A Trajectory-Driven 3D Non-Stationary mm-Wave MIMO Channel Model for a Single Moving Point Scatterer

This paper proposes a new non-stationary three-dimensional (3D) channel model for a physical millimeter wave (mm-Wave) multiple-input multiple-output (MIMO) channel. This MIMO channel model is driven by the trajectory of a moving point scatterer, which allows us to investigate the impact of a single moving point scatterer on the propagation characteristics in an indoor environment. Starting from the time-variant (TV) channel transfer function, the temporal behavior of the proposed non-stationary channel model has been analyzed by studying the TV micro-Doppler characteristics and the TV mean Doppler shift. The proposed channel model has been validated by measurements performed in an indoor environment using a MIMO radar kit operating at 24 GHz. For the measurement campaign, we used a single swinging pendulum as a model for a moving point scatterer. The trajectory of the pendulum has been captured by an inertial measurement unit attached to the pendulum and by a motion capture camera system. The measured trajectories are fed into the proposed mm-Wave MIMO channel model. The results obtained for the micro-Doppler characteristics show an excellent agreement between the proposed MIMO channel model and real-world measured channels in the presence of a moving point scatterer. We believe that our model can serve as a basis for the development of novel non-stationary MIMO channel models capturing the effects caused by moving objects and people.


I. INTRODUCTION
In recent years, millimeter wave (mm-Wave) communications has been proposed as key enabler for the development of the fifth generation (5G) communication systems [1]. Hence, mm-Wave technologies have gained a great attention from both research institutes and industries due to large bandwidth availability, which offers high transmission data rates [2]. In order to exploit important benefits of mm-Wave communications, a detailed knowledge of the corresponding channel propagation is crucial [3]. Mm-Wave radio propagation characteristics differ from channels with sub-6 GHz bands in terms of high path loss, high delay resolution, and blockage The associate editor coordinating the review of this manuscript and approving it for publication was Matti Hämäläinen . caused by humans crossing the line of sight [4]. Therefore, a number of channel models for mm-Wave bands between 6 and 100 GHz have been investigated and developed for various scenarios, such as outdoor [5]- [7], indoor [8]- [11], and vehicle-to-vehicle [12], [13]. For completeness, it should be mentioned that channel models for outdoor propagation scenarios and sub-6 GHz mobile-to-mobile (M2M) communications have also been studied in the literature. In this paper, we focus on the analysis of indoor fixed-to-fixed (F2F) communications considering systems operating in the mm-Wave bands. Therefore, the proposed model differs from M2M communications in a way that the angle-of-departure (AOD) and angle-of-arrival (AOA) of the transmitted wave and the received wave, respectively, are determined by the position of a moving point scatterer and are therefore related to each other. In an M2M communication scenarios, however, the AOD and the AOA of the wave incidents are independent random variables. Thus, by using the trajectory of a moving point scatterer, we derive a trajectory-driven channel model.
In [9], a measurement-based spatio-temporal channel model for the 60 and 70 GHz frequency bands has been presented by Haneda et al. for indoor environments, such as large office rooms, shopping malls, and stations. It has been shown that the measured spatio-temporal radio channel characteristics of the two frequency bands demonstrate similar behaviour for such environments. Maltsev et al. have developed a non-stationary mm-Wave channel model for the 60 GHz frequency band by using a quasi-deterministic modeling approach [14]. The authors have measured power delay profiles of the underlying mm-Wave channel. Rappaport et al. [8] have presented indoor and outdoor measurement-based mm-Wave channel models for 28, 38, 60, and 73 GHz frequency bands. In [11], radio propagation measurements of mm-Wave channel characteristics at the 28 GHz frequency band have been presented for in-building environments. The authors have mainly investigated the distribution of scatterers in the considered indoor area.
