ELG4179: Wireless Communication Fundamentals © S.
Loyka
Frequency-Selective and Time-Varying Channels
Amplitude fluctuations are not the only effect.
Wireless channel can be frequency selective (i.e. not flat) and time-
varying.
Frequency –flat/frequency-selective channels
• Frequency response of the channel:
FT
h ( τ ) ←→ H(f) (5.1)
Channel as a linear filter
H(f ) H(f )
x( f ) x( f )
∆f fc f fc ∆f f
H ( f ) - channel frequency response
X ( f ) - signal’s spectrum
f c - channel coherence bandwidth
∆f - signal bandwidth
a) ∆f < f c - frequency flat; b) ∆f > f c --frequency selective
Distortionless transmission:
H ( f ) = a ⋅ e − j 2πf τ (5.2)
Lecture 5 11-Oct-17 1(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Impulse Response of a Wireless Channel
The cause of frequency selective channel: delay spread. Consider
impulse response of the channel.
Given the input signal s (t ) , the signal x ( t ) at the channel output is
N
x ( t ) = ∑ Ai s ( t − τi ) (5.3)
i =1
Ai - complex amplitude, Ai = ai e jϕi
τi - delay of i-th multipath, there are N delayed components, LOS
always arrives first.
The impulse response is
N
h ( τ ) = ∑ Ai δ ( τ − τi ) (5.4)
i =1
One impulse at Tx -> many
impulses at Rx (why?)
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ELG4179: Wireless Communication Fundamentals © S.Loyka
P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.
Input-output relationship
∞ t
x (t ) = ∫ h ( τ) s (t − τ) d τ = ∫ s ( τ ) h ( t − τ ) d τ (5.5)
0 −∞
Wireless channel can be modeled as a linear system (may be time-
varying).
Delay spread is a key to FS channels. Average delay and mean-
square delay are:
∑ Pi τi ∑ Pi τi2
τ= i τ2 = i (5.6)
∑ Pi ∑ Pi
i i
where Pi - power of the i-th component.
Lecture 5 11-Oct-17 3(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Delay spread (RMS) is
2
∆τ = τ 2 − ( τ ) = ( τ − τ )2 (5.7)
i.e. the standard deviation of the delay. ∆τ characterizes time-
spreading of the pulse in the channel.
Realistic example:
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
Lecture 5 11-Oct-17 4(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
Lecture 5 11-Oct-17 5(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Frequency-Selective Properties
Compare delay spread ∆τ and symbol duration T :
Frequency-selective : T ≤ ∆τ ,
Frequency-flat: T >> ∆τ (5.8)
Coherence bandwidth ∆f c of the channel:
∆f c ~ 1/ ∆τ or ∆f c = c / ∆τ (5.9)
where c is a constant, usually c ≤ 1; e.g. c = 0.2 for 0.5
correlation.
The same can be expressed using signal (RF) bandwidth
∆f s ≈ 2 / T :
frequency-selective: ∆f s > ∆f c
frequency-flat: ∆f s ≤ ∆f c → T ≥ 10∆τ (5.10)
Error floor effect (to be discussed later).
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ELG4179: Wireless Communication Fundamentals © S.Loyka
Example: two-ray model
h ( τ ) = δ ( τ ) + aδ ( τ − ∆τ ) → H ( f ) = 1 + ae− jω∆τ
H(f) = (1 + a cos θ )2 + ( a sin θ )2 (5.11)
= 1 + a 2 + 2a cos θ
where θ = ω∆τ = 2πf ∆τ
Tap-delay model:
a
s (t ) x (t )
τ
Magnitude Channel Frequency Response
2
0
0 0.5 1 1.5 2
normalized frequency f ∆τ
a=1
a=0.5
a=0.1 H ( f ) = 1 + a 2 + 2a cos 2πf ∆τ
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ELG4179: Wireless Communication Fundamentals © S.Loyka
Consider specific cases:
(1) Frequency-flat channel:
1
2 πf ∆τ << 1 → θ ≪ 1 → f << , so that
2π∆τ
H ( f ) ≈ 1 + a - frequency-independent (flat).
1
Another criterion (less strict): 2πf ∆τ < π / 2 → f <
4∆τ
1 1
(2) Frequency-selective: f ≥ , or f ≥
2π∆τ 4∆τ
Q.: Using (5.7), find the delay spread (RMS) for the two-ray
model.
