1558 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 6, NOVEMBER 2003 A Two-Dimensional Channel Simulation Model for Shadowing Processes Xiaodong Cai, Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE Abstract—A Gaussian random process with a given power spectral density (PSD) function can be modeled as a sum of sinu- soids (SOS), and has been widely used to simulate Rayleigh-fading communication channels. The conventional one-dimensional (1-D) channel model cannot capture the spatial correlation of shadowing processes. We here develop a two-dimensional (2-D) SOS-based channel model to simulate the shadowing process. Three methods to fit the PSD of the simulated process to the true channel’s PSD are explored. Performance of the proposed channel simulator is analyzed in terms of the autocorrelation function of the simulated shadowing process. Simulations illustrate the potential of the proposed channel simulation model. Index Terms—Channel model, shadowing, spatial correlation, sum of sinusoids. I. INTRODUCTION M OBILE radio communication channels have been studied for a long period of time based on measure- ment data and analysis of radio wave propagation. Typically, the radio channel is subject to two multiplicative forms of variations: fading and shadowing. Fading is due to multipath propagation. Signals from different paths add constructively or destructively, which results in rapid fluctuation of the signal amplitude within the order of a wavelength. Fading is often modeled as a complex Gaussian random process whose autocorrelation function (ACF) is determined by its Doppler spectrum in urban areas, where there is no line of sight between the transmitter and the receiver. Shadowing, on the other hand, occurs over a relative large area with different levels of clutter on the propagation path, which is also referred to as log-normal shadowing because the signal levels (measured in dB) follow a Gaussian (normal) distribution with local mean depending on the separation distance between the transmitter and the receiver. A number of computer simulation models have been pro- posed to simulate fading channels. Basically, a stationary Manuscript received December 27, 2001; revised April 7, 2003, June 27, 2003, August 21, 2003, and September 4, 2003. This work was supported by the NSF Wireless Initiative Grant 9979443, and prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The authors are with the Department of Electrical and Computer Engi- neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: caixd@ece.umn.edu; georgios@ece.umn.edu). Digital Object Identifier 10.1109/TVT.2003.819627 Gaussian process can be generated by passing a sequence of white Gaussian random deviates through a filter whose frequency response is the square root of the Gaussian process’ power spectral density (PSD) [9, p. 403]. This method has been applied to simulate multipath fading channels in [3]. However, this filtering method usually entails considerable computational burden, especially when the bandwidth of the Doppler spectrum is narrow. An alternative method is based on the fact that a Gaussian random process can be expressed as a sum of infinite number of sinusoids with random phases, and properly selected frequencies [15], . In practice, a finite number of sinusoids can be used to approximate a Gaussian process, which reduces complexity. For this reason, the method of sum of sinusoids (SOS) to simulate fading channel has been extensively investigated [7], [8], [10]–[12], [14], [17]. In [7], a Monte Carlo method is proposed to randomly generate the sinusoids frequencies, and wide-sense stationary uncor- related scattering (WSSUS) channels are simulated. In [17], an efficient scheme that has low computational complexity is advocated to simulate WSSUS channels with correlated paths. In [11], the method of SOS is studied in more detail, and several approaches to determining the sinusoidal frequencies are devel- oped. The statistical properties of channel simulation models are presented in [12], and a fast implementation scheme based on a look-up table is advocated in [10]. The very popular Jakes’ channel simulation model [8] calculates the frequencies of the sinusoids based on a physical channel model with uniformly distributed scatterrers around a circle. By assuming that the random phases of the signals with the same Doppler frequency, but from different scatterrers are the same, Jakes came up with an efficient channel simulator. However, the channel waveform generated by the Jakes’ model is nonstationary because of the underlying random phase assumption [14]. In simulating a fading process, it is sufficient to generate a one-dimensional (1-D) random function of time because the fading waveform changes very fast even when the mobile moves a little distance. However, since shadowing effects occur over a relatively large area, it is desirable to generate a two-dimen- sional (2-D) random Gaussian process with proper spatial cor- relation. This is particularly useful when one wants to evaluate the performance of handover [13], [18], and macro-diversity al- gorithms [2]. For example, suppose a mobile is moving along a closed route rather than a straight line, and we want to study how the system handles the handover. In this case, a 1-D shadowing process cannot be used because it cannot capture the correlation of the channel along this closed route. In this paper, we propose a 2-D simulation model for shad- owing processes based on the SOS model. We first obtain the 0018-9545/03$17.00 © 2003 IEEE