辅导案例-ELEC6021
Resear h Methods, ELEC6021 (EZ619) S Chen
What's New in Communi ations
Imagine a few s enarios:
{ In holiday, use your fan y mobile phone to take pi ture and send it to a friend
{ In airport waiting for boarding, swit h on your laptop and go to your favourite
web side
{ Or play game via Internet with your mobile phone
These are not Resear h, they are not even Development, they are Developed
Do you know these words:
CDMA, multi arrier, OFDM, spa e-time pro essing, MIMO, iterative or turbo
oding, intelligent network, smart antenna
There are plenty of opportunities for R & D
In this introdu tory ourse, we only go through some A B C
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Resear h Methods, ELEC6021 (EZ619) S Chen
Wireless and Mobile Networks
Current/future: 2G GSM, 3G UMTS (universal mobile tele ommuni ation system),
and MBS (mobile broadband system) being developed for 4G mobile system
G
S
M
/
G
P
R
S
G
S
M
G
S
M
/
H
S
C
S
D
G
S
M
/
E
D
G
E
MBS
HIPERLAN
UMTS
B−ISDNISDNFixed
Movable
mobile
Solow
Fast
mobile
mobility
User
data rate
Service
(bps)
9
.
6

k
6
4

k
1
4
4

k
3
8
4

k
2

M
2
0

M
1
5
5

M
Some improved 2G, HSCSD: high-speed ir uit swit hed data, GPRS: general pa ket radio servi e, EDGE: enhan ed
data rates for GSM evolution. Also, HIPERLAN: high performan e radio lo al area network
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Resear h Methods, ELEC6021 (EZ619) S Chen
For Those Keen to Read
Any good text books on digital ommuni ations, e.g.
I.A. Glover and P.M. Grant, Digital Communi ations. 2nd edition, Pearson, 2004.
J.D. Gibson, Prin iples of Digital and Analog Communi ations. 2nd edition,
M Millan, 1993
More advan ed level, e.g.
L. Hanzo, W. Webb and T. Keller, Single- and Multi-Carrier Quadrature
Amplitude Modulation. Wiley, 2000.
L. Hanzo, M. Munster, B.J. Choi and T. Keller, OFDM and MC-CDMA. Wiley,
2003
A. Paulraj, R. Nabar and D. Gore, Introdu tion to Spa e-Time Wireless
Communi ations. Cambridge University Press, 2003
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Resear h Methods, ELEC6021 (EZ619) S Chen
General Comments on Communi ations
Aim of tele ommuni ations: to ommuni ate information between geographi ally
separated lo ations via a ommuni ations hannel of adequate quality (at ertain
rate reliably)
channel
input output
The transmission will be based on digital data, whi h is obtained from (generally)
analogue quantities by
1. sampling (Nyquist: sampling with at least twi e the maximum frequen y), and
2. quantisation (introdu tion of quantisation noise through rounding o)
Transmitting at ertain rate requires ertain spe tral bandwidth
Here hannel means whole system, whi h has ertain apa ity, the maximum rate
that an be used to transmit information through the system reliably
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Resear h Methods, ELEC6021 (EZ619) S Chen
General Transmission S heme
A digital transmission s heme generally involves:
input source
encoding
channel
encoding lation
modu
output
channel
lation
channel
decodingdecoding
source demodu
Input/output are onsidered digital (analogue sampled/quantised)
CODEC, MODEM, hannel (transmission medium)
Your 2G mobile phone, for example, ontains a pair of transmitter and re eiver
(trans eiver), onsisting of a CODEC and MODEM
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Resear h Methods, ELEC6021 (EZ619) S Chen
Sour e En oding / De oding
A digital sour e is hara terised by:
{ sour e alphabet; symbol rate; symbol probabilities; and
probabilisti interdependen e of symbols
Sour e oding is about how to ode symbols (samples) into bits; sour e de oding
is about how to ode bits ba k into symbols (samples)
Ideally, sour e en oding should remove any redundan y from the signal to be
transmitted, and the sour e de oder has to restore the original sour e signal
distortionless (lossless oding)
In pra ti e, a small (potentially imper eptible) error may be allowed in the
oding/en oding (lossy oding)
e.g. spee h oding an be lossy, but medi al diagnosis images annot tolerate lossy
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Resear h Methods, ELEC6021 (EZ619) S Chen
Channel En oding / De oding
The hannel exhibits impairments:
{ hannel distortion (e.g. multipath and fading, insuÆ ient bandwidth), and
{ noise (e.g. thermal noise in re eiver amplier ir uit)
input output
channel
Depending on the severity of impairment, transmission errors may o ur
Channel oding adds redundan y to the transmitted signal to allow error dete tion
and/or orre tion on the re eiving side
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Resear h Methods, ELEC6021 (EZ619) S Chen
Channel Chara teristi s
Channel may introdu e amplitude and phase distortion
{ Bandwidth B, signal-to-noise ratio
S
N
, and
{ maximum rate for possible error-free transmission ( hannel apa ity C)
C = B log
2

