Estimation of LEO Satellite Channels
Linda M. Davis Iain B. Collings Robin J. Evans Centre for Sensor Signal and Information Processing
Department of Electrical & Electronic Engineering University of Melbourne
Parkville, Victoria 3052 AUSTRALIA
Abstract
In this paper, we propose a new channel estimation al- gorithm for low earth orbit satellite digital communica- tions. Low earth orbit satellite channels impart severe spreading in delay and Doppler on the transmitted sig- nal. We model the channel as a tapped-delay-line lter with time-varying complex coecients. The time evolu- tion of the taps is described by an auto-regressive model.
A coupled lter using recursive prediction error methods is proposed for estimating the parameters of the chan- nel. Its performance is demonstrated via simulations.
1 Introduction
Several low earth orbit (LEO) satellite systems have been proposed for commercial mobile communica- tion services (e.g. Iridium, Orbcomm, Globalstar, and Teledesic). In this paper, we consider characterization and estimation of LEO satellite communication chan- nels.
LEO satellite communication channels are time-varying and multipath, and thus cause signi cant fading [1, 2, 3].
The motion of the land-based mobile, traversal of the satellite relative to the earth, and changing absorption, scattering, refraction, and di raction e ects of the envi- ronment cause both fast and slow fading.
Fast fading is a consequence of multipath propagation.
Measurements with mobile receivers have shown that the fast fading is Rayleigh distributed (or Rician, in the case of a line-of-sight path) [?, 2, 4, 5]. Thus, the trans- mitted signal is both Doppler and delay spread by the channel [6, 7]. For LEO satellite channels, the Doppler spreadis related to the motion of the mobile, as in terres- trial systems. (This is in addition to the large Doppler shiftwhich is imparted on the transmitted signal due to the relative motion of the satellite, up to 40 kHz.) Delay spreading is most signi cant when the satellite is at low elevation angles. However, due to the low digital symbol
yThis work is supported by the Australian Telecommunications and Electronics Research Board.
rate (e.g. 2.4 kbaud for Iridium) the delay spread causes only small amounts of intersymbol interference (ISI).
Slow fading (also known as shadowing) results from the changing terrain contours (e.g. nearby buildings, trees, mountains), and has been found to have a log-normal distribution [1, 2, 4, 5]. Shadowing is most pronounced at low elevation angles.
Optimal equalization and detection for such doubly spread and dual fading time-scale channels is non-trivial.
In this paper, we present an on-line technique for esti- mating the satellite channel to aid equalization and de- tection at the receiver. We propose a coupled estimator using recursive prediction error methods [8]. The algo- rithm is able to estimate the parameters of the delay and Doppler spread channel and also the channel mean.
Further, changes in these parameters caused by shadow- ing, variation in elevation of the satellite, and changes in mobile direction and velocity can be tracked.
For our coupled estimator, the channel is modelled as a nite impulse response (FIR) tapped-delay-line lter with time-varying taps. Since the statistics of the fad- ing of the LEO channel can be considered stationary for periods large enough to estimate the parameters, we introduce an auto-regressive (AR) model for the time- evolution of the channel taps. The model is presented in Section 2.
In Section 3, we present the coupled estimator. It con- sists of a Kalman lter for tracking the channel taps, together with an estimator for the channel mean and AR parameters. We discuss features of this algorithm relevant to the LEO satellite channel application.
The performance of our coupled estimator is demon- strated in Section 4. LEO satellite channels were sim- ulated (independently of the estimation model using Jakes spectrum for the Doppler spread channel variation [2]), and the coupled estimator was applied in a variety of scenarios. We discuss aspects of the algorithm which are important for LEO satellite communications. The paper concludes with a summary in Section 5.
-
s(t) Spectral Pulse
g(t)
- Physical Channel c(t;) h(t;)
-+h 6
y(t) WGNn(t)
-
z(t)
Figure 1: Transmission through LEO channel
2 Channel Model
In this paper, we consider the channel to consist of the physical channel in cascade with the transmitter lter (raised cosine spectral pulse), as shown in Figure 1. The channel is modelled as a tapped-delay line lter which is nite in extent, and thus has a nite impulse response.