In practice, an indoor propagation environment induces significant distortions on a signal propagating through that medium. These distortions are caused by multipath propagation and the Doppler effect arising from the motion of people within the indoor space. In this context, designing accurate and reliable channel models is of vital importance for the design and performance analysis of mm-Wave communication systems operating under real-world conditions. Therefore, several studies have been conducted to characterize the propagation delays and Doppler frequencies of the underlying channel. More importantly, propagation characteristics, such as spherical wavefronts, multipath scattering, and time-variant channel properties in space-time-frequency domains need to be thoroughly analyzed.
In general, the level of detail required for channel modeling greatly depends on the system parameters, such as the carrier frequency, the bandwidth, the propagation environment, and the configuration of the transmitter and receiver antennas. Zheng et al. [15] mentioned that the channel parameters, such as amplitude, phase, and Doppler frequency shifts, are environment-dependent and their values change with different locations and orientations. Gustafson et al. [16] presented a measurement-based multipath clustering channel model for a double-directional multiple-input multiple-output (MIMO) system. They showed that the angular cluster characteristics are related to the geometry of the room. They also discussed the impact of fixed objects on the channel characteristics. The studies in [17] demonstrated that the distribution of the coherence bandwidth depends on the velocity of the moving object and its position with respect to the transmitter antenna.
According to the aforementioned studies, designing channel models, especially for indoor communications under nonstationary conditions, requires a detailed knowledge of the dynamics of the underlying wireless channel characteristics.
In the literature, several models have been designed to analyze and represent the time-variant (TV) Doppler characteristics, which are caused by the mobility of moving objects in indoor spaces [18]- [20]. A three-dimensional (3D) nonstationary cluster-based single-input single-output (SISO) channel model has been proposed in [20] for design of human activity recognition systems. In [20], the human body motion has been modelled by a cluster of synchronized moving point scatterers. The authors of [20] have derived expressions for the TV Doppler frequencies by using the TV angles-ofmotion (AOM), TV AOA and TV AOD. In [21], a 3D stochastic double-spherical channel model has been developed for mm-Wave MIMO communication systems. To characterize human body motions, a dynamic channel model has been presented in [22], where the human body segments were modelled by using different geometrical shapes, such as cylinders and spheres.
Owing to the Doppler effect, human motion results in temporal changes of the channel characteristics. These changes are usually studied through the spectrogram of the channel and are commonly referred to as micro-Doppler signatures. Theoretical investigations of micro-Doppler signatures require a prior knowledge of the exact time-variant trajectories of the different human body segments, such as head, shoulders, upper arms, lower arms, hands, upper legs, etc. To the best of the authors' knowledge, analytical studies on the kinetic and mechanical modeling of human body motion are limited in the context of channel modeling. In this regard, several studies assume that the human body can be represented by a single point scatterer. A traditional assumption is to consider a spherical object as a single point scatterer, for which analytical expressions for the TV trajectories are available [23]. For example, the estimation and analysis of the micro-Doppler signatures of a slow-moving target have been presented in [24] by using a forward scattering SISO radar system. As a target, the authors considered a swinging pendulum that is placed on a moving trolley. The analysis of the TV Doppler characteristics of a non-stationary channel in the presence of a swinging pendulum, which is attached to the ceiling of the room, can be found in [25] and [26]. In [25], the authors presented a 3D indoor channel model for radio-frequency (RF) sensing. They described the moving object's (pendulum's) trajectory by the TV speed, TV