1 +
S
N

[bits/se ℄
ChannelTx Mod RxDem
analogue channel
digital channel
Channel hara teristi s depend on design of transmission system:
{ transmission power, bandwidth, speed (data rate), reliability (error rate)
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Resear h Methods, ELEC6021 (EZ619) S Chen
Modulation / Demodulation
Modulation requires:
{ en oding of a bit stream into a symbol stream;
{ ltering (pulse shaping) to limit the bandwidth;
{ modulation with a arrier frequen y
Demodulation requires:
{ arrier re overy: orre t arrier phase have to be found;
{ syn hronisation (timing re overy): orre t sampling instan es have
to be found;
{ equalisation: to ope with hannel distortions
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Resear h Methods, ELEC6021 (EZ619) S Chen
Human Spee h
glottal
sound
vocal tract
transmission
lip
radiation
i
n
t
e
n
s
i
t
y
frequency
vowel
F F F
F
4
321
Spee h is either voi ed or unvoi ed, whi h results in quasi-periodi or noise-like
signals, respe tively; both types have a degree of redundan y
{ Voi ed spee h: vo al folds vibrate at a fundamental frequen y (100 . . . 150 Hz
for male, 200 . . . 300 Hz for female)
{ Arti ulation: spe tral shaping through the spe i resonan es of the vo al tra t
(peaks: formant frequen ies, whi h determine the vowel /a/,/i/,et .)
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Resear h Methods, ELEC6021 (EZ619) S Chen
Sour e Coding for Spee h
Methods of sour e oding for spee h:
{ Waveform oding tries to adapt the quantiser hara teristi
! good quality, moderate ompression possible
{ Predi tive oding tries to quantise only non-redundant information (i.e. the
predi tion error and predi tive model oeÆ ients)
! good quality, reasonable ompression
{ Analysis-by-synthesis is based on a model of the vo al tra t
! up to a very high ompression ratio at an arbitrarily poor quality
Spee h ode s:
{ Waveform ode s (waveform oding method)
{ Vo oders (analysis-by-synthesis and predi tive oding methods)
{ Hybrid ode s (trade-o between waveform oding and vo oding)
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Resear h Methods, ELEC6021 (EZ619) S Chen
Quality versus Bitrate
Classi ation of spee h ode s
VOCODERS
CODECS
HYBRID CODECS
WAVEFORM
1 2 4 8 16 32 64
Complexity
Delay
Fair
Poor
Excellent
quality
Speech
Good
Bitrate
(kbps)
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Resear h Methods, ELEC6021 (EZ619) S Chen
Waveform Code s
Time domain waveform oding:
ITU standard 64 kbps PCM: sampling at 8 kHz with 8 bits quantiser
ITU standard G.721, 32 kbps adaptive dierential PCM (ADPCM)
ITU standards G.726 and G.727
{ G.726 ode s: variable-rate for rates 16 { 40 kbps, allowing the network to
adjust quality/bitrate on instantaneous requirement
{ G.727 ode s: ore-bits and enhan ement bits, allowing the network to drop
enhan ement bits under heavy loads and to keep them when in light loads
Frequen y domain waveform oding:
Sub-band oding and adaptive transform oding
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Resear h Methods, ELEC6021 (EZ619) S Chen
Analysis-by-Synthesis Coding
The spee h is divided into small segments of 20ms duration;
for ea h segment, a parametri model (ex itation and synthesis lter oe.) is
sought for the spee h generation:
s[n]
s[n]~
original
speech
synthetic
speech
voiced/
unvoiced
excitation
generator
synthesis
filter
vocal
tract
compare
ex itation generation: either noise (unvoi ed) or a pulse train (voi ed);
the synthesis lter mat hes the spe tral shape of the harmoni s for voi ed ex itation,
and the vo al tra t
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Resear h Methods, ELEC6021 (EZ619) S Chen
Linear Predi tive Coding
General aim of predi tive oding is to make predi tion error or residual as
unpredi table as possible
Linear predi tive oding (LPC) employs more omplex ex itation models than used
in analysis-by-synthesis
{ LPC oeÆ ients and predi tion residuals are quantised to give designed bitrate
{ e.g. standard 13 kbps GSM spee h ode : 8 LPC oeÆ ients are en oded with
36 bits/20 ms update interval ! 1.8 kbps
Short-term predi tion: spee h analysis segments 20 ms duration
Long-term predi tion (LTP): LTP synthesis lter models the ne stru ture of the
spee h spe trum, after short-term predi tion
{ When employing a LTP, the residual error be omes truly unpredi table ! ode
ex ited linear predi tive oding (CELP) ode s
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Resear h Methods, ELEC6021 (EZ619) S Chen
Half-Rate GSM Spee h Code
parameter bits/frame
LPC oeÆ ients 28
LPC interpolation ag 1
ex itation mode 2
mode 0:
odebook 1 index 4 7 = 28
odebook 2 index 4 7 = 28
modes 1, 2, 3:
LTPD (subframe 1) 8
LTPD (subframes 2, 3, 4) 3 4 = 12
odebook 3 index 4 9 = 36
frame energy E
F
5
ex itation gain-related
quantity [E
S
1
E
1
℄ 4 5 = 20
total no of bits 112 bits/20 ms
bitrate 5.6 kbps
5.6 kbps ve tor sum ex ited linear predi tive
(VSELP) ode , similar to CELP
Four synthesis modes, depending on grade of
voi ing dete ted in the spee h
De isions on whi h ex itation mode to use is
based on LTP gain: high { highly orrelated
voi ed; low { noise-like un orrelated unvoi ed
5 bits for overall frame energy spans a
dynami range of 64 dB, when using a
stepsize of 2 dB and 32 steps
Residual error is oded as a odeword in a
ode book
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Resear h Methods, ELEC6021 (EZ619) S Chen
Video Sour e Coding
Video signal: luminan e signal of two spatial and a temporal dimension:
n+1
n+2
+3n
+4n
f
f
f
f
n
ftime
space
f
n
is frame at time n;
there are intra- (i.e. spatial) and inter-frame orrelations, i.e. orrelations within f
n
and between f
n
and past frames f
n1
, f
n2
, et .
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Resear h Methods, ELEC6021 (EZ619) S Chen
Video Compression
A pixel in a olour image an be presented
{ An independent intensity (luminan e), and two olour ( hrominan e) signals
known as hue and saturation; or
{ Three olour signals, intensity values of red, green and blue
MPEG standards, basi blo ks are
{ Motion ompensation
{ DCT
{ Variable length oding
MPEG-1: 1.5 Mbps; MPEG-2: 2 to 8 Mbps, now upto 30 Mbps for digital TV
MPEG-4: initiated in 1993, for intera tive multimedia appli ations
MPEG-7: started in 1997, for large stored database
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Resear h Methods, ELEC6021 (EZ619) S Chen
Channel Chara teristi s (I)
Passband hannel and baseband (remove modulator/demodulator) equivalen e:
c−f fc
f
B
H (f)
f
H (f)p
2B
carrier modulationb
−B
baseband hannel bandwidth B $ passband hannel bandwidth 2B
Channel has nite bandwidth, ideally
phase is linear and amplitude is at:
Bandwidth is the most pre ise resour e
Two killer fa tors: multipath and fading
f
phase
amplitude
channel bandwidth
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Resear h Methods, ELEC6021 (EZ619) S Chen
Channel Chara teristi s (II)
Channel noise: AWGN with a onstant
power spe trum density (PSD);
N /20
f
0
Power is the area under PSD, so WN has innitely large power;
but ommuni ation hannels are bandlimited, so noise is also bandlimited and has
a nite power:
ΣTx filter Rx filterchannel
n(t)
n (t) B
n(t)
channel
B
y(t) y(k)
y(t) y(k)x(k) x(t)
Σ
Channel has ertain apa ity
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Resear h Methods, ELEC6021 (EZ619) S Chen
Digital Modulation
Aim of MODEM: transmit bit stream at ertain rate reliably
Mod. Demod.Channel
bit stream bit stream
Carrier ommuni ations
{ Low frequen y signals (baseband) annot travel far
{ Most spe trum resour es ( hannels) are in RF bands (high frequen y passband)
Carrier: A os(2f

