Future wireless communication systems will most likely be operated at carrier frequencies above 300 GHz, where the indoor radio channel behaves entirely differently compared to legacy radio communication frequencies. Being highly relevant for system performance evaluations and channel modeling, the spatial as well as the temporal dispersions are studied for a representative office wireless LAN scenario in this paper. Ray tracing serves as the means for the accurate simulation of the THz radio wave propagation. Simple stochastic models are derived to approximate and reproduce the distance-dependent behavior of the angular spread as well as of the RMS delay spread. Based on the results, the maximum symbol rates achievable without any intersymbol interference are quantified and can be shown to reach up to several 100 GSymbols/s provided that highly directive antennas are used.
In face of the almost completely regulated spectrum below 300 GHz, THz frequencies, ranging from 300 GHz up to 3 THz, hold a huge potential to host future ultra fast communication systems. Aiming at system bandwidths beyond 40 GHz and data rates of up to 100 Gbit/s, THz communication can cover a wide range of applications like extremely fast WLANs, kiosk downloads, or telemedicine applications as mentioned by Piesiewicz et al. [2007b]. Limited to short ranges by high propagation losses, THz data transmission systems will preferably be operated in indoor environments. Besides reflections, especially rough surface scattering from indoor building materials like plaster or wallpaper exerts a considerable impact on the THz indoor channel characteristics as studied by Piesiewicz et al. [2007a], causing a pulse broadening both in the angular as well as in the temporal domain. Because smart antennas are most likely to be employed in order to simultaneously achieve a high antenna gain and a steerable antenna beam, enabling the concept of directed non-line-of-sight (NLOS) communications discussed by Piesiewicz et al. , the spatial channel information is of high importance. An appropriate measure to quantify the angular dispersion of THz propagation channels is provided by the angular spread defined in Winters , which will be analyzed in this paper in an office scenario using a self-developed ray tracing tool. Up to now, comparable investigations have only been performed at far lower frequencies like, e.g., 2.5 GHz by Zwick et al. , where almost omnidirectional antennas are usually employed. At THz frequencies, the short wavelength below 1 mm facilitates small multiple-input and multiple-output (MIMO) arrays with plenty of single elements and very narrow beams, so that single propagation paths can be selected, the so-called clusters. Hence, the angular spread modeling must be conducted not only with regard to the distance between transmitter (TX) and receiver (RX) but also cluster based. This information can be used for the generation of cluster angular profiles in a THz channel model.
In consideration of the very short symbol durations far below 1 ns as required for the envisaged data rates exceeding 10 Gbits/s, the temporal channel dispersion will additionally become critical. Multipath propagation may easily cause intersymbol interference (ISI) particularly in scattering indoor environments. Here, this aspect will be evaluated in terms of the RMS delay spread similar to Hashemi and Tholl  with the aim to predict the maximum symbol rates achievable without ISI. Moreover, due to the high scattering contributions in the presence of rough materials, both the angular and the RMS delay spread will be analyzed as a function of the degree of surface roughness.
The remaining part of the paper is structured as follows. A short overview of the applied ray tracing algorithm and the investigated scenario is given in section 2. Based on the ray tracing results, a distance- and roughness-dependent angular spread model will be proposed in section 3, whereas section 4 focuses on the RMS delay spread modeling. Estimates of the maximum symbol rates will be given regarding the temporal channel characteristics before the paper is summarized in section 5.
2 Ray Tracing
A custom full 3-D ray tracing tool has been developed for an appropriate consideration of the propagation mechanisms that become relevant at THz frequencies. The output of the ray tracer covers the fully polarimetric, complex electric field strengths, the angles of arrival/departure (AoAs/AoDs), and times of arrival (ToAs) of each ray. Apart from the simple propagation phenomena like the free space loss and specular reflections, the ray tracing tool especially features an appropriate rough surface scattering model for the THz frequency range. This scattering implementation employs the Kirchhoff scattering theory, which has been validated at THz frequencies in Priebe et al. [2011c]. Moreover, the ray tracer has been programmed modularly and can be extended by further propagation phenomena such as reflections from stratified materials.
