Wireless PHY: Digital Demodulation and

Wireless Channels

Y. Richard Yang

09/13/2012

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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]1-1

g1(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)

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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 on-off 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 Transform 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

****

Matched Filter Decision

is called Matched filter.Example

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decision time

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 0

cos(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]1-1

g1(t)g0(t)

Interpretation

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For any signal s, <s, g1> computes the coordinate (projection) of s when using g1 as a base cleaner if g1 is normalized (i.e., scale g1 by

sqrt of <g1, g1>), 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]

Q: Where did we see a similar computation format for computing a00, b00?

For received signal x, computes ax=<x, cos> and bx=<x, sin> (how many correlations 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

2

1)(

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

Maximum 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?32

Spectral Density =

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

width of spectrum used

Context

Previous demodulation considers only additive noise, and does not consider wireless channel’s effects Wireless channels more than add some

noise to a signaling function g(t)

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)

d

cdtftfEd

)]/(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

reflection

scattering

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

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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

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)

d1d2

1

11 ][2cos

d

tfcd

ft2cos

2121 22)(2 21dd

c

ddfff c

dcd

2

22 ][2cos

d

tfcd

phase difference:

Assume transmitter sends out signal cos(2 fc t)

Multipath Effect (A Simple Example)

Where do the two waves totally destruct?

Where do the two waves totally construct?

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integer2121

dd

c

ddf

Sensitivity: 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

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

Backup Slides

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Dipole: Radiation Pattern of a Dipole

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