Wireless PHY: Digital Demodulation and

Wireless Channels

Y. Richard Yang

09/13/2012

2

Outline Admin and recap Digital demodulation Wireless channels

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Admin Assignment 1 posted

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Demodulation Low pass filter and FIR Convolution Theorem

Digital modulation/demodulation ASK, FSK, PSK General representation

Recap

Recap: gi() for BPSK

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1: g1(t) = cos(2πfct) t in [0, T]

0: g0(t) = -cos(2πfct) t in [0, T]

Note: g1(t) = -g0(t)

cos(2πfct)[0, T]1-1g1(t)g0(t)

Recap: Signaling Functions gi() for QPSK

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11: cos(2πfct + π/4) t in [0, T]

10: cos(2πfct + 3π/4) t in [0, T]

00: cos(2πfct - 3π/4) t in [0, T]

01: cos(2πfct - π/4) t in [0, T]

Q

I

11

01

10

00

Recap: QPSK Signaling Functions as Sum of cos(2πfct), sin(2πfct)

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11: cos(π/4 + 2πfct) t in [0, T]-> cos(π/4) cos(2πfct) + -sin(π/4) sin(2πfct)

10: cos(3π/4 + 2πfct) t in [0, T]-> cos(3π/4) cos(2πfct) + -sin(3π/4) sin(2πfct)

00: cos(- 3π/4 + 2πfct) t in [0, T]-> cos(3π/4) cos(2πfct) + sin(3π/4) sin(2πfct)

01: cos(- π/4 + 2πfct) t in [0, T]-> cos(π/4) cos(2πfct) + sin(π/4) sin(2πfct)

sin(2πfct)

11

00

10

cos(2πfct)

[cos(π/4), sin(π/4)]

01[cos(3π/4), sin(3π/4)]

[cos(3π/4), -sin(3π/4)]

[-sin(π/4), cos(π/4)]

We call sin(2πfct) and cos(2πfct) the bases.

Recap: Demodulation/Decoding

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Considered a simple setting: sender uses a single signaling function g(), and can have two actions send g() or nothing (send 0)

How does receiver use the received sequence x(t) in [0, T] to detect if sends g() or nothing?

Recap: Design

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Streaming algorithm: use all data points in [0, T] As each sample xi comes in, multiply it by a factor hT-i-1

and accumulate to a sum y

At time T, makes a decision based on the accumulated sum at time T: y[T]

xTx2x1x0

h0h1h2hT

****

Determining the Best h

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where w is noise,

Design objective: maximize peak pulse signal-to-noise ratio

Determining the Best h

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Assume Gaussian noise, one can derive

Using Fourier Analysis and Convolution Theorem:

Determining the Best h

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Apply Schwartz inequality

By considering

Determining the Best h

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Determining Best h to Use

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xTx2x1x0

gTg2g1g0

****

xTx2x1x0

h0h1h2hT

****

Summary of Progress

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After this “complex” math, the implementation/interpretation is actually the following quite simple alg: precompute auto correlation: <g, g>

compute the correlation between received x and signaling function g, denoted as <x, g>

if <x, g> is closer to <g, g> • output sends g

else• output sends nothing

Applying Scheme to BPSK

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Consider g1 alone, compute <x, g1>, check if close to <g1, g1>: |<x, g1> - <g1, g1>|

Consider g0 alone, compute <x, g0>, check if close to <g0, g0>: |<x, g0> - <g0, g0>|

Pick closer if |<x, g1> - <g1, g1>| < |<x, g0> - <g0, g0>|

• pick 1 else

• pick 0cos(2πfct)[0, T]1-1

g1(t)g0(t)

Applying Scheme to BPSK

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since g0 = -g1 <x, g0> = - <x, g1> <g0, g0> = - <g0, g1>

rewrite as if |<x, g1> - <g1, g1>| < |<x, g1> - <g0, g1>|

• pick 1 else

• pick 0

cos(2πfct)[0, T]1-1g1(t)g0(t)

