**The Wireless Optics Laboratory**

**Group of **Hans Hallen and Alexandra Duel-Hallen, Physics Department and ECE Department, North Carolina State University

Our wireless research has been based on using the latest communications techniques while keeping in mind the physical origins of the effects seen in wireless communications that they address. For example, when we first began using long range prediction to enable adaptive signaling in the 1990's, most other researchers were treating the wireless channel magnitude as a random variable with position and argued that it would not be possible to predict. We saw it as the result of an interference pattern, which can be rather complex since microwaves reflect well from many surfaces, but is continuous and predictable, at least over some range if observed for a nearby range of space (or time assuming a moving receiver). Now, long range prediction (LRP) is an integral part of 4G, and will continue to be so in 5G. We had to develop a new type of physical model to test the LRP algorithms since the real world wireless signal statistics are non-stationary -- they change as one moves in space, and because we needed an accurate model for the small scale 'fast' fading for testing, while it is usually averaged out of measurements and models aimed at long range fading for antenna siting considerations. Our model gave us important insights into the local signal variations (what is easy to predict, what makes it hard), as well as providing test datasets vetted against measurements for use in validating various adaptive signaling methods. We later used the model and its insights in the study of ultra-wideband (UWB) pulse radio. We found that the two bands of frequencies allowed by the FCC in its 'mask' should both be used in real world scenarios, since their propagation characteristics would allow a lower data rate 'safe' low-frequency-band-pulses in addition to a potentially higher rate but less reliable high-frequency-band-pulses. We also proposed detectors that would work in a variety of conditions that might be encountered in the real world. Finally, we have a reconfigurable wireless measurement system to measure in the real world, based on fast wave generation and fast detection, with software radios to provide additional, simpler, nodes.

There are two ways to view the short-range (fast) fading of a wireless channel. One, not both, should be used at any one time. The first is from the viewpoint of the mobile receiver (Rx), in the time domain. In this reference frame, the Rx is not moving, but receives signals from several different directions (from reflections and perhaps a line of sight) that are Doppler shifted in frequency. These signals add in time to give the fading signal. The (near) reversibility of the wireless signals can describe the reverse link. We often use the words from this viewpoint when discussing the long range prediction results. There is another, quite different, viewpoint that gives the same answers, but in a very different way: it is in the frame of the ground, as if viewing the Rx, Tx, and reflectors from above. In this frame, the Tx and reflectors produce signals that combine to give a spatially-dependent interference pattern. The mobile moves through this pattern to receive a time-dependent fading signal. This is the viewpoint of the physical model and where its insights arise.

Long range prediction is useful because the parameters that make up the signal, the number of reflectors, Doppler shifts f_{n}, amplitudes A_{n}, initial phases φ_{n}, etc. vary relatively slowly in time -- in fractions of seconds. The channel, c, on the other hand, is the sum of sinusoids and can vary dramatically on short (millisecond or less) time scales. Thus, it is much easier to predict and track the signal parameters rather than the channel itself. We assume that the channel is a sum of N sinusoids c(t) = ∑ A_{n} exp(i(2πf_{n}t + φ_{n})) or in an AR model with p terms as ĉ_{n} = ∑ d_{j}c˜_{n-j} where c˜ is an estimate of c. The estimate is possible at high data rates (many signal symbols per prediction point interval) and can reduce the effects of noise. This can be implemented robustly with LMS or RLS. The model is iterated for multiple step ahead prediction. The d coefficients are updated at a low sampling rate based upon prediction errors, again using LMS or RLS.

Examples of the use and properties of the long range prediction algorithm are given in the sections below.

Properties of fading signals:

- The parameters A
_{n}, f_{n}, and φ_{n}vary much slower than the actual fading coefficients c(t); - The number of significant reflectors/scatterers is often modest;
- Physical insights identify typical and challenging case environments;
- Our physical model agrees (statistically) with the actual field measured data;
- The physical model can capture the channel parameter variations and provide diverse non-stationary realistic fading modeling to test our channel prediction method.

