Detection, Channel Estimation and Power Control in

M-DPSK Detection, Channel Estimation and Power Control in LEO Satellite Communications

Fernando D. Nunes and JosC M. N. LeitZoi

Instituto de TelecomunicaqBes, Instituto Superior Tknico AV. Rovisco Pais, 1, 1049-00 I, Lisboa-Portugal,

E-mail: {nunes, jleitao} @lx.it.pt

Absrrucf- Differential detection of phase-shift keying signals entails the ability to establish a stable phase reference based on previous sym- hol(s). In low bit-rate communications strongly affected by Doppler effect, like those making use of non-geostationary satellites, significant carrier phase drifts may occur within the symbol interval; thus, large irreducible error floors result if the Doppler frequency shift (DFS) is not estimated and subtracted.

We propose a state-space based receiver for M-DPSK signals in trans- missions affected by fast fading and DFS, which evaluates the multiplica- tive distortion (amplitude and phase) of the baseband signal and uses those estimates to track the frequency offset. In addition, the proposed receiver can estimate the power of the line-of-sight component and the Ri- cian fading parameter K which allow to monitor the transmission qual- ity and control the transmitted power in adaptive systems. Simulations presented for 4-DPSK show that the performance of the new receiver is superior to that of conventional receivers in the presence of large DFS, exhibiting substantially lower error floors.

I. INTRODUCTION

Differential detection of M-ary differential phase-shift key- ing (M-DPSK) signals is often used for mobile communica- tions as its advantageous characteristics, such as non-existence of carrier recovery and resolution of phase ambiguity, lead to simple and robust receiver implementations. The penalty to be paid regarding coherent detection is a decrease in power effi- ciency.

In conventional differential detection (two-symbol observa- tion) the phase of the current symbol is compared with the phase of the previous one (reference) and a decision is made based on the phase difference [ 11. If the phase reference is not stable, due to fading, Doppler effect or poor frequency align- ment between oscillators, the detector performance degrades leading to irreducible error floors [ 2 ] .

To remedy this problem the phase reference can be averaged from more than one symbol interval [3] or multiple-symbol differential detection can be performed using maximum like- lihood sequence estimation [4]. However, both methods re- quire, in general, processing of a large number of symbols to approach the performance of coherent detection, thus implying a large architectural complexity [ 5 ] .

In this paper we address the detection of M-DPSK signals in fast fading RiceRayleigh channels affected by significant carrier frequency offsets caused by relative motion transmit- terlreceiver (Doppler effect) and/or poor frequency alignment between oscillators. This includes applications such as low bit-

$This work was supported by FCT project POSV34829/CPS12000

rate transmissions between ground transceivers and LEO/MEO (low/medium earth orbit) satellites where significant Doppler frequency shifts are responsible for non-negligible carrier phase drifts within the symbol interval [6]- [7].

It is shown that, by sampling the (baseband) incoming sig- nals N

2

3 times within the symbol interval and using a stochastic nonlinear filter to estimate the amplitude and phase, the proposed receiver can withstand large frequency offsets wd with small degradation provided that 2nwdT,

5

1 . 5 ~ rad., where T, is the symbol duration. This receiver has a low com- plexity (being suitable for an all-digital implementation) and is able to estimate the frequency offset, the power of the line-of- sight component of the incoming signal and the Rician fading parameter K . This feature is useful to monitor the channel quality such as the instantaneous signal-to-noise ratio and the probability of deep fades and enable the use of adaptive trans- mission schemes. These schemes send channel information to the transmitter through a feedback channel allowing to vary the transmitted power level or the symbol rate [8].

