Uppsala universitet


Signals and Systems, Uppsala University

Researchers: Mikael Sternad and Joachim Björsell

Previous PhD students: Rikke Apelfröjd Daniel Aronsson, Torbjörn Ekman, Lars Lindbom, and Jonas Rutström.

Prediction of Mobile Radio Channels (Research 1999-present)

Overview slides on predictor antennas (pdf)
We study and develop high-performance algorithms that predict the short-term fading of mobile radio channels. We investigate both algorithm design and the theoretic limits of performance.

Channel quality prediction is of increasing interest for a range of applications such as adaptive coding and modulation as well as allocation of time, frequency and spatial transmission resources. Channel prediction is important for advanced multi-antenna transmit schemes such as zero-forcing beamforming. Due to backhaul delays, it is crucial for joint transmission coordinated multipoint schemes.

In general, prediction of the power of a Rayleigh fading parameter is possible with useful accuracy around 0.2-0.3 wavelengths ahead in space. The attainable accuracy depends on the signal-to noise ratio, the positioning and fraction of pilots (known transmitted symbols that are used for channel estimation) as well as on the fading statistics.

The best performance that can be attained by linear predictors based on past data and channel statistics is obtained by Kalman prediction schemes. Kalman theory also provides analytical expressions for the obtainable prediction accuracy, for a given fading statistics, pilot pattern and SNR. We have studied and utilized Kalman-based adaptive predictors on real measured channel data at WCDMA as well as 3GPP LTE bandwidths. Our results show that channel predictions of sufficient accuracy for link adaptation and scheduling can be obtained in downlinks and uplinks of OFDM systems like 3GPP LTE working in the "digital dividend" band (700-800 MHz). This could enable increasing scheduling gains for wireless broadband systems such as LTE. Different frequency resource blocks can also be predicted separately, in downlinks as well as multiuser uplinks, in FDD as well as TDD systems.

In addition to Kalman predictors, we have also utilized and investigated FIR Wiener predictors and various nonlinear filtering structures. A conclusion is that nonlinear prediction of complex channel gains is nonrobust when applied to measured channel data.

More recently, we are investigating and developing the "predictor antenna concept" for use in connected vehicles: It is a simple way to improve the attainable channel prediction horizon for vehicles by at least an order of magnitude. This can be done by placing antennas on the roof of a vehicle, and letting the first antenna, in the direction of travel, perform channel prediction for the rearward antennas. See the original publication from 2012 and a more recent proposal for using the concept in 5G massive MIMO TDD links.

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References on prediction of mobile radio channels:

PhD Thesis by Rikke Apelfröjd, 2018.

Report by Rikke Apelfröjd, 2014, revised 2018, on Kalman prediction of multipoint OFDM channels.

Licentiate Thesis by Rikke Apelfröjd, 2014.

PhD Thesis by Daniel Aronsson, 2011.

Proceedings of the IEEE paper (2007, Invited Paper), giving an overview of adaptive transmission in OFDMA systems, also using channel prediction.
Licentiate Thesis by Daniel Aronsson, 2007.
IEEE PIMRC 2007 paper on Kalman predictor design for frequency-adaptive scheduling of FDD OFDMA uplinks.
EUSIPCO 2007 paper on OFDMA uplink channel prediction.
IEEE ICASSP 2005 paper on channel estimation and prediction for adaptive OFDMA/TDMA uplinks based on overlapping pilots.
IST-Summit-2005 paper on adaptive TDMA/OFDMA for wide-area coverage and vehicular velocities.
VTC 2003-paper on Channel estimation and prediction for adaptive OFDM downlinks.

PhD Thesis by Torbjörn Ekman, 2002.
VTC 2002-Fall Conference Paper on unbiased power prediction of broadband radio channels.
VTC 2001-Spring Conference Paper on long-term prediction of broadband radio channels.
Licenciate Thesis by Torbjörn Ekman, summarizing our results up to november 2000.
VTC 1999Fall Conference Paper on linear and quadratic predictors.

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Low-Complexity Tracking of Time-varying Parameters of Linear Regression Models
(Research 1990-2005, together with Lars Lindbom)

Overview Presentation slides (pdf)

We often need to construct a linear dynamic model based on data, by first selecting a suitable model structure, and then adjusting (estimating) its parameters. In situations with time-varying parameters, the parameter estimation problem becomes a parameter following (or tracking) problem.

Standard Algorithms: LMS and Windowed RLS

Ad hoc tracking schemes can be obtained by modifying algorithms for the time-invariant case, under the assumption of slow parameter variations. The modifications introduce moving data windows and non-vanishing adaptation gains. The choice of adaptation gain (or data window) is a compromise between noise sensitivity and tracking capability.

