S-72.227 Digital Communication Systems The Viterbi algorithm, optimum coherent bandpass modulation

S-72.227 Digital Communication Systems

The Viterbi algorithm, optimum coherent bandpass modulation

Timo O. Korhonen, HUT Communication Laboratory

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


Representing convolutional code compactly: code trellis and state diagram

2 1'

j j j jx m m m

2''

j j jx m m

Shift register states

0 0 1 -> 1 10 1 1 -> 0 11 1 0 -> 0 1 1 0 1 -> 0 0

...


Exercise: State diagrams

,,,,

2

3 1

'

''

'''

j j

j j j

j j j

x m

x m m

x m m


a

b

c

d

e

f

g

h

a=000

b=001

c=010

d=011

e=100

f=101

g=110

h=111

2

3 1

'

''

'''

j j

j j j

j j j

x m

x m m

x m m

111

100

011

101

' '' '''j j j

x x x

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


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 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

that is maximized by the correct path The Viterbi algorithm gets its

efficiency via concentrating intosurvivor paths of the trellis

0ln ( , ) ln ( | )jm j mjp p y x

y x

Channel output sequenceat the RX

TX Encoder output sequencefor the m:th path

2 2k L


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 canbe discarded -> all path alternatives need notto be considered

Note that in principle whole transmittedsequence must be received before decision.However, in practice storing of states for input length of 5L is quite adequate

2 branches enter each nodek

2 nodesL


Example of using the Viterbi algorithm

Assume received sequence is

and the (n,k,L)=(2,1,2) code shown below. Determine the Viterbi decoded output sequence!

01101111010001y

(Note that for this encoder code rate is 1/2 and memory depth L = 2)

states


The maximum likelihood path

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)

(1)

(1)

(0)

(2)

(1)

(1)

1

1

Smaller accumulated metric selected

First depth with two entries to the node

After register length L+1=3branch pattern begins to repeat

(Branch Hamming distancein parenthesis)


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 pointat a memory depth J

– J is a random variable whosemagnitude shown in the figure (5L) has been experimentally tested fornegligible error rate increase

– Note that this also introduces thedelay of 5L! 5 stages of the trellisJ L


Error rate determination of convolutional codes

Error rate depends on

– channel SNR

– input sequence length, numberof 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

2( )

free

e dd d

p a p d

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

that can be further simplified for low error probability channels by remembering that then the following bound works well:

Here is a table of selected convolutional codes and their associative code gains RCdf /2 (df = dfree):

2

0

2( ) b

C

Ep d Q R d

N

21( ) exp / 2

2Q x x

We return to both convolutional code and block code error rates after discussing bandpass modulation

( 0)x

2( )

free

e dd d

p a p d

21( ) exp( / 2)

2 xQ x d

where


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


CW binary waveforms


Spectral analysis of CW signals

Apply the quadrature-carrier (complex envelope) form that separates the slow and fast varying parts of the carrier:

The spectra can be decomposed by using modulation theorem

to the following four components:

( ) cos( ( ) )

( ) cos( )cos ( ) si

( )sin(

n( )sin ( )

)( )co ( )( ) s

C C C

C C C C C

C qi CC

x t A t t

x t A t

x t t

t A t t

x tx t t

2

( ) () ( ( )(4

) )i

C

q CCC qi C CG f f G f f G f f G f f

AX t

1( )cos( ) ( )exp ( )exp2C C Cv t t V f f j V f f j


M-ary signal equivalent low-pass spectrum: general expression

The respective equivalent lowpass spectra is

M-ary (M-level) baseband signal with the rate r=1/D is represented by

For which the spectra is can be shown to be

where relate to inter-symbol correlation properties for the transmitted symbols ak by

For rectangular NRZ-pulses with Fourier transform yields the PSD:

( ) ( ) ( )lp i q

G f G f G f

( ) ( )i k

kx t a p t kD

2 22 2( ) ( ) ( ) ( ) ( )i a a

nG f r P f m r P nr f nr

2 anda a

m2 2

2

, 0( ) E[ ]

, 0a a

a k k n

a

m nR n a a

m n

Pulse PSD

22 2 2 2

2

1( ) ( ) sinc sinc

D D

fP f p t D fD

r r =F

( ) / 2 / 2D

p t u t D u t D


M-ary amplitude shift keying (ASK)

Take the I-component to be an unipolar NRZ signal, hence

For this signal the mean and variance are

( ) ( ), 0,1,..., 1i k D k

kx t a p t kD a M

( 1) / 2a

m M 2 2( 1) /12a

M

22

2 11( ) ( ) sinc ( / ) ( )

