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Waveform Coding
ByMiss. Patil Apurva P.K.B.P. college of Engineering, Satara
CONTENTS1) Introduction2) Types of Signals
a. Antipodalb. Orthogonal
3) Types of Waveform Codinga. Orthogonal
Cross-Correlation Coefficientb. Biorthogonalc. Transorthogonal
4) Conclusion
IntroductionChannel coding• class of signal transformations designed • to improve communication performance • by enabling the transmitted signal to better withstand the effects of channel impairments.
Waveform Structured Sequences
M-ary signalingAntipodalOrthogonalBiorthogonalTransorthogonal
BlockConvolutional
Antipodal Signals Antipodal signals are:
- mirror images,- the negatives of each other- or 180° apart.
Orthogonal Signals
where; τ= pulse width duration, T= symbolic duration
Orthogonal signals are mutually perpendicular to one otherThe direction of the vector is the direction of the wave’s energy flow & the length of the vector is the energy E of the wave
In general, S waves are constituted as an orthogonal set ,if & only if i
Where;
jiZ ,
jiZ , =cross-correlation coefficient, i= 1,2,…….,M,M =size of codeword set
If = 0 ; orthogonal set
Orthogonal Codes
othersji
zij
01
sequence in the digits ofnumber totalntsdisagreeme digits ofnumber -agreements digits ofnumber
An orthogonal set iff
Example
Orthogonal Codes
• By using Hadamard matrix we get orthogonal codewords from message words
Hk is a Hadamard matrix which is a 2k x 2k matrix and created from Hk-1 as follows:
Probability of Bit-error
)/()1()( 0NEQMMP sE
Probability of codeword error is
The relationship between and :
EP BP
122
)()( 1
k
k
E
B
kPkP
Or 12/
)()(
MM
MPMP
E
B
From above two equation Probability of Bit error is
02)(
NEQMMP S
BOr
0
12)(NkEQkP bk
B
If then good approximation
310)( MPE
Set of M total signals.Obtained from an orthogonal set of M/2 signals
by augmenting it with the negative of each signal.
BIORTHOGONAL CODES
Two sets of orthogonal codes. Each codeword in one set has its antipodal
codeword in the other set. A combination of orthogonal and antipodal
signals.
Zij= cross-correlation coefficient
Example:Here 3-bit data set can he transformed into orthogonal
codeword set
Data set Biorthogonal codeword set
It requires one half as many code bits per codeword. One half bandwidth is required compared to
orthogonal. Antipodal signal vectors have better distance properties
than orthogonal. For equal-energy biorthogonal signals ,the probability
of codeword error is:
PE (M) ≤ (M-2)Q (√Es /No ) + Q (√2Es /No )
Becomes increasingly tight for fixed M as Eb /No is increased.
PB (M) is a complicated function of PE (M) given by:
PB (M) ≈ PE (M)/2
The approximation is quite good for M>2,therefore:
PE (M) ≤ ½[(M-2)Q (√Es /No ) + Q (√2Es /No )]
The PB performance is improved. Requires only half the bandwidth of orthogonal codes
TRANSORTHOGONAL CODES The code generated from an orthogonal set by
deleting the first digit of each codeword is called a transorthogonal or simplex code.
The code chracterized by:
Represents minimum energy equivalent. In case of error performance.
-requires the minimum Eb /No for specified symbol error rate.
For large values of M all schemes are essentially identical in error performance.
CONCLUSION The biorthogonal codes use half of bandwidth less
than orthogonal and transorthogonal codes.Bandwidth requirements for each will grow
exponentially with the value of M.So such coding schemes are attractive when only
large bandwidths are available.
Thank You!
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