Theory of stochastic local area channel modeling for wireless communications

ireless modem design efforts. 161 CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 162 7.1 Previous Measurement Campaigns This section discusses the primary contributions of this measurement campaign. Previous results for angular and delay dispersion are compared to values obtained in our campaign. 7.1.1 Contribution of this Work In this paper, peer-to-peer channel statistics for 12 local areas are derived from over 5000 ultra-wideband power delay profile (PDP) snapshots taken by a spread-spectrum sliding correlator channel sounder with 140 MHz of radio frequency bandwidth. This measurement campaign is unique and valuable for the following reasons: Joint statistics involving multipath delay dispersion and angle dispersion have been measured at each local area. A variety of transmitter-receiver configurations – outdoor-to-outdoor, outdoor-to- indoor, and indoor-to-indoor – were measured. This paper introduces a new spatial probing technique that uses directional antennas and mechanical positioning, as opposed to antenna arrays. The new technique facili- tates wider-band spatial channel sounding at potentially any carrier frequency in the UHF, microwave, or mm-wave bands. The spatial channel parameters are calculated in terms of multipath shape factors, one of the first such applications of the theory described in [41]. The results should prove valuable to designers of high-capacity wireless modems and future joint spatio-temporal channel measurement campaigns. 7.1.2 Comparison to Other Measurement Campaigns in the Literature The first documented attempt to measure both angle-of-arrival and time-delay of mobile radio multipath components was made by Japanese researchers Ikegami and Yoshida [50]. More recent spatio-temporal results have been published in [51] and [52], although these research papers measure only a few locations and are meant largely to prove new techniques and concepts in spatio-temporal channel measurement. Pedersen, et. al. have also published delay and azimuthal dispersion results for base station antennas in urban areas [53]. Beyond these publications, the joint spatio-temporal characteristics for most types of radio channels are not conclusively measured. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 163 While the bands around 1920 MHz has been measured extensively for cellular and PCS applications, measurements previously reported in the literature involve transmitters at least 5 meters above the ground – still much higher than a genuine peer-to-peer environment. The range of peer-to-peer outdoor delay spreads presented in this paper are much lower than typical delay spreads measured for mobile radio macrocells (2-3 µs, [54]) and are more similar to PCS microcell measurements (mean of 137 ns [55]). This is expected since the peer-to-peer mode of propagation is much more lossy than macrocellular configurations since both transmitter and receiver are close to the ground, buried within building, foliage, and terrain clutter. Our campaign finds a similar range of outdoor RMS delay spreads (17 ns to 219 ns) to those reported by Patwari, et. al. in [56] (25 ns to 333 ns). Unlike the measurements in [56], this paper directly measures the angles-of-arrival of multipath components. Thus, delay dispersion (Section 7.3.1), angle dispersion (Section 7.3.2), and joint angle-delay statistics (Section 7.3.3) are presented for all indoor and outdoor measurement locations. For indoor receiver locations, our measurements exhibit much lower RMS delay spreads (27-45 ns) than the outdoor locations. Devasirvatham, et. al. corroborates this result in an exhaustive indoor measurement campaign that shows delay spreads are less than 100 ns for indoor office buildings [57]. Woodward, et. al. present a set of measurements at 2.4 GHz that record angle-of-arrival and delay statistics for outdoor-to-indoor configurations. In an urban building, they show low values of RMS delay spread (mean 37 ns) but high values of RMS azimuth angle spread (89◦ which is 86% of the RMS azimuth spread for the uniform Clarke model) [58]. This compares favorably to our range of outdoor-to-indoor delay spreads 27-44 ns and our angle spread values which show angles-of-arrival spread out from 73-98% of the uniform Clarke model. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 164 7.2 Overview of Measurement Campaign This section provides a general overview of the measured locations, channel sounding hard- ware, and measurement technique used throughout the campaign. 