energy detection of a signal with random amplitude

7/29/2019 Energy Detection of a Signal With Random Amplitude

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Energy Detection of a Signal with Random Amplitude

V. I. KostylevVoronezh State University

Universitetskaya pl. 1Voronezh, 394693 Russia

Abstract- Urkowitz [1] has discussed the detection of adeterministic signal of unknown structure in the presence of band-limited Gaussian noise. That analysis we developed to thecase of a signal with random (Rayleigh, Rice, Nakagami, andother) amplitude. For such amplitude a distribution of adecision statistic of an energy detector is retrieved andexpressions for a detection probability are obtained here.

I. INTRODUCTION

In the classic paper, Urkowitz [1] discussed the detection of adeterministic signal of unknown structure in the presence of flat,band-limited Gaussian noise of known power density. The decisionstatistic is shown to have a noncentral chi-square distribution. As a

result of atmosphere turbulence, multichannel wave propagation,and other reasons, received communication signals often have arandom amplitude. In this case the distribution of energy decisionstatistic is not chi-square distribution [2].

This paper is aimed at determined the probability of energydetection of a received communication signal with a various randomamplitude.

II. ENERGY DETECTOR

In the theory of communication systems the following model of the quasideterministic signal is widely used

( ){ }0( ) Re ( ) exp 2s t AU t j f t π ϕ = + , (1)

where A is a random amplitude, U (t ) is the normalizeddeterministic complex envelope, f 0 is the carrier frequency,and ϕ is a random initial phase.

As in the classical paper [1], we assume that the carrier frequency f 0, the spectrum width ∆ f of the detected signal, and the spectralpower density N 0 of the noise are known. The processed randomsignal

( ) ( ) ( ) x t s t n t ι= + , (2)

where ι = 0, 1, is a mixture of the detected signal (1) and white noisen(t ) or only white noise n(t ) in accordance with the hypotheses H1 orH0, respectively. The a priori unknown binary parameter ι that is

used in (2) takes the values 0 or 1, which coincide with thehypothesis subscript.

As in [1], it is assumed that the processed signal (2) is prefilteredby a bandpass filter with a perfect amplitude-frequencycharacteristic to limit the average power of noise:

The research reported here was supported in part by theRussian Fund for Basic Research (project # 01-01-00356).

0 0

0

2 , / 2;( )

0, / 2.

N f f f H f

f f f

− ≤ ∆= − > ∆

(3)

The following narrow-band signal comes to the energy-receiver input from the filter output:

( ){ }0( ) Re ( )exp 2 y t Y t j f t π = , (4)

where

( )0

2( ) ( )exp ( )

AY t U t j t

N ι ϕ = + Ν , (5)

is the complex envelope of the signal y(t ) and Ν(t ) is thecomplex envelope of Gaussian noise at the filter output. It is

obvious that ( ) 0t Ν = and

* sin( )( ) ( ) 4

f t t

π τ τ

πτ

∆Ν Ν − = , (6)

where the overbar denotes averaging over the realizationensemble.

The energy receiver is intended for forming a voltage(decision statistic) at its output, which is equal to the energy

2 21 1( ) ( / )2 2 m

E Y t dt Y m f f

+∞ +∞

=−∞−∞= = ∆∆ ∑∫ (7)

of the processed realization of the output signal y(t ).However, since the infinitely proceeding of the signal y(t ) isimpossible, the energy receiver actually forms decisionstatistic Ξ that is equal to the approximation of energy (7) of the form

21

0

1( )

2

T

E Y t dt = ∫ (8)

or

2

20

1

( / )2

T f

m E Y m f f

∆

== ∆∆ ∑ , (9)

where T is the processing time of the signal y(t ) in the energyreceiver, which can always be chosen such that the productT ∆ f is an integer. Although the quantities E , E 1, and E 2 differfor the finite T , the difference disappears for T → \

The quantities E 1 and E 2 are two different approximationsfor the energy E of the realization of the signal y(t ), so thateach realization has its own value. In the majority of theoretical works, when calculating the detection

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0-7803-7400-2/02/$17.00 © 2002 IEEE



characteristics, it was assumed either explicitly that theoutput signal of the energy receiver coincides with E 2. Onlyin this case the decision statistic Ξ has exactly chi-squaredistribution under H0.

