Accurate simulation of non-isotropic fading channels with arbitrary temporal correlation

The accurate simulation of wireless channels is important since it permits the realistic and repeatable performance measurement of wireless systems. A new technique is proposed for simulating Rayleigh fading channels with isotropic or nonisotropic scattering and with arbitrary temporal correlation. Fading samples are generated by passing Gaussian samples through a spectrum shaping filter. A new iterative algorithm is then presented for designing stable complex infinite impulse response (IIR) filters with quantized coefficients. The algorithm utilizes a least-squares cost function augmented with a barrier function to ensure filter stability and to reduce quantization noise. The performance of the proposed filter design algorithm is verified with 18-bit fixed-point simulations of different fading channel scenarios including isotropic and nonisotropic scattering and the IEEE 802.11n Model F fading spectrum. I. MOTIVATION AND BACKGROUND The accurate design and verification of wireless communication systems needs faithful modeling and accurate simulation of realistic radio propagation channels. Prior to the availability of standardized channel models, wireless products needed to be verified using extensive and expensive field testing. A far less costly approach is to model the behavior of radio channels on a fading channel simulator. Two major techniques have been widely used for simulating Rayleigh fading channels. In the first approach, the so-called sum-of-sinusoids (SOS) model, the Rayleigh fading process is modeled as the superposition of a sufficiently large number of sinusoidal waves. This approach was originally proposed by Clarke [1] and later simplified by Jakes [2]. Over the past four decades several modified SOS-based models have been proposed (e.g., [3], [4]). This technique is widely used in the COST 259, COST 273, and IST-Winner fading channel models [5], [6]. The second well-known approach for fading channel simulation is to first generate a complex, zero-mean, Gaussian random process with independent samples. Then the fading process is obtained by passing the variates through a suitable spectrum-shaping filter (SSF). This technique is used to simulate the TGn fading channel models proposed for the 802.11n wireless standard [7]. The filtering process can be carried out in the time or frequency domains with digital or analog filters. In particular, one can generate the fading variates by multiplying the Gaussian samples in the frequency domain with suitable coefficients and then taking the inverse Fourier transform (IDFT) [8]. It has been shown that generating fading samples with a single Fast Fourier Transform (FFT) operation requires a relatively large memory and, hence, results in an inefficient implementation [9]. An autoregressive (AR) modeling approach has also been proposed for generating fading processes by passing the white noise samples through an all-pole infinite impulse response filter (IIR) [9]. To produce samples with accurate statistics, the AR model needs a large filter order, which greatly increases the number of required multiplications. Also, implementation of the AR fading simulator requires highly accurate variables, which makes it unappealing for compact fixed-point implementations. While the above techniques have been used to simulate isotropic scattering channels, they might not be the best candidates for simulating nonisotropic Rayleigh fading channels. In nonisotropic fading, the power spectral density (PSD) of the fading samples is asymmetric, implying an SSF with complexvalued coefficients. In conventional practice, filters are designed with standard tools (e.g., fdatool or iirlpnorm in MATLAB). While this approach is applicable to SSF design for isotropic fading channels, it is not appropriate for designing the stable complex filters required for shaping the spectrum of nonisotropic channels. Similarly, the IIR filter design procedure in [10] can be only applied to filters with real coefficients, and hence, it is not suitable for designing the complex filters required in nonisotropic channels. Our previous work focused on designing fading channel filters for the isotropic scenario (when the angle of arrival of the received multipath signals is uniformly distributed and we assume the use of an omni-directional antenna at the receiver) [11]–[15]. This work considers the more general scenario of non-isotropic scattering due to real-world antenna directivity and possible selective attenuation along some propagation directions. In this article we address the problem of designing IIR filters for the accurate simulation of nonisotropic Rayleigh fading channels with arbitrary time correlation properties. Several algorithms exist for designing stable IIR filters with real coefficients [16]–[20]. However, designing stable IIR filters with complex coefficients has been less well studied. The main contributions of this work are as follows: • A new algorithm for designing general stable IIR filters with complex and real coefficients is proposed. The maximum radius of the poles and/or zeros is limited to ensure numerical stability. The least-squares cost function is formulated in polar coordinates and augmented with a barrier function that keeps the poles (and potentially the zeros) within the unit circle. This method poses a sufficient (but not necessary) stability condition on the filter. • To minimize the computational requirements, filters are quantized for fixed-point implementation and, hence, variables are implemented with the minimum possible fixed-point wordlength. Since reducing the word-length can greatly impact the response (and potentially also the stability) of the designed filter, our proposed filter design technique searches for the best filters with fixed-point coefficients that meet our word-length budget. • Several examples are provided for designing fixed-point fading channel simulators with 18-bit IIR filters and 16-bit coefficients. Both isotropic and nonisotropic fading are simulated and the fixed-point results show a close match between several generated statistical properties and the theoretical references. The rest of this article is organized as follows. Section II reviews the nonisotropic channel model and related work on complex filter design. The new filter design technique is presented in Section III. We used our proposed complex filter design algorithm for the simulation of two different fading channel scenarios. Numerical results of these simulators are presented in Section IV. Finally, Section V makes some concluding remarks. II. FADING CHANNEL CHARACTERISTICS AND THE RELATED WORK ON COMPLEX FILTER DESIGN In wireless communication systems, the received signal strength varies significantly due to destructive and constructive interference resulting from multipath propagation [21]. Isotropic scattering refers to the case in which the incident direction of the received multipath signals, or the angle of arrival (AOA), is uniformly distributed. Assuming two-dimensional isotropic scattering with an omni-directional antenna at the receiver [2], the PSD functions associated with both the in-phase or quadrature components of a complex fading signal have the well-known Jakes’ U-shaped band-limited form [22] with independent in-phase and quadrature samples. However, such assumptions have been challenged [23] and experimentally demonstrated [24]–[26] to be inaccurate due to the selective attenuation of some propagation directions as well as antenna directivity [27]. Such effects produce a nonuniform probability density function (pdf) for the AOA at the receiver. The pdf of the AOA has a great impact on the second-order statistics of the fading process including the correlation functions, the level crossing rate (LCR) and the average fade duration (AFD) [26]. Several nonuniform pdfs have been proposed in the literature for the angle of arrival including geometrically-based AOA pdfs [28], the Gaussian pdf [29], the quadratic pdf [30], the Laplace pdf [24], the cosine pdf [31], and the von Mises pdf [32]. The von Mises pdf, which includes the uniform AOA distribution as a special case, is supported experimentally with empirical measurements of narrowband fading channels [32]. The authors in [32] argue that the von Mises pdf is preferable because it can approximate other nonuniform pdfs and it is mathematically convenient for analysis. In this article, we assume narrowband fading in which the complex envelope of the fading process is given by c(t) = cI(t) + j cQ(t)