11. PROBLEM FORMULATION

Consider the Rician land mobile satellite channel, defined by the complex vector B(t) = S ( t )

+

u l ( t )

+

ju,(t), where S ( t ) is the amplitude of the line-of-sight (or specular) compo- nent of the received signal, and u 1 ( t ) , u2(t) are independent, low-pass, zero-mean, Gaussian processes with equal variances

02, which model the diffuse (scattered) components. As in [9], we assume that u l ( t ) and u2(t) are well characterized by second-order Butterworth spectra whose -3 dB cutoff fre- quency ^fc (Hz) depends mainly on the velocity U of the ground receiver, according to ^{fc M} fow/c, where fo is the carrier fre- quency and c is the speed of light.

Fading is characterized by the Rician factor

K

= S 2 / ( 2 0 2 )

( K = 0 for the Rayleigh channel) and by the average bit signal-to-noise ratioYb = (S2+2o2)T,/(2J%~ log, M ) , where J%o/2 is the AWGN power spectral density. In LEO/MEO satellite communications the Rician factor depends on the el- evation angle of the ground antenna, Q, which is continuously varying over time; typically, the greater is ^Q, the larger is K The spectrum of the incoming signal is frequency shifted by the Doppler effect provoked mainly by the satellite veloc- ity, and widened by the time-varying multipath fading due to the land mobile and the satellite motions. We assume that the

r

^101.

receiver is driven by a free-running oscillator with frequency shift W d regarding the frequency of the incoming carrier (pos- sibly due to Doppler effect and/or poor frequency alignment between emitter and receiver local oscillators). In the sequel, we model jointly the motion effects and the oscillators insta- bilities by the phase p ( t ) , which is the sum of the time linear contribution, W d t , where ^{W d} is hereafter referred for simplic- ity as Doppler frequency shift (DFS), and a Brownian motion, P ( t ) , with E{P(t)P(7)} = q,min{t, T } .

The MPSK symbols at symbol interval [ ( k ^- 1)Ts, ICT,], k = 1 , 2 , .

. .

are a,,,[k] = exp(j27rm/M), m = O,1,.

. . ,

M - 1, and the corresponding differentially encoded symbols are b[k] = a,[k]b[k - 11, with b[O] = 1.

The received baseband signal is z ( t ) = B(t)ejt(‘)

+

v ( t ) , with [ ( t ) ⁼ p ( t )

+

B ( t ) , where the accumulated phase B ( t ) conveys information about the current and past symbols, and v(t) is AWGN with E{v(t)v*(T)} = 2&d(t - 7 ) . We have not considered herein the effect of shadowing due, for instance, to the foliage. However, the models of Suzuki or Loo could be easily included by multiplying B(t) by a lognormally dis- tributed process or by assuming that S ( t ) is lognormal [l I].

The input ~ ( t ) is sampled by integrate-and-dump circuits at a rate 4-1 = N / T s , where N

>

1 is the number of samples per symbol interval, to yield

z, = A,exp(jp,)

+

^{vr1, n} = 1 , 2 , .

. . ,

N , ₍₁₎ where z, denotes the sampled value of the generic quantity

~ ( t )

in the interval [ ( n - l ) 4 , nA]. In ( I ) , A , = lBnl = [ ( S , + ~ l l , , ) ~ + ~ ~ , , ] ~ / ~ , p, = p,+arg(B,)+B,, andv, i s a complex zero-mean white Gaussian sequence with E{vnvi} = 2r and r =

No/4.

The estimatioddetection problem consists of evaluating the channel multiplicative distortion (MD), i.e., the amplitude A , and the phase p,, in each symbol interval, and use those esti- mates to detect the symbols in some optimal way.

Asiume that B(t) and P ( t ) are slowly-varying within the symbol interval, and wd (although unknown) is constant in that interval. This scenario is consistent with practical LEOME0 satellite communications. For instance, a satellite at the alti- tude of 1000 Km exhibits typically the following values for the DFS and the Doppler rate: lwdl

5

47r x 10-5fo rad./sec.

and l b d l

5

10-6forad./sec.2 [12]. With 4-DPSK, fo = 5 GHz and a rate of 100 Kbitslsec. this corresponds to a max- imum phase drift of 47r rad./symbol. Of course, this value is excessive and must be compensated by adjusting the frequency of the receiver local oscillator. Since only a raw estimate of the DFS is often available, a desirable feature of the receiver is that it can cope with imperfect Doppler frequency compensations.