With appropriate choice of tap spacing, Bello [6] showed that a tapped-delay line accurately models most practi- cal channels with nite degrees of freedom due to lim- ited time duration, fading rate and bandwidth. Using complex notation to represent signals at baseband, the channel taps have complex coecients. For convenience, the tap spacing is the same as the input signal sampling rate. However, the algorithm presented in this paper is equally applicable to channels modelled with fractional spacing.
By representing baseband signals in complex notation, the received signal is given by:
z(t) =XL,1
=0
s(t,)h(t;) +n(t); t= 0;1;2::: (1) where L is the number of channel taps, s(t) 2 C (the set of complex numbers) is the input at time t, and h(t;) 2 C is the th channel tap at time t. The ob- servation noise, n(t), is assumed to be zero-mean white Gaussian noise, with known variance 2n (i.e. the real and imaginary components each have variance 0:52n [9]). In practice, there is some xed delay, m, from transmitter to receiver. Without loss of generality, the delay is assumed to be zero.
Each time-varying channel tap weight,h(t;), consists of a mean response component, h(), and zero-mean ran- domly time-varying component, ~h(t;):
h(t;) =h() + ~h(t;) (2) A channel with non-zero-mean taps occurs when there is a line-of-sight path or xed scatterers, giving rise to h(). The random component results from the chang- ing physical characteristics of the transmission medium.
The channel envelope has been measured to be Rayleigh or Rician [1, 2, 4, 5] and thus the random component,~ h(t;), has Gaussian real and imaginary components [10].
As with many physical channels, the statistics of the fad- ing of the LEO channel are wide-sense stationary (WSS) [6, 5]. The channel can then be considered stationary for periods large enough to estimate the parameters, and hence we introduce an AR model for the time-evolution of the channel taps. The choice of the model orderRis a trade-o between the accuracy of the model and the diculty in estimating its parameters.
Now, consider the AR equation:
~
h(t) =F1h~(t,1) ++FRh~(t,R) +u(t) (3) where ~h(t) =hh~(t;0)h~(t;L,1)i0, and0 is the trans- pose operator. The generating noise for the Gaussian random process,u(t) = [u(t;0)u(t;L,1)]0 is a zero- mean i.i.d. complex Gaussian, with known varianceu2. The AR parameters contained in the matricesF1:::FR are generally complex. (These parameters will be real if it is assumed that the real and imaginary parts ofh(t;) are independent,)
For stability and wide-sense stationarity of the AR pro- cess, the roots of the characteristic equation, j I ,
P
R
=1 F
z, j= 0, must lie inside the unit circle of the z-plane [11].
For some channels, the taps may be assumed indepen- dent. These channels are referred to as having uncor- related scatterers (US) [6]. Many radio transmission channels are assumed to be WSS{US [6, 7]. However, since we have a discrete model in which the channel con- sists of the transmitter lter in cascade with the physi- cal channel, the taps,h(t;), may be correlated even if the underlying physical channel has uncorrelated scat- terers [12]. If indeed the taps were uncorrelated, this could be built into the AR model by appropriate choice of elements in the matrices, F1:::FR (i.e. by setting cross-terms to zero).
3 Coupled Estimator
In this section, we propose a new coupled estimator for estimation of the LEO satellite channel. The estimator consists of a Kalman lter for tracking the channel taps, together with an estimator for channel mean and AR parameters. In fact, this algorithm belongs to the class of recursive prediction error (RPE) methods [8].
3.1 Channel Tap Estimation and Track-
In this section, we show that when the channel tap
ing
means,h, and the AR parameters contained in the ma- tricesF1toFRin (3) are known, a Kalman lter can be used to track the time-varying component of the channel taps.
The state vector for the Kalman lter is x(t) =
hh~(t);;h~(t,R+1)i0.
Thus, we rewrite (3) in the following form:
x(t) =Ax(t,1) +v(t); t= 0;1;2::: (4) where v(t) = [u(t);0;;0], and:
A=
2
6
6
6
6
4 F
1 F
2
F
R
I 0 0
0 ... ...
