ARC Future Fellow at The University of Melbourne Sessional Lecturer at Monash University
August 8, 2022
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022 1 / 22
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ECE5884 Wireless Communications Week 3: Wireless Channel Models
Course outline
This week: Ref. Ch. 3 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 8, 2022 2 / 22
Doppler effect
Figure 1: Illustration of the Doppler effect.
Doppler frequency: When θ is the arrival angle of the received signal relative to the direction of motion, v is the receiver velocity toward the transmitter in the direction of motion, and λ is the signal wavelength
fD=vλcosθ whereλ=fcandc=3×108m/s (1) c
So, fD is positive when the Rx is moving toward the Tx (i.e., −π/2 ≤ θ ≤ π/2). Max. Doppler spread BD = 2v/λ (2) Channel coherence time Tc ≈ 1/BD (3)
Coherence time (Tc ) is the time duration over which the channel impulse
response is considered to be not varying.
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Figure 2: Concentric ellipses model for fading channels (Tx and Rx are located at the foci of the ellipses. Considering only single bounce reflections, all paths that are associated with scatterers on the nth elliptical contour have the same delay).
1 Doppler effect. 2 Scatters.
⎡⎢ ⎛ N ( t ) − 1 ⎞ ⎤⎥ Receivedsignal r(t)=R⎢⎝ ∑ αi(t)e−jφi(t)u(t−τi(t))⎠ej2πfct⎥ (4)
αi (t ) is fading (also a function of path loss and shadowing). φi (t ) depends on delay and Doppler. These two random processes are independent.
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Fast and slow fading
1 Fast fading:
● No.of paths N ↑;
(a) A delayed signal. (b) Resultant signal. Figure 3: How signal looks like.
● Carrier frequency fc ↑⇒ λc ↓ Larger phase spread;
● Velocity v ↑⇒ Doppler spread ↑;
● Tc < Ts where Ts is the transmitted symbol duration (severe frequency
dispersion into the received signal).
2 Slow fading: Tc ≫ Ts (little frequency dispersion into the Rx signal).
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022 6 / 22
Flat and frequency-selective fading
(a) Wireless channel response. (b) Wireless channel as a filter.
1 Flat fading: the magnitude of the time-variant channel transfer function is constant (or flat) with respect to frequency; Bs ≪ Bc where Bc is the coherence bandwidth where the channel can be considered “flat”.
2 Frequency-selective fading: the magnitude of the time-variant channel transfer function is no longer flat with respect to frequency Bs ≫Bc.
● the differential path delays ∣τi − τj ∣ (Figure 2) for some i , j are sufficiently large compared to the modulation symbol period Ts .
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Intersymbol interference (ISI)
ISI is a form of distortion of a signal in which one symbol interferes with subsequent symbols.
(c) One symbol Tx (No ISI). (d) Four symbols Tx (ISI).
(e) Shorter Ts (ISI). (f) Longer Ts (No ISI).
https://www.telecomhall.net/t/what- is- isi- inter- symbol- interference- in- lte/6370
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022 8 / 22
Narrowband and wideband communications
1 Narrowband communications use a narrow bandwidth; are used in a slower form of communication (voice, RFID, GSM 900 satellite downlinks, GPS signals); have a far greater range of reception as narrower filters can be used and therefore cancel out unwanted wideband noise; the transmitted energy also concentrates on a smaller portion of the spectrum.
● The delay spread Tm of a channel is small relative to the inverse baseband signal bandwidth Bs of the transmitted signal, i.e.,Tm ≪ Ts where Bs ≈ 1/Ts for Ts the signal duration.
2 Wideband communications use a higher bandwidth; the energy of the signal is distributed across the width of the spectrum which makes the signal weaker the wider it gets; is almost exclusively done in higher frequencies (>500MHz+); common modulation technique is OFDM (Week 8); apply Wifi, 4G LTE, HSPA.
● Tm ≫ Ts .
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Fast/slow Flat/selective Narrowband/wideband
1 Slow fading: Ts > Tc , Shadowing, Log-normal
2 Fast fading: Ts ≪ Tc , Multipath fading, Next
3 Flat fading: Bs ≪ BD
4 Selective fading: Bs ≫ BD , OFDM
5 Narrowband comm.: Tm ≪ Ts, Multipath fading, Next
6 Wideband comm.: Tm ≫ Ts , OFDM
(g) Combined all.
Figure 4: Ref. Ch. 3 of [Goldsmith, 2005].
(h) Narrowband fading.
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System model
● The received signal
r(t)=hs(t)+n(t) ⎛or =
● Pt−thetransmitpower;
● h− the multipath channel gain (usually a complex number); ● s(t)− the transmit signal;
● n(t)− the additive noise.
● The received signal power, where Ps is the signal power, Pr = ∣h∣2Ps
● Multipath channel gain ● Fading channel envelop
h=hr +jhi =zejθ √
∣h∣=z= hr2+hi2
Pt hs(t)+n(t)⎞ (5)
We need Envelope (∣h∣) and Power Distributions (∣h∣2)!!! ECE5884 Wireless Communications @ Monash Uni.
August 8, 2022
Multipath fading: Rayleigh distribution
When hr and hi are two independent and identical distributed (i.i.d.) Gaussian random variables with mean zero and variance σ2, i.e.,
hr,hi ∼N(0,σ2),
1 the envelop ∣h∣ = z =
hr2 + hi2 is Rayleigh distributed; (Derive in z z2
fZ(z)= σ2e−2σ2 z2
FZ (z) = 1 − e− 2σ2 The average envelope power is Ωp = 2σ2.