For the ray tracing simulations, a realistic office room (Figure 1) has been selected with the TX being placed under the ceiling at x = 0.25 m, y = 2.5 m, and z = 2.3 m. The ceiling and the tables have a height of 2.5 and 0.7 m, respectively, whereas a nomadic device is assumed as the RX at z = 0.75 m. This setup represents a THz WLAN connection between an access point and a laptop. A frequency of 300 GHz has been chosen since it provides the lowest free space loss (FSL) within the still unregulated spectrum. 20 and 11 positions with a spacing of 0.25 m are considered in the x and y-directions, respectively. Approximately 11,000 rays are simulated at each point, yielding statistically significant results. Due to the extensive electromagnetic calculations, the computational time ranges in the order of seconds on a modern workstation for one RX position in the chosen scenario. A strong influence of the line-of-sight (LOS) path on the RMS delay and angular spread is expected, so that the scenario has been designed to have both LOS and non-line-of-sight (NLOS) conditions. Please note that only multipath components (MPCs) with a path loss of up to a maximum of 160 dB are considered throughout this paper, corresponding to a realistic noise threshold. Paths with higher losses would in practice neither be detectable nor relevant in face of the limited output powers and high thermal noise levels (cf. measurements in Priebe et al. [2011b]).
All walls, the ceiling, and the screen are covered with plaster, whereas the tables are made of wood (cf. Table 1 for the material parameters). Two different degrees of roughness, characterized by different height standard deviations σh and correlations lengths lcorr, are considered. Floor reflections are assumed negligible due to very high carpet reflection losses that have been measured by Lönnqvist et al. . In order to gain an estimate of the highest expected multipath impact, an ideal omnidirectional antenna pattern and linear θ-polarization is used. First- and second-order paths, both reflected and scattered, are taken into account only as a compromise between accuracy and computational time.
Table 1. Material Parameters
3 Angular Spread
Especially in indoor channels, where plenty of reflections and scattering processes cause a strong multipath propagation, the angular spread σϕ/θ,AoA/AoD in the azimuth/elevation and for the AoA/AoD is one of the most important channel characteristics. It is calculated as the second moment of the respective angular power profile. In order to avoid an overestimation of the angular spread due to an unambiguous definition of the angles on a circle (e.g., an AoD of −1° could be represented as 359°), complex angles must be taken as the basis like described in Fleury .
Accordingly, the angular spread gives a measure for the angular dispersion and multipath richness of the channel. Higher angular spreads basically indicate higher MIMO channel capacities, which rely on spatial multiplexing. Dependent on the chosen beamforming scheme, smart antennas can also direct the antenna main lobe at the strongest propagation path, either LOS or NLOS in case of shadowing, in order to avoid fading. The latter case is referred to as directed-NLOS communication in Piesiewicz et al. [2007b].
Figure 2a exemplarily shows the azimuth angular spread in the investigated scenario for the AoA and plaster 1. The elevation and the AoD will be discussed in the next subsection. Moderate values around 20° are observed in most of the LOS cases. Several exceptions with angular spreads of approximately 35° like in the lower left part of the room close to the screen result from individual clusters reflected at the screen and the adjacent wall. Very similar results with comparable angular spreads of several 10° have been obtained in a separate measurement campaign around 300 GHz by Priebe et al. , validating the simulations.
The reflected paths gain a higher impact on the angular spread under NLOS conditions due to the missing dominating LOS. Comparatively higher σϕ,AoA of up to 50° occur in the shadowing region. Accordingly, LOS and NLOS conditions must be considered separately for the angular spread modeling. Far higher values of up to 119° have been reported for different indoor scenarios and antenna configurations at 60 GHz in Yong et al. .
In Figure 2b, the roughness dependence of the angular spread is demonstrated additionally in terms of the difference between the angular spreads for both roughnesses. Interestingly, the higher roughness of plaster 2 leads to a decrease of σϕ,AoA by at least 10° in most regions. This is due to that a larger amount of the power is scattered diffusely for plaster 2 compared to plaster 1 and is not received any more.
