Estimation of speed, average received power and received signal in wireless systems using wavelets Rajat Bansal Sumit Laad Group Members rajat@ee.iitb.ac.in laad@ee.iitb.ac.in 01D07010 01D07011 Abstract A technique is described for estimating the speed of a mobile station and mean of received power in a wireless system. The mobile speed maps the characteristic spatial scale of the received signal into a characteristic temporal scale. The continuous wavelet transform is used to estimate the local minima. The estimate is based on samples taken at the midpoints between the local minima of the received envelope. This technique requires neither an estimate of the mobile speed nor an adaptive temporal averaging window, in contrast to other estimators. Index Terms Wavelets, continuous wavelet transform, mobile station. I INTRODUCTION A significant characteristic of wireless systems is the signal variation caused by the movement of the mobile stations. An estimate of the local mean of the received signal is useful to determine the quality of a radio link. The mean signal is also needed in handoff, power control, and channel assignment algorithms. Accurate estimates of the mean signal improve the performance of system control algorithms and thereby increase the realized system capacity. In many environments, a direct path is not present between the base station and the mobile station. The received signal consists of a sum of waves which have been reflected by objects such as mountains, trees, and buildings. The sum of many waves at the receiver gives rise to small-scale spatial variation of the received envelope (on the order of a wavelength). In situations where there is no dominant path between the base station and the mobile station, the small-scale spatial variation is called Rayleigh fading. The received signal is non stationary for distances on the order of building sizes since the mean of the small-scale variation changes considerably. This large-scale variation of the mean is known as shadowing. The mean of the shadowing also decreases as the distance between the base station and the mobile station increases. The mean signal is the local mean of the small-scale variation (up to a constant) and represents the distance-dependent trend and shadowing. The most widely used estimate of the mean signal is the average of samples of the received envelope (or logarithm of the envelope) taken at a constant temporal interval.
Due to the mobiliy of users and/or the wireless propagation environment, wireless channels are inherently time-varying and this characteristic time variance is one of the key factors to be considered to maximize the performance of wireless systems. Especially, knowing the mobile speed or more generally the fading rate of the channel is very useful in many aspects of wireless systems design. Specifically, in reality, the most widely used estimate of average received power is the average of samples of the received envelope (or logarithm of it) taken at a constant temporal interval. From speed estimates, the duration of a temporal window over which the received signal is averaged to estimate the received power can be determined. Speed estimate can also be used in the channel assignment algorithm. Depending on the mobile speed, a low speed mobile station could be assigned to a microcell and a high speed mobile station to a macrocell. Other system control algorithms such as handoff algorithms can also take advantage of accurate speed estimates. There have been proposed many mobile speed estimation techniques. Some use the statistics of the received signal such as level crossing rates or the autocovariance of the received envelope. Speed estimates are also obtained by estimating the maximum Doppler frequency using spectrum estimation methods or from squared deviations of the logarithmically compressed received envelope [2]. In [2], it is noticed that if there is a fixed time interval, the size of the deviations within the interval would increase as the maximum Doppler frequency, f D,and thus that such information could be used to estimate the maximum Doppler frequency and the mobile speed. Another interesting approach to estimating the mobile speed uses the continuous wavelet transform (CWT)[1]. It applies properties of CWT to detect the points of discontinuity in the first derivative of the received signal and relates the mean separation of the local minima in the signal to the mobile speed using the fact that the relationship between the temporal variation and the spatial variation of the received signal is determined by the mobile speed. One interesting feature of this method is that it doesn't require knowledge about the received signal power as opposed to other methods mentioned above. A new method of estimating the mobile speed and the signal mean using wavelets is described in this report. The wavelet transform at different scales corresponds to a variety of window lengths and hence eliminates the necessity of adapting the duration of a single temporal averaging window. The method presented here utilizes the fact that the smallscale spatial variation of the received envelope is dominated by the positions of the mobile and base stations. This spatial variation has a characteristic scale that is on the order of a carrier wavelength. By averaging over a fixed number of samples taken at the midpoints between the local minima of the received envelope, one can obtain an estimate of the mean signal over a spatial scale that is selected based on the variation of the distance-dependent trend and shadowing. This report is organized as follows. In Section II, a brief overview of propagation of wireless signals is presented. In Section III, CWT technique is introduced, the estimation using CWT is analyzed and summarized. Section IV speaks about the references.