... I 0
3
7
7
7
7
5
(5)
By de ning a vector of input signal samples s(t) = [s(t); s(t,1);; s(t,L+1)]0and the observation ma- trixC(t) = [s0(t);0;;0], the observation equation (1) for the system can be written:
z(t) =C(t)x(t) +s(t)h+n(t); t= 0;1;2::: (6) Assuming that the channel tap means,hare known, (6), the observation equation becomes:
e(t) =C(t)x(t) +n(t); t= 0;1;2::: (7) where e(t) =z(t),s0(t)h.
With this formulation, a complex Kalman lter [11, pg.
321] can be used to provide on-line estimates of the chan- nel tap o sets, ~h(t;), which de ne the fast fading chan- nel at each time instant.
3.2 Channel Mean & AR Parameter Es- timation
In the above section, the channel mean, h, and AR parameters, F1 to FR, were assumed known for the Kalman lter formulation. However, these parameters need to be estimated. Assuming knowledge of the chan- nel taps, ~h(t;), we use the observations, z(t), and the inputss(t), in a recursive least squares algorithm to es- timate the channel mean and AR parameters directly.
Consider the observation equation (6). By substituting (4), we can rewrite equation (6) as a linear function of a vector, , which contains the channel tap means and AR parameters:
z(t) = y(t) +n(t) (8)
= s0(t)F1h~(t,1) ++s0(t)FRh~(t,R) +s0(t)h+s0(t)u(t) +n(t)
= 0(t)+s0(t)u(t) +n(t) (9) where
0(t) 4= [s0(t);(x(t,1)s0(t))0] (10)
4= [h0;(vecF1)0;;(vecFR)0]0 (11) and denotes the Kronecker product. Note 0(t) and
2CL+RL21.
Parameter Update (RLS) Kalman Filter
-
-
- - -
-
- -
- -
6 -
?
`
`
`
`
`
`
`
`
`
`
`
`
z(t) ` s(t)
^~
h
^
Figure 2: Coupled Kalman Filter and AR Parameter Estimator
Thus:
z(t) = 0(t)+s0(t)u(t) +n(t)
= 0(t)+ ltered noise (12) This manipulation of the observation equation is the key to our estimation of the channel mean and AR param- eters, . We can now apply a complex recursive least squares (RLS) algorithm [11], using (12) to estimate.
3.3 Coupled Estimator (RPE)
Figure 2 shows the form of the coupled estimator. Given the current data, the Kalman lter estimates the chan- nel tap means and time-varying components. These are then used by the RLS algorithm to update estimates of the AR parameters. The updated AR estimates are used by the Kalman lter at the next update.
This estimator is, in fact, a version of the recursive pre- diction error algorithm [8]. The Kalman lter forms an estimate of the channel tap o sets, ^~h(t;), based on the current estimates of the channel mean and AR parame- ters, ^. The RLS algorithm uses these estimates, instead of the unknown true values. The prediction error con- sists of the di erencey(t),y^(t) and the noise contribu- tion,n(t). The parameter estimates, ^, are updated by the RLS algorithm on the basis of this prediction error.
4 Simulation
In this section, we demonstrate the performance of our coupled estimator for a LEO satellite channel.
The channel was simulated using a direct-form quadra- ture amplitude fading model [13]. The delay spread characteristics were assumed to be similar to those for typical urban GSM as cited in [13], and the Doppler spread was speci ed by Jakes spectrum [2] where the maximum Doppler spread is determined by the mobile velocity (and the carrier frequency), chosen to be 60 km/h.
For the case with no shadowing, Figure 3 shows the sim- ulated envelope of the received signal variations when
0 0.2 0.4 0.6 0.8 1
−15
−10
−5 0
Received Signal Level (dB)
Time (s)
Figure 3: Simulated received signal envelope, Rice factor
= 6.0dB
the unmodulated carrier was transmitted. The Rice fac- tor (direct-to-multipath signal power ratio) was chosen to be 6.0 dB, re ecting an urban environment at low elevation angle [4].
