Transcript S.72-227 Digital Communication Systems

S-72.227 Digital Communication Systems
The Viterbi algorithm, optimum coherent bandpass
modulation
Topics today





Revision of convolutional codes state diagrams
Viterbi decoding: phase trellis and surviving path, ending the decoding
Principle of convolutional code error rate bound determination
Bandpass digital transmission
– ASK, QAM, PSK, FSK, MSK
– waveforms (LP-presentation) and constellation diagrams
– modulator blocks
– spectral properties, transmission BW
– binary and M-ary cases
Optimum coherent detection
– Matched filter and correlator principle
– Matched filter impulse response
Timo O. Korhonen, HUT Communication Laboratory
Representing convolutional code compactly: code
trellis and state diagram
x ' j  mj2  mj1  mj
x '' j  mj2  mj
Shift register states
Timo O. Korhonen, HUT Communication Laboratory
0 0 1 -> 1 1
0 1 1 -> 0 1
1 1 0 -> 0 1
1 0 1 -> 0 0
...
Exercise: State diagrams

,,,,
xj '  mj
x j ''  m j 2  m j
x j '''  m j 3  m j 1
Timo O. Korhonen, HUT Communication Laboratory
a=000
a
b=001
b
c=010
c
d=011
d
e=100
e
f=101
f
g=110
g
h=111
h
For instance from d to h you go with
the input mj=1, thus xj’=1,
xj’’=1+d’’=0, and xj’’’=d’+d’’’=1
x j ' x j '' x j '''
111
101
100
xj '  mj
x j ''  m j 2  m j
x j '''  m j 3  m j 1
Timo O. Korhonen, HUT Communication Laboratory
011
The Viterbi algorithm



Exhaustive maximum likelihood method must search all paths in phase
trellis for 2k bits for a (n,k,L) code
By Viterbi-algorithm search depth can be decreased to comparing 2k 2 L
surviving paths where 2L is the number of nodes and 2k is the number
of branches coming to each node (see the next slide!)
Problem of optimum decoding is to find the minimum distance path
from the initial stage back to initial stage (below from S0 to S0). The
minimum distance is the sum of all path metrics
ln p(y, xm )  j0 ln p( y j | xmj )
Channel output sequence
at the RX

TX Encoder output sequence
for the m:th path
that is maximized by the correct path
The Viterbi algorithm gets its
efficiency via concentrating into
survivor paths of the trellis
Timo O. Korhonen, HUT Communication Laboratory
The survivor path




Assume for simplicity a convolutional code with k=1, and up to 2k = 2
branches can enter each stage in trellis diagram
Assume optimal path passes S. Metric comparison is done by adding the
metric of S into S1 and S2. At the survivor path the accumulated metric
is naturally smaller (otherwise it could not be the optimum path)
For this reason the non-survived path can
be discarded -> all path alternatives need not
to be considered
Note that in principle whole transmitted
sequence must be received before decision.
However, in practice storing of states for
input length of 5L is quite adequate
2L nodes
Timo O. Korhonen, HUT Communication Laboratory
2k branches enter each node
Example of using the Viterbi algorithm

Assume received sequence is
y  01101111010001
and the (n,k,L)=(2,1,2) code shown below. Determine the Viterbi
decoded output sequence!
states
(Note that for this encoder code rate is 1/2 and memory depth L = 2)
Timo O. Korhonen, HUT Communication Laboratory
The maximum likelihood path
After register length L+1=3
branch pattern begins to repeat
(1)
Smaller accumulated
metric selected
(1)
1
(1)
(1)
(2)
1
(0)
(Branch Hamming distance
in parenthesis)
First depth with two entries to the node
The decoded ML code sequence is 11 10 10 11 00 00 00 whose Hamming
distance to the received sequence is 4 and the respective decoded
sequence is 1 1 0 0 0 0 0 (why?). Note that this is the minimum distance path.
(Black circles denote the deleted branches, dashed lines: '1' was applied)
Timo O. Korhonen, HUT Communication Laboratory
How to end-up decoding?