Figure 1. (a) Standard propagation models predict log-normal fading, so use long tracks for long range fading, and average away short range effects. (b) Our model aims for an accurate physics-based prediction on short tracks.

The placement and use of the effective source positon is done by use of an effective (image) source placed behind a reflector. The position depends upon the curvature of the object. Those used to the method of images in solutions to Laplace's equation will not the difference in placement for the curved object, since here we are not solving for potential but handling reflection, which has no exact solution for a sphere -- the position depends upon the incidence angle. Hence, we sometimes employ different distances, but the R/4 seems to be a reasonably good overall value to use for typical angles of reflection encountered in use.

Figure 2. (a) Flat objects: the image source lies along the line perpendicular to the object, equidistant as the source from the object. (b) Curved objects: the effective source lies R/2 (R/4) behind the front surface, as in paraxial optics (as a better approximation for a typical angle of reflection relative to the normal).

The position immediately gives us our first insight for LRP use. The change in time of the Doppler frequencies f_{n} can be seen in Fig. 3 to be the largest near the effective source. Effective sources are closest when the objects are curved, suggesting that rounded vehicles (bubble cars) parked at the side of the road would provide a challenging case for LRP. The Doppler shifts are given approximately (v << c) by f_{Doppler} = f_{carrier} (v/c) cos(α_{n}), and the rate of change of Doppler shift depends on time derivative of this, dv/dt and (sin α) * (dα/dt).

Figure 3. Relative (to max) Doppler shift for a Rx traveling downwards near an effective source 0.5 m to the right of the image, centered vertically. .

The other major component is the Fresnel diffraction. It is described here and comes in two stages: one to provide the light level at each reflector by diffraction of the Tx signal through its (user-defined) aperture into the region of interest, and the 2nd from the Tx and all reflectors to the object through the transmitter or object aperture to the point of interest, the latter repeated for each point along the path. In the equation in Fig. 4, E_{in} is the amplitude of the electric field at the reflector, ℜ is the reflection coefficient, λ is the wavelength of the transmitted signal and φ its initial phase, **r**_{n} are position vectors defined with the same origin as the actual position of the transmitter (**r**_{trans}), the effective position of the transmitter (**r**_{effscatt}), which is calculated by the method of images, the position of the nth obstacle (**r**_{scatt}) and the position of the receiver (**r**_{ptat}) as **r**_{n} = |**r**_{ptat} - **r**_{effscatt,n}|. The phase distance **r**_{n} in the exponent is unchanged for a flat reflector, when the effective Tx position is at the other end from the Tx of the line segment bisected by the plane of the reflector. For a curved reflector with radius R a distance R_{1} from the Tx, the optimal effective Tx distance R_{2} behind the sphere’s surface (along the line from transmitter to reflector) is dependent on the scattering angle. One can use the paraxial (scattered almost straight back) R_{2} = R/2, or a better overall value R_{2} = R/4. In this case, **r**_{n} should be **r**_{n,above} + |**r**_{transm}-**r**_{scatt,n}| - R_{2,n} to correct for the proximity of the effective Tx to the reflector (the actual wave path is longer). The ratio of vector lengths corrects for the 1/r dependence of the electric field amplitude as referenced from the effective Tx, although |**r**_{transm}-**r**_{scatt,n}| needs to be replaced by R_{2,n} in the case of the curved reflector. When a 2D simulation is performed, the {}'s with the other dimension should be replaced by the complex number (1-i). Our calculation of E_{in} is more complex, using two stages of diffraction: one from the transmitter through its aperture to the reflector, and another as outlined here, so that the transmitter can enter the region of interest that is shadowed from it by some object. The Fresnel integrals and their arguments are also given in Fig. 4.