II BACKGROUND THEORY A received signal in wireless communications systems, particularly in a cellular system, generally consists of a sum of many reflected, refracted, and attenuated transmitted radio waves, thus the strength varying randomly. The signal variations are mainly due to three different types of phenomena: path loss, log-normal shadowing and multipath fading. Specifically, the received signal power falls off exponentially as the distance between the transmitter and receiver increases. This loss is inherent in the propagation of electromagnetic waves and called path loss. Its effect is put into the path loss exponent, typically ranging from two to six, in a received power model. In addition to the path loss, buildings and other obstacles in propagation paths also attenuate the signal power. This variation changes on the order of building sizes and is modeled as log-normal. Lastly, the multipath fading comes from the fact that the transmitted signal is reflected, refracted and scattered during the propagation and the sum of these multiple signals give rise to the random fluctuation on the order of a wavelength. If there is no direct path from the transmitter to the receiver, this small-scale fading is called Rayleigh fading. In the case of a dominant line-of-sight (LOS) path along with the multipath components, it is called Rician fading. Accounting for these factors, the received power at x from a base station located at x B can be modeled as [3] (1) The constant P 0 includes transmitted power and antenna gains. The second term represents the path loss where being the path loss exponent and the third term represents the log-normal shadowing L(x, x B ) which is modeled as a zero-mean Gaussian widesense stationary random process. An empirical model for the autocorrelation of the lognormal shadowing is given by [10] (2) where and are the variance and the correlation length of L(x, x B ) respectively. The power spectrum is the Fourier transform of the autocorrelation and is given by (3) where denotes spatial frequency. And if we assume there is no line-of-sight (LOS) component and the multipath components are confined to a horizontal plane and that the multipath angle-of-arrival is uniformly distributed, the in-phase and quadrature components of the received signal are independent, identically distributed wide-sense stationary, zero-mean Gaussian random processes in a small neighborhood of x. If we denote the baseband equivalent received signal as then the autocorrelation functions, and, and the corresponding power spectrum, and, of and are given by [5] (4)
(5) Here, is the total received power( in [6]). The received envelope is then Rayleigh distributed. These models are used in [1] and it is also used in this work to simulate the algorithms in Section V. A typical received signal with Rayleigh fading is shown below in Fig.1. (6) Figure 1. A typical received signal with Rayleigh fading III ESTIMATION USING CONTINUOUS WAVELETS TRANSFORM(CWT) This mobile speed estimation technique uses properties of the continuous wavelet transform to detect the points of discontinuity in the first derivatives of the received signal. One of the advantages of the technique using CWT is that it does not assume a constant velocity of the user since the different scales in the CWT can track variable mobile speed without requiring adaptation of a single estimation window [1]. Also as mentioned in the introduction, this method need not know the average received power as opposed to other techniques in the literature. One of the consequences of this advantage is that there would be no sudden change in estimated speed at a street corner where the received power changes abruptly which is known as a corner effect. To analyze the algorithm, we begin with a brief introduction to the CWT following that in [3]. Consider a "mother wavelet", where denotes the space of square integrable functions on the real line. The "time-scale" atoms by scaling by a and translating by b. of the CWT are given
(7) An important property of is that it must oscillates such that. (8) A time function for the wavelet known as "coiflet" of order 2 is shown in Fig. 2. Figure 2. Time-function of coiflet of order 2 For real, the CWT of a function g(t) is defined as The absolute value of the CWT of the signal plotted in Fig. 1 is shown below in Fig. 3. The x and y axes represent the translation/time/sample and the scale respectively. And bright color represents large magnitude and dark color represents small magnitude. (9)
Figure 3. Modulus of the CWT of the signal in Fig.1 with "coiflet" of order 2 One of the important properties of the CWT used in this technique is the scaling property, which says the CWT of is related to the CWT of g(t) by (10) This is easily seen from the definition in (8). Another important property of the CWT used is that it can characterize singularities. If g(t) has all derivatives up to order n and th derivative is discontinuous at, then for a constant, as [1]. Thus, near the point of discontinuous derivative, the CWT behaves as a 1.5. In [1], a method using the wavelet transform modulus maxima(wtmm) is used to detect the points of discontinuity in the first derivative of the signal. The singularities of a function can be located by following the maxima lines as the scale approaches zero. IV REFERENCES [1] R.Narasimhan and D.C.Cox, Speed Estimation in Wireless Systems Using Wavelets, IEEE Trans. Commun., vol. 47, no. 9, pp. 1357-1364, 1999 [2] J.M.Holtzman and A. Sampath, Adaptive averaging methodology for handoffs in cellular systems, IEEE Trans. Veh.Technol., vol. 44, pp. 59-66, Jan. 1995 [3] R.Narasimhan, Estimation of Mobile Speed and Average Received Power with Application to Corner Detection and Handoff, PhD Dissertation, Stanford University, 1999 [4] R.Narasimhan and D.C.Cox, "Estimation of Mobile Speed and Average Received Power in Wireless Systems Using Best Basis Methods," IEEE Trans. Commun., vol. 49, no. 12, pp. 2171-2183, 2001