AOM, and the TV elevation AOA. It has been shown that the proposed model allows to describe the channel state information (CSI) of measured RF data, that has been collected by using a commercial Wi-Fi system. In [26], a 3D non-stationary single-input multiple-output (SIMO) channel model has been proposed for the mobility analysis of a single swinging pendulum as a moving point scatterer. The TV spectrogram and the TV mean Doppler shift caused by the motion of the pendulum have been analyzed. The model has been validated by measured data recorded by using radar kit operating at 24 GHz. Special attention has been paid to the impact of the receiver antenna positions on the amplitude of the TV Doppler spectral components. VOLUME 9, 2021 FIGURE 1. 3D geometrical channel model for an N T × N R MIMO system with a single moving scatterer S M at the TV position (x(t ), y (t ), z(t )) and K fixed scatterers S F k at fixed The goal of this paper is to extend the analysis of a SIMO channel presented in [26] to a MIMO channel model. To this aim, we propose a method for the design of a 3D trajectorydriven non-stationary MIMO channel model for a moving point scatterer in an indoor environment. We investigate the TV micro-Doppler signature of the moving point scatterer for different MIMO links. As a mechanical model for moving point scatterer, we consider a single pendulum swinging in a fixed direction. Hence, the proposed channel model is driven by the trajectory of a swinging pendulum. By considering the trajectory of the pendulum, we derive an analytical expression for the trajectory-driven TV channel transfer function (CTF) and the corresponding complex channel gain of the MIMO system. The input signals of the trajectory-driven MIMO channel model are the TV positions of the swinging pendulum, which can be computed analytically [26]. We analyse the TV Doppler characteristics, such as the spectrograms and the TV mean Doppler shifts of different MIMO links. The theoretical results are validated through real-world measured data. Here, the comparison is made by using three different sets of measured data. Set 1 corresponds to the measured RF channel, which is obtained by using a mm-Wave MIMO radar kit (Ancortek SDR-KIT 2400T2R4) operating at 24 GHz. Set 2 and Set 3 contain the information about the TV trajectory of the moving point scatterer. The data of Set 2 has been recorded by using an inertial measurement unit (IMU) sensor [27], which was attached to the swinging pendulum. On the other hand, the data of Set 3 has been obtained from a motion capture (MOCAP) camera system [28]. In our scenario, the camera system captures the trajectories of the pendulum by using markers attached to the ball. The perfect match between the TV spectrograms of the measured RF signals and those of the proposed trajectory-driven mm-Wave MIMO channel model ascertains the validity of the model and the usefulness of the work.
Considering the MIMO channel, we studied the impact of the antenna positions (MIMO links) on the TV spectrograms and the TV mean Doppler shifts. To the best of our knowledge, the impact of different MIMO links on the micro-Doppler signatures of a moving point scatterer has not been studied yet. In this regard, the main novelties of this paper are as follows: 1) We propose a 3D non-stationary mm-Wave MIMO channel model for a moving object modelled by a point scatterer in indoor environments.

2) We derive analytical expressions for trajectory-driven
CTFs and the corresponding complex channel gains. 3) We investigate the impact of the antenna positions on the TV Doppler spectrogram and the TV mean Doppler shift. 4) We demonstrate the correctness of the concept by validating the proposed trajectory-driven channel model using measured RF data, IMU data, and MOCAP data. 5) We present a framework for pre-and post-processing of RF, IMU, and MOCAP measurement data sets. 6) We perform a qualitative and quantitative analysis to evaluate the accuracy of the proposed model.
The remaining part of the paper is structured as follows. Section II presents a 3D geometrical MIMO channel model for an indoor environment with a single moving scatterer and K fixed scatterers. The proposed trajectory-driven channel model and the spectrogram analysis are presented in Section III and Section IV, respectively. The measurement and numerical results are provided in Section V. Finally, the conclusion is drawn in Section VI.