In the previous example it was assumed that the register was finally
filled with zeros thus finding the minimum distance path
In practice with long code words zeroing requires feeding of long
sequence of zeros to the end of the message bits: wastes channel
capacity & introduces delay
To avoid this path memory truncation is applied:
– Trace all the surviving paths to the
depth where they merge
– Figure right shows a common point
at a memory depth J
– J is a random variable whose
magnitude shown in the figure (5L)
has been experimentally tested for
negligible error rate increase
– Note that this also introduces the
delay of 5L!
Timo O. Korhonen, HUT Communication Laboratory
J  5L stages of the trellis
Error rate determination
of convolutional codes




Error rate depends on
– channel SNR
– input sequence length, number
of errors is scaled to sequence length
– code trellis topology
These determine which path in trellis was followed while decoding
Assume all-zero sequence is transmitted and so far no errors have not
been occurred. Hence the maximum likelihood path having the
minimum distance dfree is followed.
Now, all the paths producing errors must have a distance that is larger
than the all-zero path distance (dfree), e.g. there exists the bound

pe   ad p2 (d )
d d
free
Timo O. Korhonen, HUT Communication Laboratory
Number of paths at
the Hamming distance d
Probability of the d:th path at
the Hamming distance d
Selected convolutional code error rates

Probability of the d:th path at the Hamming distance d depends on
decoding method. For antipodal (polar) signaling it is bounded by
 2 Eb

p
(
d
)

Q
R
d
pe   ad p2 (d ) where 2
 N C 
d d
 0


free
that can be further simplified for low error probability channels by ( x  0)
remembering that then the following bound works well:
1
Q( x) 
2


 exp(
2
/ 2)d 
x
1
Q( x)  exp  x 2 / 2 
2
Here is a table of selected convolutional codes and their associative code
gains RCdf /2 (df = dfree):
We return to both convolutional code and
block code error rates after discussing
bandpass modulation
Timo O. Korhonen, HUT Communication Laboratory
Bandpass digital transmission




Carrier wave modulation is required to transmit messages via suitable,
usually long distance medium as air, copper or coaxial cable, fiber class
or even water
The message reserves a transmission band around the allocated carrier
that depends on message bandwidth or amount of information
Discuss
– modulated carrier spectral properties
– amplitude, frequency and phase shift keying
– binary and M-ary signaling
– coherent and noncoherent detection
Compare various methods with respect of their
– spectral efficiency
– error rate performance in AWGN channel
– hardware complexity
Timo O. Korhonen, HUT Communication Laboratory
CW binary waveforms
Timo O. Korhonen, HUT Communication Laboratory
Spectral analysis of CW signals

Apply the quadrature-carrier (complex envelope) form that separates the
slow and fast varying parts of the carrier:
xC (t )  AC cos(C t   (t )   )
xC (t )  AC cos(C t   )cos  (t )  AC sin(C t   )sin  (t )
xC (t )  xi (t )cos(C t   )  xq (t )sin(C t   )

The spectra can be decomposed by using modulation theorem
v(t ) cos( C t   )  1 V ( f  f C ) exp  j   V ( f  f C ) exp   j 
2
to the following four components:
AC2
X C (t )  Gi ( f  f C )  Gi ( f  f C )  Gq ( f  f C )  Gq ( f  f C ) 
4
Timo O. Korhonen, HUT Communication Laboratory
M-ary signal equivalent low-pass spectrum:
general expression

The respective equivalent lowpass spectra is
Glp ( f )  Gi ( f )  Gq ( f )


M-ary (M-level) baseband signal with the rate r=1/D is represented by
xi (t )   ak p(t  kD)
Pulse PSD
k
For which the spectra is can be shown to be

Gi ( f )   a r P( f )  (ma r )  P(nr )  ( f  nr )
n 
2
2
2
2
where  a and ma relate to inter-symbol correlation properties for the
transmitted symbols ak by
2

