Bfsk, Dpsk,Ask,Error Control

8/10/2019 Bfsk, Dpsk,Ask,Error Control

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*Signal space diagram of QPSK signal


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BFSK

Binary Frequency Shift Keying


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*Principle of BFSK

1, 0symbols1 bit

1&0 - distinguished by transmitting waves that differ in frequency.

Tb- bit duration= symbol duration

Eb- Energy of bit = symbol energy

Transmitted signals are

where

elsewhere,0

0,2cos2

bib

b

i

TttfT

E

ts

2,1;

iT

inf

b

ci


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symbol 1f1

symbol 0f2

S1(t) -transmitted signal ,represents symbol 1

S2(t)transmitted signal , represents symbol 0

There are two basis functions written as

elsewhere,0

0,2cos2

1

1

b

b

TttfTt

elsewhere,0

0,2cos2

2

2

b

b

TttfTt

elsewhere,0

0,2cos

2

11 b

b

b

TttfT

E

ts

elsewhere,0

0,2cos

2

2 2 bb

b

TttfT

E

ts


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1 = 2 =

As a result, the signal vectors are

b

b

E

E 0and

021 ss

t1

t2


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*Signal space diagram of BFSK

Distance between two points ,

= 2


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Symbol 1- f1frequency say, f1= 2fb

Symbol 0f2frequency say, f2= fb


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TRANSMITTER

1. Generation of coherent BFSK signal / Modulator

*MODEMS of BFSK


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Onoff level encoder

Unipolar

Converts 00

1

For Symbol 1,

On off level encoder o/p = =

Upper channel o/p = m(t)

= = s1(t)Lower channel o/p = ()

=0 = s2(t)S(t) = s1(t) + s2(t) = s1(t)

t1

t1

t2

For Symbol 0

On off level encoder o/p = 0

= 0Upper channel o/p = 0= s1(t)

()= ()Lower channel o/p = ()

= s2(t)

S(t) = s1(t) + s2(t) = s2(t)

t2


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RECEIVER

2. Detection of coherent BFSK signal/ Demodulator


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Noisy received signal, = (t) + w(t)

Two matched filters with 1 and 2observationvectors

= () (t) dt = () (t) dt


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For symbol 1 ,

=

()

(t) dt =

= () (t) dt = 0 = 1 2=

For symbol 0,

= () (t) dt = 0= () (t) dt = = 1 2= -

Decision device output,

If y>0 : symbol 1

If y


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*Bandwidth of BFSK

If Tbis the bit duration & fb=

= bit rate

Symbol 1 , Centered at f12fb

Symbol 0, Centered at f22fb

Total bandwidth = 4fb


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* Probability of bit error = probability of symbol

error

Symbol 1 : x(t) = s1(t) + w(t)

Symbol 0: x(t) = s2(t) +w(t)

Two decision regions : z1and z2


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Receiver decision in favour of 1:

When 1 > 2 falls under 1region =

Receiver decision in favour of 0:

When 1 < 2 falls under 2region =


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Mean of y , when symbol 1 was transmitted,

Mean of y , when symbol 0 was transmitted,

Variance of Y,


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Conditional probability density function of y

= ?

Suppose symbol 0 transmitted,

CPDF =


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Prob of error, for 0 transmitted, 1 received

P10 =

Substitute ,


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Similarly

0

0122

1

N

EerfcP b


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Average probability of symbol error = bit error

rate for BFSK,

022

1

NEerfcP be


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Amplitude Shift Keying

(ASK)


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On- off keying

Symbol 1pulse transmitted

Symbol 0no pulse


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* Principle of ASK

Transmitted signal Symbol 1: (0t )

Symbol 0 : (0

t

)

S(t) = 0


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Orthonormal basis function:

For symbol 1:

For symbol 0:

S(t) = 0


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* Signal constellation for ASK


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* Detection of ASK signal


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Noisy received signal, = (t) + w(t)

matched filter o/p: 1 observation vector

= () (t) dt


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*Bandwidth of ASK

Bandwidth = 2fb

fb = bit rate = 1/Tb


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*Probability of error

Received signal ,

= + : 0

0

Matched filter o/p ,

= () (t) dt


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When symbol 1 transmitted,

1 =

[1] =

[1] = 0/2

When symbol 0 transmitted,[1] =0 [1] = 0/2


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Conditional probability density function or

random variable x1 = ?

Symbol 1 transmitted,

Cpdf=

Symbol 0 transmitted,Cpdf =


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Cases of Error


s/g falls under

2region

Ie : 1 <


s/g falls under 2regionIe : 1 >


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Probability of error ,

P01

=


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Substitute

= 1


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P01 = P10 =erfc(

)

= average probability of symbol error = BER


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DPSK

Non coherent version of PSK

Eliminate the need of coherent reference

signal at the receiver side

Combination of

1) Differential encoding of input binary wave

2) Phase shift keying


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Send symbol 1,

No phase change

Send symbol 0,

Phase advances by 180 degree


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DPSK transmitter


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Illustration of generation of DPSK

signal


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*detection of DPSK signal

BPF- removes undesired frequency components

Decoding : bk = + 1: xnor operation

dk-1 1 1 0 1 1 0 1 1

dk 1 1 0 1 1 0 1 1 1

bk 1 0 0 1 0 0 1 1


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Case 1: symbol 1, symbol 1

1 =

Case 2: symbol 1, symbol 0

1 = Receiver decision: 1 > 0: 1 1


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* Bandwidth of DPSK

BW= fb

Since,2Tb is the symbol duration


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Erfc(u)= 1-erf(u)


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Error control coding


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Channel encoding

Accepts msg bits

Add parity bits along with msg bit

Parity generation

Channel decoding

- Parity checking for detecting msg signal


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Channel encoder

Block code

Sending information as blocks

No memory requirement

Convolutional code

Bit by bit

Serially

With memory


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Block codes

Linear block codes

Cyclic codes


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Channel decoder

Syndrome decoder

Block codes

Viterbi decoder

Convolutional codes


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Block code

General representation

: (n, k) block codes

n ->no: of code words

k>no: of msg bits

n-k ->no: of parity bits added


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Linear block code

A code is said to be linear if any two code

words in the code can be added in modulo-2

arithmetic to produce a third code word in the

code


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Modulo-2 arithmetic

Xor operation


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Linear Block Codes

A vector notation is used for the msg bitsand codewords, Dataword m = (m0m1.mk-1)

Codeword c = (c0c1..cn-1) Parity bits b= (b0b1..bn-k-1)

the code rate, Code rate = k/n


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Linear Block Code - Example

msg length k = 4

Codeword length n = 7

This is a (7,4) block code with code rate = 4/7

For example, m = (1101), c = (0011101)


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b- linear combination of msg bits


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Structure of a lbc

Matrix representation


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Considering p as the co-efficient k x (n-k)

matrix


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Block diagram representation of generator

equation

Parity check equation for lbc


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Parity check equation for lbc

Take H, as parity check matrix orthogonal to

G(generator matrix)

Parity

check

equation


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Indicates transmitted vector received correctly

or not.

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Bfsk, Dpsk,Ask,Error Control