@G. Gong 1

5. Detection of Signals in Noise5. Detection of Signals in Noise

Detection problem: Given the observation vector x, we, then, perform a mapping (decoding) from xto an estimated of the transmitted symbol , in a way that would minimize the probability of error in the decision making process.

{ }Mimi ≤≤1|We will show that, in the case that all transmitted symbols are equally likely, then maximum likelihood detector is an optimum detector.

im̂ im

@G. Gong 2

A. Review on Bayes’ Formula

Total probability: Let events Fj, j = 1, …, n, be a partition of the sample space and E be an event. If all posteriori probabilities

of E and the probabilities of Fj’s are known, then the priori probability of E can be computed by

∑=

=n

jjj FPFEPEP

1

}{}|{}{

njFEP j ,...,1},|{ =

Bayes’ formula: with the above notation,

∑=

= n

jjj

iii

FPFEP

FPFEPEFP

1

}{}|{

}{}|{}|{

Conditional probability:

}{}|{}{}|{}{ EPEFPFPFEPEFP ==

@G. Gong 3

,...,2 ,1 ,)(

)()|()|( |

| MiP

mPmPmP iim

im ==x

xx X

X

Now we would like to apply the Bayes’ formula to the performance analysis of digital communication system. First we assume that the coordinates of observation vector x can take on a finite number of values, then given x, the probability that the symbol mi was transmitted is

(1)

where is the likely-hood function defined in Section 4, which is the pdf of X (or joint pdf of X1, …, XN ) given that the message mi is transmitted.

)|(| im mf xX

.,...,2 ,1 , )(

)()|()|( |

| Mif

mPmfmP iim

im ==x

xx

X

XX

However, the possible values of the coordinates of the observation (detection ) vector x are continuous in range. We would like to rewrite (1) into the following identity (it is a little bit awkward, and in strictest mathematical terms, it is not advisable, however, we do so here, since it has long been conventional in the data transmission literature)

(2)

@G. Gong 4

B. Decision Rules

Definition 1 (probability of error) The probability of error, is defined as the probability that decoded message is not equal to the message that was transmitted, i.e.,

m̂

}ˆ{)( iie mmPmP ≠=

The corresponding probability of being correct is therefore

}ˆ{}ˆ{1)(1)( iiieic mmPmmPmPmP ==≠−=−=

im

The optimum detector chooses to minimize , or equivalently, to maximize .

m̂ )( ie mP)( ic mP

@G. Gong 5

Maximum à posteriori probability (MAP) detector:

Thus the optimum decision device observes the particular received vector X = x and the output chooses to maximize the probability of a correct decision in (3). This quantity is referred to as àposteriori probability for the vector channel.

Mimm i ,...,2 ,1 ,ˆ ==

Definition 2. (MAP detector) The maximum a posteriori probability (MAP) detector is defined as the detector that chooses the mi to maximize à posteriori probability given a received vector x.

}|{| xX im mP

The probability of the decision be correct, given that observing vector x, is

imm =ˆ

}|{},ˆ{ | xxX X imic mPmmP === (3)

The probability of error is as follows

}|{1},ˆ{1},ˆ{ | xxXxX X imicie mPmmPmmP −===−==≠ (4)

@G. Gong 6

}|{}|{ || xx XX jmim mPmP ≥

In the following, we will use the identity (2) to rewrite (5).

)(

)()|(

)()()|( ||

xx

xx

X

X

X

X

fmPmf

fmPmf jjmiim ≥⇒

)|( )|( || jmim mfmf xx XX ≥⇒

Rule 1. (MAP detection rule)

(5) allfor }|{}|{ if ˆ || ijmPmPmm jmimi ≠≥⇒ xx XX

Thus if all transmitted symbols occur equally likely, i.e.,

MiM

mP i ,...,1 ,1)( ==

then the decision rule 1 is equivalent to the maximum likelihood (ML) decision rule.

Rule 2. (Maximum likelihood (ML) detection rule)

allfor )|( )|( if ˆ || ijmfmfmm jmimi ≠≥⇒ xx XX (6)

Remark. The conditional pdf indicates that (given that mi was transmitted) how likely we obtain the value of the observation vector x as compared with other values in the observation space.