III. DECISION STATISTIC

Under H1 the distribution of decision statistic depends ondistribution of amplitude A. It isn’t difficult to show that inthe case of deterministic amplitude the decision statistic hasnoncentral chi-square distribution [1]. In the case of randomamplitude the decision-statistic distribution can differ fromthe noncentral chi-square distribution. But the noncentral chi-square distribution may be used as a conditional (under fixed

A) distribution of the output signal from the energy receiver.

Consequently, the characteristic function Θ( jη) of thedecision statistic Ξ can be written in the form

( )2 2

/ 2

0

(1 2 ) exp ( )1 2

j A q j j W A dA

j

µ ηη η

η

∞− Θ = − − ∫ , (10)

where µ = 2(T ∆ f + 1) is a number of degrees of freedom,

22

0 0

2( )

T

q U t dt N

= ∫ , (11)

and W ( A) is a probability density function (pdf) of therandom amplitude A.

IV. DETECTION PROBABILITY

A. Rayleigh Amplitude

Substituting widely known Rayleigh pdf into (10) and thenintegrating, we obtain

( ) 121 2 ( 1) (1 2 ) B j j d jη η η− − Θ = − + − , (12)

where 2 2 2d A q= is the input signal-to-noise ratio and

B = T ∆ f is the processing base. It follows from (12) that in thecase of energy detection of a signal with Rayleigh amplitudeagainst a white-noise background, the decision statistic hasthe form

222 12( 1)

22( 1) 0

, H ;

, H ,

Bd

B

e χ

χ

+

+

+Ξ =

(13)

where ea is a random quantity having the exponential

distribution with parameter a and 2ν χ is the random quantityhaving the chi-square distribution with ν degrees of freedom.Therefore, under hypothesis H1 the decision statistic isdistributed as a sum of two independent random quantities:the exponentially distributed quantity with the parameter2(d

2 + 1) and the chi-square distributed quantity with 2 B degrees of freedom.

Writing the right-hand side of (12) as a sum of simplefractions, we obtain

( ) ( )2

1 1

; ,

k l

lm l

l m

j j mγ η α θ η λ

= =

Θ = ∑∑ , (14)

where α 11 = [(d 2 + 1)/ d

2] B, α 2m = –d –2[(d

2 + 1)/ d 2] B–m, m = 1,

2, …, B ; λ 1 = 2(d 2 + 1), λ 2 = 2; k 1 = 1, k 2 = B; and

( ) ( ); , 1 b j a b j aγ θ η η −= − , (15)

is the characteristic function of the gamma distribution withparameters a and b. Therefore, under H1 the decision statisticΞ has distribution belonging to the hypergamma-distributionclass.

From both (12) and (13) it is obvious that the probabilitydensity of the decision statistic Ξ can be obtained as aconvolution of the exponential pdf

( )exp( / )

; 1( )e

x aw x a x

a

−= (16)

and chi-square pdf

( ) / 2 1

2exp( / 2)

; 1( )2 ( / 2)

x x w x x

ν

ν χ ν

ν

− −=

Γ . (17)

Here1, 0,

1( )0, 0

x x

x

≥= <

is the unit step function, Γ ( x ) is the

gamma function [3], a is the exponential-distributionparameter, and ν is the number of degrees of freedom of thechi-square distribution. After simple but awkwardmanipulations, we obtain the pdf of the decision statistic (13)in the form

( ) ( ) ( )

( )2

12 2

212 2

0

1 ;2( 1) , ,H ;( ) 2 1

;2( 1) , H ,

B

e Bd d w x d B

p x d d

w x B χ

− + + Ρ = +

+

(18)

where Ρ (a, b) = γ (a, b)/ Γ (a) is the normalized incompletegamma function [3] and γ (a, b) is the incomplete gammafunction [3].

The detection statistic Ξ is compared with a certain level h in the threshold device. The detection probability

D = Pr{Ξ ≥ h| H1} and the false-alarm probabilityF = Pr{Ξ ≥ h| H0} characterize the detection operationefficiency. Integrating pdf (18) in the interval from h to +∞,

we obtain( )

( )

1, / 2

1

B hF

B

Γ +=

Γ +, (19)

( )

2 2

2 2 2

11 , exp ,

2 2( 1) 2 1

Bh d h hd

D B Bd d d

+ = − Ρ + Ρ + + ,(20)

where Γ (a, b) = Γ (a) – γ (a, b) is a complementary incompletegamma function [3].