Lecture 5 11-Oct-17 8(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Doppler Spread and Time-Varying
Channels
Consider moving MS:
v v
BS MS
Doppler effect: frequency shift by
v
f d = f 0 → f MS = f0 ± f d (5.12)
c
Consider moving MS at angle:
v
θ
BS MS
f d = f 0 cos θ, f MS = f 0 1 + cos θ (5.13)
v v
c c
Multipath channel:
v
i-th path: f di = f0 cos θi (5.14)
c
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ELG4179: Wireless Communication Fundamentals © S.Loyka
Time-varying frequency response
jω0t
If e is transmitted, the Rx signal (at MS) is
N
jω0 t j ( ϕi + 2 πf i t ) − jω0 τi
x (t ) = e ∑i
a e e (5.15)
i =1
v
where fi = f0 cos θi . The frequency response is
c
N
H ( f 0 , t ) = ∑ ai e j ( ϕi −ω0 τi )e j 2 πf i t ↔ h( τ, t ) (5.16)
i =1
i.e., a function of time!
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
Lecture 5 11-Oct-17 10(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Time-varying impulse response
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
If fi = 0 , the channel is fixed (not time-varying).
If fi is large, the channel is fast-fading (varying).
If fi is small, the channel is slow-fading.
How large (small) is large (small)?
Compare fd with Ts :
2πf d Ts > 1 → fast fading
(5.17)
2πf d Ts ≪ 1 → slow fading
where Ts is the symbol/block duration.
Q: how to decide fast/slow when all θi ∈ [ θ0 + ∆θ] ?
Lecture 5 11-Oct-17 11(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Coherence time of the channel
Introduce a coherence time of the channel*:
1
Tc = (5.18)
2πf d
This is the time when the channel approximately does not change,
can be considered fixed (static).
The channel is considered static for ∆t < Tc and time-varying for
∆t > Tc .
Fast/slow fading can be expressed as
Ts > Tc → fast fading
(5.19)
Ts << Tc → slow fading
Note: the error floor effect exists for both cases.
*) another definition:
π 1
2πf d Tc = → Tc =
2 4 fd
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ELG4179: Wireless Communication Fundamentals © S.Loyka
Example: two-ray model
H ( f ) = 1 + ae − jω∆τ e jωd t (5.20)
Consider ∆τ = 0 :
H ( f ) = 1 + e jωd t
(5.21)
H ( f ) = 1 + a 2 + 2a cos ω d t
i.e. H ( f , t ) . It is the same as before if θ = ω d t = 2πf d t .
Recall that ω d = ω 0v / c
Magnitude Channel Frequency Response
2
0
0 0.5 1 1.5 2
normalized timef d t
a=1
a=0.5
a=0.1
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ELG4179: Wireless Communication Fundamentals © S.Loyka
Doppler Spectrum
Consider many multipath components with uniform θ
ρθ ( θ ) = 1/ 2π, θ∈ [0,2π ] (5.22)
v
What is the pdf of f d = f 0 cos θ ?
c
a
ρθ ( θ ) d θ = ρ f ( f d ) df d → ρ f ( f d ) = (5.23)
2
fd
1−
f
d ,max
where a is a normalization constant, and f d ,max = f 0v / c is the
maximum Doppler frequency. It can be shown that Doppler power
spectrum PSD( f d ) is the same as ρ f ( f d ) (provided that
uniform angular distribution holds).
PSD ( f d )
f = fd + f0
fd
− f d ,max f d ,max
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ELG4179: Wireless Communication Fundamentals © S.Loyka
Note that a single-tone Tx signal results in spread-out spectrum at
Rx! ( ∆f d = 2 f d ,max )
Mobile wireless channel is a function of space and time!
Random channel: coherence time is defined as a time interval for
which envelope correlation ≥ 0.5 .
Coherence bandwidth: frequency interval for which envelope
correlation ≥ 0.5 .
Note: other correlation level can be used.
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ELG4179: Wireless Communication Fundamentals © S.Loyka
T.S. Rappaport, Wireless Communications, Prentice Hall, 2002
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ELG4179: Wireless Communication Fundamentals © S.Loyka
Different Forms of Fading
P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002.
Lecture 5 11-Oct-17 17(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Overview of System-Level Propagation Effects
~ 10...100's λ <λ
multipath
temporal
variations
P.M. Shankar, Introduction to Wireless Systems, Wiley, 2002. (modified)
Lecture 5 11-Oct-17 18(19)
ELG4179: Wireless Communication Fundamentals © S.Loyka
Summary
• Impulse and frequency responses of a wireless channel.
• Delay spread and frequency selective channels
• Tap-delay model. Power delay profile.
• Doppler spread and time-varying channels.
• Envelope correlation. Coherence bandwidth and coherence
time of the channel.
• Classification of fading and propagation effects
Reading:
o Rappaport, Ch. 5 (except 5.8).
o Other books (see the reference list).
Note: Do not forget to do end-of-chapter problems. Remember
the learning efficiency pyramid!
Lecture 5 11-Oct-17 19(19)