Interpretation

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For any signal s, <s, g1> computes the coordinate of s when using g1 as a base cleaner if g1 is normalized, but we do not

worry about it yet

g1=cos(2πfct)[0, T]

<g1(t), g1(t)><g0(t), g1(t)>=-<g1(t), g1(t)>

<x, g1(t)>

Applying Scheme to QPSK: Attempt 1

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Consider g00 alone, compute <x, g00> … Consider g01 alone, compute <x, g01> … Consider g10 alone, compute <x, g10> … Consider g11 alone, compute <x, g11> … Issues

Complexity:• Need to compute M correlation, where M is number of

signaling functions• Think of 64-QAM

Objective• the previous scheme is defined for a single signaling

function, does it work for M?

Decoding for QPSK using bases

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4 signaling functions g00(), g01(), g10(), g11() For each signaling function, computes

correlation with the bases (cos(), sin()), e.g., g00: [a00, b00] What is the meaning of a00, b00?

For received signal x, computes ax=<x, cos> and bx=<x, sin> (how many correlation do we do now?)

QPSK Demodulation/Decoding

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sin(2πfct)

cos(2πfct)

[a01,b01]

[a10,b10]

[a00,b00]

[a11,b11]

[ax,bx]

Q: how to decode?

Look into Noise

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Assume sender sends gm(t) [0, T] Receiver receives x(t) [0, T]

Consider one sample

where w[i] is noise Assume white noise, i.e., prob w[i] = z is

2

2

2

21)(

z

ezf

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Likelihood What is the likelihood (prob.) of observing

x[i]? it is the prob. of noise being w[i] = x[i] – g[i]

What is the likelihood (prob.) of observing the whole sequence x? the product of the probabilities

Likelihood Detection

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Suppose we know

Maxim likelihood detection picks the m with the highest P{x|gm}.

From the expression

We pick m with the lowest ||x-gm||2

Back to QPSK

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QPSK Demodulation/Decoding

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sin(2πfct)

cos(2πfct)

[a01,b01]

[a10,b10]

[a00,b00]

[a11,b11]

[ax,bx]

Q: what does maximum likelihood det pick?

General Matched Filter Detection: Implementation for Multiple Sig Func.

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Basic idea consider each gm[0,T] as a point (with

coordinates) in a space

compute the coordinate of the received signal x[0,T]

check the distance between gm[0,T] and the received signal x[0,T]

pick m* that gives the lowest distance value

Computing Coordinates

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Pick orthogonal bases {f1(t), f2(t), …, fN(t)} for {g1(t), g2(t), …, gM(t)}

Compute the coordinate of gm[0,T] as cm = [cm1, cm2, …, cmN], where

Compute the coordinate of the received signal x[0,T] as x = [x1, x2, …, xN]

Compute the distance between r and cm every cm and pick m* that gives the lowest distance value

Example: Matched Filter => Correlation Detector

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receivedsignal x

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BPSK vs QPSK

BPSK

QPSK

fc: carrier freq.Rb: freq. of data10dB = 10; 20dB =100

11 10 00 01

A

t

BPSK vs QPSK A major metric of modulation performance is

spectral density (SD)

Q: what is the SD of BPSK vs that of QPSK? Q: Why would any one use BPSK, given higher

QAM?31

Spectral Density =

bit rate-------------------

width of spectrum used

Context Previous demodulation considers only

additive noise, and does not consider wireless channel’s effects

We next study its effects

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Outline Admin and recap Digital demodulation Wireless channels

Signal Propagation

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Isotropic radiator: a single point equal radiation in all directions (three dimensional) only a theoretical reference antenna

Radiation pattern: measurement of radiation around an antenna

zy

x

z

y x idealisotropicradiator

Antennas: Isotropic Radiator

Q: how does power level decrease as a function of d, the distancefrom the transmitter to the receiver?