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Using twice the formula [3]

( 1, ) ( , ) exp( )aa z a a z z zγ γ + = − − , (21)

we transform (20) into the form

( )

2 2

2 2 2

1

exp 1,2( 1) 2 1

Bd h hd

D F Bd d d

+ = + Ρ + + +

. (22)

The dependence a detection probability from false-alarm probabilityis known as a detection characteristic. Equation (22) determinesthe energy detection characteristic in the case of randomRayleigh amplitude.

B. Rice amplitude

Substituting Rice pdf

2 2

02 2( ) exp exp

2 2

A a A aW A I A

ω ω ω

= − − , (23)

into (10) and then integrating, we obtain

( ) ( ) ( )1

1 2 1 2 exp1 2

B j j j j

j

ηλς η η ης

ης

− − Θ = − − −

, (24)

where a and ω are parameters of pdf, I ν ( x ) is the modified

Bessel function, ( ) ( )2 2 22 2 2d a aς = + + + is a weight

parameter, and ( )2 2 2 22 2 2a d d aλ = + + is a parameter of a

noncentrality. It follows from (12) that in the case of energydetection of a signal with Rice amplitude against a white-noise background, the decision statistic has the form

2 22 2, 1

22( 1) 0

, H ;

, H ,

B

B

λ χ ς χ

χ +

+Ξ =

(25)

where 2,ν λ χ is the random quantity having the noncentral chi-

square distribution with ν degrees of freedom andnoncentrality parameter λ . It isn’t difficult to see, that undercondition a = 0 (when Rice pdf transform to Rayleigh pdf)equation (25) transform to (13).

We obtain the pdf of the decision statistic (25) in the form

( )( )

( )2

0

exp / 2 1; 2

2( 1)

k

k

k

p x L w x k χ

λ ς λ µ

ς ς ς

∞

=

− − −= + −

∑ , (26)

where Lk

( x ) are Laguerre polynomials [3].

For detection probability it isn’t difficult to obtain

0

1,( )! 2

k

k

h D B k

B k

α ∞

=

= Γ + + + ∑ , (27)

where

( )exp 2 1

2( 1)

k

k k Lλ ς λ

α ς ς ς

− −= − −

. (28)

Fig. 1. Detection characteristics versus Rice parameter.

Fig. 1 shows the relationships between detectionprobability, false-alarm probability, and Rice parameter a for

the case B = 15 and 2 50d = . It is not difficult to see that inpresented case detection characteristics became better with agrowth of Rice parameter a. But such dependence take placefor a case of a relatively big detection probabilities: it is

possible that detection characteristics became worse with agrowth of Rice parameter a if signal-to-noise ratio anddetection probability are very small. On the other hand, thiscase is not interest for a practice.

C. Nakagami Amplitude

The Nakagami pdf may be written as

( )

2 1 2

22

2( ) exp

( )

m m

m

m A mAW A

Am A

− = −

Γ , (29)

where m is the parameter of distribution. It isn’t difficult tosee, that Rayleigh distribution is a particular case of

Nakagami distribution under condition m = 1.Substituting (29) into (10) and then integrating, we obtain

( ) ( ) ( )1 21 2 1 2 1 / m

m B j j j d mη η η

−− − Θ = − − + . (30)

Assume that B > m – 1, we write for Nakagami case

222( 1 ) 12(1 / ),

22( 1) 0

, H ;

, H ,

B m d m m

B

χ γ

χ

+ − +

+

+Ξ =

(31)

where γ a, b is the random quantity having the gammadistribution with characteristic function (15). Under conditionm = 1 (31) transform to (13).

Using the theoretical results of [4], we show, thatcorresponding (30) pdf of energy detector decision statistic is

( )( )

( ) ( )1 1 21 2

1( ) exp / 2F ; 1;

2 2 1!2 1

B

m B

x x x x x p x m B

d m B d m+

− = + − ++

, (32)

where 1F1(a; b; z) is the confluent hypergeometric function.

The probability of a correct detection we obtain only in theintegral form:

' ( 7 ( & 7 , 2 1

3 5

2 % $ % , / , 7 <

( ( ( (

)$/6($/$50352%$%,/,7<

3$5$0(7(5a

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( ) ( ) ( )

12

0

2

/ 2exp 1,

21

1 2 2

h

m

mm

h B m d

d m D

m B m d m

− ℘ −℘ ℘ − Γ − + ℘ + =

Γ Γ − + +

∫ . (33)

Fig. 2. Detection characteristics versus Nakagami parameter.

