同济大学：《传播信道特征估计和建模》课程教学资源（教案讲义）Chapter 05 Channel measurements.pdf

5 Channel measurements The measurements on the radio propagation channels provide deeper understanding and significant insights on the characteristics of the radio channel in different environments. These characteristics include multi-dimensional dispersion (e.g. in DoA, DoD, Doppler frequency, delay and polarizations) Fuhl et al. (1997) Steinbauer et al. (2000) Steinbauer et al. (2001) Fleury et al. (2002b) Fleury et al. (2003) Karedal et al. (2004) Bonek et al. (2006), coand cross-polarization characteristics Fleury et al. (2003) Yin et al. (2003c) Oestges (2005) and clustering effects of multiple specular path components in delay ?, delay and DoA Spencer et al. (2000), DoD and DoA ? Czink et al. (2005) as well as in delay, DoD and DoA Czink and Cera (2005). This knowledge is of paramount importance for establishing realistic channel models for system designs and optimizations. 5.1 Measurement methodologies Measurements need to be performed in many scenarios. The sampling scheme needs to be designed in such a way that the channel characteristics of interests can be extracted. We focus on the wideband channel measurements which allow jointly estimation of the channel characteristics in multiple dimensions. We describe the correct way for sampling the channel impulse response in time, space and frequency. The contents reported in Yin et al. (2003b) Pedersen and Pedersen (2004b) Pedersen et al. (2008a) are presented. Channel measurements are performed with a channel sounder. In order to estimate the bi-direction (i.e. DoD and DoA) parameters Steinbauer et al. (2001) of the propagation paths, a channel sounder needs to be equipped with antenna arrays in the Tx and the Rx. The underlying antenna array can be a virtual array generated by stepping a single antenna across specific grids in space Steinbauer et al. (2000) Czink et al. (2005). The Tx transmits the sounding signals. The Rx receives the distorted signals due to the propagation mechanism in the channel. The received signal is usually processed and the resulting data is recorded in a storage device attached with the Rx. Channel parameter estimation is performed based on the data either in real-time or off-line, depending on the complexity of the used estimation methods. The used sounding signals are usually known to the Rx. They can be various kinds of wideband waveforms, such as Phase Shift Keying (PSK) signal modulated by Pseudo-Noise (PN) sequence Stucki (2001) Kattenbach and Weitzel (2000) Rudolf Zetik and Sachs (2003), multiple frequency periodic signals with low crest factor www.channelsounder.de (n.d.) and chirp signal S. Salous and Hawkins (2002). This is a Book Title Name of the Author/Editor c XXXX John Wiley & Sons, Ltd

104 Channel measurements (a) (b) Sounding Sounding signal signal Array Array output output Tx site Tx site Rx site Rx site Figure 5.1 MIMO channel sounding systems with (a) a switched architecture and (b) a parallel architecture. 5.2 Channel sounder Current existing channel sounders equipped with multiple-element antennas in the Tx and Rx usually perform a measurement in two manners, i.e. switched channel sounding Stucki (2001) and parallel channel sounding S. Salous and Hawkins (2002) Rudolf Zetik and Sachs (2003). A channel sounder using the former technique is equipped with high speed Radio Frequency (RF) switch at both Tx and Rx. These switches connect the RF front end to individual elements in the antenna arrays. Fig. 5.1 (a) sketches a switched MIMO channel sounding system. Using a switched sounding technique lowers the costs in constructing a channel sounder. It also lowers the requirement of recording speed in real-time measurements. In the parallel sounding system, the multiple Tx antennas transmit signals simultaneously and the multiple Rx antennas receive signals simultaneously. To separate the signals, the signals transmitted from different Tx antennas should have appropriate autocorrelation and cross-correlation properties. Fig. 5.1 (b) depicts a parallel MIMO channel sounding system. Note that the switched and parallel sounding techniques can be used in combination in a channel sounder. According to S. Salous and Hawkins (2002) Rudolf Zetik and Sachs (2003), a major disadvantage of switched sounding is the trade-off between the delay estimation range and the highest absolute Doppler frequency. This tradeoff is based on a common belief that the switching sounding system can only estimate Doppler frequency with absolute value up to half the inverse of the cycle interval. Here, the cycle interval refers to the time between the starting instants of two consecutive measurement cycles. A measurement cycle is the period used to switch each pair of Tx-Rx elements once. According to this belief, in order to estimate high absolute Doppler frequency, the cycle interval has to be small. Consequently, the sensing interval within which a Rx antenna is activated decreases and thus, the estimation range of delay decreases. In the thesis, we show that with an appropriate switching strategy, the absolute Doppler frequency that a switched sounding system can estimate is up to half the inverse of the sensing interval. Thus, the above-mentioned trade-off does not exist at all. This study is reported in Chapter 5.8. 5.3 Post-processing of the measurement data In this section, we shortly describe the methods used to process the measurement data. For example, the periodogram based method, the model-based estimation method Fleury et al. (2003); Yin et al. (2003a), the spectral-based method Schmidt (1981), the subspace-based methods, such as the ESPRIT method Roy and Kailath (1989) Roy (1986), the unitary ESPRIT Haardt (1995), and the propagator method Marcos et al. (1994) Marcos et al. (1995)