II. 3D INDOOR GEOMETRICAL CHANNEL MODEL
In this section, we present a 3D indoor geometrical channel model for the propagation scenario illustrated in Fig. 1. The propagation scenario consists of N T fixed transmitter antennas A T j (j = 1, 2, . . . , N T ), N R fixed receiver antennas A R i (i = 1, 2, . . . , N R ), a moving scatterer S M , and K fixed scatterers S F k (k = 1, 2, . . . , K ). The jth transmitter antenna A T j and the ith receiver antenna A R i are set at the predefined positions Fig. 1 by a red pentagon, which has the TV coordinates x(t), y(t), and z(t) describing the 3D trajectory C(t) = (x(t), y(t), z(t)). We assume singlebounce scattering and non-line of sight conditions. As shown in Fig. 1, we denote by d T j (t) the Euclidean distance between the moving scatterer S M and the jth transmitter antenna A T j , whereas d R i (t) stands for the Euclidean distance between the ith receiver antenna A R i and the moving scatterer S M . The distances d T j (t) and d R i (t) can be expressed in terms of the TV coordinates x(t), y(t), and z(t) of S M as (2) In the propagation scenario, we also consider multiple stationary objects, such as furniture, decoration items, and walls, which are modelled by K fixed scatterers S F k (k = 1, 2, . . . , K ) as depicted in Fig. 1. The position of the kth fixed scatterer is denoted by ( (1) and (2), the Euclidean distance between the fixed scatterer S F k and the jth (ith) transmitter (receiver) antenna is denoted by and respectively.

III. MIMO CHANNEL MODEL
In this section, we present the TV-CTF of the N T ×N R MIMO channel. As depicted in Fig. 1, the radio waves travel from the jth transmitter antenna A T j to the moving (fixed) scatterer S M (S F k ) and impinges at the ith receiver antenna A R i . Hence, the TV-CTF H i,j (t, f ) of the link from A T j to A R i can be presented as: where H M i,j (t, f ) and H F i,j,k (f ) stand for the components caused by the moving scatterer S M and the kth fixed scatterer S F k , respectively. The TV-CTF determined by the moving scatterer S M can be expressed as where c M i,j , f c , θ M i,j , and τ M i,j (t) denote a fixed path gain, a carrier frequency, a phase, and a TV propagation delay of the moving point scatterer. respectively. The second term of (5) can be expressed as Analogously, the TV-CTF H F i,j,k (f ) of the kth fixed scatterer is determined by a fixed path gain c F i,j,k , a phase θ F i,j,k and a propagation delay τ F i,j,k associated with the kth fixed scatterer. It is worth noting that both phases θ M i,j and θ F i,j,k in (6) and (7), respectively, are modelled as independent and identically distributed (i.i.d.) random variables, each of them following a uniform distribution over the interval from −π to π. The TV propagation delay τ M i,j (t) of the radio wave travelling from the jth transmitter antenna A T j to the ith receiver antenna A R i via the moving point scatterer S M can be computed in terms of the Euclidean distances d T j (t) and d R i (t) as follows The TV Doppler shift corresponding to the link from A T j to A R i can be obtained by computing the derivativeτ M i,j (t) of the TV delay τ M i,j (t) as follow [25]: In the second step, we compute the complex channel gain µ i,j (t) representing the channel link from A T j to A R i by integrating h M i,j (t, τ ) over τ from 0 to the maximum propagation delay τ max , i.e., In the third step, the complex channel gain µ i,j (t) is multiplied by a sliding window function w(t), which is a positive VOLUME 9, 2021 even function with normalized energy, as follows For the sliding window function, we use the Kaiser window function that is given by where T 0 is the length of the window, and β is parameter that controls the shape of the window function. The symbol I 0 (·) stands for the modified Bessel function of the first kind of zero order. In the fourth step, we compute the short-time Fourier transform (STFT) of the signal x i,j (t , t) in (14) with respect to t , i.e., Finally, by computing the squared magnitude of the STFT X i,j (f , t) in (16), we obtain the TV Doppler spectrogram of the link from A T j to A R i as By computing the first moment of the TV Doppler spectrogram S i,j (f , t) in (17), we obtain the TV mean Doppler frequency B (1) µ i,j (t). Hence, the TV mean Doppler shift can be formulated as [25] Alternatively, the TV mean Doppler shift of the proposed trajectory-driven channel model can also be computed by using the TV Doppler shifts f M i,j (t) of the corresponding moving point scatterer S M as follows [25] B

