Digital Communications I:Modulation and Coding Course

Term 3 - 2008Catharina Logothetis

Lecture 8

Lecture 8 2

Last time we talked about:

Some bandpass modulation schemes M-PAM, M-PSK, M-FSK, M-QAM

How to perform coherent and non-coherent detection

Lecture 8 3

Example of two dim. modulation

)(1 tψ

2s 1s

sE

“00”

“11”

)(2 tψ

3s 4s“10”

“01”QPSK

3s

7s

“110”)(1 tψ

4s 2ssE

“000”

)(2 tψ

6s 8s

1s5s

“001”“011”

“010”

“101”

“111” “100”

8PSK

)(1 tψ

)(2 tψ

2s1s 3s 4s“0000” “0001” “0011” “0010”

6s5s 7s 8s

10s9s 11s 12s

14s13s 15s 16s

1 3-1-3

“1000” “1001” “1011” “1010”

“1100” “1101” “1111” “1110”

“0100” “0101” “0111” “0110”

1

3

-1

-3

16QAM

Lecture 8 4

Today, we are going to talk about:

How to calculate the average probability of symbol error for different modulation schemes that we studied?

How to compare different modulation schemes based on their error performances?

Lecture 8 5

Error probability of bandpass modulation

Before evaluating the error probability, it is important to remember that: The type of modulation and detection ( coherent or non-

coherent) determines the structure of the decision circuits and hence the decision variable, denoted by z.

The decision variable, z, is compared with M-1 thresholds, corresponding to M decision regions for detection purposes.

Nr

r1

r=∫T

0

)(1 tψ

∫T

0

)(tNψ)(tr

1r

Nr

rDecision CircuitsCompare z

with threshold.

m̂

Lecture 8 6

The matched filters output (observation vector= ) is the detector input and the decision variable is a function of , i.e.

For MPAM, MQAM and MFSK with coherent detection For MPSK with coherent detection For non-coherent detection (M-FSK and DPSK),

We know that for calculating the average probability of symbol error, we need to determine

Hence, we need to know the statistics of z, which depends on the modulation scheme and the detection type.

Error probability …

)(rfz =r

rr=z

r∠=z|| r=z

sent) |C condition satisfies Pr(zsent) | Zinside lies Pr( ii ii ssr ≡

Lecture 8 7

Error probability …

AWGN channel model: The signal vector is deterministic. The elements of the noise vector are

i.i.d Gaussian random variables with zero-mean and variance . The noise vector's pdf is

The elements of the observed vector are independent Gaussian random variables. Its pdf is

),...,,( 21 iNiii aaa=s

),...,,( 21 Nrrr=r

),...,,( 21 Nnnn=n

nsr += i

2/0N

( )

−=

0

2

2/0

exp1)(NN

p N

nnn π

( )

−−=

0

2

2/0

exp1)|(NN

p iNi

srsrr π

Lecture 8 8

Error probability …

BPSK and BFSK with coherent detection:

)(1 tψ

2s1sbE

“0” “1”

bE−

)(2 tψ

bE221 =− ss

)(1 tψ

2s

1s

)(2 tψbE

bEbE221 =− ss“0”

“1”

BPSK BFSK

2

0

=

NEQP b

B

2/

2/

0

21

−=

NQPB

ss

0

=

NEQP b

B

Lecture 8 9

Error probability …

Non-coherent detection of BFSK

∫T

0

)cos(/2 1tT ω

∫T

0)(tr

11r

12r

∫T

0

∫T

0

21r

22r

Decision rule:

)cos(/2 2tT ω

)sin(/2 2tT ω

)sin(/2 1tT ω

( ) 2

( ) 2

( ) 2

( ) 2 +

-

z

0ˆ ,0)( if1ˆ ,0)( if

=<=>

mTzmTz

m̂

212

2111 rrz +=

222

2212 rrz +=

21 zzz −=

Decision variable:Difference of envelopes

Lecture 8 10

Error probability – cont’d

Non-coherent detection of BFSK …

Similarly, non-coherent detection of DBPSK

[ ]

∫ ∫∫∞ ∞∞

=>=

>=>=

>+>=

0 2221210 2222221

2221221

112221

)|()|()|(),|Pr(

),|Pr()|Pr(

)|Pr(21)|Pr(

21

2

dzzpdzzpdzzpzzz

zzzEzz

zzzzP

z

B

ssss

ss

ss

−=

02exp

21

NEP b

B

−=

0

exp21

NEP b

B

Rayleigh pdf Rician pdf

Lecture 8 11

Coherent detection of M-PAM Decision variable:

∫T

0

)(1 tψ

ML detector(Compare with M-1 thresholds))(tr

1rm̂

Error probability ….

)(1 tψ2s1s0gE3−

“00” “01”4s3s

“11” “10”

gE− gE gE34-PAM

1rz =

Lecture 8 12

Error probability ….