But (27) is also correct under condition that (28) is replacedwith

( ) 2

2 2( ) !

k m

k

k m m d

m k m d m d α

Γ + = Γ + + . (34)

Fig. 3. Detection characteristics versus signal-to-noise ratio.

Fig. 4. Detection characteristics versus processing base.

Fig. 2 - 4 show the relationships between detectionprobability, false-alarm probability, and parameters m, B, and

d 2. Fig. 2 is calculated for the case B = 15 and 2 200d = ,

Fig. 3 – for B = 15 and m = 1, and Fig. 4 – for 2 500d = andm = 1. It is following from Fig. 2 - 4 that detectioncharacteristics became better with increase of the parameterm or the signal-to-noise ratio and with decrease theprocessing base.

D. Other Amplitude

In general case the energy detection probability may becalculate by using the equation (27), where

( )2 2 2

0

1( ) exp / 2 ( )

!2

k k k

Aq A q W A dAk

α

∞

= −∫ . (35)

We obtain:

( )

( )

12 2

2 22 2

12

1 21

k

k

d

d d α

+ −℘ = ×

−℘ + −℘

( )

2

2 1 22 2

1 4F , 1;1;

2 2 1 2

k k

d

+ ℘ × + −℘ +

(36)

for Hoyt amplitude distribution with pdf

( )( ) ( )

2 2

02 22

2( ) 1 exp

1 11

A A AW A A I

℘ = − ϒ −℘ ϒ −℘ ϒ −℘

, (37)

where ℘ and 2 A ϒ = are parameters of Hoyt distribution and

2F1(a, b; c; z) is the Gauss hypergeometric function;

( ) ( ) ( )

2 2 2 2( 1)

2 22 exp 2 2

k

k k k

q q

Lq qq

β γ γ

α β β β β

℘℘−

+℘

= − − + + + (38)

for Nakagami-Rice amplitude distribution with pdf

( ) ( )2 2

11( ) 1 exp

2 2

A AW A A I A

β γ β γ

γ β

℘ ℘

℘−℘−

= − −

, (39)

where ℘, β , and γ are parameters of distribution and ( ) ( )m

k L x are

Laguerre polynomials [3];

( )

1 2

11 2

(2 ) ( 1/ 2) ( 1/ 2) ( 1)

2 (2 1)!!(2 1)!!( )! ! 2

m l k

k k m lk

p l m m l k q

l m m l k p qα

π

+ +

+ + ++

Γ + Γ + Γ + + +=

− − + +(40)

for beta-exponential amplitude distribution with pdf

( ) ( )1

2 1 2(2 ) ( 1/ 2) ( 1/ 2)( ) 1 exp

(2 1)!!(2 1)!! !

bb p l m

W A A A pAl m bπ

++Γ + Γ += −

− −, (41)

where m, l and p are parameters of distribution and b = m + l ≥ 0;

( )

2 2 2 22 2

1 2 2 2 220

1exp

2 1 2 11

lk

y x y y x x k k

l y x

m mβ σ σ σ σ β α

σ β σ β β

∞

+=

−= − − × + ++

∑

' ( 7 ( & 7 , 2 1

3 5 2 % $ % , / , 7 <

( ( ( (

)$/6($/$50352%$%,/,7<

3$5$0(7(5m

' (

7 ( & 7 , 2 1

3 5

2 % $ % , / , 7 <

( ( ( (

)$/6($/$50352%$%,/,7<

6,*1$/7212,6(

5$7,2

' (

7 ( & 7 , 2 1

3 5 2

% $ % , / , 7 <

( ( (

)$/6($/$50352%$%,/,7<

352&(66,1*

%$6(

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( ) ( )

22( 1/ 2) ( )

2 2 2 2 22 1 2 1

yl x l k

x x y y

mm L L

σ σ σ σ β

− × − −

− + (42)

for Klovsky amplitude distribution with pdf

( )

2 2 22

2 20

( )

( ) 1 exp 2 2 (2 )!!2

y x k

k k x y y x

m A A m H

W A A k σ σ σ σ

∞

=

+ ℘= − − × ∑

2

2 2 2

1 1k k

y yk

k

y x y y

Am A I

m

σ

σ σ σ

× −

, 2 2

x yσ σ ≤ , (43)

where σ x , σ y , m x , and m y are parameters of distribution, H i( x )are Hermite polynomials, and used the following designations