t+ ), three quantities, amplitude, frequen y and phase
Basi digital modulation forms: amplitude shift keying (ASK), frequen y shift
keying (FSK), and phase shift keying (PSK)
We will onsider quadrature amplitude modulation (QAM), whi h an be viewed
as a ombination of ASK and PSK
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Resear h Methods, ELEC6021 (EZ619) S Chen
Quadrature Amplitude Modulation
( )kxq
( )kx i ( )x ti
( )x tq
( )tg
( )tg
(ω t)sin
(ω t)cos
( )ts
Σδ t-kT( s )
s/p const.
map
bit
stream
q
QAM symbol D/A conversion QAM modulation
generation
Note: e.g., odd bits go to form x
i
(k) and even bits to form x
q
(k); x
i
(k) and x
q
(k) are inphase and quadrature omponents of the
x
i
(k) + jx
q
(k) QAM symbol; x
i
(k) and x
q
(k) are M -ary symbols.
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Resear h Methods, ELEC6021 (EZ619) S Chen
Quadrature Amplitude Demodulation
( )ts
(ω t)sin
(ω t)cos
(g -t )
(g -t )
( )kx i
( )kxq
( )x ti
( )x tq
const.
map
q
Σδ t-kT( )s
LP
LP
p/s bit
stream
QAM demodulation symbol dete tion bit re overy
Basi ir uits: arrier re overy, timing-re overy, dete tion
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Resear h Methods, ELEC6021 (EZ619) S Chen
Pulse Shaping
Channel has nite bandwidth, and transmitted digital signal has to be pulse shaped
To transmit at symbol rate f
s
needs ertain bandwidth B
T
and B
T
depends on
whi h pulse shaping used
x(t) = r(t) ?
+1
X
k=1
x[k℄Æ(t kT
s
) =
+1
X
k=1
x[k℄ r(t kT
s
)
Σδ t-kT )(
k[ ]x ( )tx( )tr
Pulse shaping lter r(t) would allow to retrieve the original digital data x[k℄ from
x(t): by limiting the bandwidth of x(t) to B
T
hannel bandwidth
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Resear h Methods, ELEC6021 (EZ619) S Chen
Pulse Shaping | Time Domain
−10 −8 −6 −4 −2 0 2 4 6 8 10
−0.2
0
0.2
0.4
0.6
0.8
1
time t/T
s
f
i
l
t
e
r

i
m
p
u
l
s
e

r
e
s
p
o
n
s
e
s
sinc
square pulse
raised cosine
Impulse response of all these lters have regular symbols-spa ed zero- rossing
(Nyquist system), but dierent supports; raised osine shown is a trun ated one
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Resear h Methods, ELEC6021 (EZ619) S Chen
Pulse Shaping | Frequen y Domain
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
−70
−60
−50
−40
−30
−20
−10
0
frequency 2f/fs
f
i
l
t
e
r

m
a
g
n
i
t
u
d
e

r
e
s
p
o
n
s
e
s

/

[
d
B
]
sinc
square pulse
raised cosine
Spe trum of the square pulse produ es onsiderable ex ess bandwidth beyond the
symbol rate f
s
; sin impra ti al to realize; trun ated raised osine easy to realize
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Resear h Methods, ELEC6021 (EZ619) S Chen
Pulse Shaping | Example
Binary (1) x[k℄, ea h is transmitted as a sin pulse; the peak of dierent shifted
sin fun tions oin ide with zero rossings of all other sin s:
−5 −4 −3 −2 −1 0 1 2 3 4 5
−1.5
−1
−0.5
0
0.5
1
1.5
time t/T
x
(
t
)
At re eiver, sampling at orre t symbol rate enables re overy of transmitted x[k℄
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Resear h Methods, ELEC6021 (EZ619) S Chen
Transmit and Re eive Filters
Pulse shaping fullls two purposes: limit the transmitted bandwidth, and enable
to re over the orre t sample values of transmitted symbols; su h a pulse shaping
r(t) is alled a Nyquist system
1. sin has a (passband) bandwidth B
T
= f
s
, (innite) raised osine has f
s

