Introduction to xDSL Part II

1
IntroductiontoxDSL Part II

• Yaakov J. Stein
• Chief ScientistRAD Data Communications

2
Introduction to xDSL

• I Background
• history, theoretical limitations, applications
• II Modems
• line codes, duplexing, equalization,
• error correcting codes, trellis codes
• III xDSL - What is x?
• xI,A,S,V - specific DSL technologies
• competitive technologies

3
Introduction to xDSL II

• How to make a modem
• PAM, FSK, PSK
• How to make a better modem
• QAM, CAP, TCM, V.34, V.90, DMT
• How to make a modem that works
• Equalizers, echo, timing, duplexing
• Why it doesnt
• Noise, cross-talk

4
The simplest modem - NRZ

• Our first attempt is to simply transmit 1 or 0
  (volts?)
• (short serial cables, e.g. RS232)
• Information rate number of bits transmitted per
  second (bps)

5
The simplest modem - continued

• There are a few problems ...
• DC
• Bandwidth
• Noise
• Timing recovery
• ISI
• Actually (except the DC) these problems plague
  all modems

6
The simplest modem - DC

• Whats wrong with a little DC?
• We want to transmit information - not power
• DC heats things up, and is often purposely
  blocked
• DC is used in telephony environment for powering

7
The simplest modem - DC

• So what about transmitting -1/1?
• This is better, but not perfect!
• DC isnt exactly zero
• Still can have a long run of 1 OR -1 that will
  decay
• Even without decay, long runs ruin timing
  recovery (see below)

8
Bit scrambling

• We can get rid of long runs at the bit level
• Bits randomized for better spectral properties
• Self synchronizing
• Original bits can be recovered by descrambler
• Still not perfect! (one to one transformation)

9
The simplest modem - DC

• What about RZ?
• No long 1 runs, so DC decay not important
• Still there is DC
• Half width pulses means twice bandwidth!

10
The simplest modem - DC

• T1 uses AMI (Alternate Mark Inversion)
• Absolutely no DC!
• No bandwidth increase!

11
The simplest modem - DC

• Even better - use OOK (On Off Keying)
• Absolutely no DC!
• Based on sinusoid (carrier)
• Can hear it (morse code)

12
NRZ - Bandwidth

• The PSD (Power Spectral Density) of NRZ is a sinc
  ( sinc(x) sin(x) )
• The first zero is at the bit rate (uncertainty
  principle)
• So channel bandwidth limits bit rate
• DC depends on levels (may be zero or spike)

x
13
OOK - Bandwidth

• PSD of -1/1 NRZ is the same, except there is no
  DC component
• If we use OOK the sinc is mixed up to the carrier
  frequency
• (The spike helps in carrier recovery)

14
From NRZ to n-PAM

• NRZ
• 4-PAM
• (2B1Q)
• 8-PAM
• Each level is called a symbol or baud
• Bit rate number of bits per symbol baud rate

GRAY CODE 10 gt 3 11 gt 1 01 gt -1 00 gt -3
GRAY CODE 100 gt 7 101 gt 5 111 gt 3 110 gt
1 010 gt -1 011 gt -3 001 gt -5 000 gt -7
111
001
010
011
010
000
110
15
PAM - Bandwidth

• BW (actually the entire PSD) doesnt change with
  n !
• So we should use many bits per symbol
• But then noise becomes more important
• (Shannon strikes again!)

BAUD RATE
16
Trellis coding

• Traditionally, noise robustness is increased
• by using an Error Correcting Code (ECC)
• But an ECC separate from the modem is not optimal
  !
• Ungerboeck found how to integrate demodulation
  with ECC
• This technique is called a
• Trellis Coded PAM (TC-PAM) or
• Ungerboeck Coded PAM (UC-PAM)
• We will return to trellis codes later

17
The simplest modem - Noise

• So what can we do about noise?
• If we use frequency diversity we can gain 3 dB
• Use two independent OOKs with the same
  information
• 
  (no
  DC)
• This is FSK - Frequency Shift Keying
• Bell 103, V.21 2W full duplex 300 bps (used today
  in T.30)
• Bell 202, V.23 4W full duplex 1200 bps (used
  today in CLI)

18
ASK

• What about Amplitude Shift Keying - ASK ?
• 2 bits / symbol
• Generalizes OOK like multilevel PAM did to NRZ
• Not widely used since hard to differentiate
  between levels
• Is FSK better?

19
FSK

• FSK is based on orthogonality of sinusoids of
  different frequencies
• Make decision only if there is energy at f1 but
  not at f2
• Uncertainty theorem says this requires a long
  time
• So FSK is robust but slow (Shannon strikes
  again!)