)|(| im mf xX

In an AWGN channel, from Section 4, we have

) recall ( ..., ,1 , 1exp)()|(2

0

2/0| iii

Nim Mi

NNmf rxsrxX ==⎥

⎦

⎤⎢⎣

⎡−−= −π

[ ] MiN

NNmf iim ,...,1 ,||||1)ln( 2

)|(ln 2

00| =−−−=⇒ sxxX π

. allfor ||-|| ||-|| if ˆ 22 ijmm jii ≠≤⇒ sxsx

Thus the decision rule 2 is equivalent to the following rule.

Rule 3. (AWGN ML detection rule)

(7)

Remark. This decision rule is to choose a message point closest to thereceived signal point, which is intuitively satisfying.

@G. Gong 8

From Rule 3, the maximum AWGN ML detection rule, we have the following optimum receiver stucture.

Recall that for a correlator receiver (or equivalently a matched filter), we consider that

channel AWGN (2) ,...,, symbols sourcelikely equally (1) 21 Mmmm

C. Optimum receiver (implementation part)

( ) ,...,, )()()( 1. Step 11 Ni xxxtntstx =⇒+= x

The procedure of an optimum receiver:

performed by a correlator receiver (or a matched filter).

m̂→xStep 2. Note that

2)( 1

2

11

2

1

2 ∑∑∑∑====

+−=−=−N

jij

N

jijj

N

jj

N

jijji ssxxsxsx (8)

22 ||-|| ||-|| ki sxsx ≤ in Rule 3, is equivalent to

21

21

1

2

11

2

1∑∑∑∑====

+−≤+−N

jkj

N

jkjj

N

jij

N

jijj ssxssx

Since the first term in (8) is a common term for each i,

kEiE

k

N

jkjji

N

jijj EsxEsx

21

21

11

−≥−⇒ ∑∑==

Rule 3’.

ikEsxEsxmm k

N

jkjji

N

jijji ≠−≥−= ∑∑

==

allfor ,21

21if ˆSet

11

Selectthe

Largest

m̂

x

×

1s

+2/1E

−+

∑ jj sx 1

+

∑=

=N

j 1sum

×

2/2E−

+∑ jj sx 2

+×2/mE

−+∑ Mjj sx

2s

Ms

sum

sum

sum

Figure 1. Optimum Detector (Receiver)

@G. Gong 11

D. Decision Region for AWGN (Analysis Part)

In the case of the MAP rule, equivalently, in (7), each possible value for an N dimensional observation space maps into one of the Mpossible transmitted messages. Thus, the vector space for x is partitioned into M regions corresponding to the M possible decisions. Each region consists of points of which are closest to a transmitted signal vector si. In other words,

Definition 4 (Decision region). In an AWGN channel, the decision region using a MAP for each symbol mi is defined as

} allfor ||-|| ||-|| |{ 22 ij Z jii ≠≤= sxsxx

If iZ∈x then set imm =ˆ

Example 1. Let . We have the following signal constellations and the decision regions for the unipolar, polar and quaternary signaling (i.e., N = 1, M = 2 or 4) .

TtT

t ≤≤= 0 ,1)(1φ

)(1 tφ

0Tas

21

21 =Tas21

11 −=

Z1Z2

(a) Signal constellation and decision region for the unipolar signaling:

Ttts ≤≤= 0 ,0)(1

Ttats ≤≤= 0 ,2/)(2

)(1 tφZ1

Tas21

21 =

Z2

011 =s

Z1Z2

Ta41

(c) Quaternary case:

,)2/3()(1 ats −= ,)2/1()(2 ats −=

,)2/1()(3 ats = ,)2/3()( and 4 ats =

Tt ≤≤0for

(b) Polar case:

Ttats ≤≤= 0 ,2/)(2

Ttats ≤≤−= 0 ,2/)(1

)(1 tφZ3

Z1 Z4Z2

Ta23

−=s11 Tas21

21 −= Tas21

31 = Tas23

41 =

Ta− Ta0Figure 2.