2 the power ∣h∣2 is Exponentially distributed; (Derive in Quiz 3!) 1−t
fZ2(t)=2σ2e 2σ2 t
FZ 2 (t ) = 1 − e− 2σ2
Verify these expressions by using MATLAB simulations in Quiz 3! Use
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022
Multipath fading: Rician distribution
● The channel has a LOS component with a much larger signal power than the other multipath components.
● Then, hr and hi are two independent Gaussian RVs with non-zero mean and equal variance σ2, i.e., hr ∼ N(mr,σ2) and hi ∼ N(mi,σ2);
1 the envelop ∣h∣ = z =
hr2 + hi2 is Rician/Ricean/Rice distributed; z −(z2+s2) zs
fZ(z)=σ2e 2σ2 I0(σ2) (13) s2 = m2 + m2 and I0(x) is the modified Bessel function of zeroth order.
Verify this expression by using MATLAB simulations in Quiz 3! Use σ = 0.5 and s2 = 0.9
2σ2 is the average power in the non-LOS multipath components and s2 is the power in the LOS component.
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Multipath fading: Rician distribution
1 the envelop ∣h∣ = z =
hr2 + hi2 is Rician/Ricean/Rice distributed; z −(z2+s2) zs
fZ(z)=σ2e 2σ2 I0(σ2) (14)
2 Average envelope power: Ωp = s2 + 2σ2
3 Th●e Rice factor K (fading parameter): K = s2 2σ2
K = 0: no LoS and the envelope exhibits Rayleigh fading.
K → ∞: no scatter and the channel does not exhibit any fading.
K is a measure of the severity of the fading: a small K implies severe fading, a large K implies relatively mild fading.
2(K +1)z −K−(K+1)z2 ⎛ K(K +1)⎞ e Ωp I0 2z
fZ(z)= wheres2=KΩp andσ2= Ωp
K +1 2(K +1)
ECE5884 Wireless Communications @ Monash Uni. August 8, 2022
Multipath fading: Nakagami distribution
1 The Nakagami distribution was selected to fit empirical data and is known to provide a closer match to some measurement data than either the Rayleigh, Ricean, or log-normal distributions.
2 the envelop ∣h∣ is Nakagami-m distributed; m2m−1 2
3 Average envelope power: Ωp ● m = 1: Rayleigh distribution.
mz−mz 1 fZ(z)=2(Ω ) Γ(m)e Ωp ;m≥2
● m = 1/2: a one-sided Gaussian distribution ● m → ∞: approaches an impulse (no fading).
● m ≈ (K +1)2 : approximation for Rician distribution. (2K +1)
4 the power ∣h∣2 is Gamma distributed;
mz−mz 1 fZ2(z)=(Ω) Γ(m)eΩp;m≥2
Derive CDF FZ 2 (z ) expression in Quiz 3!
ECE5884 Wireless Communications @ Monash Uni.
August 8, 2022
Channel phase
1 The phase of the received complex envelope h = hr + jhi is
φ=tan−1(hi ) (18)
2 For Rayleigh fading; the phase φ is uniformly distributed over the interval [−π, π),
fφ(x)= 1;−π≤x≤π (19) 2π
3 For Ricean fading channels, the phase φ is not uniformly distributed and takes on a more complicated integral form.
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Additive white Gaussian noise (AWGN)
● The received signal
r(t)=hs(t)+n(t) wheren(t)isthenoise (20)
(a) Additive noise. (b) Power spectral density. Figure 5: AWGN.
● “ White” light contains components at all wavelengths (all frequencies) across the visible spectrum. The Power Spectral Density (PSD) of white noise is constant for all frequencies.
● The probability distribution of the noise samples is Gaussian with a zero mean.
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Additive white Gaussian noise (AWGN)
● The noise power is N0 Watt/Hz
● N0 = kT , k is Boltzmann’s Constant and T is the temperature in Kelvin.
● For a complex baseband signal, the thermal noise signal is a complex, white Gaussian distributed noise, with half of the power (N0) in the real component and half the power (N0) in the imaginary component (but power in itself is not a complex quantity).
(a) Single-sided PSD. (b) Double-sided PSD. Figure 6: Representations of PSD.
https://dsp.stackexchange.com/questions/54251/psd- of- complex- white- gaussian- noise
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Signal-to-noise ratio (SNR)
● The received signal
● SNR is the ratio between the power of the received signal (desired
r (t ) = h s(t ) + n(t )
signal) to the power of background noise (undesired signal).
SNR: γ = Signal power Noise power
1 For the AWGN channel (h = 1)
2 For a fading channel
γ = ∣h∣2Ps ; called the instantaneous received SNR
● If the channel bandwidth is B, the total noise power is BN0. ECE5884 Wireless Communications @ Monash Uni. August 8, 2022
SNR outage probability
● The SNR outage probability is the probability that the SNR γ falls below a certain predetermined threshold SNR γth
Pout =Pr[γ<γth] (25) = Pr[∣h∣2Ps < γth] = Pr[∣h∣2 < N0γth ] = F∣h∣2 (N0γth ) (26)
● For Rayleigh fading (use (12))
N0 γth () Ps
Pout =1−e− 2σ2
Similarly, you can evaluate the SNR outage probabilities for Rician and
Nakagami-m fading channels!
=1−e−(2σ2 Ps ) (27)
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References
A. Goldsmith, Wireless Communications, Cambridge University Press, USA, 2005.
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Thank You! See you again
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