3.1 Angular Spread Modeling
With the aim of predicting the angular spread in indoor environments, an analytical model becomes obligatory. This information can be used in a channel modeling approach for randomizing the AoA/AoD of individual rays. A comparable approach is followed in Yong et al. , where the ray AoAs/AoDs are determined according to a Gaussian distribution and the angular spreads σϕ,θ could be employed as standard deviations. Due to the complex dependences of σϕ,θ on different factors like positions of walls, materials, etc., a completely analytical derivation is hardly possible. Therefore, we apply a distance-dependent empiric approach to gain insight into the general angular spread behavior. σϕ,AoA and σθ,AoD can exemplarily be seen in Figure 3 as a function of the distance d from the TX for both LOS and NLOS conditions. First values occur for d ≈ 1.5 m due to the height difference between TX and RX.
A second-order polynomial of the form
is used to approximate the simulation results measured in degrees. χσ denotes a zero mean Gaussian random variable with the standard deviation σ, accounting for the deviations of the individual spreads from the polynomial function. With this simple function, the tendency of an angular spread decrease toward the center of the room and an increase toward the walls or vice versa can be reflected. Of course, not all specialties of the angular spread behavior can be captured with such a simple approximation, yet it is easily parameterizable and can be reused by other researchers. Despite the model simplicity, rather low standard deviations are achieved with the approximations as summarized in Table 2 along with all function parameters.
Table 2. Parameters for the Second Order Polynomials of the Angular Spreads
Only for σϕ,AoD, comparatively high deviations cannot be avoided as the TX is placed close to several walls and the ceiling, so that a strong fluctuation of σϕ,AoD occurs even for very similar distances. The AoA azimuth angular spread under LOS conditions in Figure 3b is very high close to the TX, i.e. for the shortest d. There, the multipath components are received from a broad angular range and with a moderate additional attenuation compared to the LOS. Toward the middle of the room, the indirect paths become longer relative to the LOS and are hence attenuated more, so that σϕ,AoA decreases. Close the walls, the reflected rays again gain a higher relative impact and the angular spread increases. Under NLOS conditions, the reflections are always received from a comparable angular range regardless of the distance, so that a rather flat approximation function results.
On the other hand, the elevation AoD angular spread (Figure 3b) behaves inversely in the LOS case. Smallest spreads occur closest to the TX because almost all rays depart under very similar angles. For increasing distances, the rays leave the TX toward different directions, so that σθ,AoD increases. As soon as the RX is positioned closer to the reflection points at the remote walls for largest d, the rays are again transmitted into less divergent elevation directions, which leads to a decreasing angular spread. Under NLOS conditions, several relevant indirect rays with deviating elevations are shadowed close to the screen, i.e., for distances between about 3 and 4.5 m. There, very low spreads can result. These rays can be received toward the right and the lower wall, so that the angular spread increases. Regarding the impact of the roughness (not shown), the smoother plaster 1 leads to less diffusely scattered power and hence to a higher angular spread compared to the depicted results.
Basically, all thoughts on the chosen scenario are also applicable to general environments. However, the approximation functions will strongly depend on the scenario dimensions, the positions of the TX and RX relative to the walls as well as on any further reflecting objects.
3.2 Cluster-Based Investigation
In case that a single cluster is selected with a highly focused beam for directed NLOS communication, not the overall angular spread but rather the angular spread of the respective individual cluster must be considered. Then, the cluster-based angular spread provides a measure for the broadening of the cluster due to scattering. If the specular reflections have been determined, e.g., from ray tracing, or have been modeled statistically, the angular power profile of the cluster can be randomized according to a Gaussian distribution with the standard deviations σϕ, σθ for THz channel modeling similar to Yong et al. . A distance-dependent approach is unreasonable as different clusters have individual different angular spreads for one RX position. It cannot be known a priori which cluster may be selected, so that the probability for a specific angular spread of the respective cluster must be considered instead. Figure 4 shows the histograms of the cluster-based σϕ,AoA and σθ,AoA in (a) and (b), where each cluster at each of the 220 simulation points is respected individually. It is worth mentioning that the integrals over the histograms have been normalized to 1, so that the histograms directly yield the corresponding probability density functions (PDFs). All PDFs are additionally approximated with a negative exponential distribution
described by the parameter μ. All μ and the corresponding low root mean square errors (RMSEs) can be obtained from Table 3. With the aim to separately demonstrate the roughness and ϕ, θ dependence, the approximated cumulative distribution functions (CDFs) are additionally given in Figure 4c. In contrast to the angular spreads for omnidirectional antennas before, the far higher cluster broadening for the rougher plaster 2 can easily be seen compared to plaster 1 here. A similar behavior of the scattering processes in ϕ and θ direction leads to almost identical CDFs for ϕ and θ. In case of the AoD, higher μ indicates higher angular spreads of the clusters compared to the AoA, which is again reasoned with the short distance of the TX to the ceiling and two walls.