@G. Gong 13

Example 2. One of very important signal spaces for actual applications is the set of multilevel quadrature amplitude-modulated (QAM) signals (i.e., N = 2). In this case, we have

),cos(2)(1 tT

t cωφ = )sin(2)(2 tT

t cωφ =

cA±As the signal basis functions for 0 ≤ t ≤ T. Consider M = 4 with possible amplitude of in each direction, we have four waveforms defined over 0 ≤ t ≤ T as

)sin(2)cos(2)()()(

)sin(2)cos(2)()()(

214

213

tT

AtT

AtAtAts

tT

AtT

AtAtAts

cccccc

cccccc

ωωφφ

ωωφφ

−=−+=

−−=−−=

)sin(2)cos(2)()()( 211 tT

AtT

AtAtAts cccccc ωωφφ +=++=

)sin(2)cos(2)()()( 212 tT

AtT

AtAtAts cccccc ωωφφ +−=+−=

Each of these signals has equal energy given by

2

0

2 2)( cT

ii AdttsE == ∫Thus we have the following vector representations for the set ofsignals in the two-dimensional signal space as

),( and ),,( ),,( ),,( 4321 cccccccc AAAAAAAA −=−−=−== ssss

The signal space constellation of the QAM signals is shown in Figure 3. The optimum decision regions for the equally likely symbols are just the four quadrants.

Figure 3. quadrant,fourth in the is if ˆ ; quadrant, thirdin the is if ˆ ; quadrant, second in the is if ˆ

; quadrant,first in the is if ˆSet

44

33

22

11

ZmmZmm

ZmmZmm

xxxx

==== )(2 tφ

)(1 tφ

2s

cAcA−

cA

cA−

1s

3s 4s

1Z2Z

3Z 4Z

@G. Gong 15

Extension : For 16-QAM , the signal constellation and decisionregions are shown in Figure 4.

Figure 4.

cA

3/cA−

cA−

3/cA

)(2 tφ

)(1 tφ

6. Probability of Error for Signals in AWGN6. Probability of Error for Signals in AWGN

∑=

∉=⇒M

iiie mZP

MP

1

}|{1 x

since each symbol occurs equally likely.

(1)

}|{)( iiie mZPmP ∉= x

Thus when the observing vector is x, the probability of error given that mi is transmitted is determined by

Error event: mi transmitted where received ikZk ≠∈⇒ x (due to AWGN)

ik mmm ≠=⇒ ˆ where the maximum ML detection rule is applied.

Definition 1. The average probability of symbol error, Pe, is

)(}|{)()(11

i

M

iiii

M

iiee mPmZPmPmPP ∑∑

==

∉== x

Note that in most of cases, Zi can be represented by},...,|),,,{( 11121 NNNNi bxabxaxxxZZ <<<<== L

where i = 1, 2, …, M, ai’s could be -∞, and bi’s, ∞. (here we avoid to use double indexes by setting Zi = Z). Thus we have

}|,...,{}|{ 111 iNNNii mbxabxaPmZP <<<<=∈x

(by independence)}|{}|{ 111 iNNNi mbxaPmbxaP <<<<= L

jx ∼ NjNsN ij ,...,1 ),2/,( 0 = given that mi transmitted.

We write this process in the following procedure.

The average probability of symbol correctness, Pc, is therefore

∑=

∈−=−=M

iiiec mZP

MPP

1

}|{111 x (2)

where , represented the probability of correction reception given that the message symbol mi is transmitted. This formula is frequently used in computation of probability of error.

}|{)( iiic mZPmP ∈= x

@G. Gong 18

Procedure A. Computation of probability of error

Misss iNiii ,...,2 ,1 ),,,,( 21 == Ls

and determine the decision region Zi,i = 1, …, M. In most of cases, Zi can be represented by

},...,|),,,{( 11121 NNNNi bxabxaxxxZZ <<<<== L

Step 1. Compute the message points (or the signal constellation)

Step 3. Compute ∑=

=M

iiee mP

MP

1

)(1

First write the F function in terms of Φ, the unit gaussian distribution, then transform it into erfc function. Thus

∏=

<<=N

jijjjic mbxaPmP

1

}|{)(

Step 2. For each i = 1, …, M, compute using the following sub-procedure: for each j = 1, …, N, compute

)( ic mP

)()( jj aFbF −= jx ∼ NjNsN ij ,...,1 ),2/,( 0 =}|{ ijjj mbxaP <<

)(1)( icie mPmP −=⇒

@G. Gong 19

Procedure 1. Computation of probability of error for 1-dimensional space( or M-ary PAM Signals)

Step 3. Compute

Misi ,...,2 ,1 1 =

and determine the decision region Zi,i = 1, …, M.