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Free-Space Isotropic Signal Propagation

In free space, receiving power proportional to 1/d² (d = distance between transmitter and receiver)

Suppose transmitted signal is cos(2ft), the received signal is

Pr: received power Pt: transmitted power Gr, Gt: receiver and

transmitter antenna gain (=c/f): wave length

Sometime we write path loss in log scale: Lp = 10 log(Pt) – 10log(Pr)

dcdtftfEd)]/(2cos[),(

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Log Scale for Large SpandB = 10 log(times)

Slim/Gates

~100B

Obama

~10M

~10K

1000 times

40 dB

10,000 times

30 dB

10,000 x 1,000

40 + 30 = 70 dB

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Path Loss in dBdB = 10 log(times)

source

10 W

d1

1 mW

1 uW

1000 times

40 dB

10,000 times

30 dB

10,000 x 1,000

40 + 30 = 70 dBpower

d2

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dBm (Absolute Measure of Power)dBm = 10 log (P/1mW)

source

10 W

d1

1 mW

1 uW

1000 times

40 dB

10,000 times

30 dB

10,000 x 1,000

40 + 30 = 70 dBpower

d2

40 dBm

-30 dBm

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Number in Perspective (Typical #)

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Exercise: 915MHz WLAN (free space) Transmit power (Pt) = 24.5 dBm Receive sensitivity = -64.5 dBm

Receiving distance (Pr) =

Gt=Gr=1

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Two-ray Ground Reflection Model

Single line-of-sight is not typical. Two paths (direct and reflect) cancel each other and reduce signal strength

Pr: received power Pt: transmitted power Gr, Gt: receiver and

transmitter antenna gain hr, ht: receiver and

transmitter height

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Exercise: 915MHz WLAN (Two-ray ground reflect) Transmit power (Pt) = 24.5 dBm Receive sensitivity = -64.5 dBm

Receiving distance (Pr) =

Gt=Gr=hr=ht=1

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Real Antennas Real antennas are not isotropic radiators Some simple antennas: quarter wave /4 on car roofs or

half wave dipole /2 size of antenna proportional to wavelength for better transmission/receiving

/4/2

Q: Assume frequency 1 Ghz, = ?

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Figure for Thought: Real Measurements

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Receiving power additionally influenced by shadowing (e.g., through a wall or a door) refraction depending on the density of a medium reflection at large obstacles scattering at small obstacles diffraction at edges

reflectionscattering

diffraction

shadow fadingrefraction

Signal Propagation: Complexity

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Signal Propagation: Complexity

Details of signal propagation are very complicated

We want to understand the key characteristics that are important to our understanding

48

Outline Admin and recap Digital demodulation Wireless channels

Intro shadowing

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Shadowing Signal strength loss after passing

through obstacles

Same distance, but different levels of shadowing: It is a random, large-scale effect depending on the environment

Example Shadowing Effects

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i.e. reduces to ¼ of signal10 log(1/4) = -6.02

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JTC Indoor Model for PCS: Path Loss

)(10 nLdBLogAL fA: an environment dependent fixed loss factor

(dB)B: the distance dependent loss coefficient,d : separation distance between the base station

and mobile terminal, in metersLf : a floor penetration loss factor (dB)n: the number of floors between base station

and mobile terminal

Shadowing path loss follows a log-normal distribution (i.e. L is normal distribution) with mean:

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JTC Model at 1.8 GHz

)(10 nLdBLogAL f

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Outline Admin and recap Digital demodulation Wireless channels

Intro Shadowing Multipath

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Signal can take many different paths between sender and receiver due to reflection, scattering, diffraction

Multipath

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Example: reflection from the ground or building

Multipath Example: Outdoor

ground

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Multipath Effect (A Simple Example)

d1 d2

1

11 ][2cos

dtf cd

ft2cos

2121 22)(2 21dd

cddfff c

dcd

2

22 ][2cos

dtf cd

phase difference:

Assume transmitter sends out signal cos(2 fc t)

Multipath Effect (A Simple Example) Where do the two waves totally

destruct?