2
2
 a  ma , n  0
Ra (n)  E[ak ak n ]  
2
m
,n  0

a
For rectangular NRZ-pulses with pD (t )  u t  D / 2  u t  D / 2
Fourier transform yields the PSD:
PD ( f ) = F  pD (t )
2
Timo O. Korhonen, HUT Communication Laboratory
2
1
2 f
 D sinc fD  2 sinc
r
r
2
2
M-ary amplitude shift keying (ASK)


Take the I-component to be an unipolar NRZ signal, hence
xi (t )   ak pD (t  kD), ak  0,1,..., M  1
k
For this signal the mean and variance are
1
2 f
PD ( f )  2 sinc
r
r
ma  (M  1) / 2  a2  (M 2  1) /12 hence

2
Gi ( f )   a r P( f )  (ma r )  P(nr )  ( f  nr )
n 
2
2
 M  1  ( f )
M 1
Glp ( f )  Gi ( f ) 
sinc 2 ( f / r ) 
12r
4
2
2
2
2
For high spectral efficiency
strive to get a rapid
decay
M 2
Note the carrier component that
does not convey information
Transmission BW:
BT  rb / log 2 M
Spectral width inversely proportional
to the number of bits
Spectral efficiency:
rb / BT  log 2 M
Timo O. Korhonen, HUT Communication Laboratory
Binary Quadrature Amplitude Modulation (QAM)

Note that the orthogonal branch rates are half of the data rate
Hence for ma  0, a2  1

Gi ,q ( f )   a2 rb P( f ) 2  (ma rb ) 2  P(nrb ) 2  ( f  nrb )
n 


0

 P ( f ) 2  1 sinc 2 f  P( f ) 2  1 sinc 2 f
 D
r2
r
(rb / 2) 2
rb / 2
4
2
2
2 2 f
 Glp ( f )   a rb P( f )  sinc
rb
rb

Therefore QAM is twice as spectral efficient as ASK
Also, impulse that wastes power is missing
Timo O. Korhonen, HUT Communication Laboratory
Binary phase reversal keying (PRK)



For two phases PRK is called as binary phase shift keying (BPSK)
Modulated carrier can be expressed by
xC (t )  AC  pD (t  kD)cos(C t    k )
k
This is in quadrature carrier form
 xi (t )  k pD (t  kD)cos k

 xq (t )  k pD (t  kD)sin k

xC (t )  AC cos(C t   (t )   )
xC (t )  AC cos (t )cos(C t   )  AC sin  (t )sin( C t   )
xi ( t )
xq ( t )
The phases are
2a  N 

BPSK N =0, M =2
 
, a  0,1,...M  1, N  0 or 1 
k
k
M
k
 k  0,
Note that phase shift keying has always constant envelope, still for N=1,
M=4, phase constellation of PRK and QAM are similar
 PRK has however better overall error rate performance due to missing
carrier component
Timo O. Korhonen, HUT Communication Laboratory

PRK constellations

Below PRK with M=4 and M=8 and QAM constellations
 
k
2ak N

, ak  0,1,...M  1
M
M
Timo O. Korhonen, HUT Communication Laboratory
QAM constellation
N=1, no constant envelope
Example

Draw the signal constellation and spectrum for a 2-PSK signal with
I
Q
2ak 
k 
 , M  4, ak  0,1
M
M
 I k  cos k  1/ 2
k   / 4  
Qk  sin k  1/ 2
xi (t )   pi (t  kTb ) / 2  Gi ( f )  1  ( f ), pi (t  kTb )  1
2
k
T
T
1
2 f
PD ( f )  2 sinc
r
r

2
2
Gi ,q ( f )   a2 r P( f )  (ma r ) 2  P(nr )  ( f  nr )
n
Qk  T  xq (t )dt  0, Qk  T  xq2 (t )dt  1/ 2
0
0
1
 Gq ( f ) 
2
rb
2
1
2
Pq ( f )  1
2rb
sinc 2 f / rb
 Glp ( f )  1  ( f )  1 sinc2 f / rb
2
2rb
Timo O. Korhonen, HUT Communication Laboratory
2
Note the unnecessary
DC-component
Determining variance < Qi2 > in Maple®
Timo O. Korhonen, HUT Communication Laboratory
Frequency Shift Keying (FSK)

