The Performance of DDPSK over LEO Mobile Satellite Channels

The Performance of DDPSK over LEO Mobile Satellite Channels MA CHANGNING

Department of Electronics and Information Science University of Science and Technology of China

Hefei Anhui 230027,P.R.China Email: machn@mail.ustc.edu.cn WANG DONGJIN

Department of Electronics and Information Science University of Science and Technology of China

Hefei Anhui 230027,P.R.China Email: wangdj@ustc.edu.cn

Abstract: An analysis is presented for the Doppler characterization of low earth orbit Satellite and a Doppler-time curve is given. To deal with the large Doppler shifts, several demodulation schemes are compared., which shows that the double differential PSK demodulate is insensitive to the time-variant Doppler shifts. Furthermore, the bit error probability of DDPSK over the RLN channel is computed.

Key words: LEO Satellite, Doppler shift ,Double Differential PSK, RLN Channel

1 Introduction

Low Earth Orbit Satellite seems to be most attractive constellation for mobile satellite communication.

It will play an important role in the worldwide personal communication networks for low-cost terminals and lower propagation loss and delay. Unfortunately, this type of satellite constellation generates another problem due to large and time-variant Doppler shifts. It is necessary to choose or develop a Doppler offset resistant demodulator. This paper bases on the comparison among several demodulation schemes and selects one which is insensitive to the large Doppler shifts.

2 Doppler characterization for LEO satellite

The low earth orbit satellites are placed in the range of 500∼1500km from earth surface. They are not fixed with respect to a point on earth. For these reasons Doppler is significant in LEO mobile satellite communication. The detail of Doppler characterization are carried out in [1]

with the following result:

)))) ))))

cos cos cos cos ((((

cos(cos cos(cos cos(cos cos(cos )) )) ))

(((( )) )))) ((((

cos( cos(

cos( cos(

2 2 2 2

)))) ((((

)))) ))))

cos cos cos cos ((((

cos(cos cos(cos cos(cos cos(cos )) ))

)) )) ((((

)))) ((((

sin( sin( sin(

1 sin(

1 1 1

max max max max max

max max max 11

11 0

0 0 0 22

22 222 2

max max max max max

max max max 11

11 0

00 0

θ θ

ψ ψ

ψ θ θ

ψ ψ

−

−

− +

−

− −

∆ =

−

− •

rrrr tttt rrrr

tttt rrrr

rrrr rrrr rrrr

rrrr tttt tttt rrrr

tttt rrrr

rrrr c cc ffff c

ffff

E EE E E

EE E E

EE E

EE EE E

E E E

(1)

Fig.1 shows Doppler offset in different maximum elevation angle for a satellite that follows a circular orbit of altitude 1000km and inclination 53 degree from (1).

-8 -6 -4 -2 0 2 4 6 8

-2 -1.5 -1 -0.5 0 0.5 1 1.5

2x 10^-5

Time(min)

Normalized Doppler

11.3deg 25deg 54deg 85deg

Fig.1. Doppler-time S-curve for maximum elevation angles 11.3⁰,25⁰,54⁰,85⁰

It can be deduced that if the carrier frequency is 4∼6GHz and the maximum elevation angles is 54⁰,the Doppler shifts range is 77∼116KHz and the Doppler rate range is 392∼588Hz/s.Considering a date symbol rate of 9.6kbits/s,the Doppler shift is several times wider than the symbol rate. Developing a novel demodulator maybe a necessary requirement of LEO satellite system. to resist the large Doppler shifts and fast Doppler rate.

3. DDPSK demodulator

It is either difficult or impossible to receive correctly in the presence of random Doppler frequency shifts and short-time or burst communication systems when using coherent detection. Compared with coherent detection, differential detection does not require any pilot transmission and active carrier recovery circuit .The BER performance is inferior only by approximately 1 dB in SNR to that of an ideal CPSK demodulator. However, the DPSK demodulator is quite sensitive to frequency variation in the carrier. One solution to the above sensitivity is to encode the date phases as a second-order difference phase process and use a two-stage differential detection process.

DDPSK signal s(t) in the nth transmission interval [(n-1)T,nT]is of the form:

)))) cos(

cos(

cos(

)))) cos(

(((( tttt

=

a a a a wt wt wt wt

+∆φ²²²²

ssss

^where 1111 1111 2222

2 2 2

2 =∆ −∆ ₋ = −

2 2 2 2

₋ + ₋

∆φnnnn φnnnn φnnnn φnnnn φnnnn φnnnn ⁽²⁾

An auto-correlation demodulation for DDPSK signals is illustrated in fig 2

Fig. 2 Block diagram of DDPSK demodulator We have

)))) ((((

)))) cos(

cos(

cos(

)))) cos(

(((( tttt a a a a tttt tttt tttt

x xx

x

_nnnn = ω +∆ω_nnnn +θ_nnnn +φ_nnnn +ξ_nnnn (3)