Table 3. Parameters for the Exponential Approximations of the Cluster-based Angular Spread PDFs
4 RMS Delay Spread
Previously, the angular spread has been considered as a quantification of the spatial dispersion of the angular power profile, whereas the RMS delay spread provides the analog measure for the temporal dispersion of the power delay profile (PDP). It is determined as
with τi being the delay of the ith path and being the mean delay of the PDP.
Figure 5a illustrates the RMS delay spread distribution in the office scenario. Highest RMS delay spreads of up to 3.5 ns appear in the upper left and lower right part of the room for different reasons. In the upper left part, no MPCs are shadowed by the screen. Especially the longer, twice reflected paths contribute to the high spreads within this area, whereas several longer paths are shadowed in the remaining scenario. Furthermore, the LOS power becomes increasingly dominant for longer distances from the TX, as it will be discussed in detail in the next subsection. Under NLOS conditions in the lower right corner, each MPC has a stronger relative impact on the RMS delay spread in face of the missing LOS power, resulting in higher delay spreads. Similar delay spreads of up to about 4 ns have been determined experimentally in Priebe et al.  around 300 GHz.
Regarding the roughness impact, an increase of the roughness causes a decrease of the delay spread as can be seen in Figure 5b. Such a behavior can be explained with the fact that a large part of the incident power, being reflected specularly for the lower roughness of plaster 1, is scattered out of the specular reflection direction for plaster 2. Hence, less power of the MCPs is received, leading to a reduced τRMS. This effect is particularly significant in areas with higher delay spreads. In contrast, the impact of the roughness is hardly noticeable where hardly any MPCs are received at all, like close to the right side of the screen. Please note that the difference between the roughnesses corresponds to a significant reduction of the delay spread to approximately 0.5 ns and less in most cases.
4.1 RMS Delay Spread Modeling
From the RMS delay spread, useful estimations for the maximum symbol rates achievable without intersymbol interference can be gained, so that it is worth to be modeled in addition to the spatial channel dispersion characteristics. Another example for its usefulness is an orthogonal frequency-division multiplexing (OFDM) transmission, where the symbol duration can be optimized for the intended application, if the delay can be predicted.
Figure 6 shows the RMS delay spread for plaster 1 in (a) and plaster 2 in (b). It is worth noting that a temporal resolution of 0.05 ns has been chosen for the evaluations, which relates to a realistic system bandwidth of 20 GHz. Again, a second-order polynomial
provides a good approximation of both the RMS delay spread. χτ denotes a zero mean Gaussian random variable. A summary of the determined parameters is given in Table 4.
Table 4. Parameters for the Second-Order Polynomial Approximations of τRMS
Remarkably, a second-order polynomial approximation has also been found for the delay spread of ultra wideband channels between 3 and 8 GHz within an aircraft by Jacob et al. . Compared to the aircraft environment with delay spreads of up to 22 ns, very low maximum delay spreads of 2.5 ns and 0.75 ns have been calculated under LOS conditions for plaster 1 and plaster 2, respectively. A decreasing tendency toward larger distances is observed. Due to the reflection losses and the increasing path lengths of the MPCs, weak MPCs drop below the path loss threshold of 160 dB for longer d, leaving the LOS path as the dominant contribution to the received power. Under NLOS conditions, multiple rays are shadowed near the screen, whereas they can be received for larger d, so that the delay spread rises. Please note that this effect is very specific for the chosen scenario. Generally, τRMS can be expected to decrease for larger distances due to the MPCs falling below the noise, as also observed in Jacob et al. .