Channel measurements 105 In the latest decades, high-resolution estimation methods have been proposed based on the SS model. These methods include subspace-based approaches such as the MUltiple SIgnal Classification (MUSIC) algorithm Schmidt (1986), the Estimation of Signal Parameters Via Rotational Invariance Techniques (ESPRIT) Roy and Kailath (1989) and approximations of the maximum-likelihood (ML) methods like the Expectation-Maximization (EM) algorithm Moon (1997), the Subspace-Alternating Generalized Expectation-maximization (SAGE) algorithm Fessler and Hero (1994) Fleury et al. (1999) and the RIMAX algorithm Richter et al. (2003) Richter (2004). These methods are applied to extract the parameters of the specular path components in multiple dimensions Fleury et al. (2002d) Heneda et al. (2005) Zwick et al. (2004) Steinbauer et al. (2001). According to Krim and Viberg (1996), these estimation methods can be categorized into two groups: i.e. spectralbased approaches and parametric approaches. Methods belonging to the former group estimate the channel parameters via finding maxima (or minima) of spectrum-like functions of the dispersion parameters. These techniques are usually computationally attractive as the maxima-searching can be preformed in one dimension for all paths. Techniques belonging to the latter category estimate the parameters of an underlying parametric model characterizing the effect of the propagation channel on the transmitted signal from the signals at the array output. These techniques exhibit better estimation accuracy and higher resolutions than the spectral-based techniques. However, the computational complexity is usually high due to the multi-dimensional searching required to compute the estimates. In particular, the parametric approaches are applicable in the case where the multi-path signals are coherent1 . In such a case, the spectral-based approaches usually fail or perform badly. In the sequel, a brief introduction of the classical high-resolution estimation methods is provided. The MUSIC Algorithm The MUSIC algorithm was originally proposed in Schmidt (1981), Schmidt (1986) and G. Bienvenu (1983). It was introduced in the field of array processing but has been applied since then in other applications. To explain the principle of the algorithm more easily, we consider the simple model Y = CF + W, (5.1) where Y ∈ CM×1 denotes the output signals of the M-element Rx array, C .= [c(φ1) c(φ2) . . . c(φD)] ∈ C M×D with c(φ) ∈ CM×1 denoting the array response versus the AoA φ and φd, d = 1, . . . , D representing the AoAs of the D propagation paths. The vector F ∈ CM×1 consists of the complex path weights and W ∈ CM×1 denotes the temporalspatial white circularly symmetric Gaussian noise with component variance σ 2 w. The vector Y can be visualized as a vector in M-dimensional space. The individual column c(φd), d ∈ [1, . . . , D] of C are “mode” vectors Schmidt (1986). It is apparent that the vector Y in the case with σ 2 w = 0 is a linear combination of the mode vectors. Thus, the signal-only components in Y are confined to the range space of C. Fig. 5.2 shows the graphic representation of the idea behind the MUSIC algorithm using a simple example, where M = 3, C .= [c(φ1), c(φ2)], φ1 6= φ2 and e1, e2, e3 being the eigenvectors calculated from the covariance matrix. The range space of C is a 2-dimensional subspaceof C 3 . The vector Y lies in the 3-dimensional space. The steering vector c(φ), i.e. the continuum of all possible mode vectors, lies within the 3-dimensional space. In this example, c(φ1) and c(φ2) jointly determine a 2-dimensional space that coincides with the estimated signal subspace spanned by e1 and e2. Since the estimated signal subspace is orthogonal to the estimated noise space, the steering vectors c(φ1) and c(φ2) are orthogonal to e3. Thus, the projection between the steering vector and the estimated noise eigenvector can be formulated as a criterion for parameter estimation. It is obvious that the performance of the MUSIC algorithm depends on the accuracy of the estimated signal subspace and noise subspace. Note that the true steering vectors may not exist in the estimated signal subspace in some circumstances. In this case, the projection between the true steering vector and the estimated noise eigenvector is never equal to zero. The pseudospectrum may fail to exhibit peaks in the true direction. This occurs, for instance, in the case where the matrix F are highly correlated Krim and Proakis (1994) or the propagation paths are characterized by parameters with differences less than the intrinsic resolution of the equipments Krim and Proakis (1994). The standard MUSIC algorithm consists of the following steps Schmidt (1986): 1The signals are called coherent when the signal covariance matrix is singular (Stoica and Moses 1997, Page 240)