The DDPSK auto-correlation algorithm has the form

)))) sgn( sgn(

sgn( sgn(

)))) ˆˆˆˆ ((((

)))) ((((

)))) ˆˆˆˆ ((((

)))) ((((

)))) ((((

)))) ((((

)))) ((((

)))) ((((

sgn sgn sgn sgn )))) sgn( sgn(

sgn( sgn(

_{1111 1111}

)))) 111 (((( 1

22 22 11 11 ))))

11 11 ((((

111 1 ))))

1 11

(((( 1 (((( 1111))))

2 2 2 2 1 11 1 1

1 1 1

−

−

−

−

−

−

−

− −

−

−

−

+

=





















+

=

=

∫

∫

∫ ∫

n n n n n n n n n nn n n nn nT n

nT nTnT

TT TT nnn n

nn nn nnn n nTnT

nTnT

TTT T nn nn

nnn n nn nn nT nTnT nT

T T T T n nn n

nT nT nT nT

T T T T n nn n

n n n n n nn n n

n n n n n n n

n n n

n

X X X X X X X X Y Y Y Y Y Y Y Y

dt dt dt tttt dt x xx tttt x x xx x dt dt dt tttt dt x xx tttt x x xx x

dt dt dt tttt dt xx x tttt x x xx x dt dt dt dt tttt x xx tttt x xx x x IIII

d d

d d

(4)

where

1 1 1 1 ))))

11 11 ((((

1 1 1

1(((( )))) coscoscoscos coscoscoscos ))))

(((( = ∆φ α+ξ

=

∫

−

− nT

nTnT nT

TTT T nn nn

n nn n n

n n n n n n n n

nn

n xxxx tttt xxxx tttt dtdtdtdt EEEE XX

XX ₂₂₂₂

)))) 11 11 ((((

111 1 22

22 11 11 11

11=

∫

(((( )))) (((( )))) = coscoscoscos∆φ coscoscoscosα+ξ

−

−

−

−

− nT nT nT nT

TT TT nnn n

nn nn nnn

n nn nn nnn

n xxxx tttt xxxx tttt dtdtdtdt EEEE X

XX X

For FT>>1,ξ_iiii

,,,, iiii

=

1 1 1 1 ,,,, 2 2 2 2 ,,,, 3 3 3 3 ,,,, 4 4 4 4

are Gaussian-distributed independent random variables with variances

2 )))) 2 2 1 2 11 1 ((((

)))) ((((

)))) ((((

)))) ((((

))))

(((( _{1111 2222 3333 4444 0000} ⁰⁰⁰⁰

E EE E FTFT FTFT NN NN ENENEN EN DD

DD DD

DD DDD

D DDD

D ξ = ξ = ξ = ξ = +

Letting vvvv₁₁₁₁ = ((((XXXXnnnn +XXXXnnnn₋₁₁₁₁))))²²²²+((((YYYYnnnn +YYYYnnnn₋₁₁₁₁))))²²²² vvvv₂₂₂₂ = ((((XXXXnnnn −XXXXnnnn₋₁₁₁₁))))²²²² +((((YYYYnnnn−YYYYnnnn₋₁₁₁₁))))²²²²

Algorithm (4) is equivalent to sign I=

sgn( sgn( sgn( sgn( v vv v

₁₁₁₁−

vv v v

₂₂₂₂

))))

.On the above assumptions that

v vv v

₁₁₁₁ will have a generalized Rice distribution and

v vv v

₂₂₂₂ will have Rayleigh distribution.

2 )))) 22 exp( 2 exp(

exp(

exp(

)))) ((((

)))) ////

((((

2 )))) 22 exp( 2 exp(

exp(

exp(

))))

(((( ₂₂₂₂

2 2 2 2 22 22 2

22 2 22 22 2 2 2 2 2

22 2 1 1 1 1 0 00 2 0 22 2 2 22 2 2 22 2 11 11 2

2 2 2 11 11 1 1 1

1 σ ρ σ σ

σ

ρ σvvvv vvvv mmmm IIII mvmvmvmv vvvv ^{vvvv vvvv} v

vv

v = − + = −

Where mmmm EEEE

E E E E FT FT FT FT N N N EN N EN EN

EN )))) 2222 2

22 1 2 11 1 ((((

2 2 2

2 ₀₀₀₀ ⁰⁰⁰⁰

2 2 2

2 = + =

σ

E E E E a a a a

²²²²

T T T T 2 2 2 2 1 1 1

=

1

is the energy in the signal.

The bit error probability is

1]]]]

11 )))) 1 00 00 2 ////

2 2 2 ((((

000 0 exp[

exp[

exp[

exp[

5 5 5 5 ....

0 0 0 0

2)))) 22 4 2 44 4 2 ////

2 2 exp( 2 exp(

exp(

exp(

*

*

*

* 5 55 5 ....