V. MEASUREMENTS AND NUMERICAL RESULTS
This section presents the measurement setup for capturing the trajectory of a moving point scatterer in an indoor environment. The analyses of the results obtained from the measurements and the analytical model are also discussed. The indoor propagation environment is a room with a width of 6 m, length of 11.5 m, and height of 2.5 m. In the room, there exist fixed objects, such as chairs, tables, boards, and computers and a moving point scatterer. As a point scatterer, we consider a single swinging pendulum, which is a medicine ball weighing 3 kg. We attach the pendulum to the ceiling of the laboratory room by using a rope with length L = 1.52 m. From the geometrical representation of the swinging pendulum shown in Fig. 3, we can compute the trajectory (displacements) x(t), y(t), and z(t) of the pendulum as follows [26]: where g and x max represent the gravity acceleration and the maximum displacement in x-direction, respectively. The TV positions x(t), y(t), and z(t) characterize the TV trajectory of the pendulum C(t) = (x(t), y(t), z(t)). The TV trajectory C(t) is the input signal of the trajectory-driven channel model, which we refer hereafter as the analytical model. Starting from the mathematical trajectory C(t) (pendulum model), we can compute the TV spectrogram S i,j (t, f ) as outlined in Section IV and depicted in Fig. 4(a).

A. MEASUREMENT SCENARIO FOR RADAR SENSING
The mobility of the pendulum is captured with the MIMO radar sensing system SDR-KIT 2400T2R4 from Ancortek operating at 24 GHz. The radar system has two transmitter and four receiver directional antennas. The transmitted chirp signals have a bandwidth of 250 MHz with 1 ms chirp duration. While employing a full MIMO setup, i.e., 2 × 4 antenna configurations, we experience a mixture of micro-Doppler signature patterns of the swinging pendulum corresponding to several MIMO sub-links. To avoid this subchannel cointerference, we used only the 2 × 2 MIMO configuration, in which this mixture of patterns is not negligible. Moreover, to reduce the impact of the subchannel co-interference, we subtracted the TV spectrograms of different links from each other as follows: 1 (f , t)) represents the TV spectrograms of the link A T consisting of the patterns of spectrograms from the other link . The symbols α and β are weighting factors associated with the spectrograms of the interfering subchannel. Thus, we compute the TV spectrogram of the desired A T 1 −A R 1 link S 1,1 (f , t) as follows 1). (26) If α and β are very small compared to 1, i.e., α 1 and β 1, then we can write 1 (f , t) to zero for all values of f and t for which the difference in (27) is negative. Thus, we obtain The same approach can be applied on the other links, which results in S 2,1 (f , t) ≈Ŝ 2,1 (f , t), S 1,2 (f , t) ≈Ŝ 1,2 (f , t), and S 2,2 (f , t) ≈Ŝ 2,2 (f , t). To form communication distinct links, we placed the antennas in a distributed manner as shown in Fig. 2. The antennas are connected to the radar kit with cables of length 3 m each. The transmitter and the receiver antennas are set at the positions ( , respectively, as illustrated in Fig. 3. The coordinates of the antennas and pendulum parameters, such as the maximum displacement x max , and the length of the rope L are listed in Table 1. The emitted waves from the jth transmitter antenna will encounter the pendulum, and the scattered waves impinge at the ith receiver antenna. Subsequently, the received signals are stored and processed for further analysis. Then, we analyse the TV Doppler spectrogram of each link as illustrated in Fig. 4(d).