Antenna effects can be incorporated by noting that the irradiance, given by the time average of the Poynting vector S is related to the power as S = cε_{0}E^{2}/2 for c the speed of light. The received signal power is only from the LOS path and E_{rx} = E_{0} λ/(4πr_{0}). In this case, the received power follows the free-scale path loss expression and is given by P_{rx} = S A_{effective} = S A_{0}G_{rx} = Sλ^{2}G_{rx}/(4π) for A the area of the antenna, A_{0} the effective area of a perfectly isotropic antenna, and G_{rx} the receiver antenna gain. A similar relation, S_{r} = G_{rx}P_{in}/(4πr^{2}), can be used to find the electric field at the reflector from the transmitted power for free space propagation.

Figure 4. (left) The diffraction equations. (right) The parameters used to calculate the amplitude from one reflecting object, shown as a dotted rectangle. Also shown is the aperture chosen.

Figure 5 shows the reflection from just one reflector, with contributions from the transmitter and other reflectors removed. It demonstrates the qualitative features of a diffracted wave. The reflector is only a few wavelengths across, but still exhibits a strong specular lobe. The size is evident in the diffraction lobes on either side, and the broadening of the central lobe with propagation distance.

Figure 5. The amplitude, A, after reflecting from a small object about three wavelengths in size. The centroid follows the specular reflection while spreading and decaying.

Figure 6 shows a simple test case in which a square region, rather than just a line, is calculated and displayed as an image. One large reflector simulates a building, while several medium-sized, curved reflectors simulate vehicles parked on a roadside in front of it. The building effective reflector is far behind the building face, while the curved vehicles have effective sources within them. The image shows the strong variations in signal (in dB) over rather short distances (in meters) close to the reflectors. One of the paths has been extracted from the image and is shown in profile.

Figure 6. The interference pattern and a line-cut from it, which is challenging for LRP. Parameters: f_{carrier} = 1 GHz, v = 60 miles/hr, max. Doppler shift = 90 Hz, sampling rate = 1000 Hz.

The measured dataset was provided by Ericsson Inc. and contains 100,000 samples. The carrier frequency f_{c} = 1877.5 MHz, sampling rate fs = 1562.5 Hz, vehicle speed: 0 ~ 50 km/h (but mostly 30km/h), and environment: low density urban Stockholm. We find that it is nonstationary, as the statistics vary over different portions of the path. The physical model can create scenarios for the different portions of the path that match the statistics for each, giving insights into the environments and showing the capabilities of the model. Performance of the LRP on the model also mimics the performance on the measured data, but both differ from that created by a stationary model.

It is not particularly useful to try to match a measured channel point by point. This would require placement of the correct number of reflectors at interferometrically (within a small fraction of a wavelength) accurate positions. Instead, forming scenarios that match the statistics -- several at once if possible, can show the utility of the model. Figures 7 and 8 show two very different environments within the dataset.

Figure 7. Measured data and a physical model in a curved reflector environment. The large subtended angle towards the efective sources within the reflectors causes a large Doppler shift range, which shows up in the statistics as a narrow main lobe with large sidelobes and a small coherence time.

Figure 8. Measured data and a physical model in a flat reflector environment. The small subtended angle towards the effective sources far behind the reflector surfaces causes a small Doppler shift range, which shows up in the statistics as a wider main lobe with small sidelobes and a long coherence time.

The comparisons between the measured and modeled methods continues with LRP results on the datasets. Figure 9 shows the prediction mean-square error as a function of prediction range. The measurements and physical model produce performance that follows a similar trend, quite different from the (stationary) Jakes model. Since they are non-stationary, they are harder to predict than Jakes model for short distances. At longer prediction ranges, the Jakes model becomes more difficult. This can be understood as follows: the Jakes model always has all reflectors contribute equally to the wireless channel. In reality, the signal level decreases with distance from the reflectors, especially if they are small or curved -- and these are the ones that make it more challenging. Thus, they become simpler at longer range, while the Jakes model artificially does not.