0 00 0 0

0 0

0 )))) 2222

222 (((( 2 11 11 1)))) 11 (((( 1 2))))

22 2 11 11 ((((

+ −

=

−

∫ =

∞ ∞∫

=

<

=

N NN N E E E E

FT FT FT FT N

N N N E E E E

m m m m vzzzz

vv v

dv dv dv dv v vv v dv

dv dv dv v vv v v

vv v v vv v P P P er P er erer P P P

P ρ ρ σ

(5)

Clearly from (5) ,the BER performance is not affected by the unknown frequency offset. In fig. 3 we compare the BER of CPSK,DPSK and DDPSK when FT=1.In fig. 4 we plot the bit error probability for DDPSK when FT=1,2,6.We can see that DDPSK is inferior by approximately 3dB in SNR relative to optimum detection of DPSK But it is effective against the large and time- variant Doppler shifts caused by the LEO mobile satellite.

Fig. 3 BER for CPSK,DPSK,DDPSK Fig.4 BER For DDPSK(FT=1,2,6) 4. BER over RLN channel

In the satellite scenario, Corazza presented a Rice-lognarmal(RLN) model and evaluated average error probability in a few significant cases.[2].Now we calculate the BER of DDPSK over RLN channel.

The probability density function(p.d.f) of received signal envelope, r, is given by

∞

∫

=

00 00

)))) ((((

))))

||||

((((

))))

((((rrrr pppp rrrr SSSS pppp SSSS dSdSdSdS p

pp

p_{rrrr ssss} (6)

4 44 4 ))))

111 (((( 1

1 1 1 1 2

22 2 1 1 1 1 1

1 1 1 3

33 3 ))))

11 11 ((((

1 11

1(((( )))) coscoscoscos sinsinsinsin (((())))ˆˆˆˆ (((( )))) sinsinsinsin sinsinsinsin ˆˆˆˆ

))))

(((( = ∆φ α+ξ = = ∆φ α+ξ

=

∫ ∫

−

−

−

−

−

−

−

nT nT nT nT

TT TT nnn n

n n n n n

nn n n nn n n

n n n nT

nTnT nT

TTT T nnn n

n nn n n

nn n n n n n n

n n

n xxxx tttt xxxx tttt dtdtdtdt EEEE YYYY xxxx tttt xxxx tttt dtdtdtdt EEEE YY

YY

0 2 4 6 8 10 12 14 16

10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

FT=1 FT=2 FT=6

0 2 4 6 8 10 12 14 16

10^-6 10^-5 10^-4 10^-3 10^-2 10^-1

CPSK DPSK DDPSK

In (5) S is lognormal distribution with p.d.f

2 2 2

]]]]

2

////

)) )) )) (((( )) [(ln

[(ln [(ln [(ln

*

*

*

* 5 5 5 .... 5 0 0 0 0 exp(

exp(

exp(

exp(

2 2 2 2

1 1 1 )))) 1

((((

_{ssss ssss}

ssss

ssss

S S S S h h h h h h h h

S S S S h h h h S

S S S p p p

p

µ α σ

σ

π − −

=

))))

||||

(((( rrrr ssss p p p

p

is a Rice p.d.f conditioned on a certain shadowing ,S

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

2 2 (((( 2 ]]]]

////

)))) 1 1 1 (((( 1 exp[

exp[

exp[

)))) exp[

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

||||

((((

= + ₂₂₂₂ − + ^{2222 2222} − ₀₀₀₀

K K K K K K K K

+

ssss

IIII rrrr K K K K S S S rrrr S K K K S K

S S S K rrrr K K K S

S S rrrr S p p p p

where the parameter

K K K K ,,,,

µ_ssss

,,,,

σ_ssssdepend on the actual environmental characteristics. The symbol error probability for transmission in channels affected by time and frequency nonselective fading can be written as

∫

∞

=

=

0 00 0

)]}

)]})]}

|||| )]}

((((

[[[[

{ { { )))) {

((((

))))

||||

((((eeee rrrr PPPP rrrr drdrdrdr EEEE EEEE PPPP eeee rrrr P

P P P P

P P

P_{eeee rrrr SSSS RRRR} (7)

By substituting (5) into(7) we have the probability of error In the RLN channel. The bit error probability for DDPSK modulation has been evaluated at different elevation angles(see fig. 5)

Fig.5 BER for DDPSK over RLN channel

5 Inclusion

Compared with the CPSK and DPSK,DDPSK is absolutely insensitive to the large Doppler shifts and fast Doppler rate. In addition, the demodulator does not require any preamble transmission for the carrier recovery and has simple structure It is known the DDPSK demodulator is one of choice to compensate the Doppler shifts in the LEO mobile satellite communication.

References

[1] Irfan Ali. “Doppler Characterization for LEO Satellites”. IEEE Trans.Commu. March 1998,46(3):309-313.

[2] Corazza G E,Vatalaro F. “Probability of Error and Outage in a Rice-Lognormal Channel for Terrestrial and Satellite Personal Communications”. IEEE Trans., Commun., August 1996,44(8):pp921∼924