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
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