Two frequency modulation methods can be used:
Discrete generator M-ary FSK


Continuous phase FSK
M-ary FSK signal is defined by
xC (t )  AC  cos(C t     d ak t ) pD (t  kD) d  2 f d
k
Adjacent frequencies are space by 1/Ts=2fd
f k  fC  f d ak , ak  1, 2,...  (M  1)

Phase continuity can be obtained by selecting generator frequencies as
multiples of data rate r=1/D:  D   N
d
2 f d D   N
D / TS  N , TS  1/(2 f d )
Timo O. Korhonen, HUT Communication Laboratory
Example of continuous discrete generator M-ary
FSK signals
D  Tb
fk
f k  f C  f d ak
f1
ak  1, 2,...  ( M  1)
D / TS  N , TS  1/(2 f d )
f2
2 f d  4,6,8Hz
f3
Timo O. Korhonen, HUT Communication Laboratory
Binary FSK (Sunde’s FSK)


For Sunde’s FSK select M  2, D  1/ rb , N  1, 2 f D  rb
Assume rectangular data-pulses:
pD (t )  u(t )  u(t  kTb )

The lowpass i and q components are obtained from the general FSK
expression (constant envelope!):
xC (t )  AC  cos( C t     d ak t )
k


 AC  cos( C t   )cos( d ak t )  sin( C t   )sin( d ak t ) 
k


x
xq
xi (t )  AC  cos( d ak t ), f d  rb / 2 constant rate: produces at two
k
sided spectra impulses at  rb / 2
d t
i
xq (t )  AC  sin( d ak t )
k
 AC  ak sin  rbt  pD (t  kTb ), ak  1  Qk  0, Qk2  1
k
Qk
Timo O. Korhonen, HUT Communication Laboratory
p (t )
Sunde’s FSK PSD

PSD was defined by Glp ( f )  Gi ( f )  Gq ( f )

Gq ( f )   a rb P( f )  (ma rb )  P(nrb )  ( f  nrb )
n
2
2
2
2
0
where now
p (t )  sin  d t  pD (t  kTb ), f d  rb / 2 & modulation theorem 
P( f ) 
2
1 
f  rb / 2
f  rb / 2 
sinc

sinc

4rb2 
rb
rb

4  cos  f / rb 
 2 2
2
 rb   2 f / rb   1
2
2
For high spectral efficiency
strive to get a rapid decay
r
r 
1 
2

Glp ( f )    f  b     f  b    rb P( f )
4 
2
2 

Timo O. Korhonen, HUT Communication Laboratory
Continuous phase FSK

The baseband waveform is defined by

x(t )   ak pD (t  kD), ak  1, 2,...  ( M  1)
k 0
and the modulated carrier is
xC (t )  AC cos C t     d


x( )d  

0

t
Why does this term
enables continuous phase?
Substituting x(t) into the integral yields then by using piecewise
integration

a0 t , 0  t  D


x( )d   
a0 D  a1 (t  D ), D  t  2 D
0
 k 1
a j D  ak (t  kD ), kD  t  (k  1) D
 
j o


t
 
Thus the CPFSK can be expressed as
xC (t )  AC  cos C t    k   d ak (t  kD)  pD (t  kD) k   d D  a j

k 1
k 0
j 0
Timo O. Korhonen, HUT Communication Laboratory
Minimum-shift keying (MSK)

Analyze CFSK by MSK that is its frequently used form. Now
2 f d  rb / 2, ak  1, k 

k 1
 aj
2 j 0
and its PSD can be shown to be
f  rb / 4
f  rb / 4 
1
Glp ( f )  sinc
 sinc
rb 
rb / 2
rb / 2 

2
Note that continuous carrier phase can be illustrated as a phase trellis:
Timo O. Korhonen, HUT Communication Laboratory
Coherent binary systems: Error rate analysis







