Lecture 1 Wireless Channel I-Hsiang Wang ihwang@ntu.edu.tw - - PowerPoint PPT Presentation

Lecture ¡1 Wireless ¡Channel

I-Hsiang Wang ihwang@ntu.edu.tw 2/20, 2014

Wireless ¡channels ¡vary ¡at ¡two ¡scales

• Large-scale fading: path loss, shadowing, etc.
• Small-scale fading: constructive/destructive interference

2

Time Channel quality

Large-­‑Scale ¡Fading

• Path loss and Shadowing
• In free space, received power
• With reflections and obstacles, can attenuate faster than
• Variation over time: very slow, order of seconds
• Critical for coverage and cell-cite planning

3

∝ 1 r2 1 r2

Small-­‑Scale ¡Fading

• Multipath fading: due to constructive and destructive

interference of the waves

• Channel varies when the mobile moves a distance of the
• rder of the carrier wavelength
• Typical carrier frequency ~ 1GHz
• Variation over time: order of hundreds of microseconds
• Critical for design of communication systems

4

λ

= ⇒ λ ≈ c/f = 0.3m

Plot

• Understand how physical parameters impact a wireless

channel from the communication system point of view. Physical parameters such as

• Carrier frequency
• Mobile speed
• Bandwidth
• Delay spread
• etc.
• Start with deterministic physical models
• Progress towards statistical models

5

Outline

• Physical modeling of wireless channels
• Deterministic Input-output model
• Time and frequency coherence
• Statistical models

6

Physical ¡Model: ¡ Warm-­‑up ¡Examples

Physical ¡Model: ¡Simple ¡Example ¡1

8

d

Transmitted Waveform (electric field): cos 2πft

r

Received Waveform (path 1): α r cos 2πf ⇣ t − r c ⌘ Received Waveform (path 2): − α 2d − r cos 2πf ✓ t − 2d − r c ◆ = ⇒ Received Waveform (aggregate): α r cos 2πf ⇣ t − r c ⌘ − α 2d − r cos 2πf ✓ t − 2d − r c ◆

Physical ¡Model: ¡Simple ¡Example ¡1

9

d

Transmitted Waveform (electric field): cos 2πft

r

Received Waveform (aggregate): α r cos 2πf ⇣ t − r c ⌘ − α 2d − r cos 2πf ✓ t − 2d − r c ◆ Phase Difference between the two sinusoids: ∆θ = ⇢2πf(2d − r) c + π

• − 2πfr

c = 2π (2d − r) − r c f + π = ( 2nπ, constructive interference (2n + 1)π, destructive interference

Delay Spread:

difference between delays

Td

Delay ¡Spread ¡and ¡Coherence ¡Bandwidth

• Delay spread

: difference between delays of paths

• If frequency f change by
• , then the combined

received sinusoid move from peak to valley

• Therefore, the frequency-variation scale is of the order of
• Coherence bandwidth

10

1/(2Td) 1 Td Td Wc := 1 Td

Physical ¡Model: ¡Simple ¡Example ¡2

11

v d

Received Waveform (path 1): α r(t) cos 2πf ✓ t − r(t) c ◆ Transmitted Waveform (electric field): cos 2πft Received Waveform (path 2): − α 2d − r(t) cos 2πf ✓ t − 2d − r(t) c ◆ = ⇒ Received Waveform (aggregate): α r(t) cos 2πf ✓ t − r(t) c ◆ − α 2d − r(t) cos 2πf ✓ t − 2d − r(t) c ◆ = α r0 + vt cos 2πf h⇣ 1 − v c ⌘ t − r0 c i − α 2d − r0 − vt cos 2πf ⇣ 1 + v c ⌘ t − 2d − r0 c

• r(t) = r0 + vt

Physical ¡Model: ¡Simple ¡Example ¡2

12

v d

Approximation: distance to mobile Rx ⌧ distance to Tx Time-invariant shift

• f the original input

waveform Time-varying amplitude = ⇒ Received Waveform (aggregate): = α r0 + vt cos 2πf h⇣ 1 − v c ⌘ t − r0 c i − α 2d − r0 − vt cos 2πf ⇣ 1 + v c ⌘ t − 2d − r0 c

• ≈

2α r0 + vt sin 2πf ✓vt c + r0 − d c ◆ sin 2πf ✓ t − d c ◆

Difference of the Doppler shifts of the two paths, cause this variation

• ver time.