7.2.1 Measured Locations The measurement campaign was performed during November of 1999 on the campus of Virginia Tech. A total of 12 locations were measured during this campaign using the peer- to-peer configuration of transmitter and receiver antennas (both set 1.5m height above ground). The first set of 6 measurements were outdoor-to-outdoor locations which emphasized long-distance, obstructed links with up to 1km of transmitter-receiver (TR) separation distance. The signal had to propagate over irregular campus terrain which included hills, multi-story buildings, and leafless trees. Then, 3 indoor-to-indoor locations were measured inside a four-story office building. The same transmitter – placed inside a fourth-floor office – was used for all 3 indoor-to-indoor measurements. One receiver location was measured on the same floor as the transmitter, on the opposite side of the building. Another receiver location was measured on the ground floor, on the same side of the building as the transmitter. A third location was measured on a different floor than the transmitter and on the opposite side of the building. Finally, 3 outdoor-to-indoor locations were measured using a transmitter placed 330m away from the four-story office building. One receiver location was measured on the ground floor, on the same side of the building as the transmitter. Another receiver location was measured on the fourth floor, also on the same side of the building as the transmitter. A third location on the back side of the building was measured as well. A graphical summary of the 6 indoor local area configurations may be found in Figure 7.1. 7.2.2 Channel Sounding Hardware A spread spectrum sliding correlator approach was used to sound the channel (see [59] for a description). A transmitted carrier frequency of 1920 MHz was used throughout the cam- paign. The transmitted spread spectrum signal had a radio frequency (RF) bandwidth of 200 MHz, allowing for theoretical resolution of multipath with as little as 10ns of propaga- tion delay. In practice, the resolution was slightly worse since we filtered the signal with a 140 MHz passband filter to remain within our licensed band of 1850-1990 MHz. Since this is an active PCS band in the U.S., the interference rejection capability of the sliding CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 165 TX R 12X R 11X R 10x Outdoor-to-Indoor Configuration 330 m TX R 7X R 8X R 9X Side View of Building Indoor-to-Indoor Configuration Figure 7.1: The transmitter-receiver configurations for the 6 local areas measured indoors. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 166 correlator was particularly useful. In order to send the channel sounding signal through the lossy peer-to-peer environment, the transmitter used a 20-watt wideband RF amplifier. Special precautions were taken to control the temperature of the RF hardware and to limit human RF exposure. Back-to-back system calibrations were performed on the transmitter and receiver units at the beginning and end of each measurement day to ensure system stability. 7.2.3 Automated Antenna Positioning These measurements used a precise automated positioning system to place the receiver antenna along a linear track and, for a directional antenna, to orient the antenna with respect to azimuth. The antenna platform is positioned using stepper motors that drive a rotating table and a long serpentine-drive track. The positioning error for placement along the track is ±10µm and for rotation about an axis is ±0.01◦ . A laptop computer simultaneously coordinated the movement and rotation of the re- ceiver antenna and the data acquisition from the channel sounder. This system was used to take two types of measurement sequences: track measurements and rotational measure- ments. Track Measurements: For the track measurement sequence, an omnidirectional antenna is mounted atop the positioning table. Two linear track measurements are performed, each using different track orientations. The first orientation aligns the track along an axis (referred to as the x-axis) parallel to the direction of the transmitter location. The second orientation aligns the track along an axis (referred to as the y-axis) transverse to the direction of the transmitter location. The orientations are depicted in Figure 7.2. For each linear track measurement, snapshots of the channel PDP are taken along the length of the measurement track (about 9 wavelengths at 1920 MHz). Each PDP snapshot is spaced 0.25 wavelengths apart, producing a total of 36 snapshots per linear track measurement. Thus, a measurement along two orthogonal tracks produces a total of 72 PDP snapshots. A photograph of a track measurement is shown in Figure 7.3. Rotational Measurements: The second measurement sequence at a location is a ro- tational measurement using a directional antenna. PDP snapshots are recorded from the channel sounder as the test antenna is rotated in steps across the entire horizon in evenly- spaced 10◦ increments. Thus, a single sweep in the rotational measurement results in a total of 36 PDP snapshots. The antenna platform is then moved along the track by 2.67 wavelengths (0.42 meters) and another series of 36 rotational PDP snapshots are recorded. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 167 x-axis y-axis Transmitter Figure 7.2: In a local area, power delay profiles are measured along two orthogonal linear tracks using an omnidirectional antenna. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 168 Figure 7.3: A track measurement is made with an omnidirectional antenna in a campus parking lot. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 169 This procedure is repeated until a fourth rotational measurement is made. Figure 7.4 illus- trates this sequence of measurements. In all, the rotational measurement sequence results in a total of 144 PDP snapshots. A photograph of a rotational measurement is shown in Figure 7.5. 7.2.4 Antenna Specifications All antennas used during this campaign had sufficient bandwidth over the 1850-1990 MHz frequencies to transmit and receive the wideband spread spectrum signal without distortion in space or time. An omnidirectional PCS base station antenna made by Andrew Corpora- tion was used at the transmitter. This antenna had a gain of 8 dBi along the horizon – the direction of peak gain for the elevation pattern. This same type of antenna was also used at the receiver for the track measurements. The rotational measurements used a directional horn antenna. The horn antenna had a gain of 15 dBi and a half-power beamwidth of 30◦ . All antennas used at the receivers are elevated to 1.5m height using poly-vinyl chloride masts to minimize scattering from the positioning track to the receiver antenna. 7.2.5 Sources of Error in the Experiment Great care was taken to minimize (but not necessarily remove) the primary sources of error in this experiment. These sources of error are Channel Transients: Objects moving within the channel during the acquisition time of a PDP may distort the channel measurements. This effect is most pronounced outdoors, where even a slight breeze will rustle the leaves of trees and introduce small-but-noticeable fluctuations to all of the measured multipath components. In November, however, all of the deciduous trees had lost their leaves so this effect was largely absent from the data set in this paper. Self-Scattering Effects: A large number of people and measurement equipment in the immediate area of the receiver may scatter multipath power to the receiver antenna that would be unrealistic for a real-life scenario. To remove this effect from each of the measurements, all personnel were evacuated from the immediate receiver area, the channel sounding hardware was kept as low to the ground as possible, and the receiver antenna was elevated above the positioning system so as to operate in an area of uncluttered free space. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 170 Figure 7.4: In a local area, power delay profiles are measured by spatially averaging angular sweeps with a directional antenna. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 171 Figure 7.5: A rotational measurement is made with a directional horn antenna in the lobby of an office building. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 172 Finite Sampling: Due to finite sampling of PDP snapshots in a local area, all dispersion statistics derived from these measurements will be approximate. However, no single delay statistic presented in this paper was calculated from fewer than 36 PDP snapshots and no single angle-of-arrival statistic from fewer than 144 snapshots. The large number of snapshots per statistic produces reliable results. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 173 7.3 Results This section presents the delay dispersion, angle dispersion, and joint angle and delay statistics calculated from the measurement campaign. For a mathematical description and explanation of each channel parameter discussed, refer to Appendix 7.A. 7.3.1 Delay Dispersion Results Table 7.1 records all of the dispersion results for the 12 measured locations. This table records delay spread, centroid jitter, centroid standard deviation, timing jitter, and timing standard deviation for PDP’s measured along orthogonal tracks. The results are calculated from the numerous series of PDP snapshots calculated during a track measurement. One such collection of snapshots is