lec 13 error rate performance

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What are we studying ?Digital Communications Systems

Base Band Modulation Schemes Baseband Detection/Demodulation Bandpass Modulation Schemes

BPSK, QPSK,QAM Inter-Symbol Interference (ISI) Eye Diagram Analysis Pulse Shaping Equalization Modulation Schemes Detection of Modulation Schemes

BPSK ; ASK ; FSK ; M-PSK etc

Error Performance

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4.7.1 Probability of Bit error for coherently detected BPSK

Received signal: r(t) = s i(t)+n(t) The antipodal signals s 1(t) and s 2(t) can be characterized in a one-

dimensional signal space as:

Decision is made on the basis:

Probability of bit error P B,

1 1

2 1

( ) ( )0

( ) ( )

s t E t t T

s t E t

otherwiset s

T z if t s

)(

0)()(

2

01

0

21

)2/(

2

22exp

21

021

aaQdu P

aa B



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For equal-energy antipodal signaling, the receiver output components are

Then 2211 s for E aand s for E a bb

du P N E B b

0/2

2

2exp

2

1

0

2 b B

E P Q

N

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4.7.2 Probability of Bit error for coherently detectedDifferentially Encoded BPSK

00

21

22 N

E Q N

E Q P

bb

B

Fig 4.25

This formulae is forreference only



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4.7 Error Performance For BinarySystems

11:52AM

6

Passband Baseband

Modulation P B P B Modulation

PSK(Coh)

Bipolar

DPSK (DiffCoh)

Bipolar

Orthogonal FSK(Coh)

Unipolar

Orthogonal FSK (NonCoh) Unipolar

QPSKPBPSK =PBPSK

QPSK

0

2 b E Q N

0

2 b E Q N

0

1exp2

b E N

0

b E Q N

0

1 exp2 2

b E N

0

b E Q N

0

2b

E Q N

0

b E Q N

0

2 b E Q N

0

2 b E Q N

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4.7.3 Probability of Bit error for coherently detected BinaryOrthogonal FSK

A general treatment for binary coherent signals (not limited to antipodalsignals) yields

For orthogonal, BFSK = /2; thus =0 and:

0

2

(1 ) /

1exp ;

22 b B E N

P du

du P N E B b

0/

2

2exp

2

1

0 N

E Q P b B

cos



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4.7.4 Probability of Bit error for Noncoherently detectedBinary Orthogonal FSK

02exp

21

N E P b B

E b / N 0 penalty of the simplernoncoherent detection is onlyabout 1dB at practical bit errorrates

As a result, the simpler,noncoherent FSK forms the basisof many low end (e.g. 1200 bps)telephone and radio modems in

the market-place



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4.8 M-ary Signaling and Performance4.8.1 Ideal Probability of Bit Error Performance

Typical probability of error versus Eb/N 0 curve has a waterfall like shape The ideal curve displays the characteristics as the Shannon limit The limit represents the threshold Eb/N 0 below which reliable

communication cannot be maintained Ideal Curve:

For all values of Eb/N 0 above the Shannon limit of -1.6db, P B isarbitrarily small

Once Eb/N 0 is reduced below the Shannon limit, P B degrades to worsecase value of



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4.8.2 M-ary Signaling

In M-ary signaling the processor, considers k bits at a time and modulatesone of M=2 k waveforms

K=1 implies binary signaling Trick question:

Does M-ary signaling improve or degrade error performance?

Lets review the next two figures before answering the question



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Figure 4.28: Bit error probability for Coherently detected M-ary

orthogonal signaling.11:52 AM Adv Digital Comm - Dr. M. ArifWahla 13


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Figure 4.29: Bit error probability forcoherently detected multiple phase

signaling

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It appears that M-ary signaling produces improved error performance withorthogonal signaling and degraded error performance with multiple phasesignaling

This result poses two more questions: Why is multiple phase PSK signaling used in systems if it provides

degraded error performance compared to binary PSK signaling? Is error probability Eb/N 0 , the only performance criterion?