Time-variation scale: (ms)

Physical ¡Model: ¡Simple ¡Example ¡2

13

t

Time-varying envelope 2α r0 + vt sin 2πf ✓vt c + r0 − d c ◆

Time-variation scale: (seconds or minutes), much smaller than that of the second term r0/v c/fv

Doppler Spread Ds = 2fv c

Doppler ¡Spread ¡and ¡Coherence ¡Time

• Mobility causes time-varying delays (Doppler shift)
• Doppler spread : difference between Doppler shifts of

multiple signal paths

• If time t change by
• , then the combined received

sinusoidal envelope move from peak to valley

• Therefore, the time-variation scale is of the order of
• Coherence time

14

1/(2Ds) 1 Ds Ds Tc := 1 Ds

What ¡we ¡learned ¡from ¡the ¡examples

• Delay spread/coherence bandwidth and Doppler spread/

coherence time seem fundamental

• However, it is difficult to derive the explicit received

waveform mathematically.

• Out of scope – EM wave theory
• Instead, we construct useful input/output models, and

take measurements to determine the parameters in the models

15

Physical ¡Model: ¡ Input/Output ¡Relations

Physical ¡Input/Output ¡Model

• Wireless channels as linear time-varying systems:
• Recall Example 2:

17

y(t) = X

i

ai(t)x (t − τi(t)) ai(t): gain of path i τi(t): delay of path i v d r(t)

x(t) = cos 2πft a1(t) = |α| r0 + vt τ1(t) = r0 + vt c a2(t) = |α| 2d − r0 − vt τ2(t) = 2d − r0 − vt c − π 2πf

Physical ¡Input/Output ¡Model

• Wireless channels as linear time-varying systems:
• Impulse response:
• Frequency response:

18

y(t) = X

i

ai(t)x (t − τi(t)) ai(t): gain of path i τi(t): delay of path i h (τ, t) x(t) y(t) = X

i

ai(t)x (t − τi(t)) h(τ, t) = X

i

ai(t)δ (τ − τi(t)) H(f; t) = X

i

ai(t)e−j2πfτi(t)

Passband–Baseband ¡Conversion

• Communications takes place in a passband
• Carrier frequency
• Bandwidth
• Real signal

19

1 S( f ) f –fc – W 2 fc – W 2 – fc W 2 + W 2 fc +

fc W < 2fc s(t)

Passband–Baseband ¡Conversion

20

1 S( f ) f –fc – W 2 fc – W 2 – fc W 2 + W 2 fc + W 2 Sb ( f ) f W 2 – 2 √

Sbf = √ 2Sf +fc f +fc > 0 f +fc ≤ 0

Passband–Baseband ¡Conversion

21

X X X X

ℜ[sb(t)] ℑ[sb(t)] ℜ[sb(t)] ℑ[sb(t)]

–√2 sin 2π fc t –√2 sin 2π fc t √2 cos 2π fc t √2 cos 2π fc t s(t)

–W 2 W 2 –W 2 W 2

1 1 +

st = 1 √ 2

• sbte j2fct +s∗

bte−j2fct

= √ 2ℜ

• sbte j2fct

Baseband ¡System ¡Architecture

22

X X X X

ℜ[xb(t)] ℑ[xb(t)] ℜ[yb(t)] ℑ[yb(t)]

–W 2 W 2 –W 2 W 2

1 1 + x(t) y(t) h(τ, t) –√2 sin 2π fc t –√2 sin 2π fc t √2 cos 2π fc t √2 cos 2π fc t

yb(t) = X

i

ab

i(t)xb (t − τi(t)) ,

where ab

i(t) := ai(t)e−j2πfcτi(t)

Continuous-­‑time ¡Baseband ¡Model

• Complex baseband equivalent channel:
• Frequency response: shifted from passband to baseband
• Each path is associated with a delay and a complex gain

23

xb(t) hb (τ, t) yb(t) = X

i

ab

i(t)xb (t − τi(t))

hb(τ, t) = X

i

ab

i(t)δ (τ − τi(t)) ,

where ab

i(t) := ai(t)e−j2πfcτi(t)

Hb(f; t) = H(f + fc; t)

Modulation ¡and ¡Sampling

• Modern communication systems are digitized, (partially)

thanks to sampling theorem

• Our baseband signal can be represented as follows:

24

xb(t) = X

n

x[n]sinc(Wt − n), x[n] := xn(n/W), sinc(t) := sin πt πt

Modulation ¡and ¡Sampling

25

X X X X

ℜ[x[m]]

sinc (Wt – n)

ℑ[x[m]]

sinc (Wt – n) h(τ, t) 1 –W W –W W 1 +

ℜ[xb(t)] ℑ[y[m]]

ℜ[y[m]]

ℜ[yb(t)] ℑ[yb(t)]

y(t) x(t)

ℑ[xb(t)]

2 2 2 2 –√2 sin 2π fc t –√2 sin 2π fc t √2 cos 2π fc t √2 cos 2π fc t

y[m] = X

l

hl[m]x[m − l], where hl[m] := X

i

ab

i(m/W)sinc [l − τi(m/W)W]

Discrete-­‑Time ¡Baseband ¡Model

• Discrete-time channel model
• Note: the l-th tap hl contains contributions mostly for the

paths that have delays that lie inside the bin (roughly)

• System resolves the multipaths up to delays of

26

y[m] = X

l

hl[m]x[m − l] hl[m] x[m] hl[m] := X

i

ab

i(m/W)sinc [l − τi(m/W)W]

1 W  l W − 1 2W , l W + 1 2W

Multipath ¡Resolution

• sinc(t) vanish quickly outside of

the interval [-0.5, 0.5] (roughly)

• The peak of the i-th translated

sinc lies at

• To contribute significantly to hl,

the delay must fall inside

27

1 W

Main contribution l = 0 Main contribution l = 0 Main contribution l = 1 Main contribution l = 2 Main contribution l = 2 i = 0 i = 1 i = 2 i = 3 i = 4

1 2 l

hl[m] := X

i

ab

i(m/W)sinc [l − τi(m/W)W]

τi  l W − 1 2W , l W + 1 2W

Time ¡and ¡Frequency ¡ Coherence

Varying ¡Channel ¡Tap

• The discrete-time baseband channel model is the

equivalent one in designing communication systems

• It only matters how the taps hl[m] vary over time m and

carrier frequency fc

• l-th tap of the discrete-time baseband channel model

29

hl[m] := X

i

ab

i(m/W)sinc [l − τi(m/W)W]

= X

i

ai (tm) e−j2πfcτi(tm)sinc [l − τi(tm)W] tm := m

W

≈ X

i∈l-th delay bin

ai (tm) e−j2πfcτi(tm) Difference in phases (over the paths that contribute significantly to the tap), causes variation of the tap gain

Frequency ¡Variation

• Delay Spread
• Coherence Bandwidth
• For a system with bandwidth W

30

hl[m] ≈ X

i∈l-th delay bin

ai (tm) e−j2πfcτi(tm) Td := max

i,j |τi(t) − τj(t)|

Wc := 1 Td Wc W = ) single tap, flat fading Wc < W = ) multiple taps, frequency-selective fading

Flat ¡and ¡Frequency-­‑Selective ¡Fading

• Effective channel depends on both physical environment

(Wc) and operation bandwidth (W)

31

10 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 –60 –50 –40 –30 –20 –10 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.45 –10 –20 –0.001 –0.0008 –0.0006 –0.0004 –0.0002 0.0002 0.0004 0.0006 0.0008 0.001 50 100 150 200 250 300 350 400 450 500 550 –30 –40 –50 –60 –70 –0.006 –0.005 –0.004 –0.003 –0.002 –0.001 0.001 0.002 0.003 0.004 50 100 150 200 250 300 350 400 450 500 550 0.5

(d)

Power spectrum (dB) Power specturm (dB) Amplitude (linear scale) Amplitude (linear scale)

(b)

Time (ns) Time (ns)

(a) (c)

40 MHz Frequency (GHz) Frequency (GHz) 200 MHz

Larger bandwidth, more paths can be resolved

Time ¡Variation

• Doppler Spread
• Coherence Time
• For a system with delay requirement (application

dependent) T

32

hl[m] ≈ X

i∈l-th delay bin

ai (tm) e−j2πfcτi(tm) Ds := max

i,j fc|τi 0(t) − τj 0(t)|

Tc := 1 Ds Tc T = ) slow fading Tc < T = ) fast fading

Representitive ¡Numbers

33

Key channel parameters and time-scales Symbol Representative values Carrier frequency fc 1 GHz Communication bandwidth W 1 MHz Distance between transmitter and receiver d 1 km Velocity of mobile v 64 km/h Doppler shift for a path D = fcv/c 50 Hz Doppler spread of paths corresponding to a tap Ds 100 Hz Time-scale for change of path amplitude d/v 1 minute Time-scale for change of path phase 1/4D 5 ms Time-scale for a path to move over a tap c/vW 20 s Coherence time Tc = 1/4Ds 2.5 ms Delay spread Td 1s Coherence bandwidth Wc = 1/2Td 500 kHz