2

2 2

1 12 x x

x y

m σ σ σ

℘= −

and

2 2

2

2 2 2 2

y

x y x y

d m m

σ β σ σ

=+ + +

;

( )2 2

2 / 2 / 0

( 1)1 2( ) /

2 ! 2 !

k l l

k k k l ll

q qk l

k c l cα

∞

℘ ℘=

−= Γ + + ℘∑ (44)

for lognormal amplitude distribution with pdf

( )( )

2

2

ln1( ) 1 exp

22

AW A A

A σ πσ

−℘= −

, (45)

where 2 2 2ln Aσ = −℘ and ln A℘= are parameters of

distribution;

( ) ( )

( )

2( )

0

11 ( 1)

2 ! 2 !

l k l

k k ll

k l k ld

k lα

+∞

=

Γ ℘+ + Γ + +−=

℘ Γ ℘∑ (46)

for K - amplitude distribution with pdf

( )( )

( )( 1) / 2

1

4( ) 1 2

( )

AW A A K A

℘+℘

℘−

℘ ϒ= ℘ ϒ

Γ ℘, (47)

where ℘ and 2 A ϒ = are parameters of distribution and K ν ( z) is

Macdonald function [3];

( ) ( )1 2

2( )2 2 2

0

1 ( 1) 11 exp

! ! 2 1

k llk l

k

l

aq

k l

α σ + +∞ +

=

− −℘= −℘ − ×

+℘ ∑

( )2

22

0

11

2 1

nn

n

n

aε

∞

=

−℘× ℘ −℘ × +℘ ∑ (48)

( )3/ 22

21 1

0

1(3 3 1)F 2( ) 1;2 1;

! ( 1) 2 1

m

m

am nk l n m n

m m n

∞

=

−℘Γ + + × ℘ + + + + + Γ + + +℘

∑

for Beckman amplitude distribution with pdf

( )2 2 4

2 2

(1 ) / ( ) 1 exp

2 1

A a AW A A

σ

σ

−℘ + = − × −℘

(49)

2 4

2 220

/ 1,12 1

k k k

k

A A

I I a

σ ε σ

∞

=

℘ −℘ × +℘−℘ ∑

0 1ε = , 0 2nε > =

where a, ℘, and σ 2 are parameters of distribution;

( )

( )

( )

( )

22( )

/ 2

/ 2 / 2 20 / 2

2 / 21 ( 1)

2 ! 2 ! / 2 2

l k l k lk l

k k ll

D pk lq

k l D p

σ σ α

σ

+ +∞ − − −℘

= −℘

−Γ + +℘−=

Γ ℘ −∑

(50)

for parabolic cylinder amplitude distribution with pdf

( )

( )

4 21

4

2

14 / 2 42

21 exp

4( )

2 / 2 exp82

A pA A A

W A

p p D α

σ

σ σ σ

℘−

℘

−−℘

−−

=

− Γ ℘

, (51)

where p, ℘, and σ 2 are parameters of distribution and Dν ( z) isthe function of parabolic cylinder [3]; and

2( )

2( )2( ) 2

2( )

20 1

1 ( 1) 2exp

2 ! 2 ! 2

k l

k lk ll

k l

k k ll m

m mq

k l m

ηγ α

η

++

+∞+

= =

− −= ℘ +

∑ ∑

(52)

for S L amplitude distribution with pdf

( )

( )

( )

221( ) exp ln

22

AW A A

A

η η γ

ηπ

−℘ = − + −℘

−℘

, (53)

where ℘, γ , and η are parameters of distribution.

ACKNOWLEDGMENT

The author is indebted to Prof. A.P. Trifonov forinstructive and stimulating discussions of these problems andtechniques.

REFERENCES

[1] H. Urkowitz, “Energy Detection of UnknownDeterministic Signals,” Proc. IEEE, vol. 55, pp. 523-531, April 1967.

[2] V. I. Kostylev, “Characteristics of Energy Detection of Quasideterministic Radio Signals,” Radiophysics and Quantum Electronics, vol. 43, pp. 833-839, October2000.

[3] M. Abramovitz and I. A. Stigan, eds., Handbook of Mathematical Functions. New York, NY: Dover, 1970.

[4] V. I. Kostylev, “The Distribution of Two IndependentGamma-Statistics Sum,” Journal of Communication

Technology and Electronics, vol. 46, pp. 530-533, May2001.

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energy detection of a signal with random amplitude

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