OPTIMUM DEMODULATOR FOR ACOUSTIC SIGNALS' WAVEFORMS

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OPTIMUM DEMODULATOR FOR ACOUSTIC SIGNALS’ WAVEFORMS

B. Lu, J. Shen, Y. He

To cite this version:

B. Lu, J. Shen, Y. He. OPTIMUM DEMODULATOR FOR ACOUSTIC SIGNALS’ WAVEFORMS.

Journal de Physique Colloques, 1990, 51 (C2), pp.C2-797-C2-800. �10.1051/jphyscol:19902186�. �jpa-

00230491�

OPTIMUM DEMODULATOR FOR ACOUSTIC SIGNALS' WAVEFORMS

B. LU, &. SHEN and Y. HE

Department of Electronic Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083. P.R. China

ABSTRACT- L'sing statistic performance of practical telemetry signals ,the paper discussed the Demodulator for FDM telemetry System and proposed Optimum third order Phase-locked loop (PLL) Demodulator .

1.

IWTRODUCTIOB

At present ,generally monofrequency sinusoid waveforms were used as a test signal, corresponding to this the Demodulator is always traditional pulse averager or second order phased-lock~d loop which is a fairly rerent development. Hokever the majority of telemetry signals are stochastic signals with changing amplitude,therefore it can not be described with mathematical formulas completely.

If

the telemetry signals here simulated by sinusoid waveform signals,it kou!d let

t o

a great difference from the practice. Owing to the situation,t!le Demodulator

of FW

carriers-third PLL Demodulator which is more accurately appronchi

rig

to the practice

j

s deduced based on Optimum receiving theory and associated statislic data.

T t is well known that the maximum-frequency of telemetry signals S(t) which we want to transmite is specified,that is a band-limited signal

S ( t ) +t ~ ( ~ z c ) =

{do

^I"' " , l W m

! w ~ , w ,

where is a maximum angular frequency of telemetry signal. According to Sampling theorem, using rectangular sampling function

S . ( x ) . S ( t 1

can be represented as the following expansion

where

7 ,

is a sampling period,

7 , = w , / 2 r .

The cut-off angular frequency of ideal lowpass filter is

w , = w ,

, we note that

S . ( x )

is a energy signal(energy limited) ,but general telemetry signal is a capacity signal(capacity limited),so that we can not simulate the signal s(t) with the signal

S , ! x ) . .

Optimum receiving system can be deduced from Wiener filttering theory,then using statistic performance of telemetry signal we can get Optimum PLL Demodulator

2.

LINEAR OPTIMUM SYSTEM

Consider the linear causal system which is shown i n Fig.1

Fig.1 Linear Causal System

suppose that the input is a sum of statistic independent stochastic signal

S , ( t )

and noise

n,itj

then the output is also a statistic independent stochastic process

* r

~ , ( t ) = I ~ , ( t - r ) h ( t ) d t , n o ( t ) = j n , ( t - T ) / I ( t ) d r

jo 0

the system error is

~ [ t ) = S o i t ) - S l ( t ) + n o [ t j

According to minimum mean-squere error criteria,we can get optimum

h O p t ( t )

with

E f e 2 ( t ) j - min

Obviously,

S o ( t ) - S , ( t l

is statistically independent of

n o ( t i .

It can be shown

J [ E ~ ( ~ ) ] = ~ . ~ l 0 ) - 2 / ~ ~ . . ( r ) h ( r ) d z ⁺ L - L - [ R . , ( r

-

a ) + R . . ( T - a ) ] h ( a ) h ( ~ ) d a d ~ (21

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902186

COLLOQUE DE PHYSIQUE

where Re,(.) and R.,l.) a r e autocorrelation functions of input S,CtJ and n l ( f ) respectively.

Let E [ E ~ ( ~ ) ]

-

^{J (h), Rct(r)+ Ra(r)- R C r ) I}

Then the equation(2) becomes

J ( h ) = R , ( O ) - 2 j a ~ . , ( r ) h ( r ) d r + k ' k ' ~ ( r - a ) h ( a ) h ( r ) d o d r (3)

Obviously J(h) i s functional of h(7)

.

I t can be shown that there exists a Frechet differential for the functional' J(h):

s J ( h , ~ ) = d ( J ( h + B ~ l l l d B

Is_,,

Let B,[h.c)=O

We get 2 [ [ ~ ~ ( r - o ) h ( a ) d a - R ~ , ( r ) s ( r ) d r = O

I

Because € ( a ) is arbitrary,therefore L m R ( r

-

a ) h ( a ) d a = R . , ( r )

This equation i s satisfied by h ( t ) .The Fourier transform of equation(4) can be written a s

@ ( w ) H ( i w ) = @ , , ( w ) the total input power spectral density O ( w ) is

@ ( w ) = F ( R ( . t ) ) = @ , * ( w ) + @",(W> ( 5 ) so that we get the linear system H o ( j w l = +,(w)/+(wl which makes E [ E ' ( ~ ) ]

-

^{m i n}

However t h e H o ( j t u ) i s physically unrealizable system. Suppose the noise power spectral density i s constant N(white noise ),that i s

@ " * ( w ) =

N ,

W E ( - = , ~ ) ( 6

1

we can get physically realizable optimal system a s

-

Here suppose c ( j w I = ( @ , , I w l +

~r

Therefore c(/w)c'(fw)

-

^@(w)

3.LINEAR MODEL OF BASEBAND FM SYSTEM

We analyse baseband requency modulation(FM) and demodulation(PLL) system

.