B
T
2f
s
depending on roll-o fa tor
2. a Nyquist time pulse have regular zero- rossing at symbol-rate spa ings to avoid
interferen e with neighboring pulses at orre t sampling instan es
The Nyquist system r(t) is separated into transmit lter g(t) and re eive lter
g(t) (square-root Nyquist systems)
1. the lter g(t) in the re eiver is also alled a mat hed lter (to g(t)); g(t) and
g(t) are basi ally identi al (square-root of r(t))
2. this division of r(t) enables suppression of out-of-band noise and results in the
maximum re eived SNR
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Resear h Methods, ELEC6021 (EZ619) S Chen
QAM Modulator / Demodulator
Re all modulator and demodulator of the QAM s heme:
(ω t)sin
(ω t)cos(ω t)cos
(ω t)sin
( )ts
( )x tq( )x tq
( )x ti( )x ti LP
LP
i( )x t
q( )x t
( )ts
c
c c
c
Modulation of in-phase and quadrature omponents to arrier frequen y !

:
x
i
(t) os(!

t) and x
q
(t) sin(!

t)
The transmitted signal is: s(t) = x
i
(t) os(!

t) + x
q
(t) sin(!

t)
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Resear h Methods, ELEC6021 (EZ619) S Chen
QAM | Demodulation
To explain the demodulation, assume perfe t transmission ^s(t) = s(t)
Demodulation for the \in-phase" omponent:
^x
0
i
(t) = s(t) os(!

t) = (x
i
(t) os(!

t) + x
q
(t) sin(!

t)) os(!

t)
= x
i
(t)
1
2


1 + os(2!

t)

+ x
q
(t)
1
2
sin(2!

t)
If the lowpass lter LP is sele ted appropriately ( ut-o frequen y !

), the
omponents modulated at frequen y 2!

an be ltered out, and hen e:
^x
i
(t) = LP

^x
0
i
(t)

=
1
2
x
i
(t)
A similar al ulation an be performed for the demodulation of ^x
q
(t):
^x
0
q
(t) = x
i
(t)
1
2
sin(2!

t) + x
q
(t)
1
2


1 os(2!

t)

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Resear h Methods, ELEC6021 (EZ619) S Chen
Modulation | Complex Notation
The modulation/demodulation an be expressed in omplex notation | in-phase
and quadrature omponents are \real" and \imaginary" part of the signal:
x(t) = x
i
(t) + j x
q
(t)
The transmitted signal is obtained by taking the real part only of a omplex arrier
(e
j!

t
) modulated signal:
s(t) = Refv(t)g = Refx(t) e
j!

t
g = x
i
(t) os(!

t) + x
q
(t) sin(!

t)
Flow graph:
-





-
Refg
-
6
x(t) s(t)
v(t)
e
j!

t
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Resear h Methods, ELEC6021 (EZ619) S Chen
Demodulation | Complex Notation
Flow graph for the omplex demodulation s heme:
-





-
LP
-
6
^s(t) ^x(t)
^x
0
(t)
e
j!

t
The demodulated signal:
^x
0
(t) = e
j!

t
s(t)
= ( os(!

t) + j sin(!

t)) (x
i
(t) os(!

t) + x
q
(t) sin(!

t))
= x
i
(t)
1
2

1 + os(2!

t) + j sin(2!

t)

+
jx
q
(t)
1
2

1 os(2!

t) j sin(2!

t)

The lowpass lter (LP) will again remove omponents modulated at 2!

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Resear h Methods, ELEC6021 (EZ619) S Chen
Bits to Symbols
The bit stream to be transmitted is serial / parallel multiplexed onto a stream z[k℄
of symbols of q bits/symbol (dis rete 2
q
levels)
Example for q = 2 bits/symbol: 4-ary modulation
(0,0)
(0,1)
(1,0)
(1,1)
0
1
1 1 1 1 10 0 0 0 0 time
time
z(k)
bit stream
symbol stream
Symbol rate is half of bit rate, and required bandwidth is half (but transmit power
has to be in reased); symbol stream is then pulse shaped ...
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Resear h Methods, ELEC6021 (EZ619) S Chen
Mapping to Constellation Pattern
The symbols z[k℄ are translated into values for the in-phase and quadrature
omponents, x
i
[k℄ and x
q
[k℄, by assigning them to points in a onstellation pattern
Example for a ase of q = 2 bits/symbol:
i( )x k
q( )x k
(1,1) (1,0)
(0,1) (0,0)
From the onstellation pattern, the values x
i
[k℄ and x
q
[k℄ are determined
In the re eiver, the onstellation point (and therefore the transmitted symbol) is
determined from ^x
i
[k℄ and ^x
q
[k℄
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Resear h Methods, ELEC6021 (EZ619) S Chen
Phase Shift Keying (PSK)
Phase shift keying (PSK): arrier phase is used to arry symbol information
Example of QPSK (minimum phase separation

2
) onstellation pattern and
transmitted signal s(t):
i( )x k
q( )x k
(1,0)
(1,1)
(0,1)
(0,0)
(00) (01) (11) (00) (10)

! time t
There are means to smooth dis ontinuities in the signal s(t)
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Resear h Methods, ELEC6021 (EZ619) S Chen
Amplitude Shift Keying (ASK)
Amplitude shift keying (ASK): arrier amplitude is used to arry symbol information
Example of 4-ary onstellation pattern and transmitted signal s(t):
i( )x k
q( )x k
(0,0)(0,1)(1,0)(1,1)
(00) (01) (11) (00) (10)