For example, the use of modified stochastic approximation gives LMS while Gauss-Newton schemes lead to e.g. windowed Recursive Least Squares. These two algorithms are in frequent use. They in general work well if the regressors data provides sufficiently rich information, and if the time-variations are slow. "Slow" is here a relative concept. It depends on the speed of parameters drifts as compared to noise power, and on the number of parameters to be estimated.

[LMS tracking performance with low and high step length]

The example above illustrates the tracking performance of the LMS algorithm (solid line). The dashed line represents a time-varying scalar "true" parameter. A too low adaptation gain (left) will result in bad tracking, while a high gain (right) increases the noise sensitivity of the estimate.

For fast time-variations, the performance of both LMS and RLS tracking schemes can be poor; We may find no reasonable adjustment of the data window, since both types of algorithms will, in general, have a suboptimal structure.

The most efficient way of improving the tracking performance is to utilize any existing prior information about the nature of the time variations. For example, it may be known that the parameters behave approximately as sinusoids. Such information about smooth variations can be included into algorithms based on Kalman filters, either in the form of stochastic models or as functional series. In the latter case, the parameters are modelled within the data window e.g. as ramps, parabolas or sinusoids with adjustable parameters. Other types of schemes, using fault detection or hidden Markov models, are used when rare but abrupt parameter changes can be expected.

Tracking Time-varying Mobile Radio Channels

This was, roughly, the state of the art for tracking problems when we faced the challenging problem of estimating models of time-varying radio channels in digital cellular systems. It may be of interest to briefly describe this problem.

A communication channel for baseband signals can be represented by a FIR-filter with time-varying parameters (taps). In present TDMA systems, digital data are transmitted in bursts of fixed length. Within each burst, a small fraction of the transmitted data (the training sequence) is known to the receiver. The channel can be estimated by correllating the known channel input with the received output signal. A detector (equalizer) is then designed by using the model, with the aim of retrieving the remaining, unknown, symbols. This procedure is repeated for each data burst.

The time dispersion of the channel is caused by multipath propagation. Due to reflections in the environment, the receiver is reached by the superposition of several delayed versions of the transmitted signal. The time variability, fading, is due to the mobile moving through standing wave patterns. These wave patterns are also a result of the multipath propagation.

Among other factors, the properties of discrete-time channels depend on the sampling rate, which is related to the symbol transmission rate. For example, with the high symbol rate of the GSM system, FIR channels of length 3-5 are obtained, which are almost time-invariant over the duration of one burst. A channel model estimated from the training sequence will therefore be almost correct within the whole data burst. One equalizer (or Viterbi detector) with fixed parameters can then be used within the whole burst, but it may be of interest to robustify the design with respect to somewhat time-varying channel dynamics.

In contrast, the low symbol rate utilized in the North American D-AMPS standard causes the discrete-time FIR channel to be very short, having just one or two coefficients. These coefficients will vary appreciably over time-windows of 10 samples, while a data burst consists of 168 symbols. The time-variation is even more rapid in systems with carrier frequencies around 1.8 GHz. A model based on the training sequence can therefore not be relied upon over the whole duration of a burst. Adaptation of a time-varying model is required.

[Symbol estimation in feedback with channel estimation]
A recursive algorithm for simultaneous adaptation and detection is illustrated above. It is initialized with a model based on the training sequence { t = 1 ... Ntr }. "Decision-directed" adaptive modelling is then applied on the remaining received data { y }, with detected symbols used as substitutes for the unknown channel input. The use of an adaptation algorithm that attains small tracking errors is of crucial importance in this type of receiver.

When we studied the IS-136 scenario a few years ago, existing methods for analysis and design turned out to be unsuitable, for the following reasons:

  • The parameter variations of the FIR-channel can not be regarded as ``slow".

  • The performance of LMS and RLS- algorithms was unacceptable, so a scheme for using a priori information about the nature of the time-variations had to be developed. Such information was available from physical considerations, but

  • the use of Kalman-based algorithms was out of the question, due to severe limitations on the allowable computational complexity.

A Method for Systematic Design of Adaptation Algorithms with Fixed Gains

As a response to the problem formulated above, we have developed a novel class of algorithms for tracking smooth but fast variations in the parameters of linear regression models, in particular FIR models. The novel design technique is the subject of a PhD Thesis by Lars Lindbom. The performance of a resulting low-complexity algorithm, the Simplified Wiener-LMS or SWLMS scheme, is illustrated below. It is compared to a well-tuned LMS algorithm for a time-varying parameter typical of the IS-136 environment. The (patented) SWLMS algorithm works well in IS-136 equalizers also in difficult situations with flat fading, i.e. for single-tap FIR channels.