Q: where do the two waves construct?

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integer2121

dd

cddf

Option 1: Change Location If receiver moves to the right by /4:

d1’ = d1 + /4; d2’ = d2 - /4;

->

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21

21

21

2

)4/(4/22

''2

dd

dd

dd

By moving a quarter of wavelength, destructiveturns into constructive.Assume f = 1G, how far do we move?

Option 2: Change Frequency

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Change frequency:

2121'

ddcff

2121 22 ddcddf

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Multipath Delay SpreadRMS: root-mean-square

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Multipath Effect(moving receiver)

d1 d2

1

11 ][2cos

dtf cd

ft2cos

example

2

22 ][2cos

dtf cd

Suppose d1=r0+vt

d2=2d-r0-vtd1d2

d

Derivation

62

])[sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

)sin()sin(2

])[2cos(])[2cos(

0

0

0

0000

020020

00

2

2)2(

22

2][2][2

2][2][2

2

cvrd

cvf

cd

cdvtr

cd

cdvtr

cd

cvtrdvtr

cvtrdvtr

tftftftf

cvtrd

cvtr

ttf

ftf

ftf

ftf

tftf

cvtrd

cvtr

cvtrd

cvtr

See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation

63

])[sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

)sin()sin(2

])[2cos(])[2cos(

0

0

0

0000

020020

00

2

2)2(

22

2][2][2

2][2][2

2

cvrd

cvf

cd

cdvtr

cd

cdvtr

cd

cvtrdvtr

cvtrdvtr

tftftftf

cvtrd

cvtr

ttf

ftf

ftf

ftf

tftf

cvtrd

cvtr

cvtrd

cvtr

See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation

64

])[sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

)sin()sin(2

])[2cos(])[2cos(

0

0

0

0000

020020

00

2

2)2(

22

2][2][2

2][2][2

2

cvrd

cvf

cd

cdvtr

cd

cdvtr

cd

cvtrdvtr

cvtrdvtr

tftftftf

cvtrd

cvtr

ttf

ftf

ftf

ftf

tftf

cvtrd

cvtr

cvtrd

cvtr

See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation

65

])[sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

)sin()sin(2

])[2cos(])[2cos(

0

0

0

0000

020020

00

2

2)2(

22

2][2][2

2][2][2

2

cvrd

cvf

cd

cdvtr

cd

cdvtr

cd

cvtrdvtr

cvtrdvtr

tftftftf

cvtrd

cvtr

ttf

ftf

ftf

ftf

tftf

cvtrd

cvtr

cvtrd

cvtr

See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation

66

])[sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

)sin()sin(2

])[2cos(])[2cos(

0

0

0

0000

020020

00

2

2)2(

22

2][2][2

2][2][2

2

cvrd

cvf

cd

cdvtr

cd

cdvtr

cd

cvtrdvtr

cvtrdvtr

tftftftf

cvtrd

cvtr

ttf

ftf

ftf

ftf

tftf

cvtrd

cvtr

cvtrd

cvtr

See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

Derivation

67

])[sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

])[2sin(])[2sin(2

)sin()sin(2

])[2cos(])[2cos(

0

0

0

0000

020020

00

2

2)2(

22

2][2][2

2][2][2

2

cvrd

cvf

cd

cdvtr

cd

cdvtr

cd

cvtrdvtr

cvtrdvtr

tftftftf

cvtrd

cvtr

ttf

ftf

ftf

ftf

tftf

cvtrd

cvtr

cvtrd

cvtr

See http://www.sosmath.com/trig/Trig5/trig5/trig5.html for cos(u)-cos(v)

68

Waveformv = 65 miles/h, fc = 1 GHz: fc v/c =

10 ms

deep fade

Q: How far does a car drive in ½ of a cycle?

