t r vt r / c) Received sinusoid has frequency f(1 v/c)! o Doppler shift = -fv/c
9 Phase difference between two waves Odd multiple of destructive interference Even multiple of constructive interference Reflecting wall, fixed antenna Assume static Rx; perfectly reflecting wall Ray tracing model Assumption: received signal superposition of two waves r d c r d t f r c r t f t f E r 2 ) ) / (2 ( cos 2 ) / ( cos 2, ) ( 4 2 ) (2 2 r d c f c fr c r d f
10 Illustration The blue waveform is the superposition of the two black waveforms
11 Coherence distance Coherence distance is distance from peak to valley x c = /4 Distance over which signal does not change appreciably
12 Delay spread and coherence bandwidth Delay spread (T d ) difference in propapgation delays along two signal paths o T d = ((2d-r)/c r/c) Let s go back to our reflecting wall example We calculated the phase difference to be: 2 f (2d r) 2 fr 4 f ( d r) c c c Changing f to f ± 1/2T d changes by move from constructive to destructive interference or vice-versa We call W c = 1/2T d coherence bandwidth Bandwidth over which signal does not change appreciably
13 Reflecting Wall, moving antenna E r f, t cos 2 f ((1 v / c) t r vt 0 r / c) cos 2 f ((1 v / c) t ( r0 2d r vt 0 0 2d) / c) Doppler spread D s = fv/c (-fv/c) = 2fv/c Example: v=60 km/h, f=900mhz => Doppler spread = 100 Hz Coherence Time = 1/2D s ; time to travel from a peak to a valley In our example: 5 ms
14 Doppler Shift Effect Assume mobile very close to the wall d ~ r 0 + vt vt r d c d r t c v f vt r c r t c v f t f E r ) ) / 2 ( ) / ((1 cos 2 ) / ) / ((1 cos 2, vt r c d t f c d r c vt f t f E r 0 0 ) / ( )sin 2 ) / ( / ( sin 2 2,
15 Reflecting Wall, moving antenna Multipath fading: Moving between constructive and destructive interference of two waves Once every 5 ms Significant changes in path length (denominator) occur on time scales of several seconds or minutes
16 Summary We applied physical model in a toy example to understand multipath fading in terms of: Delay spread & Coherence bandwidth Doppler spread & Coherence time But, we want to model a wireless channel in a world with many obstacles, reflectors etc. We are more interested in aggregate behavior of channel Not so much detailed response on each path
17 Continuous-time input/output model of wireless channels Wireless channels can be modeled as linear timevarying systems: where a i (t) and i (t) are the gain and delay of i-th path and w(t) is the thermal noise Note the linear relationship between input-output The time-varying impulse response is: where (x) is the Delta function and is 1 when x = 0 and 0 elsewhere
18 Continuous-time input/output models of wireless channels In terms of time-varying impulse response, y(t) can be expressed as: In practice, we convert a continuous time signal of bandwidth W at passband to o a baseband signal of bandwidth W/2; and o Take samples at integer multiples 1/W (sampling theorem)
19 Discrete Time input/output model Discrete-time (sampled) channel model: where h l is the l-th complex channel tap. and the sum is over all paths that fall in the delay bin System resolves the multipaths up to delays of 1/W.
20 Flat and Frequency-Selective Fading Fading occurs when there is destructive interference of the multipaths that contribute to a tap.
21 The channel as seen by the communication system depends on: ophysical environment (in terms of delay spread) ocommunication bandwidth
22 Time Variations f c i (t) = Doppler shift of the i-th path
24 Types of Channels
25 Statistical Models Consider the l-th complex channel tap To completely specify h, we need to know each path that contributes to the tap In practice, this is difficult We are only interested in the sum of all the paths at each tap When there are many reflectors in environment Many many paths between transmitter and receiver Many many paths contributing to each channel tap
26 Central Limit Theorem Sum of large number of iid random variables converges (in distribution) to the Gaussian Complex channel tap modeled as a complex Gaussian random variable This is the Rayleigh fading model
27 Rayleigh Fading h - magnitude (amplitude) of complex filter tap Rayleigh random variable with density 2 x 2 exp( x / 2 ), x 0 2 Squared magnitude (power density) h 2 exponentially distributed with density 1 2 exp( x / ), x 2 This model appropriate when there are many small reflectors 0
28 Rician Fading Appropriate when there is a large line of sight path as well as independent paths Often more accurate than Rayleigh model Pdf does not have a simple form and not easy to work with Has found limited applicability in research community
29 Large scale fading: path loss P r d P d t n n estimated by measurement. T.S. Rappaport, Wireless Communications Depends on density of obstacles, their locations, absorption behavior etc. Not much to do about it.
30 Shadowing Example: d Same distance, different power. Due to shadowing. Modeled as a random power distribution (log-normal) Capture randomness in environment d 5 n 2 Pt 2 d db P r 5dB
31 Fading and Diversity Improve performance of wireless systems by transmitting symbols over multiple paths that fade independently Reliable communication possible as long as at least one strong path exists This technique is called Diversity
32 Diversity: Motivating Example Probability of unusable link: p = 0.1 = 10% of the time. Probability of unusable link: = (p)(p)(p)(p)(p) = = 0.001% of the time
33 Wireless Channel Cellular Link Buildings Countryside Wireless LAN Customers/Baristas Traffic The channel is dynamic and it is a function of time. The reliability of the link depends upon the gain of the channel, which is a random variable always less than one.
34 Three Types of Diversity Time Diversity Space Diversity 1. Antenna Arrays 2. MIMO Systems Frequency Diversity 1. WiFi 2. Cell Phones All forms of diversity have their pros and cons.
35 Time Diversity All the obstacles between the cell phone to base station is modeled by a channel gain, h(t). This includes the mobility of the phone/user.
36 Time Diversity Receiver and Transmitter protocol states that each bit will be transmitted L times. (called repetition coding) Important that the coherence time of the channel elapses before retransmitting.
37 Time Diversity v By pausing for the coherence time, this provides the cell phone with five separate links to the base station. The drawback, of course, is a reduction of the rate, Rnew = Rold/L. (TCP uses time diversity)
38 Space Diversity Channel gain not only function of time, but also of position, h(t, r) Transmitter must travel far enough to see different channel Far enough? Same as coherence time for time diversity
39 Space Diversity Far Enough = / h 2 (t) h 1 (t) Most cell towers have multiple transmit cells serving one cell Down link channel-miso system. Antennas must be spaced at least /2 apart to see independent channels.
40 Space Diversity Probability all four channels in deep fade much smaller than probability any one channel in deep fade h 2 (t) h 3 (t) h 1 (t) h 4 (t) Multiple antennas at phone, base station - MIMO system Cost is receiver/transmitter complexity Benefit - BER approximately 1/SNR L
41 Frequency Diversity Channel gain is a function of frequency as well, h(t,r,f) Different frequencies see different channels, this provides independent links. For example: AM: FM: UHF TV: Cell Phones: WiFi: GPS: khz MHz MHz GHz Cloud Radar: > 30 GHz
42 Frequency Diversity If signal does not reach the base station, try: Transmitting different frequency, at least coherence bandwidth away
43 Conclusion Physical models of wireless channels Good for preliminary understanding of fading Hard to study real-world systems Input/Output models Wireless channel as LTV system Statistical models Rayleigh and Rician models Diversity Time Transmit at different times. Space Transmit at different points in space. Frequency Transmit at different frequencies, or over a bandwidth of frequencies.