}|{ 11 bxaxZi <<=

Step 2. For each i = 1, …, M, compute :)( ic mP

)()( aFbF −= 1x ∼ )2/,( 01 NsN i}|{)( 1 iic mbxaPmP <<=

First write the F function in terms of Φ, the unit gaussian distribution, then transform it into erfc function.

Step 1. Compute the message points:

∑=

=M

iiee mP

MP

1

)(1

)(1)( icie mPmP −=⇒

@G. Gong 20

Example 1. Compute the probability of error for the signals given in Example 1 in Sec. 5.

Solution. We will use Procedure 1 for this question since N = 1.

(a). Uni-polar case

Step 1. Tassmm21 0, 1,0 211121 ==⇒==

⎭⎬⎫

⎩⎨⎧ <= TaxxZ

41| 111

⎭⎬⎫

⎩⎨⎧ ≥= TaxxZ

41| 112

This step can be replaced by drawing the signal constellation.

Step 2. Compute

}0|)4/1({)0( 1 TaxPPc <= 1x ∼ )2/,0( 0NN

⎟⎟⎠

⎞⎜⎜⎝

⎛Φ=

042

NTa

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

04211

NTaerfc

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

0421)0(

NTaerfcPe

}1|)4/1({)1( 1 TaxPPc ≥=

1x ∼ )2/,)2/(( 0NTaN

Similarly, we get

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0421)1(

NTaerfcPe

Step 3.

⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

04))1()0((

21

NTaerfcPPP eee

(The same result as in Sec.2 in Chapter 3).

@G. Gong 21

(b). Polar case

Step 1. TasTasmm21 ,

21 1,0 211121 =−=⇒==

{ }0| 111 <= xxZ { }0| 112 ≥= xxZ

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

0221)0(

NTaerfcPe

Step 3.

⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

02))1()0((

21

NTaerfcPPP eee

}1|0{)1( 1 ≥= xPPc

1x ∼

Similarly, we get

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0221)1(

NTaerfcPe

)2/,)2/1(( 0NTaN

Step 2. Compute

}0|0{)0( 1 <= xPPc

1x ∼ )2/,)2/1(( 0NTaN −

⎟⎟⎠

⎞⎜⎜⎝

⎛Φ=

022

NTa

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

02211

NTaerfc

@G. Gong 22

(c). Quaternary case

Step 1. See Figure 2 in Sec. 5 using Gray code.

⎟⎟⎠

⎞⎜⎜⎝

⎛Φ=

022

NTa

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

02211

NTaerfc

1x ∼ )2/,)2/3(( 0NTaN −

}00|{ 1 TaxP −<=

Step 2. Compute

}00|{)00( 11 ZxPPc ∈=(i)

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

0221)00(

NTaerfcPe

(1)

1x ∼ )2/,)2/1(( 0NTaN −

}01|0{ 1 ≤<−= xTaP

}01|{)01( 21 ZxPPc ∈=(ii)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

02)01(

NTaerfcPe

Similarly, we can derive

(2)

(iii) }11|{)11( 31 ZxPPc ∈=

}11|0{ 1 TaxP ≤<=

1x ∼ )2/,)2/1(( 0NTaN

Similarly, we get

⎟⎟⎠

⎞⎜⎜⎝

⎛=

02)11(

NTaerfcPe

(3)

@G. Gong 23

⎟⎟⎠

⎞⎜⎜⎝

⎛=+++=

0243))10()11()01()00((

41

NTaerfcPPPPP eeeee

Step 3.

(iv) )00(}10|{)10( 41 cc PZxPP =∈=

⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒

0221)10(

NTaerfcPe

@G. Gong 24

Procedure 2. Computation of probability of error for 2-dimensional space

Step 3. Compute

and determine the decision region Zi,i = 1, …, M.

Miss iii ,...,2 ,1 ),,( 21 ==s

} ,|),{( 2121 dxcbxaxxZZi <<<<==

Step 1. Compute the message points (or the signal constellation)

∑=

=M

iiee mP

MP

1

)(1

}|{}|{)( 21 iiic mdxcPmbxaPmP <<<<=

Step 2. For each i = 1, …, M, using Step 2 in Procedure 2, compute:

}|{ 1 imbxaP << }|{ 2 imdxcP <<and

Then

)(1)( icie mPmP −=⇒

(Example on applying Procedure 2 will be given in the tutorial and in the next chapter.)