])[sin(])[2sin(2 02cvrd

cvf

cd ttf

109 * 30 / 3x108 = 100 Hz

69

Multipath with Mobility

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Effect of Small-Scale Fading

no small-scalefading

small-scalefading

71

signal at sender

Multipath Can Spread Delay

signal at receiver

LOS pulsemultipathpulses

LOS: Line Of Sight

Time dispersion: signal is dispersed over time

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JTC Model: Delay SpreadResidential Buildings

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signal at sender

Multipath Can Cause ISI

signal at receiver

LOS pulsemultipathpulses

LOS: Line Of Sight

Dispersed signal can cause interference between “neighbor” symbols, Inter Symbol Interference (ISI)

Assume 300 meters delay spread, the arrival time difference is 300/3x108 = 1 ns if symbol rate > 1 Ms/sec, we will have serious ISI

In practice, fractional ISI can already substantially increase loss rate

74

Channel characteristics change over location, time, and frequency

small-scale fading

Large-scalefading

time

power

Summary: Wireless Channels

path loss

log (distance)

Received Signal Power (dB)

frequency

75

Preview: Challenges and Techniques of Wireless Design

Performance affected

Mitigation techniques

Shadow fading(large-scale fading)

Fast fading(small-scale, flat fading)Delay spread (small-scale fading)

received signal

strength

bit/packet error rate at deep fade

ISI

use fade margin—increase power or reduce distance

diversity

equalization; spread-spectrum; OFDM;

directional antenna

76

Representation of Wireless Channels

Received signal at time m is y[m], hl[m] is the strength of the l-th tap, w[m] is the background noise:

When inter-symbol interference is small:

(also called flat fading channel)

Backup Slides

77

Received Signal

78

2

2

1

1 )]/(2cos[)]/(2cos[),(d

cdtfd

cdtftfEd

cfd

cfd 12 22diff phase

d2

d1 receiver

cddf )(2 12

79

Multipath Fading with Mobility: A Simple Two-path Example

r(t) = r0 + v t, assume transmitter sends out signal cos(2 fc t)

r0

80

Received Waveform

v = 65 miles/h, fc = 1 GHz: fc v/c = 109 * 30 / 3x108 = 100 Hz

10 ms

Why is fast multipath fading bad?

deep fade

81

Small-Scale Fading

82

signal at sender

Multipath Can Spread Delay

signal at receiver

LOS pulsemultipathpulses

LOS: Line Of Sight

Time dispersion: signal is dispersed over time

83

Delay Spread RMS: root-mean-square

84

signal at sender

Multipath Can Cause ISI

signal at receiver

LOS pulsemultipathpulses

LOS: Line Of Sight

dispersed signal can cause interference between “neighbor” symbols, Inter Symbol Interference (ISI)

Assume 300 meters delay spread, the arrival time difference is 300/3x108 = 1 msif symbol rate > 1 Ms/sec, we will have serious ISI

In practice, fractional ISI can already substantially increase loss rate

85

Channel characteristics change over location, time, and frequency

small-scale fading

Large-scalefading

time

power

Summary: Wireless Channels

path loss

log (distance)

Received Signal Power (dB)

frequency

86

Dipole: Radiation Pattern of a Dipole

http://www.tpub.com/content/neets/14182/index.htmhttp://en.wikipedia.org/wiki/Dipole_antenna

Free Space Signal Propagation

87

1 0 1

t

1 0 1

t

1 0 1

t

at distance d?

Why Not Digital Signal (revisited) Not good for spectrum usage/sharing The wavelength can be extremely large

to build portal devices e.g., T = 1 us -> f=1/T = 1MHz ->

wavelength = 3x108/106 = 300m

88

Exercise Suppose fc = 1 GHz

(fc1 = 1 GHz, fc0 = 900 GHzfor FSK)

Bit rate is 1 Mbps Encode one bit at a time Bit seq: 1 0 0 1 0 1

Q: How does the wave look like for?

89

11 10 00 01

Q

I

11

01

10

00

A

t