For channel estimation purposes, a state-space approach is adopted. The state vectorx, = [q,,,

. . .

x5,,IT, n = 1,

. . . ,

N , is defined in each symbol interval by the stochastic difference equation [ 131

and w C = 27rfc. The dynamics noise is a zero-mean, white Gaussian sequence w, = [wl,,, WZ,,, w3,,IT, with covari- ance matrix diag(q1,

Q,

q3}, where q1 = q 2 = f i o 2 / ( 7 r fc4) and 93 = q c 4 .

111. PROPOSED SOLUTION A. state vector estimation

Based on the dynamics model (2)-(3) and the set of observa- tions Z, = { Z I

,

^22,

. . . ,

z,}, obtained according to model (l), the estimation of x, is a stochastic nonlinear problem. The off- the-shelf solution consists in applying the extended Kalman- Bucy filter [14], but this solution is strongly dependent on the filter’s initial conditions which may lead to large acquisi- tion intervals (hangup effect). A better approach consists of propagating iteratively the probability density functions (pdfs) P, 3 p(x,lZ,-1) and F, p(xnlZn) which are associ- ated respectively with the prediction and filtering steps [ 141.

The filter’s initial conditions correspond to set the pdf P I . In general, it is impossible to propagate exactly the filtering and prediction densities; thus, some type of approximation (repre- sentation) has to be used.

In [13], a sub-optimal nonlinear filter (NLF) was proposed for this class of problems. The prediction and filtering den- sities are generically represented by

N ( %

- q,, V n ) 7 ( 2 5 -

[,,

^T,), where

M(%

- q,, V,) is a Gaussian pdf with mean q, = [VI,,,

. . . ,

^r/4,,IT and covariance matrix V,, and 7 ( 2 5

- L, m)

= exp[y, ~ 0 4 x 5 - ~ , ) ] / [ 2 7 r ~ o ( m ) l , with 7,

>

0 and Jz51

5

^{7 r ,} is a Tikhonov pdf. I,,(.) is the v.th-order modified Bessel function of the first kind.

For the sake of brevity we omit here the NLF equations:

they can be found in [13]. The most significant difference be- tween the NLF herein considered and the one proposed in [ 131 is in the prediction step where we have now exp(j<z+l) = e x p ( j G 2 ) 4 ) e x p ( j J c ) . The indices P and F refer, respec- tively, to the prediction and filtering steps, and

62)

is the cur- rent estimate of W d (corresponding to the symbol interval k ) .

The MD vector is estimated as

(4) The DFS estimate is updated in the current symbol interval according to

( 5 )

- ( k - 1 )

G y

= (1

-

p)w,

+

^pi&,

where the optimal value of the updating factor p is adjusted to minimize the probability of bit errors. Assuming that the major contribution to the change of phase pn within the sym- bol interval is due to the DFS, the local frequency estimate is computed as

which results from applying the Kay algorithm [ 151 to the filtering-step estimates of (P,.

B. fading parameters estimation

To estimate the fading parameters S and K we assume that both quantities are slowly-varying. Namely, we consider that

s^

^{= i?!,} and

k

⁼

i?i

(constant) in each interval

Z i .

Its duration is made much larger than 27r/w, in order to ensure that the estimation errors are small.

The amplitude A, is Rician-distributed with the I.th moment given by [ 161

E ( A ~ ) = (2a2)1/2e-Kr

(

¹

⁺

^-

;)

^1Fl(1+

^;’

^{1; K )}

^’

(7) where

r(z)

is the gamma function and l F ~ ( a ,

p;

x) is the confluent hypergeometric function. In particular, E ( A 2 ) = S 2

+

2 a 2 and E(A4) = S4

+

^8S2a2

+

^{8a4. Thus, S4 =}

2E2 ( A 2 ) - E(A4). and we estimate Si and Ki as

The local estimates are

(9) where and E(2:) result from averaging over the inter- val

&.

