Andrea Goldsmith

100 Pages·2010·12.13 MB·English

by  Gerhard Kramer

Andrea
 Goldsmith

Stanford
 University

2010 School of Information Theory
University of Southern California
Aug. 5,
 2010
Future
 Wireless
 Networks

Ubiquitous
 Communica/on
 Among
 People
 and
 Devices

Next-­‐genera4on
 Cellular

Wireless
 Internet
 Access

Wireless
 Mul4media

Sensor
 Networks

Smart
 Homes/Spaces

Automated
 Highways

In-­‐Body
 Networks

All
 this
 and
 more
 …
Challenges

  Fundamental
 capacity
 limits
 of
 wireless
 networks

are
 unknown
 and,
 worse
 yet,
 poorly
 defined.

  Wireless
 network
 protocols
 are
 generally
 ad-­‐hoc

  Applica4ons
 are
 heterogeneous
 with
 hard

constraints
 that
 must
 be
 met
 by
 the
 network

  Energy
 and
 delay
 constraints
 change
 fundamental

design
 principles
Fundamental
 Network
 Capacity

The
 Shangri-­‐La
 of
 Informa/on
 Theory

  Much
 progress
 in
 finding
 the
 capacity
 limits
 of

wireless
 single
 and
 mul4user
 channels

  Limited
 understanding
 about
 the
 capacity
 limits
 of

wireless
 networks,
 even
 for
 simple
 models

  System
 assump4ons
 such
 as
 constrained
 energy
 and

delay
 may
 require
 new
 capacity
 defini4ons

  Is
 this
 elusive
 goal
 the
 right
 thing
 to
 pursue?

Shangri-La is synonymous with any earthly paradise;
a permanently happy land, isolated from the outside world
Wireless
 Channel
 and
 Network
 Capacity

Fundamental
 Limit
 on
 Data
 Rates

The set of simultaneously achievable rates with P →0
e
R
3
R
R
2
1
(R R …,R )
12, 13,, 1n
  Main
 drivers
 of
 channel
 capacity

  Bandwidth
 and
 power

  Sta4s4cs
 and
 dynamics
 of
 the
 channel

  What
 is
 known
 about
 the
 channel
 at
 TX
 and/or
 RX

  If
 feedback
 is
 available

  Number
 of
 antennas

  Single
 or
 mul4-­‐hop
In
 the
 beginning...

  Shannon’s
 Mathema/cal
 Theory
 of
 Communica/on
 derived

fundamental
 data
 rate
 limits
 of
 digital
 communica4on
 channels.

 
 Shannon
 capacity
 is
 independent
 of
 the
 transmission
 and

recep4on
 strategy;
 Depends
 only
 on
 channel
 characteris4cs

  Shannon
 capacity
 for
 sta4onary
 and
 ergodic
 channels
 is
 the

channel’s
 maximum
 mutual
 informa4on

 
 Significance
 of
 capacity
 comes
 from
 Shannon's
 coding
 theorem

and
 converse

  Show
 capacity
 is
 the
 channel’s
 maximum
 “error-­‐free”
 data
Shannon
 Capacity

  Discrete
 memoryless
 channel
 w/
 input
 X∈X,
 output
 Y∈ Y

has
 mutual
 informa/on
 (MI)

  Related
 to
 the
 no4on
 of
 entropy:

  Shannon
 proved
 DMC
 has
 capacity
 equal
 to
 it’s
 maximum
 MI

  Maximum
 is
 taken
 over
 all
 possible
 input
 distribu4ons
Coding
 Theorem

  Coding
 theorem:
 A
 code
 exists
 with
 rate
 R=C-­‐∈
 with

probability
 of
 error
 approaching
 zero
 with
 blocklength

  Shannon
 provided
 a
 random
 code
 with
 this
 property

  Decoding
 is
 based
 on
 the
 no4on
 of
 typical
 sets:

Sets
 where
 the
 probability
 of
 the
 realiza4on
 of
 xn,
 yn,
 (xn,yn)
 approximates

 
that
 of
 their
 entropies

Decode
 sequence
 xn
 that
 is
 jointly
 typical
 with

 received
 sequence
 yn

 
For
 codes
 with
 large
 blocklengths,
 probability
 of
 error
 approaches
 0

 
  Shannon’s
 “Communica4on
 in
 the
 presence
 of
 noise”

 P

 +

 N

uses
 geometry
 for
 AWGN
 coding
 theorem
 proof

  Input
 occupies
 sphere
 of
 radius
 P

  Output
 occupies
 sphere
 of
 radious
 P+N

  Capacity
 corresponds
 to
 number
 of
 input

messages
 that
 lie
 in
 nonoverlapping
 output
 spheres
Converse

  Oben
 based
 on
 Fano’s
 inequality
 for
 message
 W
nR=H(W)=H(W|Yn)+I(W;Yn)
≤ H(W|Yn)+I(Xn(W);Yn) since W=f(Xn)
≤  1+P (n)nR+ I(Xn(W);Yn)≤ H(W|Yn)+nC
e
  This
 implies

P (n)≥1-1/(nR)-C/R as n→∞
e
Error
 bounded
 away
 from
 zero
 for
 R>C
  Fano’s
 inequality
 based
 on
 cutset
 bound

 
 Loose
 for
 many
 channels
 /networks
Capacity
 of
 AWGN
 Channels

n[i]
x[i] y[i]
+
  Discrete-­‐4me
 channel;
 x[i]
 is
 input
 at
 4me
 i,
 y[i]
 is
 output,
 n[i]

is
 iid
 sample
 of
 white
 Gaussian
 noise
 process
 w/
 PSD
 N
0

  Channel
 bandwidth
 is
 B,
 received
 power
 is
 P

  Received
 signal-­‐to-­‐noise
 power
 ra4o
 (SNR)
 is
 SNR=γ=P/(N B)

0
  Maximum
 mutual
 informa4on
 achieved
 with
 Gaussian
 inputs

C=Blog (1+SNR) bps
2