! time t
Note: (i) the quadrature omponent is not used; (ii) this is not \purely" ASK, as
a phase shift of is exploited in the modulation s heme.
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Resear h Methods, ELEC6021 (EZ619) S Chen
Combined ASK / PSK
QAM: ombines features of PSK and
ASK, and is bandwidth very eÆ ient
Example of 16-QAM (4 bits per
symbol):
q( )x k
i( )x k
Depending on the hannel quality, 64-QAM (6 bits/symbol), or 256-QAM (8
bits/symbol) are possible
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Resear h Methods, ELEC6021 (EZ619) S Chen
Gray Mapping
If noise or distortions are likely
to ause mis lassi ation in the
re eiver, Gray ode mapping an
minimize the bit error rate:
q( )x k
i( )x k
(0000) (0001)
(0101) (0111) (0110)
(0010)(0011)
(1100) (1101)
(1000) (1001)
(1111) (1110)
(1011) (1010)
(0100)
Adja ent onstellation points only vary in a single bit (minimum Hamming distan e)
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Resear h Methods, ELEC6021 (EZ619) S Chen
Carrier Re overy | Phase Oset
Previously, we assume
^s(t) = s(t) = x
i
(t) os(!

t) + x
q
(t) sin(!

t)
so that we an use e
j!

t
to remove arrier in demodulator
Most likely, the transmitted signal having traveled to the re eiver will a umulate
a phase oset ':
^s(t) = x
i
(t) os(!

t+ ') + x
q
(t) sin(!

t+ ')
Thus, the re eiver has to \re over" the arrier e
j(!

t+')
(in fa t the phase ') in
oder to demodulate the signal orre tly
Usually, this is done by means of some phase lo k loop based arrier re overy
39
Resear h Methods, ELEC6021 (EZ619) S Chen
Carrier Re overy | Frequen y Oset
Tx and Rx frequen y generators are unlikely to mat h exa tly. Consider
demodulation with a Rx lo al \ arrier" having a frequen y oset !

:
-





-
LP
-
6
^s(t) ^x(t)
^x
0
(t)
e
j(!

+!

)t
Even assuming ^s(t) = s(t), the demodulated signal prior to sampling is ^x(t) =
x(t) e
j!

t
, not ^x(t) = x(t)
The ee t of arrier frequen y mismat h is !

t and, like the phase dieren e ',
it has to be ompensated at the re eiver
!

t+' is alled arrier oset, and has to be \re overed" in order to demodulate
the signal orre tly
40
Resear h Methods, ELEC6021 (EZ619) S Chen
Syn hronisation
The pro ess of sele ting the orre t sampling instan es is alled syn hronisation,
also known as timing or lo k re overy
Tx and Rx lo ks are likely to have mismat h, lo k re overy tries to syn hronise
the re eiver lo k with the symbol-rate transmitter lo k to obtain samples at
appropriate instan es
This is equivalent to repla ing the impulse train
P
Æ(t kT
s
) by
P
Æ(t kT
s
)
with 0 T
s
:
kTs - τ
k[ ]x( )tx
The demodulated signal an be oversampled, and from the distribution (histogram)
of the sample sets for dierent , the one with the smallest deviation from dis rete
levels (depending on the QAM mode, 16-QAM, 64-QAM, et .) is hosen
41
Resear h Methods, ELEC6021 (EZ619) S Chen
Eye Diagram | Perfe t Channel
We are looking at sta ked 2 symbol period intervals of the demodulated signal
^x
i
(t) in a QPSK s heme (^x
i
(t) is BPSK):
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
time /symbol periods
x
i
(
t
)
This is alled an eye diagram; ideal sampling of ^x
i
(k) will sample the rossing
points ^x
i
(t) = 1 ! lo k/timing re overy ( 0:85T
s
)
42
Resear h Methods, ELEC6021 (EZ619) S Chen
Eye Diagram | Noisy Channel
With hannel noise at 3dB SNR, the eye diagram looks dierent:
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
time /symbol periods
x
i
(
t
)
But as long as the sampling points an be learly determined and the eye is \open",
^x
i
[k℄ will orre tly resemble x
i
[k℄. At higher noise levels, mis lassi ations an
o ur if the eye is \ losed"
43
Resear h Methods, ELEC6021 (EZ619) S Chen
Eye Diagram | Distorting Channel
The hannel is non-ideal with an impulse response (t) = Æ(t)
1
2
Æ(t T
s
=4),
where T
s
is the symbol period:
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time /symbol periods
x
i
(
t
)
The eye diagram is distorted; this together with noise ee t will make the eye
ompletely losed, leading to mis lassi ation due to intersymbol interferen e
44
Resear h Methods, ELEC6021 (EZ619) S Chen
Intersymbol Interferen e (ISI)
The response of an ideal pulse shaping lter with regular symbol-spa ed zero
rossings, and the same system in ombination with the hannel impulse response
(t) = Æ(t)
1
2
Æ(t T
s
=4):
0 1 2 3 4 5 6 7 8
−0.2
0
0.2
0.4
0.6
0.8
1
time / symbol periods
t
x
r
x

f
i
l
t
e
r

*

c
h
a
n
n
e
l
The system, Tx-lter { hannel { Rx lter, has lost the property of a Nyquist system;
peaks of the fun tion no longer oin ide with zero rossings of neighbouring pulses
45
Resear h Methods, ELEC6021 (EZ619) S Chen
Equalisation
For an ideal hannel without ISI, sampled re eiver output is
^x[k℄ = x[k ℄ + n[k℄
If the hannel has sever amplitude and phase distortion, sampling alone is unable
to re over the orre t symbols; This is be ause now (note the ISI)
^x[k℄ =
n