The updating factor 0

<

p‘

<

1 prevents the estimated quantities from exhibiting large fluctuations.

C. symbol detection

Let

E,

⁼ Gik’(n - 1 / 2 ) 4 be the estimated phase contribu- tion of the DFS to

&.

The phase (modulo 27r) at the iteration n, contributed by the fading and the current symbol, is

with

lihk)1 <

7r. The average phase (modulo 2n), over the symbol interval, due to the fading and the current symbol, is

Considering that the fading is slowly-varying in the symbol interval, the proposed detection algorithm consists of evaluat- ing the modulo 27r phases

and mapping them into the corresponding symbol. An example is given in Fig. 1 for M = 4, where Gray coding is assumed.

Since N decisions are taken, the symbol is selected by major- ity. Note that this algorithm relies on the ability of the stochas- tic nonlinear filter to track the phase even during the symbol transitions.

symbok01

symbol=l o

Fig. 1 . Mapping the phase tik’ into the detected symbols for A4 = 4

IV. SIMULATION RESULTS

Tests on the receiver’s bit error performance were conducted with 4-DPSK, T, = 20 p e c . (rate of 100 kbitslsec.), Rician fading characterized by K = 2 or K = 10, and w,T, = 0.105 rad./symbol (corresponding to a ground vehicle speed of 180 Km/h and a carrier frequency of 5 GHz). Several values of the normalized DFS, wdT,, were used. The results depicted in Figs. 2 and 3 were obtained with N = 3 samples per sym- bol interval (larger values of N yield essentially the same re- sults). For the sake of comparison, the theoretical bit error probabilities of the conventional differential detector (see, for instance, [ 161, [ 171) were also plotted for wdT, = 0 (ideal case) and w ~ T , = 0 . 2 ~ . These probabilities were evaluated using equation (A. I ) of the Appendix. Even for small values of w ~ T , the performance degradation of the conventional receiver is very significant leading to large error floors. In contrast, the proposed receiver can withstand significant DFS (typically

W d T ,

5

1 . 5 ~ ) with substantially lower error floors. However, this behavior is not maintained for wdTs x 27r, as the receiver exhibits a poor performance associated with large error floors.

In Fig. 4 we depict the DFS estimation for initial values of 0, 2.5 and 5 radhymbol. The jump of 1 rad./symbol in each curve is readily tracked by the DFS estimator. The estimate

100

-

proposed receiver K=2

- _ _ _ _

I

10-1

K w m

10 -2

1 0 . ~

20 25 30

15

-

10-4

lL

Y,,

dB

10.’

10.2

a m w

1 0 . ~

3 _K=10

-

proposed receiver conventional receiver

U T =0.2n d s

Fig. 2. BER for the proposed receiver and the conventional receiver with Fig, 3. BER for the proposed receiver and the conventional receiver with

K = 2 K = 10

corresponding to the upper solid line is slightly biased as the DFS approaches the limit of 2n rad./symbol.

An important feature of the proposed receiver is its ability to estimate the fading parameters S and K and to adapt itself to their variations. In Fig. 5 we show an example of the K estimation for initial values of K = 2, 10 and 50. Each curve exhibits a discontinuity corresponding to a jump to twice the initial value of K . The length of the fading parameters esti- mation interval, Zi, is made equal to lo4 symbols in Figs. 2-5, corresponding to update the parameters estimates each 200 ms.

