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_大学文库

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},

Channel measurements 109 θS˜: the entries in θ with indices listed in the complement of S, XS : the hidden-data space selected for θS, x S : a realization of XS , S i : the index set selected in the ith iteration of the SAGE algorithm. The space XS associated with θS is an admissible hidden data if the following condition is satisfied Fessler and Hero (1994) f(y, xS ; θ) = f(y|x S ; θS˜)f(x S ; θ). (5.2) The above equation implies that the conditional distribution f(y|x S ; θ) coincides with f(y|x S ; θS˜). In the SAGE algorithm proposed for channel parameter estimation Fleury et al. (1999), the parameter subsets S i contain one element. Thus, the multiple-dimensional maximization in the ML estimation reduces to an onedimensional search in each SAGE iteration. The number of elements in S i can be larger than one, depending on the definition of the hidden-data space XS i . For example, when two paths are closely-spaced with separation below the resolution of the equipment, the admissible hidden-data should be defined to embody the sum of the signal contributions of the two paths. In this case, S i contains more than one element and the maximization step becomes computationally more “expensive”. Fig. 5.3 shows the flow graph of the SAGE algorithm. One iteration of the SAGE algorithm consists of two major steps: expectation (E-) step and maximization (M-) step. In the E-step, the expectation of the loglikelihood function of admissible hidden data for the current parameter vector θS is computed based on the observation and the parameter estimates from the previous iteration. This expectation is an objective function that is maximized with respect to the parameter vector θS in the M-step. To further reduce the complexity, in the case where the parameter vector θS contains more than one entry the coordinate-wise updating procedure Fleury et al. (1999) can be used to estimate these parameter entries sequentially. Thus, the multiple-dimensional optimization problem is solved using 1-dimensional searches. This coordinate-wise updating procedure still belongs to the SAGE framework Fleury et al. (1999), i.e. updating each parameter entry can be viewed as one SAGE iteration. The RIMAX Algorithm The RIMAX algorithm Richter (2004), Richter and Thoma (2005) and Richter et al. (2003) can be viewed as an extension of the SS-model-based SAGE algorithm Fleury et al. (1999). The RIMAX algorithm can be used for joint estimation of the parameters characterizing specular propagation paths and dispersion of distributed diffuse scatterers. The contribution of the distributed diffuse scatterers to the received signal is called dense multipath components in the papers. The power delay profile of these components is characterized using a one-sided exponential decaying function. This function can be described by three parameters: the time of arrival and the delay spread of the dense multipath components, as well as the average power of these components. In the RIMAX algorithm, unknown parameters are grouped into two sets. One set consists of the parameters of the specular paths and the other set contains the parameters characterizing the dense multipath components. In the Msteps of the algorithm, the gradient based methods, such as Gauss-Newton or Levenverg-Marquardt Marquardt (1963) algorithm, are implemented. For each specular path, an approximation of the Hessian is computed to be the estimate of the Fisher information matrix of the parameter estimates Richter and Thomä (2003). The diagonal elements of the inverse of this matrix provide estimates of the variances of the estimated parameters. In this algorithm, the variance estimates are used to describe the reliability of the corresponding parameter estimates. A specular path is dropped when its parameter estimates are considered to be “unreliable” Richter et al. (2003). In the RIMAX algorithm, individual dominant path components are treated as contributions of specular paths. In the estimation schemes proposed in this thesis (Chapter ??), the dominant path components are treated as dispersed path components. Similar to the RIMAX algorithm, these estimation schemes can be extended to include the estimation of the characteristics of (largely-)distributed diffuse scatterers Richter and Thomä (2003). 5.4 Evaluation of the measurement efficiency The following words should go to the beginning of the chapter. It is always necessary to evaluate whether the measurements have been taken effectively. In this section, we describe the impact of the inaccurate calibration on the