12 4lp i

MMG f G f f r f

r

2/ log

T bB r M

2M

Spectral efficiency:

2/ log

b Tr B M

Note the carrier component that does not convey information

2 22 2( ) ( ) ( ) ( ) ( )i a a


2 2

2

1( ) sinc

D

fP f

r rhence

Transmission BW:

Spectral width inversely proportional to the number of bits

For high spectral efficiencystrive to get a rapiddecay


Binary Quadrature Amplitude Modulation (QAM)

Note that the orthogonal branch rates are half of the data rate Hence for

Therefore QAM is twice as spectral efficient as ASK Also, impulse that wastes power is missing

20, 1a a

m 2 22 2

,

2 22 2

2 2

0

( ) ( ) ( ) ( ) ( )

1 1( ) sinc ( ) sinc

( / 2) / 2

i q a b a b b bn

D

b b

G f r P f m r P nr f nr

f fP f P f

r r r r

22 24 2( ) ( ) sinc

lp a b

b b

fG f r P f

r r


Binary phase reversal keying (PRK)

For two phases PRK is called as binary phase shift keying (BPSK) Modulated carrier can be expressed by

This is in quadrature carrier form

The phases are

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

( ) ( )cos( )C C D C k

kx t A p t kD t

( ) ( )cos

( ) ( )sin

i D kk

q D kk

x t p t kD

x t p t kD

2, 0,1,... 1, 0 or 1k

k k

a Na M N

M

BPSK =0, =2

0,k

N M

( ( ))

( ) cos( ( ) )

( ) cos ( )cos( ) sin ( )sin( )C C C

C C C C C

i qx tt x

x t A t t

x t A t t A t t


PRK constellations

Below PRK with M=4 and M=8 and QAM constellations

2, 0,1,... 1k

k k

a Na M

M M QAM constellation

N=1, no constant envelope


Example

Draw the signal constellation and spectrum for a 2-PSK signal with

2 , 4, 0,1k

k k

aM a

M M

cos 1/ 2 / 4

sin 1/ 2

k k

k

k k

I

Q

1( ) ( ) / 2 ( ) ( ), ( ) 12i i b i i bk

x t p t kT G f f p t kT

1 2 1 2

0 0

( ) 0, ( ) 1/ 2T T

k q k qQ T x t dt Q T x t dt

221( ) ( ) sinc /2 2

bq q b

b

rG f P f f rr

21 1( ) ( ) sinc /2 2lp bb

G f f f rr

Note the unnecessary DC-component

2 22 2

,( ) ( ) ( ) ( ) ( )

i q a an

G f r P f m r P nr f nr

2 2

2

1( ) sinc

D

fP f

r r

I

Q


Determining variance < Qi2 > in Maple®


Frequency Shift Keying (FSK)

Two frequency modulation methods can be used:

M-ary FSK signal is defined by

Adjacent frequencies are space by 1/Ts=2fd

Phase continuity can be obtained by selecting generator frequencies asmultiples of data rate r=1/D:

Discrete generator M-ary FSK Continuous phase FSK

( ) cos( ) ( )C C C d k D

kx t A t a t p t kD

, 1, 2,... ( 1)k C d k k

f f f a a M

2

/ , 1/(2 )

d

d

S S d

D N

f D N

D T N T f

2d d

f


Example of continuous discrete generator M-ary FSK signals

2 4,6,8Hzd

f

bD T

/ , 1/(2 )S S d

D T N T f

1, 2,... ( 1)k C d k

k

f f f a

a M

kf

1f

2f

3f


Binary FSK (Sunde’s FSK)

For Sunde’s FSK select Assume rectangular data-pulses:

The lowpass i and q components are obtained from the general FSK expression (constant envelope!):

2, 1/ , 1,b

M D r N 2D b

f r

( ) cos( )

cos( )cos( ) sin( )sin( )i

C C C d kk

C C d k C d kk

qx x

x t A t a t

A t a t t a t

( ) cos( ), / 2i C d k d b

k

dt

x t A a t f r

2

( )