illustrated in Figure 7.6. PDP Snapshots Along a Linear Track Figure 7.6: A series of PDP snapshots along a track, measured with an omnidirectional receiver antenna. For outdoor-to-outdoor links, the delay spreads in Table 7.1 range as low a 17 ns to as high as 219 ns. Low values for delay spread are found at location 4, the only line-of-sight link in the outdoor-to-outdoor measurements, as well as locations 5 and 6. It should be C H A P T E R 7. S P A T IO -T E M P O R A L M E A S U R E M E N T S 174 T able 7.1: Sum m ary of dispersion statistics calculated from track m easurem ents. Parallel Track Transverse Track Delay Centroid Centroid Timing Timing Delay Centroid Centroid Timing Timing Location Spread Jitter Std. Dev. Jitter Std. Dev. Spread Jitter Std. Dev. Jitter Std. Dev. 1 164 ns 118 ns 34 ns 128 ns 37 ns 138 ns 95 ns 22 ns 146 ns 30 ns 2 196 89 22 90 20 148 101 23 101 26 3 185 115 23 53 13 219 84 24 71 18 4 55 17 4 24 6 22 15 4 20 5 5 51 38 11 51 12 40 25 7 35 9 6 25 8 2 16 4 17 6 2 10 3 7 30 33 8 13 4 29 26 6 18 4 8 45 29 5 24 6 45 29 7 20 5 9 44 40 11 25 6 42 31 8 25 6 10 28 10 2 25 6 31 47 12 39 11 11 35 37 10 46 10 44 46 12 44 12 12 43 39 9 40 8 27 31 7 31 7 Average 75 48 12 44 11 67 45 11 47 11 Std. Dev. 62 36 9 32 9 62 30 7 39 8 CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 175 noted that the values in Table 7.1 are comparable to delay spreads measured by Patwari in [56] (from 25 ns to 333 ns), despite the increased average link distances. In fact, the correlation between delay spread and TR separation is weak. For example, the longest obstructed link – location 5 with 910 m of TR separation – has one of the smallest delay spreads at an average of 46 ns. Delay spreads for indoor receivers demonstrate much more homogeneous behavior, inde- pendent of whether the transmitter is indoors or outdoors. The twelve indoor delay spreads (2 for each track for locations 7-12) fall within a 27-45 ns range. Thus, a much simpler equalizer may be used if the radio is guaranteed to be operating indoors. A final observation should be made about the variability of the delay spread about its mean value within a local area. The value Timing Jitter represents the difference between the smallest and largest delay spreads measured along a track. The value Timing Standard Deviation measures the mean-squared variation about the average delay spread in a local area. Both Timing Jitter and Timing Standard Deviation measure the variability of delay spread within a local area, but the Timing Standard Deviation may appear to be a better measure of typical behavior since it de-emphasizes the extreme cases of measured delay spread (see Appendix 7.A). However, as Table 7.1 shows, the Timing Jitter is almost always related to the Timing Standard Deviation by a factor of 4. This relationship is completely independent of delay spread or measurement environment. A similar relationship holds between the Centroid Jitter and the Centroid Standard Deviation. 7.3.2 Angle Dispersion Results A number of other multipath parameters may also be calculated from the measured track and rotational data. Table 7.2 records transmitter-receiver separation distance, path loss with respect to 1m free space, the angular spread, and the peak multipath direction of arrival. An example of a measured angle-delay profile is shown in Figure 7.7. One trend that is apparent from Table 7.2 is in the peak direction of multipath arrival. This parameter measures the direction in azimuth that the horn antenna was pointing when the maximum total power was received. According to Table 7.2, the peak direction of multipath arrival is almost always in the direction of the transmitter (corresponding to 0◦ ). This is true even for obstructed receiver locations. The one exception in Table 7.2 is location 9, where the peak direction of multipath arrival is 140◦ . However, location 9 is an indoor location corresponding to an indoor transmitter that is nearly directly above the receiver, with two floors in between. Given this unique location, the deviation from the trend is understandable. This property of the peak multipath arrival angle implies the C H A P T E R 7. S P A T IO -T E M P O R A L M E A S U R E M E N T S 176 T able 7.2: Sum m ary of spatial m ultipath param eters calculated from spatially-averaged azim uthal sw eeps of a horn antenna. Path Loss TR Sep Angular Angular Angle of Peak Mult. Location w.r.t. 1m FS Dist. Spread Constriction Max. Fading Arrival 1 90 dB 770 m 0.82 0.69 44◦ 0◦ 2 51 550 0.65 0.44 -63◦ 0◦ 3 39 240 0.91 0.51 36◦ 20◦ 4 81 585 0.52 0.77 89◦ 0◦ 5 72 910 0.46 0.70 76◦ -10◦ 6 83 410 0.36 0.76 85◦ 0◦ 7 70 29 0.73 0.52 84◦ -10◦ 8 82 33 0.74 0.50 30◦ 30◦ 9 76 15 0.90 0.33 -74◦ 140◦ 10 65 340 0.78 0.72 85◦ 0◦ 11 84 365 0.86 0.19 -46◦ 10◦ 12 85 340 0.98 0.42 -38◦ -50◦ Average 73 – 0.73 0.55 – 11◦ Std. Dev. 15 – 0.19 0.19 – 43◦ CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 177 PDP Snapshots for Azimuthal Sweep Figure 7.7: A local area angle-delay spectrum as measured from a set of rotational mea- surements. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 178 effectiveness of simple direction-finding algorithms. Another trend may be observed in the angular spread data. For indoor receivers at locations 7-12, the angular spread falls within the range 0.73-0.98. Thus, indoor angular spread values are almost always near the maximum value of 1.00. For outdoor receivers at locations 1-6, the angular spread falls within a lower range of 0.36-0.91. Thus, an omnidirectional fading model for narrowband fading such as the Clarke model in [13] is not accurate for long-distance peer-to-peer links. The increase in angular spread indoors as opposed to outdoors may be explained by the increased density of scatterers (doors, walls, shelves, etc.) in all directions in an indoor environment. Still another trend may be observed in the angular constriction data. From Table 7.2, we see that the average value for angular constriction, γ, is 0.55. This fairly large value indicates that multipath power is clustering about a few directions instead of being uniformly spread out. This is another indication that idealized, uniform multipath models may not characterize the spatial fading behavior for the peer-to-peer channel. 7.3.3 Joint Angle-Delay Statistics The graph in Figure 7.8 shows angular spread vs. delay spread for the 12 measured local areas. We would expect that higher delay spreads indicate more multipath components from a larger variety of scattering mechanisms; under these circumstances, the angular spread should increase as well. While counterexamples certainly exist, most of the measured points in Figure 7.8 follow this basic trend. Note that the rate at which angular spread increases as a function of delay spread depends heavily on whether the receiver is indoors or outdoors. To study this effect quan- titatively, we propose the following empirical guideline for angular spread, Λ, as a function of delay spread, στ : Λ ≈ exp ( −σc στ ) (7.1) The critical delay spread, σc, is the key parameter in Eqn (7.1) for determining the rate of angular spread increase. The critical delay spread may be calculated using linear regression on a set of measurement points, (− ln Λ, 1στ ). The six indoor points produce a critical delay spread value of 7.4 ns, while the six outdoor points produce a much larger σc of 32.5 ns. Plots of Eqn (7.1) for these two critical delay spreads are shown in Figure 7.8. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 179 0.0 50.0 0.20 0.40 0.60 0.80 1.00 100.0 150.0 200.0 250.0 300.0 Indoor Rx Outdoor Rx A n g u la r S p re ad ,  Delay Spread, (ns)  Angular Spread vs. Delay Spread Figure 7.8: The trend between angle spread and delay spread for indoor and outdoor receivers. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 180 7.4 Conclusions This chapter presented a novel technique and quantitative results for measuring 1920 MHz peer-to-peer spatio-temporal radio channels. Angle and delay dispersion data were recorded for outdoor-to-outdoor, outdoor-to-indoor, and indoor-to-indoor transmitter-receiver con- figurations. This data set illuminates many of the following previously unmeasured, hypo- thetical trends in the peer-to-peer radio channel: 1. RMS delay spread for outdoor-to-outdoor channels locations is an order of magnitude lower than PCS macrocellular radio channels, primarily due to the reduced transmitter antenna height. 2. RMS delay spread for outdoor-to-indoor and indoor-to-indoor channels drops to around 35ns – much lower than the outdoor-to-outdoor case. 3. Angular spread as measured at the receiver is higher for indoor receivers and lower for outdoor receivers. 4. A distinct, quantified trend emerged that shows how angular spread increases as delay spread increases. 5. Angular constriction is relatively high for all peer-to-peer channels, indicating that much of the multipath power is arriving from several discrete directions in azimuth instead of across a smooth, wide continuum of azimuthal angles. These quantified data trends will help engineers develop new spatio-temporal channel mod- els and design wireless modems for radios operating in the peer-to-peer network configura- tion. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 181 7.A Description of Measured Parameters This appendix defines the terminology and parameters used in the analysis of the wideband, space-varying channels. The appendix provides a quick, comprehensive summary of all measured parameters and terminology. 