X
i=0

i
x[k i℄ + n[k℄
An equaliser is required to ombat the hannel distortion, and a typi al linear
equaliser is dened by:
y[k℄ =
n
w
X
l=0
w
l
^x[k l℄
The equaliser soft output y[k℄ is used to determine the transmitted symbol x[k℄
46
Resear h Methods, ELEC6021 (EZ619) S Chen
Equalisation Design
Let the z-transforms of the hannel and equaliser be C(z) and W (z); Similarly
dene X(z),
^
X(z) and Y (z)
-
X(z)
C(z)
-
W (z)
-
^
X(z)
Y (z)
Zero-for ing: we want to nd an equaliserW (z) su h that Y (z) is a delayed version
of the transmitted signal, Y (z) = z

X(z)
{ The solution is
W (z) C(z) = z

or W (z) = z

C
1
(z)
{ Completely eliminate ISI, but amplify the noise too mu h
Minimum mean square error: hoose the equaliser W (z) to minimise
MSE = E [jx[k ℄ y[k℄j
2

47
Resear h Methods, ELEC6021 (EZ619) S Chen
Adaptive Equalisation
Training: periodi ally provide the re eiver with x[k℄ so that equaliser an adjust its
weights w
i
using fx[k℄; ^x[k℄g
{ This happens for example in your GSM mobile phone
{ Training requires extra bandwidth and may not always be possible
Blind: equaliser has to adjust its weights having only the re eived signal f^x[k℄g
{ Constellations of a
blind equaliser's input
^x[k℄ and output y[k℄
-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20
I
m
Re
-20
-15
-10
-5
0
5
10
15
20
-20 -15 -10 -5 0 5 10 15 20
I
m
Re
48
Resear h Methods, ELEC6021 (EZ619) S Chen
User / Channel Separation
System has large apa ity and supports many users; dierent users or \ hannels"
mu h separate in some way:
{ FDMA: separation is in frequen y domain, dierent users have dierent arriers
and hannels o upy dierent frequen y bands
{ TDMA: same arrier or frequen y band, but separation is in time domain,
dierent users o upy dierent time slots
{ CDMA: no physi al separation, i.e. same arrier and time slot, but virtual
separation is in \ ode" domain, dierent users have dierent spreading odes
{ SDMA: if no separation in frequen y or time or ode domain, separation an still
be a hieved in spatial domain { this further enhan es spe trum eÆ ien y
CDMA is a key te hnology in 3G systems, and SDMA has an important role in 4G
systems
49
Resear h Methods, ELEC6021 (EZ619) S Chen
DS-CDMA System
DS-CDMA uplink, supporting K users
x (t)1
s (t)1 cos( t)ωc
τ1X X
cos( t)ωc
XX
s (t)
x (t)
2
2

.
.
.
cos( t)ωc
XX
cos( t)ωc
X
s (t− )1 τ1
x (t)1^
X
x (t)K^x (t)K
s (t)K
τK
n(t)