106 Channel measurements e1 e2 e3 c(φ1) c(φ2) c(φ) O Figure 5.2 Graphic representation of the signal subspace and the noise subspace. The vectors c(φ1) and c(φ2) represent two steering vectors. The eigenvectors e1 and e2 are an orthonormal basis of the range space of the matrix [c(φ1) c(φ2)]. 1. Calculate the sample covariance matrix and its eigen-value decomposition; 2. Find the orthonormal basis of the estimated noise subspace; the number of specular path components D in the received signal needs to be either known in advance or estimated when this number is unknown; 3. Calculate the pseudo-spectrum, i.e. the inverse of the Euclidean distance (squared) between the estimated noise subspace and the steering vector c(φ) with respect to φ; 4. Find the D arguments of the pseudo-spectrum leading to the D highest peaks in the pseudo-spectrum. The MUSIC algorithm can be easily extended to jointly estimate multiple-dimensional parameters, such as delay, angular parameters and Doppler frequency. A typical example of the extension is the Joint Angle and Delay Estimation (JADE) MUSIC algorithm Vanderveen (1997). The ESPRIT algorithm The ESPRIT algorithm Roy and Kailath (1989) and the propagator method Marcos et al. (1994) Marcos et al. (1995) are two classical algorithms based on the shift-invariance-property. Both algorithms exploit a translational rotational invariance among signal subspaces induced by an array. In these algorithm, parameter estimates can be computed analytically. Thus, when applicable, these methods exhibit significant computational advantages over the methods that relies on solving optimization problems by exhaustive searching. As discussed in the previous section, the SAGE and the MUSIC algorithms need the knowledge of the array manifold, while the ESPRIT does not. However, the ESPRIT algorithm needs multiple sensor doublets. The elements in each doublet must have identical radiation patterns and are separated by known constant spacings. Apart from that requirement the radiation patterns can be arbitrary. The fundamental idea of the ESPRIT algorithm can be simply explained as follows. The underlying antenna array is divided into two subarrays. Each subarray consists of the same number of elements. The element spacing in each subarray and the spacing between the subarrays are known. The mth element in the 1st subarray and the mth element in the 2nd subarray compose a “doublet”. In this case, the array response C1(φ) of the 1st array and the array response C2(φ) of the 2nd array can be related by C1(φ) = C1(φ)Φ(φ), where Φ(φ) = diag[exp{jψ1}, . . . , exp{jψD}]. Here, ψd is the phase difference between the signals received at the two elements in each doublet for the dth path. This phase difference is a known function of the AoA φd. It can be shown that the columns in C1(φ) and the columns

Channel measurements 107 in C2(φ) span the same space. Based on this property, the estimate of the matrix Φ(φ) can be obtained based on the estimated signal subspace computed using the sample covariance matrix. Since the relation between the elements of Φ(φ) and φ is known, the estimate of φ can be calculated from the estimate of Φ(φ) in close-form. The implementation of the ESPRIT algorithm based on a sample covariance matrix can be summarized as follows: 1. Calculate the sample covariance matrix of the signals at the output of the M-element array and compute the eigenvalue decomposition. We assume that M is even. 2. Estimate the number of the specular path components. Find the orthonormal basis of the estimated signal subspace. Decompose the basis into two parts, say, Ex and Ey which contain the first M/2 rows and the rest M/2 rows respectively. 3. Compute the eigenvalue decomposition of the matrix EH x EH y ExEy = EΛE H . 4. Decompose the matrix E into four D × D matrices: E = E11 E12 E21 E22 . 5. Calculate the eigenvalues of E12E −1 22 or E21E −1 11 and compute the azimuth estimates according to either φˆ d = cos−1 λ arg{λd(E12E −1 22 )} 2π∆ or φˆ d = cos−1 −λ arg{λd(E21E −1 11 )} 2π∆ . In the above expression, λ represents the wavelength, arg(·) denotes the complex argument, λd(·) is the d th eigenvalue of the given matrix and ∆ is the distance between the two antennas in one doublet. Similar to the MUSIC algorithm, the performance of the ESPRIT algorithm in parameter estimation depends on the accuracy of the estimation of the signal subspace. In the case where the signals contributed by different propagation paths are correlated or the difference of the path parameters is less than the intrinsic resolution of the measurement equipments, the ESPRIT algorithm fails to resolve the paths accurately. The Unitary ESPRIT algorithm Haardt (1995), which incorporates spatial smoothing techniques, is applicable in the scenario where the path components are correlated. The ESPRIT algorithm can also be extended to estimate multiple dimensional parameters of specular paths. For example, the two-dimensional Unitary ESPRIT algorithm Haardt et al. (1995) Fuhl et al. (1997) is applicable for joint estimation of the azimuth and elevation at one end of the link. A three-dimensional Unitary ESPRIT algorithm has been proposed in Richter et al. (2000) to estimate the delay, DoA and DoD of individual paths. The SAGE Algorithm The ML estimation method provides the optimum unbiased parameter estimates from a statistical perspective. However, it is computationally cumbersome due to the exhaustive multi-dimensional searches required for calculation of the estimates. As an alternative, the SAGE algorithm is proposed in Fessler and Hero (1994) as a low-complexity approximation of the ML estimation. In recent years, this algorithm has been successfully applied for different application purposes such as parameter estimation in channel sounding Fleury et al. (1999) and joint data-detection and channel-estimation in the receivers of wireless communication systems Kocian et al. (2003). The SAGE algorithm updates the estimates of the unknown parameters sequentially by alternating among subsets of these parameters Fessler and Hero (1994). To explain the idea of the algorithm, we introduce the following notations: y: the observed signal, θ: the parameter vector belonging to a p-dimensional space, θS: the entries in θ with indices specified in a subset of {1, . . . , p},