Types ¡of ¡Channels

• Typical channels are underspread
• Coherence time Tc depends on carrier frequency and

mobile speed, of the order of ms or more

• Delay spread Td depends on distance to scatters and cell

size, of the order of ns (indoor) to µs (outdoor)

34

Types of channel Defining characteristic Fast fading Tc delay requirement Slow fading Tc delay requirement Flat fading W Wc Frequency-selective fading W Wc Underspread Td Tc

Stochastic ¡Models

• Continuous-time Passband ⟶ ¡Discrete-time Baseband:
• Continuous-time Passband (Real)
• Continuous-time Baseband (Complex)
• Discrete-time Baseband (Complex)

Recap: ¡Deterministic ¡Modeling

36

h (τ, t) x(t) y(t) < {xb(t)} = {xb(t)} √ 2 cos 2πfct − √ 2 sin 2πfct √ 2 cos 2πfct − √ 2 sin 2πfct Filter ⇥ − W

2 , W 2

⇤ Filter ⇥ − W

2 , W 2

⇤ = {yb(t)} < {yb(t)} sinc(Wt − n) sinc(Wt − n) < {x[m]} = {x[m]} 1 W 1 W < {y[m]} = {y[m]}

y(t) = X

i

ai(t)x (t − τi(t)) ai(t): gain of path i τi(t): delay of path i

yb(t) = X

i

ab

i(t)xb (t − τi(t)) ,

ab

i(t) := ai(t)e−j2πfcτi(t)

y[m] = X

l

hl[m]x[m − l], hl[m] := X

i

ab

i(m/W)sinc [l − τi(m/W)W]

Up-conversion Down-conversion Modulation Sampling

• Thermal noise at the receiver:
• Continuous-time Passband (Real)
• Continuous-time Baseband (Complex)
• Discrete-time Baseband (Complex)

Noise

37

h (τ, t) x(t) y(t) < {xb(t)} = {xb(t)} √ 2 cos 2πfct − √ 2 sin 2πfct √ 2 cos 2πfct − √ 2 sin 2πfct Filter ⇥ − W

2 , W 2

⇤ Filter ⇥ − W

2 , W 2

⇤ = {yb(t)} < {yb(t)} sinc(Wt − n) sinc(Wt − n) < {x[m]} = {x[m]} 1 W 1 W < {y[m]} = {y[m]} w(t)

ai(t): gain of path i τi(t): delay of path i y(t) = X

i

ai(t)x (t − τi(t)) + w(t) yb(t) = X

i

ab

i(t)xb (t − τi(t)) + wb(t),

ab

i(t) := ai(t)e−j2πfcτi(t)

y[m] = X

l

hl[m]x[m − l] + w[m], hl[m] := X

i

ab

i(m/W)sinc [l − τi(m/W)W]

Additive ¡White ¡Noise ¡Model

38

• Additive White Gaussian Noise (AWGN)
• Standard modeling for thermal noise
• {w(t)}: zero-mean (real) white Gaussian process with spectral

density N0/2

• In other words,
• Discrete-time baseband equivalent noise:

E [w(0)w(t)] = N0 2 δ(t)

√ 2 cos 2πfct − √ 2 sin 2πfct Filter ⇥ − W

2 , W 2

⇤ Filter ⇥ − W

2 , W 2

⇤ 1 W 1 W w(t) < {w[m]} = {w[m]}

• System is linear ⟶ can

separate the noise out

• Rectangle filter in frequency

⟺ sinc in time

Equivalent ¡Discrete-­‑Time ¡Baseband ¡Noise

39 √ 2 cos 2πfct − √ 2 sin 2πfct Filter ⇥ − W

2 , W 2

⇤ Filter ⇥ − W

2 , W 2

⇤ 1 W 1 W w(t) < {w[m]} = {w[m]}

1 W 2 −W 2 f

rect ✓ f W ◆ rect ✓ f W ◆

F −1

− − − → Wsinc (Wt) < {w[m]} = Z ∞

−∞

w(t) hp 2W cos (2πfct) sinc (Wt m) i | {z }

ψm,1(t)

