ARC Future Fellow at The University of Melbourne Sessional Lecturer at Monash University
August 1, 2022
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022 1 / 23
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ECE5884 Wireless Communications Week 2: Wireless Channel (Path Loss and Shadowing)
Course outline
This week: Ref. Ch. 2 of [Goldsmith, 2005]
● Week 1: Overview of Wireless Communications
● Week 2: Wireless Channel (Path Loss and Shadowing)
● Week 3: Wireless Channel Models
● Week 4: Capacity of Wireless Channels
● Week 5: Digital Modulation and Detection
● Week 6: Performance Analysis
● Week 7: Equalization
● Week 8: Multicarrier Modulation (OFDM)
● Week 9: Diversity Techniques
● Week 10: Multiple-Antenna Systems (MIMO Communications) ● Week 11: Multiuser Systems
● Week 12: Guest Lecture (Emerging 5G/6G Technologies)
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022 2 / 23
Normal/Gaussian distribution (recap)
Normal/Gaussian distribution: X is continuous probability distribution for a real-valued random variable (RV) with the mean or expectation μ, the standard deviation σ and the variance σ2.
Notation ∶ PDF ∶
CDF ∶ Error fun ∶
CEF ∶ Q−fun∶
X ∼ N(μ,σ) or X ∼ N(μ,σ2)
(x−σ)2 fX(x)=√2πσe− 2σ2 ;−∞
1 log(x)−μ
FX(x)= 2[1+erf( √2σ )] ECE5884 Wireless Communications @ Monash Uni.
August 1, 2022
Transmit signal model (recap)
A bandpass signal s(t) at carrier frequency fc:
s(t ) = sI (t ) cos(2πfc t ) − sQ (t ) sin(2πfc t ) (12)
where sI(t) (in-phase component of s(t)) and sQ(t) (quadrature component of s(t)) are real lowpass (baseband) signals of bandwidth B << fc .
A complex lowpass signal:
u(t) = sI(t) + jsQ(t), so sI(t) = R{u(t)} and sQ(t) = I{u(t)}. (13) Then,
s(t) = R{u(t)}cos(2πfct) − I{u(t)}sin(2πfct) (14) = R{u(t)ej2πfc t } (15)
This is the complex lowpass representation of the bandpass signal s(t), and the baseband signal u(t) is called the equivalent lowpass signal for s(t) or its complex envelope.
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022 5 / 23
Receive signal model (recap)
The received signal is the convolution of s(t) with the channel impulse response h(t) plus an additional noise component n(t) introduced by the channel:
r(t) = s(t) ∗ h(t) + n(t) (16) =R{v(t)ej2πfct}+n(t); wherev(t)=u(t)∗c(t) (17)
and c(t) is the equivalent lowpass channel impulse response for h(t). Note: we will neglect the random noise component n(t) in our analysis in
Week 2 and Week 3.
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022 6 / 23
Radio wave propagation
We characterize the primary phenomena that affect signal propagation: path loss, shadowing, signal reflection, diffraction and scattering.
Figure 1: Multipath propagation due to reflection, diffraction and scattering. https://openclipart.org/detail/194650/multipath- propagation
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● Path loss characterizes how a signal’s received power decreases with transmit-receive distance; occurs over long distances (100-1000 m).
The linear path loss is the ratio of transmit power to receive power:
PL = Pt (18)
The linear path loss in decibels
PL(dB) = 10 log10 ( Pt ) (dB) (19)
= 10 log10 (Pt ) (dBm) − 10 log10 (Pr ) (dBm) (20)
The dB path gain is defined as the negative of the dB path loss:
PG = −PL(dB) = 10 log10 (Pr )(dB) (21)
● 15 dBm (32 mW) Typical wireless LAN transmission power in laptops. ● 27 dBm (500 mW) Typical cellular phone transmission power
● 33 dBm (2 W) Maximal output from a UMTS/3G mobile phone
● 46 dBm (40 W) Maximum allowed output of a single port LTE BS
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022 8 / 23
Line-of-sight (LOS) propagation
Free-space signal propagation: There are no obstructions between the transmitter and receiver. Then, the signal propagates along a straight line between Tx and Rx.
● Line-of-sight (LOS) channel
Figure 2: Line-of-sight (LOS) propagation.
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022 9 / 23
Free-space path loss
● The received signal [Parsons, 2000]
r(t) = R[(λ√GtGru(t −τl)e−j2πd/λ )ej2πfct] (22)
● Gt−thetransmitantennapowergain;
● Gr−thereceiveantennapowergain;
● τl− the signal propagation delay;
● e−j2πd/λ− the phase shift due to the distance d.
● Thereceivedsignalpower(usereiθ =r(cosθ+jsinθ)and∣reiθ∣=r)
Pr = ( Pt Gt ) ( λ2 Gr ) The Friis formula [Friis, 1946] (23)
4πd2 4π Free-space path loss
PL = −10log10 [GtGr ( λ )2] in dB 4πd
ECE5884 Wireless Communications @ Monash Uni.
August 1, 2022
Two-Ray multipath model
Figure 3: A simple point-to-point wireless communications system.
● The received signal consists of two components:
1 The LOS component
2 A reflected component which is the signal reflected off the ground.
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022 11 / 23
Two-Ray multipath (mathematical) model
Figure 4: Two ray model. Thereceivedsignal(√Gt0Gr0 =√G0and√Gt1Gr1 =√G1):
λ √G0u(t − τ0)e−j2πd0/λ √G1u(t − τ1)e−j2πd1/λ j2πf t
r(t)= R[( + )e c ] (26)
4πd0 d1 Thereceivedpower(d≫0,u(t−τ0)≈u(t−τ1),√G0 =√G1 =√G,etc.):
Pr≈Pt( t r)∝d−4 (27)
The received power is inversely proportion to d4.