7.A.1 Noncoherent Channel Measurements The wideband radio channel is measured as a function of space in this campaign using a single-channel, noncoherent receiver. Below is a list of terminology used to describe the types of channels measured during this campaign. Channel Impulse Response (CIR): The CIR is the complex baseband representation of the radio channel, h˜(τ, r). The CIR varies as a function of time-delay, τ , and vector position of receiver antenna in space, r. If a directional antenna is used at the receiver, the CIR may also depend on the azimuthal orientation of the antenna, θ. Power Delay Profile (PDP): A noncoherent channel measures a PDP instead of a CIR. The PDP has units of power time−1 and is defined as the magnitude-squared of the CIR: p(τ, r) = |h˜(τ, r)|2 (7.2) Written without a θ-dependence, it may be assumed that Eqn (7.2) represents a measurement with an omnidirectional antenna. Power Angle Profile (PAP): The PAP is the spatial equivalent of a PDP. The PAP has units of power radian−1 and is defined as p(θ,r) = |h˜(θ,r)|2 (7.3) Written without a τ -dependence, Eqn (7.3) represents the angle-of-arrival character- istics of a narrowband channel. Angle-Delay Profile (ADP): When a directional antenna at position, r, and azimuthal orientation, θ, is connected to a wideband noncoherent channel sounder, an ADP is measured. The ADP is defined as p(τ, θ, r) = |h˜(τ, θ, r)|2 (7.4) CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 182 7.A.2 Power Spectra When power delay or angle profiles are linearly averaged in space, an estimate of a power spectrum is produced. Averaging various power profiles will produce the following power spectra: Delay Spectrum: Spatially averaging a collection of PDP’s measured within the same local area produces an estimate for the delay spectrum, Sh˜(τ), of the channel: Sh˜(τ) ∼= 1 N N∑ i=1 p(τ, ri) (7.5) where {ri} is the set of measurement positions. The delay spectrum characterizes the frequency selectivity of the stochastic, time-varying radio channel [60]. Angle Spectrum: Spatially averaging a collection of PAP’s measured within the same local area produces an estimate for the angle spectrum, Sh˜(θ), of the channel: Sh˜(θ) ∼= 1 N N∑ i=1 p(θ,ri) (7.6) The angle spectrum characterizes the spatial selectivity of the stochastic, space- varying radio channel [39, 41]. Angle-Delay Spectrum: Spatially averaging a set of full ADP’s within the same local area produces an estimate for the joint angle-delay spectrum, Sh˜(τ, θ): Sh˜(τ, θ) ∼= 1 N N∑ i=1 p(τ, θ, ri) (7.7) This joint power spectrum characterizes the full spatio-temporal randomness radio channel. Note that the use of power spectra in channel modeling assumes radio channels that are wide-sense stationary in space and frequency over basic intervals of interest. The frequency interval of interest is the RF bandwidth of the transmitted signal and the spatial interval of interest is the local area. Definitions for the coherence bandwidth and the coherence distance of a fading radio channel are based on delay and angle spectra, respectively. 7.A.3 Time Delay Parameters A number of delay dispersion parameters are measured in this campaign and reported in this paper. Each parameter is calculated for a measured local area and is defined below: CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 183 Centroid, τ¯ : The centroid is the first moment of a delay spectrum. The nth moment of a spectrum is defined as τn = +∞∫ −∞ τnSh˜(τ) dτ +∞∫ −∞ Sh˜(τ) dτ (7.8) Thus, the centroid is Eqn (7.8) evaluated for n = 1. RMS Delay Spread, στ : The RMS delay spread is the second centered moment of a delay spectrum, defined mathematically as στ = √ τ2 − (τ )2 (7.9) The RMS delay spread is related to the frequency selectivity of a wideband multipath channel [61]. Centroid Jitter: If the centroid were calculated from a single PDP instead of the delay spectrum using the definition in Eqn (7.8), then each local area would have a set of centroids as a function of space, {τ (ri)}. The centroid jitter is the maximum centroid value minus the minimum centroid value measured in the same local area. This value measures range of possible centroid fluctuations within a local area. Centroid Standard Deviation: This measure is defined similar to centroid jitter except it is the standard deviation of the set of centroids, {τ (ri)}, measured within the same local area. This measure is less sensitive to pathological instances of centroid measured at one or two points. Timing Jitter: This parameter, defined by Devasirvatham