[LMS versus SWLMS]

While the novel technique for analysis and design has been inspired by mobile radio applications, the results have wider significance. The key tool is a Wiener filtering approach to the design of tracking algorithms. From this viewpoint, the understanding of algorithms such as LMS, momentum LMS and signed regressor LMS can be improved and a general class of algorithms with time-invariant gains can be proposed and optimized.

The new class of algorithms combines low computational complexity with the possibility for a significant performance increase as compared to LMS/windowed RLS. This is attained by introducing stochastic "hypermodels" which describe the second order properties of the vector of time-varying parameters:

h(t+1) = F h(t) + C e(t) , or

D h(t) = C e(t) .

Here, h(t) is the parameter column vector, while F, C and D = qI - F are polynomial matrices in the shift operator q and e(t) is a white noise column vector. The optimal structure of an adaptation algorithm with constant adaptation gain will then be as follows:

hh(t+k|t) = F hh(t+k-1|t-1) + G fi(t) eps(t) , (a)


hh( . ) is the parameter estimate (column vector)
fi(t) is the transposed regressor matrix
eps(t) is the prediction error, or output error
F is obtained directly from the hypermodel, while
G is in general a matrix of transfer functions.
The LMS algorithm is obtained if F = I and if G is a diagonal matrix containing adaptation gains. This structure is optimal only if the true parameter vector h(t) has random walk statistics h(t+1) = h(t) + e(t).

The polynomial matrix F will be given directly by the hypermodel. The adjustment of G which provides a minimal sum of squared parameter errors can be obtained by solving a linear (Wiener) design problem. This problem is illustrated below, where R is the covariance matrix of the regressors and the leftmost block represents the hypermodel. The rational matrix G in the adaptation law (a) is uniquely determined by the resulting optimal stable rational matrix Lk below.

[ Fig. of h<sub>h</sub>(t+k|t) = L (R h(t) + n(t))  ]

The Wiener problem can in general be solved via a spectral factorization and a bilateral Diophantine equation. In simplified but powerful variants, the design equations become trivial, and no equations need to be solved. The design can also be made robust against uncertainties in the hypermodels.

Adaptation laws for parameter prediction (k>0), filtering (k=0) and fixed lag smoothing (k<0) can be derived. The use of smoothing improves the attainable performance, while multistep prediction of channel coefficients is of interest e.g. in adaptive Viterbi receivers for TDMA mobile radio systems, or in power control algorithms in CDMA.

In digital communications, the resulting tracking algorithms can be applied to the multi-input multi-output FIR channels appearing in multiuser detection, in CDMA systems as well as in OFDM systems. In other applications, the method can be applied to time-varying output error models and various functional series models. A restriction is that analysis and design has not yet been developed for models with lagged output data as regressors, such as ARMA models.

This work has also resulted in an analysis leading to exact expressions for parameter error covariances for fast time-variations, in situations of relevance for mobile communications.

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References on Model-based Design and Analysis of Adaptation laws:

Overview Presentation slides (pdf).

Paper 1 on design of constant-gain adaptation algorithms. (Also IEEE ICASSP 2001)
Paper 2 on analysis of stability and performance. (Also IEEE ICASSP 2001,2002)
Paper 3 on the Wiener LMS adaptation algorithm, a special case. (Also IEEE VTC 2000)
Paper 4 on a case study on IS-136 channels. (Also IEEE VTC 2000)

Conference paper in IEEE VTC 2003-Fall on ODFM channel adaptation with the constant-gain tracking algorithms.
Conference paper in IEEE VTC 2002-Fall on gain adaptation in WLMS tracking algorithms.
Conference paper at the European Control Conference, 2001.
Robust design of adaptation laws with constant gains, IFAC Como 2001.

Licentiate Thesis by Jonas Rutström, 2005.
PhD Thesis by Lars Lindbom 1995.
Licentiate Thesis by Lars Lindbom, 1992.
Conference paper on the simplest variant, SWLMS, and a channel tracking example.
Master Thesis on the use of a fixed grid of algorithms in IS-136.

Conference paper in IEEE ICASSP'93 on an early version of the method.
Deterministic modelling of time-variations in digital mobile radio channels.
PhD Thesis by Kenth Öhrn, May 1996, on robust filtering and Kalman tracking.