1. Wireless PHY: Digital Demodulation and Wireless Channels
Y. Richard Yang
09/13/2012

2. Outline
Admin and recap
Digital demodulation
Wireless channels

3. Admin
Assignment 1 posted

4. Recap Demodulation Low pass filter and FIR Convolution Theorem
Digital modulation/demodulation
ASK, FSK, PSK
General representation

5. Recap: gi() for BPSK 1: 0: Note: g1(t) = -g0(t)
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
g0(t)
g1(t)

6. Recap: Signaling Functions gi() for QPSK
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

7. Recap: QPSK Signaling Functions as Sum of cos(2πfct), sin(2πfct)11 00 10
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]
sin(3π/4) sin(2πfct)
01: cos(- π/4 + 2πfct) t in [0, T]
sin(π/4) sin(2πfct) 01
[cos(3π/4), sin(3π/4)]
[cos(π/4), sin(π/4)]
cos(2πfct)
[cos(3π/4), -sin(3π/4)]
[-sin(π/4), cos(π/4)]
We call sin(2πfct) and cos(2πfct) the bases.

8. Recap: Demodulation/Decoding
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?

9. Recap: Design xT x2 x1 x0 h0 h1 h2 hT *
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] xT x2 x1 x0 h0 h1 h2 hT *

10. Determining the Best h where w is noise,
Design objective: maximize peak pulse signal-to-noise ratio

11. Determining the Best h Assume Gaussian noise, one can derive
Using Fourier Analysis and Convolution Theorem:

12. Determining the Best h
Apply Schwartz inequality
By considering

13. Determining the Best h

14. Determining Best h to UsexT x2 x1 x0 h0 h1 h2 hT * xT x2 x1 x0 gT g2 g1 g0 *

15. Summary of Progress
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

16. Applying Scheme to BPSK
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)

17. Applying Scheme to BPSK
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)

18. Interpretation
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
<x, g1(t)>
g1=cos(2πfct)[0, T]
<g0(t), g1(t)>
<g1(t), g1(t)>
=-<g1(t), g1(t)>

19. Applying Scheme to QPSK: Attempt 1
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?

20. Decoding for QPSK using bases
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?)
Question: what is the meaning of a00, b00

21. QPSK Demodulation/Decoding
sin(2πfct)
[a00,b00]
[a01,b01]
[ax,bx]
cos(2πfct)
[a10,b10]
[a11,b11]
Q: how to decode?

22. Look into Noise 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

23. 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

24. Likelihood Detection 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

25. Back to QPSK
Yry: Ignored noise effect

26. QPSK Demodulation/Decoding
sin(2πfct)
[a00,b00]
[a01,b01]
[ax,bx]
cos(2πfct)
[a10,b10]
[a11,b11]
Q: what does maximum likelihood det pick?

27. General Matched Filter Detection: Implementation for Multiple Sig Func.
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

28. Computing Coordinates
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

29. Example: Matched Filter => Correlation Detector
received signal x

30. BPSK vs QPSK fc: carrier freq. Rb: freq. of data 10dB = 10; 20dB =10011 10 00 01

31. Spectral Density = bit rate ------------------- width of spectrum used
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?
Spectral Density = bit rate width of spectrum used

32. Context
Previous demodulation considers only additive noise, and does not consider wireless channel’s effects
We next study its effects

33. Outline
Admin and recap
Digital demodulation
Wireless channels

34. Signal Propagation

35. Antennas: Isotropic Radiator
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 z y z
ideal
isotropic
radiator y x x
Q: how does power level decrease as a function of d, the distance
from the transmitter to the receiver?

36. 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(2ft), 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)

37. Log Scale for Large Span
dB = 10 log(times)
Log Scale for Large Span
~100B
10,000 times
10,000 x 1,000
40 dB
= 70 dB
~10M
1000 times
30 dB
~10K
Slim/Gates
Obama

38. Path Loss in dB dB = 10 log(times) 40 dB 40 + 30 = 70 dB 30 dB 10 W
10,000 x 1,000
40 dB
= 70 dB
power
1 mW
1000 times
30 dB
1 uW
source d1 d2

39. dBm (Absolute Measure of Power)
dBm = 10 log (P/1mW)
dBm (Absolute Measure of Power)
10 W
40 dBm
10,000 times
10,000 x 1,000
40 dB
= 70 dB
power
1 mW
1000 times
30 dB
1 uW
-30 dBm
source d1 d2

40. Number in Perspective (Typical #)
Data rate (Mbps)
Receive threshold (dBm)
Signal/Noise (dB) 6
-82
6.02 9
-81
7.78 12
-79
9.03 18
-77
10.79 24
-74
17.04 36
-70
18.8 48
-66
24.05 54
-65
24.56

41. Exercise: 915MHz WLAN (free space)
Transmit power (Pt) = 24.5 dBm
Receive sensitivity = dBm
Receiving distance (Pr) =
Gt=Gr=1

42. 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

43. Exercise: 915MHz WLAN (Two-ray ground reflect)
Transmit power (Pt) = 24.5 dBm
Receive sensitivity = dBm
Receiving distance (Pr) =
Gt=Gr=hr=ht=1