s (t− )2 τ2
X
.
.
.
K τK
x (t)2^
s (t− )
50
Resear h Methods, ELEC6021 (EZ619) S Chen
OFDM
Channel is divided into many sub hannels, ea h supports a sub arrier; Data stream
is divided into several bit streams, whi h are used to modulate several arriers
S ,0 S ,1 ...SN−1 S ,0 1...SN−1
fN−1demodulator
demodulator
demodulatormodulator
modulator fN−1
f 0
f1modulator
.
.
.
.
.
.
S/P P/S
SN−1
S
S0
1
S
S
SN−1
1
0
0f
1f
S ,
OFDM trans eiver
N−1SS1...S0
S0
S1
SN−1 .
.
.
S
/
P
I
F
F
T
P
/
S
s0
s1
sN−1
.
.
.
add
C.Ext.
cos(2 f t)pi c
picos(2 f t)c
S
/
P
F
F
T
P
/
S
rem.
C.Ext.
r
r
N−1r
1
0
.
.
.
N−1R
R
R
0
1
.
.
.
51
Resear h Methods, ELEC6021 (EZ619) S Chen
MIMO
SDMA indu ed MIMO system Spa e-time equalisation
user 1 modulator Σ
modulatoruser 2
.
.
.
modulatoruser K
Σ
Σ
n (t)
n (t)
1
2
L
x (t)1
x (t)L
2x (t)
n (t)
.
.
.
R
e
c
e
i
v
e
r
XXX
...
...
...+
+ +
X
...
+
...
...
+
+
XX
.
.
.
.
.
.
x (k)
x (k)L
x (k)1
∆ ∆ ∆
∆∆∆
∆ ∆ ∆
w
XX
1,0
* w1,1
* w1,D
*
w w w2,0 2,1 2,D
***
w w
w
L,D
L,0 L,1
* *
*
y(k)
+
X
2
52
Resear h Methods, ELEC6021 (EZ619) S Chen
Channel Capa ity and Channel Coding
Even in ase of a noisy hannel, there is no obstru tion of reliable transmission,
but only a limitation of the rate at whi h transmission an take pla e
Reliable transmission an be established by introdu ing redundan y into the
transmitted message to a hieve arbitrarily low bit errors
The hannel apa ity C forms a limit for the oding rate R:
R =
K
N
C
R = K=N is the ratio between the number of true (information) bits (K) and the
number of bits after redundan y has been introdu ed by hannel oding (N)
Example: binary symmetri hannel, P (1) = P (0) with p
e
= 0:1 ! C 0:5
! N 2K, i.e. at least 100% redundan y needs to be in orporated into the
transmitted bit stream in order to obtain an error-free re eption
53
Resear h Methods, ELEC6021 (EZ619) S Chen
Convolutional Coding (CC)
A onvolutional oder CC(N;K;L) takes a K-bit information symbol, whi h is
shifted into an L-bit shift register (holding a memory of L+ 1 bits!) and forms by
onvolution over this memory N output bits. Hen e oding rate R = K=N
Example: blo k diagram of a CC(2,1,2) half-rate (R =
1
2
) systemati oder:
Cn1
C2n
Bn
1Sn Sn
2
Output bits are dened by N generator polynomials; here: g
1
= [1 0 0℄, g
2
= [1 1 1℄
54
Resear h Methods, ELEC6021 (EZ619) S Chen
CC | State and State Transition Diagrams
With the previous CC(2,1,2) example, a state diagram (left) an be asso iated
00
11
01 10
Bn
Bn
=0
=1
Bn
Bn
S2n+1S1n+1S1nS2n Cn1C2n
01 01
00
11
10
00
10
11
=0
=1
11
00
10
01
01
00
10
11
The state transition diagram (right) indi ates the time dimension, and the ode
output with ea h iteration of the CC
55
Resear h Methods, ELEC6021 (EZ619) S Chen
Trellis Path
Consider the data sequen e B
n
= [0 0 1 0 1℄
Channel oding with previous (2,1,2) oder gives, starting from state S
1
n
S
2
n
= 00:
nS1 nS2
nB
Cn1C2n
01
10
11
00
0 0 1 0 1
00 00 11 01 10
This an be exploited for the hannel de oding
56
Resear h Methods, ELEC6021 (EZ619) S Chen
Viterbi De oding (I)
In our example, the transmitted bit sequen e is C
1
n
C
2
n
= [00 00 11 01 10℄
Assume, the re eived sequen e is
~
C
1
n
~
C
2
n
= [00 10 11 11 10℄, i.e. two bit errors have
o urred
Viterbi de oding inspe ts all possible paths in the trellis diagram
A metri to sele t the best (or \survivor") path is the minimum Hamming distan e
(number of dierent bits)
Therefore, starting from state 00, for ea h possible path the Hamming distan e is
evaluated and a umulated
If paths are joined, the ones with the larger Hamming distan e are dis arded; for
equal distan e, a random de ision is made
57
Resear h Methods, ELEC6021 (EZ619) S Chen
Viterbi De oding (II)
Trellis graphs for re eived sequen e
~
C
1
n
~
C
2
n
= [00 10 11 11 10℄:
nS1 nS2
Cn1C2n
nB
01
10
11
00 0
00 11 101110
0 1 1
1
0
2
2
2
1
2
4
2 3
0
1
1
2
20
1
1
3
0
1
3
0 2
1
2
0
21
3
0
1 3
2
1 4
~ ~
0 0 1 0 1
path
survivor
58
Resear h Methods, ELEC6021 (EZ619) S Chen
Viterbi De oding (III)
The nal survivor path determines the \most likely" transmitted sequen e; hen e
the Viterbi de oder is referred to as a maximum likelihood (ML) dete tor
If the orre t sequen e is yielded, the umulative Hamming distan e of the survivor
path gives the number of o urred bit errors
If too many bit errors have o urred or bit errors are awkwardly distributed, Viterbi
de oding will fail
The above hard-de ision de oding an be improved by soft-de ision de oding,
whereby \ner" levels of probability are assigned to the bran hes in the trellis
diagram
More generally, demodulator an output soft-de ision, thus de oder has soft-input
and an output soft-de ision (SISO)
59
Resear h Methods, ELEC6021 (EZ619) S Chen
Turbo Coding
Turbo en oder, assuming
systemati C(2; 1; 2)
Turbo de oder
iterative de oding
output bitspuncturing
and
multiplexing
encoder 1
encoder 2
input bits
interleaver
decoder 1 deinterlv
deinterlvdecoder 2
+
+




systematic
parity 1
parity 2
outputs
channel
soft
interleaver
60
Resear h Methods, ELEC6021 (EZ619) S Chen
Adaptive Equalisation
Re all the framework of adaptive equalisation with two operation modes, training and de ision-
dire ted, where the hannel model is:
r(k) =
n

X
i=0

i
s(k i) + n(k)
symbols are N -ary
s(k) 2 fs
i
= 2iN1; 1 i Ng
and AWGN n(k) has varian e
2
n
^s(k) r(k) r(k) s(k-d)
s(k-d)
-
Σ
Σ
n(k)
^
ddelay
y(k)
channel equaliser decisioncircuit
We will rst dis uss symbol-de ision equalisers and follow it by an introdu tion to the MLSE
An equaliser de ision delay d is ne essary for oping with non-minimum phase hannels
The zero-for ing equaliser H
E
(z) inverses the hannel H
C
(z): H
C
(z)H
E
(z) z
d
Solving this gives the linear equaliser's weights. Although this zero-for ing equaliser ompletely
eliminates ISI, it suers from a serious noise enhan ement problem
The most popular designs are the linear equaliser and de ision feedba k equaliser based on the
mean square error riterion
61
Resear h Methods, ELEC6021 (EZ619) S Chen
Linear Transversal Equaliser
The linear equaliser is given by:
y(k) =
M
X
i=0
w
i
r(k i) = w
T
r(k)
where r(k) = [r(k) r(k M)℄
T
and
M is the equaliser order
r(k-1) r(k-M)
...
s(k-d)
z z
^
filtering
r(k)
-1 -1
y(k)
circuit
decision
Typi al design is based on mean square error with the MMSE solution:
^
w = R
1
p,
where R = E[r(k)r
T
(k)℄ and p = E[r(k)s(k d)℄
Adaptive implementation typi ally adopts the LMS:
~
w(k+1) =
~
w(k)+r(k)e(k) with e(k) =

y(k) s(k d); training
y(k) ^s(k d); de ision-dire ted
62
Resear h Methods, ELEC6021 (EZ619) S Chen
De ision Feedba k Equaliser
The DFE onsists of a feedforward lter
and a feedba k lter:
y(k) = w
T
r(k) + b
T
^
s(k d)
=
M
X
i=0
w
i
r(ki)+
n
f
X
i=1
b
i
^s(kdi)
The DFE generally outperforms the LTE
in terms MSE and BER
z z
z z
z
f
. . .
. . .
s(k-d)decisionfiltering
r(k) r(k-1)
s(k-d-2) s(k-d-1)
-1 -1
-1 -1 -1
^
^^ ^
r(k-M)
circuit
s(k-d-n )
y(k)
Assuming equaliser de isions
^
s(k d) are orre t, the feedba k lter b
T
^
s(k d) eliminates a
large proportion of ISI without enhan ing noise and the feedforward lter w
T
r(k) takes are the
remaining ISI
Error propagation. O asionally error o urs in symbol dete tion, i.e. ^s(k d) 6= s(k d), it
is fed ba k and will ae t subsequent symbol dete tions ! further burst errors
Choi e of stru ture parameters. A simple pra ti al rule: feedforward lter overs entire hannel
dispersion, i.e. M = n

