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!