6. Probability of Error for Signals in AWGN (Cont.)

Some Remarks on Computation of PeA. Invariance of probability of error to rotation and translation

The partition of the observation space depends on the signal constellation, or equivalently, the basis functions in the AWGN ML detection. The changes in the orientation of the signal constellation with respect to both the coordinate axes and origin of the signal space do not affect the probability of symbol error Pe. This result follows from the following two facts:

Fact 1. In ML detection, the probability of symbol error Pe depends only on the relative Euclidean distances between the message points in the constellation (Rule 3 and Definition 4 in Sec. 5).

Fact 2. The AWGN is spherically symmetric in all directions in signal space.

@G. Gong 26

Since the signal constellation is the vector representation of the signals under the basis functions, thus the effect of a rotation applied to all the message points in a constellation is equivalent to multiplying si by an N by N orthonormal matrix for all i.

A N by N matrix Q is orthonormal if it satisfies

NIQQ =τ

where IN is the identity matrix.

Figure 1. Rotational invariance example

)(2 tφ

)(1 tφ

2s

cAcA−

cA

cA−

1s

3s 4s

(a)

)(2 tφ

)(1 tφ2s

cA21s

3s

4s

cA2−

cA2−

cA2

(b)

@G. Gong 27

Rotational invariance: If a signal constellation is rotated by an orthonormal matrix Q, that is,

then the probability of error of the ML detector remains unchanged on an AWGN channel.

MiQ iroti ,...,1 ,, == ss

Translational invariance: If a signal constellation is translated by a constant vector v, that is,

then the probability of error of the ML detector remains unchanged on an AWGN channel.

Miitrani ,...,1 ,, =+= vss

@G. Gong 28

B. Minimum Energy Translate

The minimum energy translate of a signal constellation is defined as that constellation obtained from each signal vector si in the given constellation by subtracting the constant vector E[s], the average energy of the signal, defined by Def. 2 in Sec. 2.

Figure 2. Translational invariance example

)(1 tφTas

23

11 −= Tas21

21 −= Tas21

31 = Tas23

41 =

0

(a)

)(1 tφ011 =s

Tas =21Tas 231 = Tas 341 =

0

(b)

In the following example, the constellation (a) has the minimum energy, whereas that of (b) does not.

@G. Gong 29

C. Union Bound

In some cases, a signal constellation is complicated. Therefore, it is not easy to get the probability of error. Some bound on the probability of error is useful in this case. One of the most frequently used bound is called the union bound since it is derived by considering a union event. Here, we list this bound without proof.

Definition (minimum distance, dmin) The minimum distance, dmin, of the signal set, is defined as the minimum distance between any two signal points in a signal constellation, i.e.,

jid jiji , allfor ||,||minmin ss −= ≠

Union bound: The probability of error for the ML detector on the AWGN channel is bounded by

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−≤

0

min

221

NderfcMPe

7. Matched Filters and Signal-to-Noise Ratio (SNR) Maximization

7. Matched Filters and Signal-to-Noise Ratio (SNR) Maximization

We have shown that the matched filter receiver is equivalent to the correlative receiver. In this section, we will establish that the matched filters, shown in Figure 1, satisfy the SNR maximization property.

+ )(th)(ts

n(t)AWGN:

)(ty)()( tTsth −=SNR is max when

t = T

r(t)

Figure 1. SNR maximization by matched filter

(A filter whose impulse response , where s(t) is assumed to be confined in the interval , is called the matched filter to the signal s(t).)

)()( tTsth −=Tt ≤≤0

@G. Gong 31

The output of the filter:

where and

)()()( tnTstr oo +=

)()()( thtstso ∗= )()()( thtntno ∗=

)]([|)(|

)( 2

2

tnETs

SNRo

oo =

2|)(| Tsowhere is the instantaneous power in the output signal so(t) measured at time t = T and , the average power of the output noise.

)]([ 2 tnE o

Recall that the signal-to-noise ratio (Sec.2 in Chapter 3) is defined as

@G. Gong 32

Schwarz inequality: for any a pair of real-valued signal x(t) and y(t) with finite energy,

The equality holds if and only if

where k is a constant.

||)((||||)(|||)(),(| tytxtytx ⋅≤><

)()( tkxty =

@G. Gong 33

Proof.