B. TRAJECTORY CAPTURING BY USING THE IMU
To obtain measurement of the pendulum trajectory, we use MetaMotionR sensor fusion [27]. The IMU sensor is attached to the swinging pendulum as shown in Fig. 2. By using a smartphone, we control the IMU and record the data files via Bluetooth. The data files contain quaternions and linear accelerations of the pendulum. They are applied to compute the Euler angles to rotate the linear accelerations as shown in Fig. 4(b). Then, we have employed a quadratic regression method to smooth the rotated linear accelerations. Thereafter, we have computed the velocities and the trajectory (positions) by integrating linear accelerations. Due to an experimental error, there are linear and quadratic drifts in the velocity and the trajectory. These drifts have been compensated by using the zero-update (ZUPT) algorithm [29]. The TV drift-free horizontal displacement x(t) and the vertical displacement z(t) of the recorded data from the IMU are compared with those of the pendulum model given in (20) and (22), respectively. It is worth noting that to fit the displacements of the IMU data to those of the pendulum model, it is required to calibrate the IMU data by shifting and scaling the graphs in both axes. In our analysis, we have normalized the trajectories obtained from the IMU data as follows x IMU (t) (29) y(t) = 0 (30) Alternatively, the z(t) displacement can be computed by using the x(t) displacement of the IMU data by means of the relationship in (22). However, this is only true if an analytical solution (a mathematical model) is known. Therefore, it is important to develop a channel model that can be used as a reference model to replicate the real channel. Thus, the final obtained (normalized) trajectoryC(t) = (x(t),ỹ(t),z(t)) will be used as the input signal of the IMU data-driven channel model and to compute its corresponding TV micro-Doppler signaturesS i,j (f , t) as illustrated in Fig. 4(b).

C. TRAJECTORY MEASUREMENTS USING A MOTION CAPTURE (MOCAP) CAMERA SYSTEM
To validate the correctness of the proposed channel model, we also recorded the trajectory of the swinging pendulum by using the MOCAP camera system. The adopted MOCAP camera system is based on the Qualisys capture system [30]. This MOCAP system features the highest-performing technology that is currently available on the market. The setup of the Qualisys MOCAP system is shown in Fig. 5, where 8 Miqus hybrid cameras are used. The Miqus hybrid cameras allow us to capture the mobility of the object by using either a marker-based tracking or a marker-less tracking system. The Miqus hybrid cameras [31] are dual mode cameras capable of streaming full high definition (HD) color video at 85 frames per second or streaming marker data at 340 frames per second. In our preliminary experiment, a collection of MOCAP data is conducted by using Qualisys Track Manager (QTM) [32] based on a marker tracking system. The tracking can be performed in real-time or by post-processing. In the proposed experiment, markers have been placed on the swinging pendulum. Based on the selected markers, a virtual reference frame is established in the center of mass (CoM) of the medicine ball. The QTM's advanced 3D six-degree of freedom (6DOF) tracker computes 3D and  6DOF data by using two-dimensional (2D) data points with minimal latency. Thus, the obtained trajectory of the pen-dulumC(t) = (x(t),ȳ(t),z(t)) will be used to compute the MOCAP data-driven channel model. For reasons of comparison, we can then compute TV micro-Doppler signaturē S i,j (f , t), as shown in Fig. 4(c).
Finally, by using the TV micro-Doppler signature, we can obtain the TV mean Doppler shift. We conducted another experiment with the swinging pendulum by using simultaneously the IMU sensor, the MOCAP camera system, and the radar kit configured for the SISO communication link. To investigate the performance of the MOCAP camera    system, we computed the TV-CTFH i,j (t, f ) and its corresponding mean Doppler shiftB (1) H i,j (t) by using the trajectory obtained from the MOCAP data. Figures 6 and 7 illustrate x(t) and z(t) displacements obtained from the mechanical model, the MOCAP data, and the IMU data. We can see that all models match well and demonstrate a good agreement. To investigate the accuracy of the conducted data, we computed a similarity score between the mechanical model and the recorded data. The similarity FIGURE 9. Similarity score (in %) of the z(t ) displacement of the mechanical model, the measured IMU data, and the MOCAP data. score can be obtained by using where a and a stand for the measured and analytical data, respectively. By replacing a and a with x(t) andx(t), respectively, we can compute the similarity score between VOLUME 9, 2021 the displacements of the mechanical (pendulum) model and the recorded trajectories. Hence, the similarity score of the displacements x(t) and z(t) obtained from the MOCAP data, the IMU data, and the mechanical model can be seen in Figs. 8 and 9. We can observe that the MOCAP data obtained from the QTM system outperforms the one of the IMU data when comparing to the mechanical (pendulum) model.