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022
Simplified path-loss model
A general received signal due to multiple rays:
⎡⎢⎛ Nr Nd Ns ⎞ ⎤⎥ r(t)=R⎢⎝LoS+∑reflectedi +∑diffractedj +∑scatteredk⎠ej2πfct⎥ (28)
⎣i=1 j=1 k=1 ⎦
● The complexity of signal propagation makes it difficult to obtain a single model that characterizes path loss accurately across a range of different environments and frequencies.
● Best to use a simple model that captures the essence of signal propagation, e.g., single-slope model:
Pr = Pt K ( dr ) (29)
● dr is a reference distance for the antenna far field, α is the path-loss exponent, and K is a unit-less constant.
K , dr , and α can be obtained to approximate either an analytical or empirical model (Lab 2: Path loss measurement using NI-USRP).
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● Shadowing is the attenuation caused by obstacles between the transmitter and receiver that absorb the transmitted signal. shadowing occurs over distances that are proportional to the length of the obstructing object (10-100 m).
Figure 5: Path loss, shadowing and fading vs distance.
● Random blockages due to location, size, and dielectric properties of
the blocking objects cause the random attenuation, and are unknown.
● Statistical models must be used to characterize this attenuation.
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Log-Normal shadowing
● The ratio of transmit-to-receive power ψ = Pt is a RV with a log-normal
distribution:
2 (10 log10 x −μψdB )
PDF: fψ(x)=√2πσψdBxe ψdB ;x>0 (30)
1 μ−ξln(x)
CDF: Fψ(x)=2erfc( √2σ ) (31)
● ξ=10/ln(10),
● μψdB is the mean of ψdB = 10log10(ψ) in dB, and ● σψdB is the standard deviation of ψdB in dB.
ψdB is then a Gaussian distribution with mean μψdB and standard deviation σψdB .
(x −μψdB ) 1 −2σ2
ψ ∼N(μ ,σ ); f (x)=√ e ψdB (32)
dB ψdB ψdB ψdB
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022
Combined path loss and shadowing
Pr (dB)=Path-loss(dB)−Shadowing(dB) (33) Pt
=10log10(K)+10αlog10(dr )−ψdB (34) d
For the simplified path-loss model,
Pr (dBm) = Pt (dBm) + 10 log10(K ) (dB) − 10α log10 ( d ) (dB) − ψdB (35) dr
RV Fixed for a given d, ≜ C(d)
Pr (d) = C(d) − ψdB (36)
ECE5884 Wireless Communications @ Monash Uni. August 1, 2022
Outage probability
Wireless systems typically require a target minimum received power level Pmin (or equivalently a minimum signal-to-noise ratio (SNR)). Performance of a wireless network becomes unacceptable below this threshold Pmin.
Outage probability Pout (Pmin , d ): The probability at which the received power value falls below a power threshold Pmin at distance d.
Outage probability for the the simplified path-loss model and log-normal shadowing with ψdB ∼ N (0, σψdB )):
Pout (Pmin , d ) = Pr [Pr (d ) < Pmin ]
= Pr [C(d) − ψdB < Pmin] = Pr [ψdB > C(d) − Pmin]
= Q (C(d) − Pmin ) σψdB
(37) (38) (39)
ECE5884 Wireless Communications @ Monash Uni.
August 1, 2022
Coverage area
Figure 6: Telstra coverage maps (access on 29/07/2022).
https://www.telstra.com.au/coverage- networks/our- coverage
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Coverage area
Figure 7: The evolution of wireless cellular coverage boundaries.
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Cell coverage percentage
The expected percentage of locations within a cell where received power exceeds Pmin (the fraction of useful service area).
Coverage area πR2
2π dθ R Pˆout(r)r dr
=∫0 ∫0 (41)
Pˆout(Pmin,r) = Pr(Pr(r) ≥ Pmin) = 1−Pout(Pmin,r) (42)
⎛Pmin −Pt −10log (K)−10αlog (dr )⎞
10 10 r (43)
⎛Pmin −(Pt +10log (K)+10αlog (dr )+10αlog (R))⎞
10 10 R 10 r ⎝σψdB ⎠
⎛P −(P +10log (K)−10αlog (R)) ⎞
min t 10 10 10αlog (r)
dr + 10R⎟ σψdB ⎠
ECE5884 Wireless Communications @ Monash Uni.
August 1, 2022 20 / 23
Cell coverage percentage
a= min t 10 10 dr andb=10αlog10(e)
P −(P +10log (K)−10αlog (R)) σψdB
R Q(a+bln(r ))rdr ∫0 R
2−2ab =Q(a)+e b2
Q(2−ab) (47) b
Pˆout(Pmin,r)=Q(a+bln(Rr ))
ECE5884 Wireless Communications @ Monash Uni.
August 1, 2022
References
A. Goldsmith, Wireless Communications, Cambridge University Press, USA, 2005. D. Parsons, The Mobile Radio Propagation Channel, 2nd Ed., Wiley, , 2000.
H. T. Friis, A note on a simple transmission formula, Proc. IRE, vol. 34, no. 5, pp. 254-256, May 1946.
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Thank You! See you again
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