The transmitter used a raised cosine pulse with roll-o factor, = 0:5. The time-varying impulse response of the channel (physical channel in cascade with the trans- mitter lter), h(t;) is shown in Figures 4 and 5. The carrier frequency was chosen to be fc= 1:6 GHz and the symbol ratef = 2:4 kHz, as for the Iridium system.
Note that the taps have been spaced at T=2 (where T is the symbol interval) in .
When estimating the channel, it was assumed that an estimate of the xed delay from transmitter to receiver was available. Thus the region of delays at which signif- icant taps occur was known, and the only taps in this region were estimated. ThreeT=2 spaced taps were es- timated from a random 4-PSK training sequence, s(t), and the received signal, z(t), assuming that the re- ceiver was synchronized. A second order AR process (R = 2) was used to model the time-evolution of the taps. Figures 6 and 7 show the converging estimates of the real and imaginary components of the channel tap means. The estimated AR parameters are not shown, but the corresponding Doppler spread (most signi cant tap) is shown in Figure 8 where the estimated channel (dashed line) is overlayed on the simulated data Doppler spread (solid line). As expected, the Doppler spread is
fDmax = (v=c)fc = 89 Hz, wherev is the mobile ve- locity and c is the speed of light. (The additional low frequency components in the simulated Doppler spread are caused by the implementation of Jakes spectrum in the channel simulator.)
To approximate the dynamic behaviour of the log- normal shadowing, the received signal was multiplied by a shadowing factor which is constant within a shad- owing interval, but changes from one interval to the next [4]. The shadowing rate is in the order of a few Hertz [5]. Figure 9 shows how the coupled estimator copes with tracking the change in the channel tap means, for a shadowing rate of 2 Hz. In a similar manner, the cou- pled estimator is also able to track changes in the
0
0.5
1
0 1 2 x 10−3
0 1 2 3
ε t
Re{h(t,e)}
Figure 4: Simulated channel response, no shadowing, Refh(t;)g
0
0.5
1
0 1 2 x 10−3
−1
−0.5 0 0.5 1
ε t
Im{h(t,e)}
Figure 5: Simulated channel response, no shadowing, Imfh(t;)g
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 1.2
t
Re{h(e)}
Figure 6: Estimated channel means, no shadowing, Refh()g
0 0.2 0.4 0.6 0.8 1
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3
t
Im{h(e)}
Figure 7: Estimated channel means, no shadowing, Imfh()g
−2000 −100 0 100 200 5
10 15 20 25 30
Hz
dB
Figure 8: Doppler spread '-' simulated, '-.-' estimated
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 1.2
t
Re{h(e)}
Figure 9: Estimated channel means (real part, shadow- ing)
Doppler spread of the channel resulting from changing velocity of the mobile, and changing propagation con- ditions. Tuning of the algorithm's tracking ability is controlled by the forgetting factors in the RLS part of the algorithm.
Receiver synchronization is particularly dicult for LEO satellite channels. The receiver must tolerate the large (and changing) Doppler shifts caused by the rela- tive motion of the satellite as well as the Doppler spread which we are estimating here. Since the large Doppler shift is related to the perceived orbit of the satellite by the mobile, it can be predicted and somewhat compen- sated at the receiver. Deviations from this predicted Doppler shift arise from orbit perturbations and land terrain contours. Figure 10 shows that the algorithm was still able to sensibly estimate the Doppler spread, despite a shift in the received signal by 3%.
5 Conclusion
In this paper, we have presented a new on-line algorithm for estimating the satellite channel in order to aid equal- ization and detection at the receiver. The technique es- timates the parameters of the delay and Doppler spread channel and also the channel mean. Further, changes in these parameters are tracked. We have considered the performance of the algorithm for a variety of scenarios including shadowing and during errors in receiver syn- chronization. Considering the adaptive nature of this
−2000 −100 0 100 200
5 10 15 20 25 30
Hz
dB
Figure 10: Doppler spread '-' with receiver synchro- nized, '-.-' with Doppler shift in received signal
technique, its application in bursty noise conditions (a major problem in many communication systems) is the focus of on-going research.
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