Coherent systems utilize carrier phase information to recover data, thus
 optimum error rate can be obtained
 carrier reconstruction required at the receiver
 carrier reconstruction must be precise
Non-coherent systems decode data without carrier phase reference, thus
 error rate is deteriorated
 detection easier when carrier phase recovery related circuits omitted
A good compromise of the coherent and non-coherent techniques are the
differentially coherent systems
Concentrate first on AWGN system only
Focus on OOK, FSK, PSK
Band-limited channels are considered later. Then techniques are introduced to
alleviate or remove produced Inter Symbolic Interference (ISI)
Important special case are fading channels that are characterized by statistical
multipath propagation
Timo O. Korhonen, HUT Communication Laboratory
Optimum binary detection

Any bandpass signal can be presented by


xC (t )  AC  I k pi (t  kTb )cos(C t   )   Qk pq (t  kTb )sin(C t   )
k
k

0
This can be expressed by using different waveforms for ‘0’ and ‘1’ bits:
xC (t )   sm (t  kTb ), m  0,1
k
sm (t )  AC I k pi (t ) cos( C t )  Qk pq (t )sin( C t )

0
Received waveforms, that indicate the transmitted bits, are recovered
coherently by using matched filtering or correlation receiver:
Timo O. Korhonen, HUT Communication Laboratory
Bases of optimum detection

Received signal consist of bandpass filtered signal and noise, that is then
sampled at the time instants tk :
Y  y(tk )  zm  n

Assuming that the BPF has the impulse response h(t), the signal at the
sampling instant is then expressed by
zm  sm (t  kTb )  h(t ) t t
k

( k 1) Tb
kTb


Tb
0

sm (  kTb )h(tk   )d 
sm ( )h(Tb   )d 
( x  y (t ) 
 x( ) y(t   )d  )
A
Note how this expression shows
the MF and correlator reception!
How the bandpass filter impulse response should be selected to
maximize received SNR at the time instant of sampling? Let’s first have
a look on optimum binary error rate:
Timo O. Korhonen, HUT Communication Laboratory
Optimum binary error rate

Assuming ‘0’ and ‘1’ reception is equally likely, error happens when
the H0 signal hits the dashed region (or H1 its left-hand side). The
decision threshold is at Vopt  ( z1  z0 ) / 2
For optimum performance
we wish to maximize the
SNR
 z1  z0 / 2 
Therefore for equally likely ‘0’ or/and ‘1’ the error rate is
1
pe 
 2
Timo O. Korhonen, HUT Communication Laboratory


Vopt
exp    z0  / 2 2  d   Q  z1  z0 / 2 
2
2
Impulse response of matched filtering

The signal part of the SNR expression is the difference signal after the
bandpass filter (z1 and z0 are convoyed by s1 and s0 respectively):
z1  z0 
2




 s ( )  s ( ) h(T   )d 
1
b
The noise component of the SNR after the bandpass filter is
 
2

0
2

2


h ( ) d 
2

And the SNR after the matched filter is:
z1  z2

2
4
2



 s ( )  s ( ) h(T   )d 
1
0

2


2
b
h(Tb   ) d 
2
sm (t )  AC I k pi (t )cos(C t )  Qk pq (t )sin(C t ), m  0,1
Timo O. Korhonen, HUT Communication Laboratory
Using Schwarz’s inequality for optimum filtering

Schwarz’s inequality:




2
2
V ( )W ( )d 



W ( ) d 
z1  z2

2
4
2

2






2
V ( ) d 
2
 s ( )  s ( ) h(T   )d 
1
0

2


b
h(Tb   ) d 
provided that W ( )  KV ( )
2
K
2



 s ( )  s ( )  d 
2
1
0
that holds when W ( )  KV ( )  h(Tb   )  K  s1 ( )  s0 ( )

Therefore SNR is maximized at the time instant of sampling by using
hopt (t )  K  s1 (Tb  t )  s0 (Tb  t ) 
Timo O. Korhonen, HUT Communication Laboratory
Matched filtering and correlator reception

Note that both circuits fulfil the expression zm 

Tb
0
sm ( )hopt (Tb   )d 
hopt (t )  K  s1 (Tb  t )  s0 (Tb  t ) 
Matched filter
Correlator
Timo O. Korhonen, HUT Communication Laboratory