dt = {w[m]} = Z ∞

−∞

w(t) h

• p

2W sin (2πfct) sinc (Wt m) i | {z }

ψm,2(t)

dt = hw(t), ψm,1(t)i = hw(t), ψm,2(t)i

Equivalent ¡Discrete-­‑Time ¡Baseband ¡Noise

40

< {w[m]} = Z ∞

−∞

w(t) hp 2W cos (2πfct) sinc (Wt m) i | {z }

ψm,1(t)

dt = {w[m]} = Z ∞

−∞

w(t) h

• p

2W sin (2πfct) sinc (Wt m) i | {z }

ψm,2(t)

dt = hw(t), ψm,1(t)i = hw(t), ψm,2(t)i

• forms an orthogonal set of waveforms.

Fact. {ψm,1(t), ψm,2(t) | m ∈ Z}

• The real and the imaginary parts of w[m] are
• Both Gaussian with zero-mean and variance WN0/2
• Independent and identically distributed (i.i.d.) over time (m)
• “White” discrete-time processes

Circular ¡Symmetric ¡Complex ¡Gaussian

41

• is a zero-mean white circular symmetric complex

Gaussian process with auto-correlation function Fact. {w[m] | m ∈ Z} R[m] := E [w[n + m]w[n]∗] = WN0δ[m] w[m] = < {w[m]} + j= {w[m]} < {w[m]} ⇠ N ✓ 0, WN0 2 ◆ , = {w[m]} ⇠ N ✓ 0, WN0 2 ◆ (< {w[m]} , = {w[m]}) : independent Fact. X is circular symmetric if eiθX

d

= X for all θ ⇐ ⇒ w[m] ∼ CN (0, WN0) : circular symmetric complex Gaussian

Discrete ¡Baseband ¡Model ¡with ¡Noise

42

h (τ, t) x(t) y(t) < {xb(t)} sinc(Wt − n) sinc(Wt − n) = {xb(t)} < {x[m]} = {x[m]} √ 2 cos 2πfct − √ 2 sin 2πfct √ 2 cos 2πfct − √ 2 sin 2πfct Filter ⇥ − W

2 , W 2

⇤ Filter ⇥ − W

2 , W 2

⇤ = {yb(t)} < {yb(t)} 1 W 1 W < {y[m]} = {y[m]} w(t)

≡

hl[m] x[m] y[m] = X

l

hl[m]x[m − l] + w[m] w[m] ∼ CN (0, WN0) , where N0 2 is the spectral density of white Gaussian process {w(t)}

Please ¡refer ¡to ¡Appendix ¡A ¡ for ¡a ¡review ¡on ¡complex ¡ Gaussian ¡random ¡variables ¡ and ¡random ¡vectors.

Fading

• Additive noise w[m]
• Essentially completely random, no correlation over time
• Largely depends on nature
• Can be dealt with using digital (wired) communication techniques
• Filter taps hl[m]
• Varying over time and frequency
• Largely depends on nature
• Why not use stochastic models for taps as well?

44

y[m] = X

l

hl[m]x[m − l] + w[m], where hl[m] := X

i

ab

i(m/W)sinc [l − τi(m/W)W]

w[m] ∼ CN (0, WN0)

Rayleigh ¡Fading

• Many small scattered paths for each tap:
• Phase for each path is uniformly distributed over [0, 2π]
• For each path it is a circular symmetric random variable
• Each tap: sum of many small independent circular

symmetric random variables

• By Central Limit Theorem (CLT), we can model
• Zero-mean because of rich scattering

45

hl[m] ≈ X

i

ai (tm) e−j2πfcτi(tm) hl[m] ∼ CN

• 0, σ2

l

Magnitude ¡and ¡Phase ¡of ¡Raylegh ¡Fades

• Magnitude
• Phase

46

h ∼ CN

• 0, σ2

∠h ∼ Unif (0, 2π) |h|2 ∼ Exp ✓ 1 σ2 ◆

Rician ¡Fading

• If there is a strong line-of-sight path, then model it as
• K-Factor :
• The larger it is, the more deterministic the channel will be.

47

hl[m] = r κ κ + 1σlejθ + r 1 κ + 1CN

• 0, σ2

l

• Line-of-sight

Scattered Multipath

κ