Figure 9. Prediction MSE vs. prediction range for measurements and models. Non-stationary models and measurements require a little higher sampling rate to track channel parameter variations. Model order p = 40, maximum Doppler shift f_{dm} = 46 Hz, sampling rate f_{s} = 1562.5 Hz.

The remarkable ability of the model to predict the channel with a modest number of slowly varying parameters is shown in Fig. 10. Here, the model order p = 30, observation interval = 100 samples with pre-training. The prediction range = 5.12 ms ahead, f_{s} = 1562.5 Hz, and f_{dm} = 46 Hz.

Figure 10. The caption.

We now demonstrate that the model insights are accurate in practice. Two paths from Fig. 6, the challenging case shown and a further away, typical, case are predicted with LRP. The prediction error is shown as a function of distance along the track. The prediction range p=38, observation interval = 80 samples, 1-step ahead prediction (2.92ms), and f_{dm} = 90 Hz.

Figure 11. Prediction error along a track is lower for a typical case than for a challenging case, validating projections from physical insights as applied to the LRP algorithm.

After understanding the error performance of LRP in relation to real and physically modeled scenarios, it is time to make it perform useful activities. We discuss its application to several adaptive signaling methods that it enables.

In this signaling method, adapted power control is used to provide constant power to the receiver, unless the signal level is too low, in which case no signal is sent (truncated). Two different prediction ranges are used for comparison. Here, actual measured data is used for the simulation, the model order p = 30, observation interval is 200 samples, and max. Doppler frequency shift f_{d} = 1562.5 Hz. Use of delayed rather than predicted channel state information significantly reduces performance.

Figure 12. LRP performance in driving truncated channel inversion.

Here, the LRP algorithm decides which of two antennas will have a better channel and switches the signal to that antenna. Again, the LRP is very effective, measured by the SNR required for a certain bit error rate, or the error rate at a given SNR.

Figure 13. LRP performance in driving selection transmitter diversity.

In adaptive modulation the transmission parameters are adapted to the current (so it needs to be predicted so that the transmitter and receiver can agree as to what will be done) channel conditions and to achieve the full potential of channel capacity. Figure 14 shows a quadrature amplitude modulation (QAM) example. The better the channel, the larger a constellation of sub-channels that can be supported, so the higher the data rate. The constraint is a limit on the bit error rate, usually taken to be 0.001 before error correction. Actual measured data is used for these studies, with target BER = 10^{-3}. Switching thresholds are calculated based on perfect CSI.

Figure 14. (left) A sample channel vs. time is broken into regions by signal strength. A larger constellation, also shown, will be used when the signal is larger, so more bits per symbol. (right) The system performance is dramatically improved by LRP.

Our proposed adaptive long range channel prediction algorithm can accurately predict the channel behavior far ahead. This opens up a new way to cope with multipath fading. The joint fading prediction and adaptive transmission could provide a basis for wireless communications to fully exploit the potential capacity of the mobile radio channel. Realistic physical model gives the insights into the signal fading and provides the diverse non-stationary testing datasets for our prediction method and adaptive transmission techniques.

Ultra-wideband radio is pulse radio, because the FCC rules require use of a broad frequency range at low power. The wide band means that it is likely that the channel will vary within the frequency band. Frequency-dependent distortion of individual multipath components is investigated for impulse radio UWB channels. In particular we focus on outdoor or sensor network UWB applications, where line-of-sight (LOS) or non-distorted reflected signals might not be available at the receiver. In such scenarios, the dominant propagation mechanisms are shadowing (diffraction) and/or reflection by small objects (e.g. street signs or lamp-posts). We use our physical model at many frequencies combined with Fourier methods as described below to investigate the position-dependent distortion of the UWB pulse, and it is demonstrated that shadowing and reflection by small objects (relative to the transmit signal wavelengths) cause per-path distortion. Design of correlation receiver per-path templates is investigated for distorted channels. We show that receivers that choose from a set of templates given by fractional derivatives of the transmitted pulse are near-optimal in terms of energy capture while a simple transmit pulse template provides excellent complexity-performance trade-off for most practical scenarios. Moreover, the effectiveness of the iterative correlation receiver is demonstrated in the presence of overlapping multipath components. Finally, a large gap between the propagation gains of the transmit pulses in the lower and upper bands of the FCC spectrum is characterized for several propagation mechanisms, and the implications for adaptive transmission methods are discussed.