; de ision delay is set to d = n

; and feedba k lter order n
f
= n

63
Resear h Methods, ELEC6021 (EZ619) S Chen
Blind Equalisation
In blind equalisation, there is no training, an equaliser has to estimate the
transmitted symbols and/or hannel based only on the re eived samples r(k)
There are three lasses of blind equalisation algorithms
{ Joint data and hannel estimation: e.g. using blind or super trellis sear h
te hniques. This produ es the best results but an be omputationally prohibitive
{ Higher-order statisti s based methods: to identify the hannel using r(k)
only, 2nd order statisti is insuÆ ient as it is phase blind. Higher-order statisti s
based methods an over ome this problem. This approa h produ es very good
results but omputational ost an be very expensive
{ Bussgang-type adaptive FIR lters: optimize some non-MSE type ost
fun tions using sto hasti gradient, omputationally very simple
We will dis uss the 3rd lass. Sin e there is no desired response s(k d) for the
adaptive lter, one has to \invent" some substitute ! the resulting non-MSE ost
fun tions generally have lo al minima, and this often auses problems
64
Resear h Methods, ELEC6021 (EZ619) S Chen
Constant Modulus Algorithm
We will onsider the general QAM ase and use omplex notations, e.g. the hannel
taps
i
=
R;i
+ j
I;i
, the re eived signals r(k) = r
R
(k) + jr
I
(k), the symbols
s(k) = s
R
(k) + js
I
(k), and the equaliser weights w
i
= w
R;i
+ jw
I;i
Dene the onstant
2
= E[js(k)j
4
℄=E[js(k)j
2
℄, and onsider adaptive lter or
blind equaliser:
y(k) = w
T
r(k)
with w = [w
0
w
M

T
and r(k) = [r(k) r(k M)℄
T
Although QAM symbols do not fall on the onstant modulus ir le of radius
p

2
,
by penalizing equaliser output y(k) whi h deviates from this ir le, the orre t
symbol onstellation an be restored
This leads to the CMA, whi h is the most popular blind equaliser for high-order
QAM signalling, as it has simple omputational requirements similar to those of
the LMS
65
Resear h Methods, ELEC6021 (EZ619) S Chen
CMA ( ontinue)
The CMA an be viewed to adjust w by minimizing the non- onvex ost fun tion

J
CMA
(w) = E[(jy(k)j
2

2
)
2

using a sto hasti gradient method, i.e. a tually through minimizing (jy(k)j
2

2
)
2
At sample k, given y(k) = w
T
(k)r(k), the equaliser weights are updated using:
(k) = y(k)(
2
jy(k)j
2
)
w(k + 1) = w(k) + (k)r

(k)

where is a very small positive adaptive gain and r

(k) is the onjugate of r(k)
Compare this with the LMS, where (k) = s(k d) y(k)
There are many solutions w
s
that minimize the ost fun tion

J
CMA
(w). One of them, w
opt
,
restores the orre t signal onstellation and is orresponding to the MMSE solution
The weight ve tors that minimize

J
CMA
(w) are thus
w
s
= exp(j)w
opt
; 0 < 2
This undesired phase shift annot be resolved by the CMA (all blind equalisers suer more or less a
similar problem), and must be eliminated by other means, e.g. using dierential en oding
66
Resear h Methods, ELEC6021 (EZ619) S Chen
Complex Variable Derivative
Complex-valued variable derivative is dened as
J(w)
w
i
=
1
2

J
w
R;i
+ j
J
w
I;i

Note that y(k) = w
0
r(k) + + w
M
r(k M) and
J(w) =
1
2
(jy(k)j
2

2
)
2
=
1
2
(y(k)y

(k)
2
)
2
Hen e we have
J
w
i
=
1
2
2(y(k)y

(k)
2
)
y(k)y

(k)
w
i
= (jy(k)j
2

2
)

y(k)
w
i
y

(k) + y(k)
y

(k)
w
i

Note
y(k)
w
i
=
1
2

y(k)
w
R;i
+ j
y(k)
w
I;i

y(k)
w
R;i
= r
R
(k i) + jr
I
(k i) and
y(k)
w
I;i
= r
I
(k i) + jr
R
(k i)
This leads to
y(k)
w
i
= 0
67
Resear h Methods, ELEC6021 (EZ619) S Chen
Complex Variable Derivative ( ontinue)
Note
y

(k)
w
i
=
1
2

y

(k)
w
R;i
+ j
y

(k)
w
I;i

y

(k)
w
R;i
= r
R
(k i) jr
I
(k i) and
y

(k)
w
I;i
= r
I
(k i) jr
R
(k i)
This leads to
y

(k)
w
i
= r

(k i)
Therefore, the gradient
rJ(w) =

J(w)
w

T
= y(k)(jy(k)j
2

2
)r

(k)
Using
w(k + 1) = w(k) + (rJ(w(k)))
leads to
w(k + 1) = w(k) + y(k)(
2
jy(k)j
2
)r

(k)
68
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