( ) sin( )

sin ( ), 1 0, 1

q C d kk

C k b D b k k kk

kQ p t

x t A a t

A a r t p t kT a Q Q

( ) ( ) ( )D b

p t u t u t kT

/ 2b

rconstant rate: produces at two sided spectra impulses at


Sunde’s FSK PSD

PSD was defined by

where now

2 22 2

0

( ) ( ) ( ) ( ) ( )q a b a b b b


( ) ( ) ( )lp i q

G f G f G f

2

2

2

2

22 2

( ) sin ( ), / 2 & modulation theorem

1 / 2 / 2( ) sinc sinc

4

4 cos /

2 / 1

d D b d b

b b

b b b

b

b b

p t t p t kT f r

f r f rP f

r r r

f rr f r

21( ) ( )

4 2 2b b

lp b

r rG f f f r P f

For high spectral efficiencystrive to get a rapid decay


Continuous phase FSK

The baseband waveform is defined by

and the modulated carrier is

Substituting x(t) into the integral yields then by using piecewise integration

Thus the CPFSK can be expressed as

0( ) ( ), 1, 2,... ( 1)

k D kk

x t a p t kD a M

0( ) cos x( )

C C C d

t

x t A t d

0

( ) cos ( ) ( )C C C k d k D

kx t A t a t kD p t kD

1

0

k

k d jj

D a

0

0 1

10

, 0

x( ) ( ), 2

( ), ( 1)k

j kj o

ta t t D

d a D a t D D t D

a D a t kD kD t k D

Why does this term enables continuous phase?


Minimum-shift keying (MSK)

Analyze CFSK by MSK that is its frequently used form. Now

and its PSD can be shown to be

Note that continuous carrier phase can be illustrated as a phase trellis:

1

02 / 2, 1,

2

k

d b k k jj

f r a a

2

/ 4 / 41( ) sinc sinc

/ 2 / 2b b

lp

b b b

f r f rG f

r r r


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


Optimum binary detection

Any bandpass signal can be presented by

This can be expressed by using different waveforms for ‘0’ and ‘1’ bits:

Received waveforms, that indicate the transmitted bits, are recovered coherently by using matched filtering or correlation receiver:

0 0

( ) ( )cos( ) ( )sin( )C C k i b C k q b C

k kx t A I p t kT t Q p t kT t

( ) ( ), 0,1

( ) ( )cos( ) ( )sin( )

C m bk

m C k i C k q C

x t s t kT m

s t A I p t t Q p t t


Bases of optimum detection

Received signal consist of bandpass filtered signal and noise, that is then sampled at the time instants tk :

Assuming that the BPF has the impulse response h(t), the signal at the sampling instant is then expressed by

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:

( )k m

Y y t z n

( 1)

0

( ) ( )

( ) ( ) ( ( ) x( ) ( ) )

( ) ( )

b

b

b

m m b

m k

m b

k

b

k T

kT A

T

t tz s t kT h t

s h t d x y t y t d

s h d

kT

T

Note how this expression shows

the MF and correlator reception!


Optimum binary error rate

Assuming ‘0’ and ‘1’ reception is equally likely, error happens whenthe H0 signal hits the dashed region (or H1 its left-hand side). The decision threshold is at 1 0( ) / 2optV z z

2 20 1 0

1exp / 2 / 2

2opt

e

V

p z d Q z z

Therefore for equally likely ‘0’ or/and ‘1’ the error rate is

For optimum performancewe wish to maximize the

SNR

21 0 / 2z z


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):

The noise component of the SNR after the bandpass filter is

And the SNR after the matched filter is:

2

2

1 0 1 0( ) ( ) ( )

bz z s s h dT

22 ( )2

h d

2

21 0

1 2

22

( ) ( ) ( )

4 ( )2

b

b

s s h dz z

h T d

T

( ) ( )cos( ) ( )sin( ) , 0,1

m C k i C k q Cs t A I p t t Q p t t m


Using Schwarz’s inequality for optimum filtering

Schwarz’s inequality:

Therefore SNR is maximized at the time instant of sampling by using

2

21 0

21 2

1 022

( ) ( ) ( )( ) ( )

4 ( )2

b

b

s s h dz zK s s d

h T d

T

2

22

2

( ) ( )( )

( )2

V W dV d

W d

1 0( ) ( ) ( )

opt b bh t K s T t s T t

provided that ( ) ( )W KV

1 0that holds when ( ) ( ) ( ) ( ) ( )

bW KV h T K s s


Matched filtering and correlator reception

Note that both circuits fulfil the expression0

( ) ( )b

m m opt b

T

z s h dT

Correlator

Matched filter

1 0( ) ( ) ( )

opt b bh t K s T t s T t

Documents

S-72.227 Digital Communication Systems The Viterbi algorithm, optimum coherent bandpass modulation