44. Real Antennas Q: Assume frequency 1 Ghz,  = ?
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,  = ?

45. Figure for Thought: Real Measurements

46. Signal Propagation: Complexity
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
diffraction
reflection
refraction
scattering
shadow fading

47. 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

49. 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

50. Example Shadowing Effects
i.e. reduces to ¼ of signal 10 log(1/4) = -6.02

51. JTC Indoor Model for PCS: Path Loss
Shadowing path loss follows a log-normal distribution (i.e. L is normal distribution) with mean:
A: an environment dependent fixed loss factor (dB)
B: the distance dependent loss coefficient,
d : separation distance between the base station and mobile terminal, in meters
Lf : a floor penetration loss factor (dB)
n: the number of floors between base station and mobile terminal

52. JTC Model at 1.8 GHz

53. Outline Admin and recap Digital demodulation Wireless channels Intro
Shadowing
Multipath

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

55. Multipath Example: Outdoor
Example: reflection from the ground or building
ground

56. Multipath Effect (A Simple Example)
Assume transmitter sends out signal cos(2 fc t) d1 d2
phase difference:

57. Multipath Effect (A Simple Example)
Where do the two waves totally destruct?
Q: where do the two waves construct?

58. Option 1: Change Location
If receiver moves to the right by /4: d1’ = d1 + /4; d2’ = d2 - /4;
->
By moving a quarter of wavelength, destructive turns into constructive.
Assume f = 1G, how far do we move?

59. Option 2: Change Frequency

60. Multipath Delay Spread
RMS: root-mean-square

61. Multipath Effect (moving receiver)
example d d1 d2
Suppose d1=r0+vt
d2=2d-r0-vt
d1d2

62. Derivation
See for cos(u)-cos(v)

63. Derivation
See for cos(u)-cos(v)

64. Derivation
See for cos(u)-cos(v)

65. Derivation
See for cos(u)-cos(v)

66. Derivation
See for cos(u)-cos(v)

67. Derivation
See for cos(u)-cos(v)

68. Waveform v = 65 miles/h, fc = 1 GHz: fc v/c =
109 * 30 / 3x108 = 100 Hz
10 ms
deep fade
Q: How far does a car drive in ½ of a cycle?

69. Multipath with Mobility

70. Effect of Small-Scale Fading
no small-scale
fading

71. Multipath Can Spread Delay
signal at sender
LOS pulse
Time dispersion: signal is dispersed over time
multipath
pulses
signal at receiver
LOS: Line Of Sight

72. JTC Model: Delay Spread
Residential Buildings

73. Multipath Can Cause ISI
Dispersed signal can cause interference between “neighbor” symbols, Inter Symbol Interference (ISI)
Assume 300 meters delay spread, the arrival time difference is /3x108 = 1 ns
if symbol rate > 1 Ms/sec, we will have serious ISI
In practice, fractional ISI can already substantially increase loss rate
signal at sender
LOS pulse
multipath
pulses
signal at receiver
LOS: Line Of Sight

74. Summary: Wireless Channels
Channel characteristics change over location, time, and frequency
Received
Signal
Large-scale
fading
Power
power
(dB)
path loss
log (distance)
time
small-scale fading
signal at receiver
LOS pulse
multipath
pulses
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
use fade margin—increase power or reduce distance
bit/packet error rate at deep fade
diversity
equalization; spread-spectrum; OFDM; directional antenna
ISI

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)

77. Backup Slides

78. Received Signal d2 d1
receiver

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
More detail see page 16 Eqn. (2.13):

80. Received Waveform v = 65 miles/h, fc = 1 GHz:
10 ms
deep fade
v = 65 miles/h, fc = 1 GHz:
fc v/c = 109 * 30 / 3x108 = 100 Hz
Why is fast multipath fading bad?

81. Small-Scale Fading

82. Multipath Can Spread Delay
signal at sender
LOS pulse
Time dispersion: signal is dispersed over time
multipath
pulses
signal at receiver
LOS: Line Of Sight

83. RMS: root-mean-square
Delay Spread
RMS: root-mean-square

84. Multipath Can Cause ISI
dispersed signal can cause interference between “neighbor” symbols, Inter Symbol Interference (ISI)
Assume 300 meters delay spread, the arrival time difference is /3x108 = 1 ms
if symbol rate > 1 Ms/sec, we will have serious ISI
In practice, fractional ISI can already substantially increase loss rate
signal at sender
LOS pulse
multipath
pulses
signal at receiver
LOS: Line Of Sight

85. Summary: Wireless Channels
Channel characteristics change over location, time, and frequency
Received
Signal
Large-scale
fading
Power
power
(dB)
path loss
log (distance)
time
small-scale fading
frequency

86. Dipole: Radiation Pattern of a Dipole

87. Free Space Signal Propagation1 t
at distance d ?

88. 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

89. Exercise Suppose fc = 1 GHz (fc1 = 1 GHz, fc0 = 900 GHz for FSK)
Bit rate is 1 Mbps
Encode one bit at a time
Bit seq:
Q: How does the wave look like for? 11 10 00 01 Q I A t