20
PSK

• What about sinusoids of the same frequency but
  different phases?
• Correlations reliable after a single cycle
• So lets try BPSK
• 
  1 bit / symbol
• or QPSK
• 
  
  2 bits /
  symbol
• Bell 212 2W 1200 bps
• V.22

21
PSK - Eye diagrams

• PSK demodulator extracts the phase as a function
  of time
• Proper decisions when eye is open
• Eye will close because of
• Timing errors
• Channel distortion
• Noise

22
QAM

• Finally, we can combine PSK and ASK (but not FSK)
• 2 bits per symbol
• V.22bis 2W full duplex 2400 bps used 16 QAM (4
  bits/symbol)
• This is getting confusing

23
The secret math behind it all

• The instantaneous representation
• x(t) A(t) cos ( 2 p fc t f(t) )
• A(t) is the instantaneous amplitude
• f(t) is the instantaneous phase
• This obviously includes ASK and PSK as special
  cases
• Actually all bandwidth limited signals can be
  written this way
• Analog AM, FM and PM
• FSK changes the derivative of f(t)
• The way we defined them A(t) and f(t) are not
  unique
• The canonical pair (Hilbert transform)

24
The secret math - continued

• How can we find the amplitude and phase?
• The Hilbert transform is a 90 degree phase
  shifterH sin(f(t) ) cos(f(t) )
• Hence
• x(t) A(t) cos ( 2 p fc t f(t) )
• y(t) H x(t) A(t) sin ( 2 p fc t f(t) )
• A(t) x2(t) y2(t)
• f(t) arctan( y(t) x(t) )

25
Star watching

• For QAM eye diagrams are not enough
• Instead, we can draw a diagram with
• x and y as axes
• A is the radius, f the angle
• For example, QPSK can be drawn (rotations are
  time shifts)
• Each point represents 2 bits!

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QAM constellations

• 16 QAM V.29 (4W 9600
  bps)
• V.22bis 2400 bps Codex
  9600 (V.29)
• 2W
• first non-Bell modem
  (Carterphone decision)
• Adaptive equalizer
• 
  Reduced PAR constellation
• 
  Today - 9600 fax!
• 8PSK
• V.27
• 4W
• 4800bps

27
Voicegrade modem constellations
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QAM constellations - continued

• What is important in a constellation?
• The number of points
  N
• The minimum distance between points
  dmin
• The average squared distance from the center E
  ltr2gt
• The maximum distance from the center
  R
• Usually
• Maximum E and R are given
• bits/symbol log2 N
• PAR R/r
• Perr is determined mainly by dmin

29
QAM constellations - slicers

• How do we use the constellation plot?
• Received point classified to nearest
  constellation point
• Each point has associated bits (well thats a
  lie, but hold on)
• Sum of errors is the PDSNR

30
Multidimensional constellations

• PAM and PSK constellations are 1D
• QAM constellations are 2D (use two parameters of
  signal)
• By combining A and f of two time instants ...
• we can create a 4D constellation
• From N times we can make 2N dimensional
  constellation!
• Why would we want to?
• There is more room in higher dimensions!
• 1D 2 nearest neighbors 2D 4 nearest
  neighbors
• ND 2N nearest neighbors!

How do I draw this?
31
Duplexing

• How do we send information in BOTH directions?
• Earliest modems used two UTPs, one for each
  direction (4W)
• Next generation used 1/2 bandwidth for each
  direction (FDD)
• Alternative is to use 1/2 the time (ping-pong)
  (TDD)
• Advances in DSP allowed 4W technology to be used
  in 2W
• V.32 used V.33 modulation with adaptive echo
  canceling

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Multiplexing Inverse multiplexing
data streams
physical line
data stream
physical lines

• Duplexing 2 data streams in 2
  directions on 1 physical line
• Multiplexing N data streams in 1
  direction on 1 physical line
• Inverse multiplexing 1 data stream in 1
  direction on N physical lines
• Inverse multiplexing (bonding) can be performed
  at different layers

33
Modern Voice Grade (not DSL) Modems

• V.34 (lt33.6 Kbps)
• Line probing and adaptive (water pouring?)
  spectral allocation
• Multidimensional QAM
• Huge constellations
• Laroia precoding
• Shell mapping (noninteger bits/symbol)
• V.90, V.92 (lt 56 Kbps)
• Asymmetric rates (V.90 uses V.34 for upstream)
• Downstream PCM(G.711) not analog modem
• Spectral shaping to overcome effects of D/A,
  XMFRs, etc.