We compute the SNR at sample time t = T as follows (note this is an approach from the time domain. In the text book, this result is proved from the frequency domain.

22 ]|)()([|)(| Tto thtsTs =∗=2

)()( ⎟⎠⎞

⎜⎝⎛ −= ∫

∞

∞−dttThts

2])(),([ >−<= tThts

)]([)]([ 22 TnEtnE oo =Note that (why?)

(1)

Theorem 1. (SNR maximization) For the system shown in Figure 1, the filter that maximizes the output signal-to-noise ratio, by matched filter . )()( tTsth −=

A. SNR Maximization

@G. Gong 34

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−= ∫ ∫

∞

∞−

∞

∞−

dssThsndttThtnETnE o )()()()()]([ 2

∫ ∫∞

∞−

∞

∞−

−−−= dtdstThsThstN )()()(2

0 δ(Using the same techniques that used in Sec. 4)

∫∞

∞−

−= dttThN

)(2

20

20 ||||2

hN

=

Then (1) and (2) produce that

(2)

2

2

0 ||)(||])(),([2)(

thtThts

NSNR o

>−<=

@G. Gong 35

with equality if and only if )()( tTksth −=

Applying the Schwarz inequality, we obtain that

(3)2

0||)(||2)( ts

NSNR o ≤

Thus (3) is maximized over all choices for h(t) when. The filter is matched to s(t), and the

corresponding maximum SNR (for any k) is )()( tTsth −=

2

0max ||)(||2)( ts

NSNR = (4)

@G. Gong 36

0max

2)(

NE

SNR s=

2||)(|| tsE s =

Remark.

where , the energy of the signal s(t).

@G. Gong 37

filter. matched a isdetector optimum e that thimplies This

) )()( ( )( ofversion delayed and reversed time the tomatched

is (SNR) maximizes(which detector optimum theof response impulse The

tTsthts

o

−=

T

a Tatts =)(

T

a)(th

@G. Gong 38

B. Frequency domain interpretation of the matched filters

Property 1. For the matched filter ,)()( tTsth −=

fTjefSfH π2* )()( −=

(We have seen this property in Chapter 4.)

Proof.

dtetTsfH ftjT

π2

0

)()( −∫ −= fTjfjT

edes πτπ ττ 22

0

)( −

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∫ fTjefS π2* )( −=

Remark. The matched filter has a frequency response which is the complex conjugate of the transmitted signal spectrum multiplied by the phase factor , which represents the sampling delay of T.

fTje π2−

Property 2. fTj

o efSfS π22|)(|)( −=

0 T

(a). Noise-like input signal

)(ts

T0

(b). Matched filter output

Figure 2.

)(tso

@G. Gong 40

1. Conversion of waveforms to vector space representation

- Inner product of signals- Norm- Gram-Schmidt orthogonalization algorithm (GSOA)

2. Geometric interpretation of signals

- Signal constellation: computed by the GSOA- Some properties: invariance of inner product, energy identities

3. Demodulation Correlative demodulation ⇔ Matched filter demodulation

(This is the process from the received signal waveform to reconstruct the vector representation of the signal.)

Summary of Chapter 5

@G. Gong 41

4. Response of bank of correlators to noisy input

)()()( tntstr ii +=

Received signal:

where n(t): AWGN

)()( trtx i=We write . The demodulator produces

),...,()( 1 Nxxtx =↔ x where >=< )(),( ttxx jj φ

(1) Any component xj is a Gaussian random variable with meansij and variance N0/2

(2) Any two different components xj and xj are independent.

Likelihood function: )|(| iM mf xX

5. Detection of Signals in Noise

- Decision rules: Rule 1(MAP detection rule, in general)

→ Rule 2 (ML, the symbol occurs equally likely),

→ Rule 3 (AWGN ML )

- Structure of optimum receiver (from Rule 3)

- Decision region (from Rule 3)

6. Probability of error for signals in AWGN

∑=

=M

iiee mP

MP

1

)(1

where Zi is the decision region.

Conditions: (1) the symbol occurs equally likely; (2) AWGN

}|{1}|{)( iiiiie mZPmZPmP ∈−=∉= xxwhere

@G. Gong 43

- Computation of the probability of error of the M-aryPAM signals, especially for M = 2, and 4.

- Procedures A and 1

7. Matched filters and SNR maximization

- SNR maximization

- Frequency response