D. TV SPECTROGRAM AND MEAN DOPPLER SHIFT
The TV spectrograms for the MIMO links A T  Doppler frequency has positive (negative) values, when the pendulum swings towards (away from) the transmitter antenna A T 1 . When the swinging pendulum reaches its local minimum and maximum displacements as shown in Figs. 6 and 7, then its speed of motion reaches 0 m/s. Hence, the TV Doppler frequency corresponding to the motion of the pendulum approaches 0 Hz. Similar, observations can be seen in Figs. 11, 12, and 13 for the links respectively. From the results in Figs. 10−13, the impact of the transmitter and the receiver antenna positions on the spectrograms in terms of the amplitude of the Doppler frequency can be seen. In fact, the wave travel distance has an impact on the corresponding propagation delay, which is directly related to the Doppler shift.
We have also investigated the TV mean Doppler shiftŝ B (1) i,j (t),B (1) i,j (t), and B (1) i,j (t) of the measured channel, the IMU-driven channel, and the analytical channel models, respectively. In Figs. 14−17, we have illustrated the TV mean Doppler shifts for the different MIMO links. Similar to FIGURE 13. SpectrogramsŜ 2,2 (f , t ),S 2,2 (f , t ), and S 2,2 (f , t ) of the (a) measured channel, (b) IMU data-driven model, and (c) analytical model, respectively, for the A T 2 −A R 2 link.    Furthermore, we can observe an excellent match between the TV mean Doppler shift of the measured channel, the IMU data-driven channel, and the analytical channel model. This confirms the usefulness of the proposed model for all MIMO links considering different antenna positions. As we mentioned earlier, we also conducted MOCAP data for the trajectories of the pendulum, while configuring the radar kit as VOLUME 9, 2021 (t ) of the SISO channel.
a SISO system and activating the IMU sensor too. Fig. 18 shows the corresponding TV mean Doppler shift for this experiment. We can see that the results of all models exhibit an excellent agreement.

VI. CONCLUSION
In this paper, we have proposed a 3D trajectory-driven nonstationary mm-Wave MIMO channel model for a moving point scatterer. We have presented the TV-CTF capturing the effect of the mobility of the moving point scatterer.
We have studied the TV Doppler characteristics of the proposed channel model, i.e., the TV Doppler spectrogram and the TV mean Doppler shift. The TV micro-Doppler signature of the proposed channel model has been validated by measured data, which are obtained by using a MIMO radar operating at 24 GHz. The obtained results have shown excellent agreement between the measured data and the proposed trajectory-driven channel model. Moreover, we have captured the TV trajectory of the pendulum by using both the IMU sensor and the QTM-based MOCAP camera system. The accuracy of the proposed non-stationary channel model has been confirmed by demonstrating a perfect match between the TV Doppler spectrograms of the measured (radar, IMU, MOCAP) data and the analytical model. We have also investigated the impact of the different MIMO links on the Doppler spectrogram of the received signal. The results demonstrate the importance of the antenna elements position for the analysis of the micro-Doppler signatures caused by the mobility of an object. Furthermore, the comparison analysis of the trajectories obtained from the MOCAP and IMU sensor data show that the MOCAP system outperforms the IMU sensor with respect to the analytical trajectory-driven model. It is important to obtain a set of data that can accurately replicate the fingerprints of real-life activities. We believe that our trajectory-driven channel model can serve as a basis for the derivation of a generic indoor channel model reproducing the Doppler characteristics caused by a moving point scatterer. As a future work, this model can be extended by considering multiple moving scatterers.