110 Channel measurements estimation results. We will present how to avoid or correct the estimation results for example by introducing some lookup-table based methods or use analytical methods to correct the estimation results. The impact of inaccurate calibration data on the estimation results are described (Käske et al. 2009). The influence of the phase noise on the estimation results is also shown (Taparugssanagorn and Ylitalo 2005) (Taparugssanagorn et al. 2007a) (Taparugssanagorn et al. 2007b). 5.5 Impact of phase noise in a TDM channel sounding system 5.5.1 Introduction Most available channel sounders for measuring MIMO spatial channel adopt the time-division-multiplexing mode. Examples of these channel sounders are the PROPSound, the RUSK sounder, and the sounder designed by the CRC, Canada. The propagation channels between any pair of the Tx antenna and Rx antenna are measured sequentially in timeslots. During the measurements, two switches are applied in the Tx and Rx for connecting the RF transceiving chains to the specific Tx and Rx antennas respectively in the dedicated timeslots. As indicated in (Taparugssanagorn et al. 2007a), the switching rate should be selected to satisfy two criteria: i) all subchannels between the Tx antennas and the Rx antennas are measured within the channel coherence time; ii) enough samples of the channels are collected, which offer sufficiently high intrinsic resolution for parameter estimation. In the TDM-sounding scheme, phase noise can be generated in the following items, i.e. the local phased locked oscillators at the Tx and the Rx, the switches at the Tx and the Rx, and the calibration errors of the antenna responses. The latter errors attribute to the low antenna gain due to the directional characteristics of the antennas, the coupling among antennas, and the impact of the scatterers appearing in the near field in the vicinity of the antenna array. Phase noise generated by the local phase locked oscillators in the Tx and the Rx can have impact on the MIMO channel capacity estimation (Baum and Bölcskei 2004; Pedersen et al. 2008b; Taparugssanagorn and Ylitalo 2005), and on channel parameter estimation (Taparugssanagorn et al. 2007a,b). The long-term slowly varying phase noise can be modeled as the zero-mean non-stationary infinite power Wiener process Almers et al. (2005). Usually the longterm varying phase noise can be corrected or mitigated by adding synchronization devices, such as the Rubidium Clock, in the Tx and the Rx, and thus, it is not necessary to consider the long-term varying phase noise in the MIMO channel sounder. The short-term varying phase noise, appearing within the time period much smaller than 1 second, usually has significant impact on the high-resolution parameter estimation. For this reason, we concentrate on investigating the behavior of short-term varying phase noise. In the following subsections, we will show the impact of the phase noise on the estimation and the techniques proposed to mitigate the impact of the phase noise. 5.5.2 Behavior of the short-term phase noise The short-term phase noise, observed in less than 1 s, may be modeled as an autoregressive integrated moving average (ARIMA) process Taparugssanagorn et al. (2007a). Normally, for a MIMO channel sounder, a measurement cycle is usually less than 1 s. For instance, for the measurement campaigns in Chapter 9 address in Czink (2007), 50 × 32 MIMO matrix has been adopted for 5.25 GHz. A subchannel is sounded within a period of 510µs. Thus, for one cycle of measurement, the time is about 8.42 ms. In the case where 4 cycles are combined and the data is processed as one snapshot, the time duration is 67 ms. So for this scenario, the properties of the phase noise with the time less than 100 ms are of interest to investigate. The Allan variance is applied to characterize the time-domain statistical behavior, which is calculated as σ 2 y (τ) = E (¯yk+1 − y¯k) 2 2 (5.3) where y¯k = 1 τ Z tk+τ tk y(t)dt = φ(tk + τ) − φ(tk) 2πfcτ . (5.4)

Channel measurements 111 Figure 5.4 The measurement setting for measuring the phase noise The term y(t) in (5.4) represents the instantaneous normalized frequency deviation from the carrier frequency fc, which is computed as y(t) = 1 2πfc · dφ(t) dt (5.5) with φ(t) denotes the instantaneous phase variation. Assuming that the sampling rate of the phase 1 T . The samples of the Allan variance at τ = mT can be estimated as ˆσ 2 y (mT ) = 1 2(N − 2m)(2πfcmT ) 2 N X−2m i=1 (φ(ti+2m) − 2φ(ti+m) + φ(ti))2 , (5.6) with m = 1, . . . , N−1 2 and N denoting an odd number referring to the total number of samples of the phase. An example of measuring the phase noise is illustrated as follows. Fig. 5.6 depicts the measurement setting used to measure the phase noise of a single-input single-output channel sounder. An RF cable connects the Tx and the Rx with a 50 dB fixed attenuator. The Tx and the Rx each equipped with individual clock operate in the same way as in real field measurements. The Allan deviation, i.e. σˆy(mT ), is computed and illustrated in Figure 5.5. In this figure, both the sample Allan variance and their asymptotic characteristics computed from the measured phase noise sequence are depicted. Furthermore the curves computed based on proposed models are also illustrated. It can be observed that an ARMA model with model parameters computed based on the sample Allan variance can be used to describe the behavior of the Allan variance. Furthermore, the short term phase noise component predominates within the range of τ ∈ [0, 200µs]; for τ > 200µs, the Allan variance of the phase noise is identical with that obtained from a white phase noise, as suggested in Characterization of frequency and phase noise (1986); for τ > 1s, the phase noise can be described using random walk models. 5.5.3 Mitigation of the impact on the high-resolution parameter estimation A method proposed in Taparugssanagorn et al. (2007d) for mitigating the impact of phase noise on the parameter estimation performance is to consider multiple consecutive snapshots of measurement data as the observation of the same channel. This method is called “sliding window”. For the considered TDM-based sounding system, this sliding window solution is extended to the spatial domain, i.e. by considering more antennas. In general, the sliding window function is to average over multiple channel observations, such that the impact of the phase noise can be compromised to certain degree. Fig. 5.7 (a) and (b) depict respectively the comparison of the estimation results obtained by using the SAGE algorithm with and without using the sliding window solution over 20 cycles of data. Similarly results obtained with 8 × 8 and 4 × 4 MIMO channel matrices are also illustrated in Figure 5.8 and 5.9 respectively. It can be observed that by using the sliding window function, it is possible to reduce the probability of generation of artifact estimates. Another method introduced in Taparugssanagorn et al. (2007b) is to modify the specular-path SAGE algorithm Fleury et al. (1999) to include a whitening function based on the known covariance matrix of the phase noise. The modified SAGE algorithm is derived based on the signal model introduced in Section 3.3. For simplicity, the bidirection-delay-Doppler frequency generic specular path model is modified to a narrowband channel model based on the assumption that the signal bandwidth is much smaller than the channel bandwidth. Here, the channel bandwidth is referred to as the inverse of delay spread of the channel. The TDM sounding scheme is considered