V. FINAL REMARKS

The results of the previous section show that the conven- tional differential detection, although yielding very simple re- ceiver implementations, is largely penalized for the presence of Doppler frequency shifts (DFS) that occur, for instance, in non-geostationary satellite mobile communications. In this pa- per we proposed a state-space based open-loop receiver (with a local free-running oscillator) where the fading and the phase drift are estimated by a stochastic nonlinear filter. The filter’s outputs are used for symbol detection and estimation of the DFS and the Rician fading parameters S and K . These pa- rameters are important to evaluate the quality of the transmis- sion in diversity or handoff schemes or in adaptive transmis- sion schemes (using a feedback channel) which aim at con- trolling the transmitted power and/or the bit rate. By track- ing the frequency shift, the bit error performance of the pro- posed receiver approaches that of the conventional differential detection (without frequency offset) for a wide range of val- ues of DFS. Wherever this range is insufficient, side informa- tion about the satellite trajectory may be utilized to provide a

6 -

(/I

k 1 - 0 -

0 2e+05 4e+05 6e+05 8e+05 le+06

symbol no.

Fig. 4. DFS estimation with K = 2 and Tb = 15 dB

coarse tuning of the receiver’s local oscillator. Alternatively, a scheme consisting of a bank of stochastic nonlinear filters, each one driven by a local oscillator with a different frequency, can be envisaged.

APPENDIX

Let vo be the carrier frequency error of the incoming 4- DPSK signal regarding the conventional receiver’s local os- cillator. The baseband signal is z o ( t ) = B(t) exp[j(2nvot

+

a ) ] a ( t )

+

^no(t), where the fading vector B(t) is characterized by the Rician parameter K = S 2 / 2 a 2 , ^(Y is the phase error at t ⁼ 0,

00

a=--

represents the digital modulation, and no(t) is AWGN statisti- cally equivalent to v(t). Assuming without loss of generality

- Y = xexp[-j(27rvTs

+

^7r/4)].

..._..._...

100: I

Quantities pxx = pyy and pxy are given by

?-. Y -

U

correct U

t+T, 0 2

I - estimated KJ

0 pxy = - e j T / 4

lTs 1

p,(r)ejzT”‘ drdt,

2e+05 4e+05 6e+05 8e+05 le+06 T,”

symbol no.

Fig. 5. Rician parameter estimation with Yb = 15 dB

that all ai = 1

+

^{j 0} and that the pulse p ( t ) is rectangular, the normalized integration of z o ( t ) is

X(v,) = ZO(iT,) =

-

1 iT,

zo

( t )

dt T~ A i - I ) T s

and the symbol decision variable of the conventional receiver is X ( ~ o ) Y + ( v O ) e j ~ / ~ , where Y(v0) = zo[(i

-

l)Ts] [17].

Since X and Y are complex Gaussian RVs (conditioned on the transmitted symbols) it is easy to show that the probability of bit error is

1

pb = -[prob 2 { D ( v o )

<

0}

+

prob {D(-vo)

<

O}], (A.1)

where p , ( ~ ) = fiexp(-fi7rfcIrl) cos(fi7rfclrl - 7r/4) is the normalized correlation of the fading components u l ( t ) and uz ( t ) , which have 2.nd-order Butterworth spectra (see section 11).

[41 Dl

with D ( v ) = X ( v ) Y * ( v )

+

X * ( v ) Y ( v ) . The probabilities of the right member of (A.l) are determined by [I61

[71 prob { D ( v )

<

0} = & ( a , b )

where &(a, b) is the Marcum’s Q-function, a = [2v~0z(p1v2 -

h ) ] ’ / 2 / l ~ ~ +

^{w2I and b =} [20121;(P1~1

+

P2)]’/2/(v~

+

^2121.

Quantities v1 and ^v2 are obtained from

/?2 = 2Re{x‘Y}

-

where pzz = (1/2)E{IX

- XI2},

pxy = (1/2)E{(X - X)(Y

- y ) * } ,

and

- sin(nvTs)

X = S e x p [ j ( x v ( 2 i - l)Ts + a ) ]

rvT, ’

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