The typical system is shown in Fig.2

"%./

VCO,]

I {

^{BPF PD}

I 1 I

Fig.2 Typical baseband FM and demodulation( PLL) system

According to t h e following conditions,the mathematical model can be derived : (1) Almost all of the useful frequency portion passed through BPF and LPF,therefore we can neglect them in analysis.

(2) The transfer functions of V C O , and V C O , a r e respectively:

K , ( j w ) = K o , / j w ( 8 )

Where K O , hnd K o z a r e gains of V C O , and V C O , respectively.

(3) The transfer function of linearized PD i s K , D [ j w ! = K , where K , i s constant

.

(4) The transfer function of LF is F(jw)

(5) Suppose the noise n'(t) i s white Gaussian noise,power spectral density is

@,.(w]=N,/2 w ~ [ - m , m ]

Thus we get the linear model which is shown in Fig.3.

I I

Fig.4 The simplified linear phase model Here the power spectral of n ( t ) is

@ , ( w ) = N , / ~ K $ , w E ( - ~ , ~ ) ( 1 0 ) and the power spectral of 6it) is

@ , ( w ) = l K , ( i w >

i 2

. S , ( w ) ( 1 1 ) 4. THE OPTIMUM PLL DEMODULATOR

Substituting Eq.(lO) and Eq(l1) into Eq.(7),we get the optimum system 2K:

I t i s described in Fig.4

First we determine +,(w).It i s mentioned in the introduction,the mathematical description of telemetry signal s(t) i s not perfect,majority of them a r e simulated with mono frequency sine-wave signals. However we can describe s(t) with statistical characteristic. The studies indicate that using such telemetry signal s(t) for shudder,vibration ,voice,temperature and so on ,the amplitude distribution approximates to Gaussian distribution,and t h e power spectral preserlts low-pass performance(see Fig.5 and Fig.6)

Fig.5 The amplitude distribution Fig.6 The power spectral density of telemetry signal s ( t ) of telemetry signal s(t)

We can see from Fig.6,that the break frequency is located a t t h e point w = ( 1 / 2 ) w , . ,where w , i s the maximum frequency deviation of signal, The curve decreases with -12 dB/oct. Thus this curve can be approximated by second order Butterworth spectral

The spectrum .P,(w.nl a r e presented in Fig.7

.

The break frequency i s located a t the point w=k,when w>k t h e amplitudes decrease with 6ndB/oct,when n=2,k = ( 1 / 2 ) w , and P = k p , / z ,then we can use .P,(w.2) to describe +,(w) perfectly. From Eq.(13) we get

7

COLLOQUE DE PHYSIQUE

Fig.7 Butterworth spectral

The formulation of optimum PLL Demodulator can be derived from t h e formulation of @ , ( w ) Using Eq.(S).Eq(ll) and Eq.(l4),we have

Consider Eq.(lO).~q.(l5) and Eq.(S),we have

N

w 6 + p w 2 + q

@ ( w ) = @ , ( w ) +

N = L

2 ~ 2 ' w 2 ( w 4 + p ) ( 1 6 )

Through computation we can find G ( j w )

c ( j w ) = ( I W ) ~ + a ( j w 1 2 + b ( j t u ) + c

2 ~ : ( j w ) i ( ~ ~ ) ~ + ~ G ( j w ) +

G ]

⁽¹⁷⁾

Where

a =

G +

J 2 J m - y / 2

Finally t h e Physically realizable optimum t r a n s f e r function can be derived from Eq.(12) a s follows .---- -

H O p t ( j w ) = ( a - 4 J 4 P ) ( i w ) 2 + ( b - G ) ( j w ) + c

( j w l 3 + ~ ( j w ) ~ + b ( j w ) + c ^{( 1 9 )}

I t is clear from Eq.(17) t h a t all t h e coefficients i n Eq.(19) a r e positive a n d H , , r [ j w l i s stable system

.

The o ~ t i m u m filter F o . t ( ~ w l can be derived from linear PLL moded.that is

I t is seen from Eq.(19) and Eq.(20) ,for the telemetry signal whose power spectral density has distribution like Fig.6,using F M modulation and PLL demodulation ,the optimum PLL should be third order.

REFERENCES

[ I ] Zheng Junli, Signal and System ,The People's Education Publishing House ,1981,China.

[ 2 ] H.L.Van Trees,Detection,Estimation and Modulation Theory ,John Wiley and Sons,Inc ,1971.

131 A.Papoulis,Probability,F&ndom Variables,and Stochastic Processes,McGrall-Hi11,1984.