EE Large Scale Path Loss Log Normal Shadowing. The Flat Fading Channel

Transcription

1 EE447- Large Scale Path Loss Log Normal Shadowing The Flat Fading Channel The channel functions are random processes and hard to characterize We therefore use the channel correlation functions Now assume: The channel impulse response is a random variable We describe the channel at any time t using a pdf Consider a flat fading channel where the delay spread is small compared with the symbol duration.

2 The Flat Fading Channel FLAT FADING CHANNEL The delay spread does not effect the received signal The channel delay function is reduced to the mean delay τ δt-τ The channel ehibits a time-varying gain gt gt has a short term fading component Zt due to multipath. It is modeled statistically by a Rayleigh, Rician, or Nakagami disitribution and is independent of the distance between the transmitter and the receiver. gt also has a long-term path loss component that is the mean of gt 3 Large scale path loss 4

3 The Flat Fading Channel The dashed line is the mean m path loss. The variation about the mean is described by the Normal Distribution 5 Plane Earth Loss*-again Antennas and Propagation for Wireless Communication Systems by Saunders 6 3

4 Log-Distance PL with Shadowing mean path loss Lp d d d 0 k, for d d d Lp Lp d0 + 0kLog0 db, for d d d0 d 0 km for macrocells,m indoors 0 0 Typical Path Loss Eponents for Different Environments Environment Path loss Eponent, κ free space urban cellular radio.7 to 3.5 shadowed urban cellular radio 3 to 5 in building with LOS.6 to.8 obstructed in building 4 to 6 7 Log-Distance PL with Shadowing Statistical component to loss The total path loss L p d with shadowing is then: Statistical path loss component 8 4

5 Log-Distance PL with Shadowing The total path loss L p d with shadowing is then: L d L + ε ε L p db p is the statistical variation of the path loss p p L d 0 db db + 0kLog 0 d d 0 is the mean pathloss L d L p p ma for the TR link to perform correctly note that for L p, d 0 and k may be adjusted to model Friss loss, plane earth loss, or any of the other models for mean path loss at a given frequency. 9 Log-Distance PL with Shadowing Shadowing: When the line of site path is blocked by an obstruction such as a building or a hill that is much larger than the λ of the signal Long term fading is then a combination of the log-distance path loss and the log-normal shadowing -that is statistical. Let εdb be a zero-mean Gaussian distributed random variable d db with a standard deviation σ ε in db. The pdf of εdb is given as: 0/ ln 0 σε fε e πσε A variable transform of will give ε in a linear scale and is is sad to follow a log-normal distribution with pdf: f y ε 0log σ 0/ ln 0 ε e π yσ ε 0 y 0 5

6 Log Normal Shadowing* *Rappaport Log-Distance PL with Shadowing The statistical term is a log-normal distribution This distribution is fully defined by the mean, m0, and the standard deviation, σ ε in db What we want to determine is the Pε db >ε dbma f y ε 0log σ 0/ ln 0 ε e π yσ ε 0 y σ ε is approimately 8 to0 db outdoors, and 4 to 6 db in a typical room. refer to the class discussion for the calculation of σ ε from field data. 6

7 EE447 Propagation Small Scale Multipath Fading 3 Review: The Normal, Gaussian, Distribution PDF : f CDF : Φ Φ e πσ P X t m σ normalized, m 0, σ : X X X f X e π t dt dt 4 7

8 8 5 Review: The Normal Distribution < Φ > e Q or e Q Q Q Q Q Q X P X π Bounds : Upper The Q-function as the area under the tail of the Normal pdf 6 Review: The Normal or Gaussian Distribution > > σ π π m Q X P erf erfc Q Q d e erfc d e Q X P Q calulator function check your is the error where erf function: complementary error

9 The Log Normal Distribution in path loss Given a Normal Distribution with meanm and standard deviation σ : m P X > Q σ m is the mean path loss as shown in previous slides σ ε is approimately 8 to0 db outdoors, and 4 to 6 db in a typical room. refer to the class discussion for the calculation of σ ε from field data. 7 Review - The Q function 8 9

10 Log-Normal PL with Shadowing-Eample Eample: It has been determined that a link will operate as long as the path lossdoes not eceed the mean path loss by more than 5 db. The standard deviation of the path loss variation has been determined to σ ε 5dB. f ε Ε What is the probability that the path loss will eceed Lmean+5 db? P L p > Lpma What is the probability that the path loss will eceed Lmean+0 db? Lp Lp ma L p + ε 9 Log-Distance PL with Shadowing-Eample Eample: It has been determined that a link will operate as long as the path loss does not eceed the mean path loss by more than 5 db. The standard deviation of the path loss variation has been determined to σ ε 5dB. What is the probability that the path loss will eceed Lp+5 db? What is the probability that the path loss will eceed Lp+0 db? P ε > 5dB ε ε 5 P ε > 5 Q Q 5 σ ε σ ε P ε > 5 Q from the Q table given σ 5dB is Q % ε P ε > 0dB given σ 5dB is ε ε 0 P ε > 0 Q Q 5 σε σ ε P ε > 0 Q from the Q table Q % This is why links are often designed so that the mean received power is about 0 db above the minimum power required for proper operation 0 ε 0

11 Multipath Propagation 3 Multipath Propagation 4

12 Multipath Propagation Amplitude and Phase of the Received Signal Vary 5 Multipath Propagation 6

13 Time-Variant Transfer Function Impulse Response Phase may change more rapidly than Amplitude 7 Time-Variant Transfer Function Impulse Response 8 3

14 Time-Variant Transfer Function Impulse Response 9 Small-Scale Multipath Fading 30 4

15 Small-Scale Multipath Fading Given a channel with N scatterers, each with gain α n t and delay τ n t Consider a digital transmission with carrier f c and a symbol interval >> Δτ the delay spread 3 Small-Scale Multipath Fading Let Zt Z c t jz s t 3 5

16 .5 Rayleigh Fading - NLOS 33 Rayleigh Fading - NLOS 34 6

17 Rician LOS Propagation 35 Rician LOS Propagation 36 7

18 Small-Scale Multipath Fading Rayleigh 37 EE447 Propagation LCR and AFD 38 8

19 Other Statistics The pdfs of the amplitude distortion f α and the phase distortion f θ eplain how the signal will behave at each instant in time. They do not tell us how they change with time. We need to know how fast the channel fading changes with time. LCR- The Level Crossing Rate AFD- The Average Fade Duration LCR and AFD describe the frequency of fading 39 LCR- Level Crossing Rate R is the chosen threshold The observation time is [0,T] The number of positive crossings is M T 5 N R M T /T # per second 40 9

20 LCR 4 LCR 4 0

21 LCR 43 LCR 44

22 LCR ma The maimum LCR is at ρ -3 db because the pdf of α is maimized at threshold 45 AFD- Average Fade Duration 46

23 AFD- Average Fade Duration 47 AFD- Average Fade Duration 48 3

24 AFD- Average Fade Duration χ R 49 AFD- Average Fade Duration 50 4