Bandwidth: Another performance criterion

In Figure 4.28, increase in k increases the required bandwidth For M-ary multiple phase curves in Figure 4.29, increase in k implies that

larger bit rate can be transmitted within the same bandwidth (or for fixeddata rate, the required bandwidth is decreased)



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4.8.4 BPSK and QPSK have the same Bit Error Probability General relationship between Eb/N 0 and S/N isWhere

S: is the average signal powerR: is the bit rate

Eb/N 0 characteristics of theorthogonal BPSK channels

making up QPSK signal canbe shown to be equivalentto the above Eb/N 0

R

W

N

S

N

E b

0

RW

N S

RW

N S

N E b

000 2/2/

Figure 4.31 11:52 AMAdv Digital Comm - Dr. M. Arif Wahla 17


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4.8.5 Vectorial View of MFSK Signaling

Figure 4.32: MFSK signal sets for M=2,3



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Basic mapping relation:

Where W is the detection bandwidth. Since

Where T is the symbol duration, then

For FSK signaling, the detection bandwidth W (in Hz) is typically equal invalue to the symbol rate 1/T; or WT 1, therefore

RW

N S

N E b

0

T k T M R 2log

k

WT

N

S

M

WT

N

S

N

E b20 log

k N

S

N

E b 1

0



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Figure 4.33: Symbol error probability versus SNR for coherent FSKsignaling11:52 AM Adv Digital Comm - Dr. M. ArifWahla 20


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4.9 Symbol Error Performance For M ary Systems 4.9.1 Probaility of Symbol Error for MPSK

For large energy to noise ratios, the symbol error performance P E(M), for

equally likely, coherently detected M-ary PSK signaling:

where PE(M) : is the probability of symbol error

Es=Eb(log2M) : is the energy per symbolM=2 k : is the size of the symbol set

Symbol error performance for differentially coherent detection of M-ay

DPSK (for large E b/N 0) is :

M N E

Q M P s E

sin2

2)(0

M N E

Q M P s E 2

sin2

2)(0



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Figure 4.35: Symbolerror probability forcoherently detected

multiple phasesignaling

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Figure 4.36: Symbol error probability for coherently detected M-aryorthogonal signaling11:52 AMAdv Digital Comm - Dr. M. Arif

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Figure 4.37: Symbol error probability for noncoherently detected M-aryorthogonal signaling

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4 9 3 Bit E P b bilit S b l E P b bilit f


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4.9.3 Bit Error Probability versus Symbol Error Probability forOrthogonal Signal

The ratio is:

In the limit as k increases, we get

Example of P B verses P E(Figure 4.38)

1

2/

12

2 1

M

M

P

P k

k

E

B

21

lim E

B

k P P

0

00

0

1

1

1

1

0

01

1

0

0

1

1

0

10

1

0

1

0

1

Bit Position

Transmitted

Symbol


4 9 4 Bi E P b bili S b l E P b bili f


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4.9.4 Bit Error Probability versus Symbol Error Probability forMultiple Phase Signaling

For the case of MPSK signaling, P B is less than or equal to P E The difference is:

For orthogonal signaling, selecting any one of the (M-1) erroneoussymbols is equally likely

For MPSK signaling, each signal vector is not equidistant from allothers

111

001

000

110101

100

011

010Transmitted

Symbol 001

000

101111

110

010

011

100

Figure 4.3911:52 AMAdv Digital Comm - Dr. M. Arif Wahla 27


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Advantages and Disadvantages of FSK

Advantages FSK is a constant envelope modulation

hence insensitive to amplitude (gain) variations in the channel hence compatible with non-linear transmitter and receiver systems

Detection of FSK can be based on relative frequency changes betweensymbol states and thus does not require absolute frequency accuracy in

the channel (FSK is thus relatively tolerant to LO drift and Doppler Shift)

Disadvantages FSK is less bandwidth efficient than ASK or PSK The bit/symbol error rate performance of FSK is worse than PSK In case of FSK, increasing the number of frequencies can increase the

occupied bandwidth



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Advantages and Disadvantages of PSK

Advantage:

Bandwidth Efficiency In order to improve on the bandwidth efficiency of bandpass data

transmission, we can increase the number of symbol states

A reduction in bandwidth by a factor of k

M k

k

B B Binaryary M 2 _ log

bary M

b Binary kT

BthenT

Bif 1

,1

_



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Disadvantages: Reduced immunity to noise

As a general rule, we know that as the number of symbolstates is increased, the tolerance to noise is reduced

Two exceptions to this rule, QPSK and orthogonal MFSK

Decreased immunity to noise compared to binary Increased transmission power compared to binary Increased complexity compared to binary Lower transmission quality compared to binary


SNR d E /N


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where,Rb = bit rate in bits/secondEb = Energy per bit in Joules/bitS = Total Signal power in Watts

Introducing the noise power

This equation implies that the SNR will be more than E b/N o bya factor R b (if Rb > 1 bit/symbol)

Increasing the data rate will increase the SNR, however , increasing Rb will also cause

more noise and noise term also increases ( due to ISI intersymbol interference , sincemore bits are packed closer and sent through the channel).

So we cannot increase SNR by simply increasing Rb. We must strike a compromisebetween the data rate and the amount of noise our receiver can handle.

SNR and E b/N o

lec 13 error rate performance

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