4.2 Cluster-Based Investigation
Alike the cluster-based investigation of the angular spread, also the temporal channel dispersion must be considered on a cluster basis in case of directed NLOS communication with very directive antenna beams. Under the assumption that exactly one cluster can be selected, extremely low RMS delay spreads of no more than 0.03 ns (plaster 1) and 0.1 ns (plaster 2) occur. Typically, one cluster is not wider than a few degrees, and antennas with a half power beam width of several degrees suffice for the suppression of the other clusters. Relevant MPCs from one cluster arrive within 1 ns or less as found by Priebe et al. [2011a]. Note that Priebe et al. [2011a] also provide a more detailed amplitude- and time-of-arrival-modeling of individual clusters, which reaches beyond the scope of this paper.
The simulated PDFs are demonstrated Figures 7a and 7b including an approximation with a negative exponential distribution (cf. Table 5 for the parameters). p0 denotes the probability for cluster delay spread of 0 ns. This case corresponds to a temporal cluster broadening below the virtual temporal system resolution of 0.05 ns, so that the RMS delay spread cannot be evaluated reasonably.
Table 5. Parameters for the Exponential Approximations of the Cluster-based RMS Delay Spread PDFs
For a better comparison, the approximated CDFs are additionally given in part (c) of the figure. It can be seen that the surface roughness has a high impact on the cluster-based delay spread. The implications of these findings on the possible symbol rates will be discussed next.
4.3 Maximum Achievable Symbol Rates
With the aim to avoid ISI in a time dispersive channel, the symbol duration must be sufficiently long, so that the maximum symbol rate is limited by
where K denotes a constant between 0 and 1. According to Rappaport , K is dependent on various factors like the modulation scheme, equalizers, etc., and cannot be determined analytically. Here, K = 1 is chosen in order to estimate the maximum possible symbol rates. For other K, the obtained rs,max just must be multiplied with the respective K value. All delay spreads obtained from the ray tracing simulations (cf. Figure 6) are considered to determine empiric CDFs for the achievable symbol rates in the entire scenario according to equation (5). Figure 8a illustrates the results. Again, both LOS and NLOS cases as well as plaster 1 and 2 are distinguished between. The highest symbol rates between 1.5 and 5 GSymbols/s become possible for the LOS link and for the higher roughness of plaster 2. This behavior can be explained with the less severe temporal dispersion due to the lower amplitudes of the specular reflections compared to plaster 1. For plaster 1, only between 0.5 and 1 GSymbols/s can be achieved. Accordingly, spatial filtering with smart antennas will be required to reach several 10 GSymbols/s. Therefore, the maximum symbol rates are additionally considered cluster based in Figure 8b. The selection of a single cluster allows for symbol rates between 30 and 104 GSymbols/s under consideration of the mere limitation due to temporal dispersion, whereas the less rough plaster 1 permits a performance better by a factor of approximately 8. Here it is worth mentioning that obviously not the temporal dispersion but the roughness-dependent propagation loss of the NLOS path effectively limits the maximum data rates as discussed in Piesiewicz et al. [2007b], which is not in the focus of the investigations presented in this paper. Regarding the temporal characteristics, no limitations are to be expected even for data rates of up to a few 100 Gbit/s under the realistic presumption of highly focusing antenna arrays.
Spatial and temporal channel properties have been investigated at 300 GHz in an indoor office scenario with respect to the operation of future THz WLAN systems. Ray tracing simulations have been evaluated in order to determine the angular and RMS delay spreads. The distance-dependent behavior of these parameters has been analyzed and modeled.
Angular spreads of up to 50° and RMS delay spreads of up to 3.5 ns have been observed for omnidirectional antennas, whereas values of up to 4° and 0.06 ns occur for individual clusters. It has become obvious that the surface roughness exerts a considerable influence on both the spatial and temporal channel characteristics. A higher roughness is favorable in view of the temporal and spatial dispersion due to the attenuation of the disturbing MPCs under LOS conditions. On the other hand, less rough surfaces allow for a better directed NLOS performance. Then, THz channels facilitate symbol rates of up to several 100 GSymbols/s by employing narrow antenna beams without any ISI to be expected.