34
The simplest modem - Timing

• Proper timing
• Provided by separated transmission
• uses BW or another UTP
• Improper timing
• causes extra or missed bits, and bit errors

35
Timing recovery

• How do we recover timing (baud rate) for an NRZ
  signal?
• For clean NRZ - find the GCF of observed time
  intervals
• For noisy signals need to filter b T / t
• t a t (1-a) T/b
• PLL
• How can we recover the timing for a PSK signal?
• The amplitude is NOT really constant (energy
  cut-off)
• Contains a component at baud rate
• Sharp filter and appropriate delay
• Similarly for QAM
• BUT as constellation gets rounder
• recovery gets harder

36
Carrier recovery

• Need carrier recovery for PSK / QAM signals
• How can we recover the carrier of a PSK signal?
• X(t) S(t) cos ( 2 p fc t ) where S(t)
  /- 1
• So X2(t) cos2 ( 2 p fc t )
• For QPSK X4(t) eliminates the data and emphasizes
  the carrier!
• Old QAM saying
• square for baud, to the fourth for carrier

37
Constellation rotation recovery

• How can we recover the rotation of the
  constellation?
• Simply change phase for best match to the
  expected constellation!
• How do we get rid of 90 degree ambiguity?
• We cant! We have to live with it!
• And the easiest way is to use differential
  coding!
• DPSK NPSK Gray code
• 000 100 110 010 011 111 101 001 000
• QAM put the bits on the transitions!

00
10
01
11
38
The simplest modem - ISI
39
QAM ISI

• The symbols overlap and interfere
• Constellations become clouds
• Only
  previous symbol
• Moderate ISI
• Severe ISI

40
Equalizers

• ISI is caused by the channel acting like a
  low-pass filter
• Can correct by filtering with inverse filter
• This is called a linear equalizer
• Can use compromise (ideal low-pass) equalizer
• plus an adaptive equalizer
• Usually assume the channel is all-pole
• so the equalizer is all-zero (FIR)
• How do we find the equalizer coefficients?

41
Training equalizers

• Basically a system identification problem
• Initialize during training using known data
• (can be reduced to solving linear algebraic
  equations)
• Update using decision directed technique (e.g.
  LMS algorithm)
• once decisions are reliable
• Sometimes can also use blind equalization
• e e (ai)

e
42
Equalizers - continued

• Noise enhancement
• This is a basic consequence of using a linear
  filter
• But we want to get as close to the band edges as
  possible
• There are two different ways to fix this problem!

noise
channel
modulator
equalizer
demodulator
filter
43
Equalizers - DFE

• ISI is previous symbols interfering with
  subsequent ones
• Once we know a symbol (decision directed) we can
  use it
• to directly subtract the ISI!
• Slicer is non-linear and so breaks the noise
  enchancement problem
• But, there is an error propogation problem!

linear
slicer
out
equalizer
feedback
filter
44
Equalizers - Tomlinson precoding

• Tomlinson equalizes before the noise is added
• Needs nonlinear modulo operation
• Needs results of channel probe or DFE
  coefficients
• to be forwarded

noise
Tomlinson precoder
channel
modulator
demodulator
filter
45
Trellis coding

• Modems still make mistakes
• Traditionally these were corrected by ECCs (e.g.
  Reed Solomon)
• This separation is not optimal
• Proof incorrect hard decisions - not obvious
  where to correct
• soft decisions - correct symbols
  with largest error
• How can we efficiently integrate demodulation and
  ECC?
• This was a hard problem since very few people
  were expert
• in ECCs and signal processing
• The key is set partitioning

46
Set Partitioning - 8PAM
Final step
First step
Original
Subset 0
Subset 1
00
01
10
11
47
Set Partitioning - 8PSK
48
Trellis coding - continued

• If we knew which subset was transmitted,
• the decision would be easy
• So we transmit the subset and the point in the
  subset
• But we cant afford to make a mistake as to the
  subset
• So we protect the subset identifier bits with
  an ECC
• To decode use the Viterbi algorithm

49
Multicarrier Modulation

• NRZ, RZ, etc. have NO carrier
• PSK, QAM have ONE carrier
• MCM has MANY carriers
• Achieve maximum capacity by direct water pouring!
• PROBLEM
• Basic FDM requires guard frequencies
• Squanders good bandwidth

50
OFDM

• Subsignals are orthogonal if spaced precisely by
  the baud rate
• No guard frequencies are needed

51
DMT

• Measure SNR(f) during initialization
• Water pour QAM signals according to SNR
• Each individual signal narrowband --- no ISI
• Symbol duration gt channel impulse response time
  --- no ISI
• No equalizer required