Coherent detection of M-PAM …. Error happens if the noise, , exceeds in amplitude

one-half of the distance between adjacent symbols. For symbols on the border, error can happen only in one direction. Hence:

( )( ) ( )gMMege

gmme

ErnPErnP

MmErnP

−<−==>−==

−<<>−==

ssss

ss

111111

11

Pr)( and Pr)(

;12for ||||Pr)(

−

−=0

22

1log6)1(2)(

NE

MMQ

MMMP b

E

gbs EMEME3

)1()(log2

2−==

mrn s−= 11

Gaussian pdf with zero mean and variance 2/0N

( ) ( ) ( )

( )

−=−=>−=

−<+>+>−==

∫

∑∞

=

01

1111

2)1(2)()1(2Pr)1(2

PrM1 Pr

M1||Pr2)(1)(

1 NE

QMMdnnp

MMEn

MM

EnEnEnMMP

MMP

g

E ng

ggg

M

mmeE

g

s

Lecture 8 13

Error probability …

Coherent detection of M-QAM

∫T

0

)(1 tψML detector1r

∫T

0

)(2 tψML detector

)(tr

2r

m̂Parallel-to-serialconverter

s) threshold1 with (Compare −M

s) threshold1 with (Compare −M

)(1 tψ

)(2 tψ

2s1s 3s 4s“0000” “0001” “0011” “0010”

6s5s 7s 8s

10s9s 11s 12s

14s13s 15s 16s

“1000” “1001” “1011” “1010”

“1100” “1101” “1111” “1110”

“0100” “0101” “0111” “0110”

16-QAM

Lecture 8 14

Error probability …

Coherent detection of M-QAM … M-QAM can be viewed as the combination of two modulations on I and Q branches, respectively. No error occurs if no error is detected on either the I or the Q

branch. Considering the symmetry of the signal space and the

orthogonality of the I and Q branches:

PAM−M

branches) Q and Ion detectederror noPr(1)(1)( −=−= MPMP CE

( )( )22 1I)on error Pr(no

Q)on error I)Pr(noon error Pr(nobranches) Q and Ion detectederror noPr(

MPE−==

=

−

−=

0

2

1log3114)(

NE

MMQ

MMP b

E Average probability of symbol error for PAM−M

Lecture 8 15

Error probability …

Coherent detection of MPSK

Compute Choose smallest2

1arctanrr φˆ

|ˆ| φφ −i

∫T

0

)(1 tψ

∫T

0

)(2 tψ)(tr

1r

2r

m̂

3s

7s

“110”)(1 tψ

4s 2ssE

“000”

)(2 tψ

6s 8s

1s5s

“001”“011”“010”

“101”

“111” “100”

8-PSK

Decision variable r∠== φˆz

Lecture 8 16

Error probability …

Coherent detection of MPSK … The detector compares the phase of observation vector to M-1

thresholds. Due to the circular symmetry of the signal space, we have:

where

It can be shown that

φφπ

π φ dpPPM

MPMPM

Mc

M

mmcCE )(1)(1)(11)(1)(

/

/ ˆ11

∫∑ −=

−=−=−=−= ss

≈MN

EQMP sE

πsin22)(0

( )

≈MN

EMQMP bE

πsinlog22)(0

2or

2|| ;sinexp)cos(2)( 2

00ˆ

πφφφπ

φφ ≤

−≈NE

NEp ss

Lecture 8 17

Error probability …

Coherent detection of M-FSK

Mr

r1

r=∫T

0

)(1 tψ

∫T

0

)(tMψ)(tr

1r

Mr

rML detector:

Choose the largest element

in the observed vector

m̂

Lecture 8 18

Error probability …

Coherent detection of M-FSK … The dimension of the signal space is M. An upper

bound for the average symbol error probability can be obtained by using the union bound. Hence:

or, equivalently

( )

−≤

0

1)(NEQMMP s

E

( ) ( )

−≤

0

2log1)(N

EMQMMP bE

Lecture 8 19

Bit error probability versus symbol error probability

Number of bits per symbol For orthogonal M-ary signaling (M-FSK)

For M-PSK, M-PAM and M-QAM

21lim

12/

122 1

=

−=

−=

∞→

−

E

Bk

k

k

E

B

PP

MM

PP

Mk 2log=

1for < <≈ EE

B PkPP

Lecture 8 20

Probability of symbol error for binary modulation

EP

dB / 0NEb

Note!• “The same average symbol

energy for different sizes of signal space”

Lecture 8 21

Probability of symbol error for M-PSK

EP

dB / 0NEb

Note!• “The same average symbol

energy for different sizes of signal space”

Lecture 8 22

Probability of symbol error for M-FSK

EP

dB / 0NEb

Note!• “The same average symbol

energy for different sizes of signal space”

Lecture 8 23

Probability of symbol error for M-PAM

EP

dB / 0NEb

Note!• “The same average symbol

energy for different sizes of signal space”

Lecture 8 24

Probability of symbol error for M-QAM

EP

dB / 0NEb

Note!• “The same average symbol

energy for different sizes of signal space”

Lecture 8 25

Example of samples of matched filter output for some bandpass modulation schemes