in [62], is based on the set of instantaneous RMS delay spreads, {στ (ri)}, calculated from PDP’s in a local area. The timing jitter is the maximum delay spread minus the minimum delay spread measured in the same local area. This value measures the range of possible RMS delay spread fluctuations within a local area. Timing Standard Deviation: This measure is defined similar to timing jitter except it is the standard deviation of the set of RMS delay spreads, {στ (ri)}, measured within the same local area. This measure is less sensitive to pathological instances of RMS delay spread measured at one or two points. CHAPTER 7. SPATIO-TEMPORAL MEASUREMENTS 184 7.A.4 Angle-of-Arrival Parameters There are also several angular dispersion parameters that are measured in this campaign and reported in this paper. Each parameter is also calculated for a measured local area and defined below: Peak Angle of Arrival, θpeak: This parameter, calculated from the estimate of angle spectrum Sh˜(θ), is the azimuthal angle in which the largest average multipath power is received. Angular Spread, Λ: This parameter ranges between 0 and 1 and describes how multi- path power concentrates about a single direction-of-arrival in space, with 0 denoting perfect concentration in one direction and 1 representing no clear directional bias in arriving multipath power. The angular spread is calculated from complex Fourier coefficients of the angle spectrum: Fn = 2π∫ 0 Sh˜(θ) exp(jnθ) dθ (7.10) and is mathematically defined as [40]: Λ = √ 1− |F1| 2 F 20 (7.11) This definition of angular spread is directly related to spatial selectivity in a narrow- band multipath channel: larger values of angular spread imply higher spatial selectiv- ity [63]. Angular Constriction, γ: This parameter also ranges between 0 and 1 and describes how multipath power concentrates about two directions-of-arrival in space, with 1 denoting perfect concentration in two directions and 0 representing no clear bias in two directions. The angular constriction is mathematically defined as [63]: γ = |F2F0 − F 21 | F 20 − |F1|2 (7.12) Angular constriction is also directly related to spatial selectivity in a narrowband multipath channel: larger values of angular constriction imply spatial selectivity in a local area that is anisotropic, depending on the orientation of movement in space. Angle of Maximum Fading, θmax: This parameter represents the azimuthal direc- tion in space that a receiver must move to experience the maximum possible spatial selectivity. It is defined to be θmax = 1 2 arg(F2F0 − F 21 ) (7.13) Chapter 8 Conclusions This dissertation has presented a broad overview of stochastic local area channel modeling techniques. This work may be divided into roughly two parts. The first part, Chapter 2- Chapter 3, outline the basic theory and physics required to develop local area channel models that vary as a function of time, frequency, and space. The second part, Chapter 4-Chapter 7, presents many applications of this theoretical and physical framework. Applications include development of small-scale fading distributions characterization of multipath using shape factors development of spatio-temporal second-order channel statistics presentation of results from a spatio-temporal measurement campaign Each of these applications represents a unique, original contribution to the field of channel modeling research. There is still a tremendous amount of research to be done in the field of stochastic local area channel modeling. The theoretical framework presented in this work will serve, hopefully, as a springboard for future ideas and applications of basic channel modeling theory. 185 CHAPTER 8. CONCLUSIONS 186 8.1 Future Areas of Research Let us now close with the 5 key areas of future research in the area of stochastic local area channel modeling. Any number of these areas are capable of generating valuable dissertation topics for researchers. 8.1.1 Theoretical Framework The basic theoretical framework in this dissertation deals with a unified description of wireless channels as a function of time, frequency, and position in space. There are two additional dependencies that must be added to complete the treatment: Transmitter Position: The spatial position dependence that has been discussed in this work is the position of the receiver antenna. The most sophisticated spatial treatment of the wireless channel, however, should also include the transmitter antenna as well. Thus, we could write a more complete spatio-temporal channel model as h˜(f, t, rR, rT ) where rR is receiver position and rT ) is transmitter position in 3D space. This framework would allow a more in-depth study of multiple-input, multiple-output (MIMO) space-time coding techniques. Antenna Orientation and Polarization: Another interesting addition to the the- oretical framework could be the introduction of directional antennas capable of arbitrary orientation in space. While characterization of channels with directional receiver antennas is possible under the existing framework, a key assumption is that the orientation of the antenna in space remains constant. If the antenna is allowed to rotate in azimuth or tilt in space, the gain and polarization interaction with the antenna and the impinging multipath waves will change dramatically. An excellent example this type of treatment may be found in [64]. 8.1.2 Specific Analytical Problems The solid theoretical framework provided in the beginning chapters should provide the impetus for higher-level analytical work in channel modeling theory. Three novel examples of analytical “discoveries” that directly result from the theoretical framework were already presented: 1) the TIP PDF, 2) shape factor theory, and 3) a more powerful analysis of the level-crossing problem and unit autocovariance for Rayleigh channels. Numerous others are also possible. CHAPTER 8. CONCLUSIONS 187 For example, additional analytical research could be performed on fading PDF’s, includ- ing improved representations of the TIP PDF. Deriving level-crossing statistics for Rician channels is another excellent example of much-needed analytical research. Developing unit autocovariance expressions for non-Rayleigh channels is yet another example of an unsolved problem in channel modeling, although Karasawa, et. al. have recently made some impres- sive progress in [65]. All of these analytical problems have more than just scholarly value, too. Their solutions directly impact the design and performance of many types of wireless technologies. 8.1.3 Applications to Wireless Technology There are numerous sophisticated technologies that are ushering in the new era of broad- band wireless communications that have channel-dependent performance. Exciting contri- butions are possible by relating basic channel statistics to the performance of these new technologies. For example, space-time coding techniques are related to spatial correlation properties which, in turn, are related to the shape factors of the multipath channel [3]. Thus, a researcher who couches the performance of space-time systems in terms of mul- tipath geometry stands to make a tremendous contribution to wireless communications. Likewise, new broadband transmission schemes such as orthogonal frequency division mul- tiplexing (OFDM) are directly affected by the second-order statistics of fading as a function of frequency [48]. A researcher who couches the performance of OFDM systems in terms of frequency level-crossing rates or correlation properties will have a powerful research legacy in wireless. 8.1.4 Measurement Theory Of course, countless more spatio-temporal measurement campaigns could be performed to augment our current understanding of wireless channels in a variety of environments and transmitter-receiver configurations. Moreover, the theory of channel measurement itself could be made more rigorous. The science of channel measurement too often is dominated by ad-hoc experimentalists who measure first and ask questions later. Invention of new techniques and rigorous statistical analysis of existing methodologies are needed. The spatial probing system and analysis in Chapter 7 is just the proverbial “tip of the iceberg”. 8.1.5 Computer Simulation Throughout this treatment, an important topic – computer simulation of spatio-temporal channels – has been intentionally omitted. This omission was not meant to diminish the CHAPTER 8. CONCLUSIONS 188 importance of simulation research. Rather, the science of computer modeling is a potentially rich and rewarding field that deserves a tremendous amount of detail and exposition for proper treatment. Many dissertations could be written about using the general theory developed in this work to reconstruct realistic spatio-temporal channels in the laboratory to assist wireless modem design efforts. Chapter 9 Vitae Gregory David Durgin was born in Baltimore, MD, on October 23, 1974. He received the B.S.E.E. and M.S.E.E. degrees from Virginia Tech in 1996 and 1998, respectively. He is currently working toward the Ph.D. degree at the Mobile and Portable Radio Research Group (MPRG) at the same university as a Bradley Fellow. Since 1996, he has been a Research Assistant at MPRG, where his research focuses on radio wave propagation, channel measurement, and applied electromagnetics. 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