The FCC defined rules for UWB signalling in 2002, Figure 15 (a). It allows low power/frequency, but a large frequency range (with the GPS frequencies protected). This lends itself to pulse radio. We have found simple functions that fill these masks well. The nth order Gaussian monocycle pulse is defined as GMC_{n}(t) = d^{n}/dt^{n}(exp(-2πt^{2}/t_{p}^{2})), where t_{p} is a parameter that controls the bandwidth of the pulse, and n corresponds to the shift in the ‘mode frequency’
of the spectrum, or the peak of the power spectral density (PSD), given by t_{p}^{-1}√(n/π). Figure 15 shows that these functions in 2nd order with t_{p} = 2.5 ns, and 8th order with t_{p} = 0.25 ns fit the lower and higher, respectively, FCC bands for UWB very well. We use them in our simulations. The GMC functions give a wavelength weighting at different frequencies. We multiply this by the calculated, wavelength-dependent, channel coefficients (that also vary in space, so it has to be done for each point along the path), then take an inverse Fourier transform of the obtained wavelength-dependent function to calculate the time-dependent UWB signal. This can be used to study capture against various template waveforms as described below.

Figure 15. PSD of several UWB pulses and the FCC spectral masks.

To see what happens, we need to create a scenario for the model calculations. The interesting effects are within shadows, where the strongly frequency-dependent diffraction is important, and in regions that are illuminated by reflection from small reflectors, where again strong wavelength effects are at play. The first case is shown in Fig. 16, with a path that proceeds from deep shadow to line-of-sight and a 'removable' reflector that is small enough to have interesting wavelength dependence.

Figure 16. A simple geometry for the UWB physical model.

The channel in line-of-sight gain from the Fig. 16 scenario with no reflector is shown in Fig. 17. Within the line-of-sight region, the gain is independent of frequency. This changes after the receiver enters the shadowed region, where a frequency, f, dependence of Cf^{α} is found. For deep shadow by a simple edge, α = -0.5, and that is observed here. The figures below do not show the wavelength dependence, but the path dependence. The form of this depends upon the same Fresnel integrals, which have argument d√(2f/cr), where c is the speed of light, d the Tx-Rx distance, and d the distance along the aperture of the line between the Tx and Rx where d=0 defines where the line hits an aperture edge. This clearly varies along a path depending upon the geometry, and we see the dependence becoming power law (straight line on log (dB) plot). The distance between the two curves can be understood by the 1/√f dependence and the peak frequencies of the pulses in the two bands.

Figure 17. Propagation gain comparison for pulses in the lower and upper bands of the spectral mask. Receiver moves along line A in Fig. 16, from deep shadow to LOS; Reflector is not present.

Once the small reflector in Fig. 16 is added, we obtain the channel gains in Fig 18. The longer wavelengths within the lower band pulse are not influenced much by the small reflector, so a small effect is observed. The high band pulse's wavelength is much more strongly matched to the reflector size, and a sharp specular reflection is observed along the path. This illustrates the different qualitative behavior in the different bands. The lower band diffracts more easily, but is not strongly influenced by small reflectors. It has specular reflections only from larger ones. The higher frequency band does not diffract strongly, but responds strongly to small reflectors, and sharp structures along the path can develop. This situation is even stronger in the mm-wave ands proposed for 5G. I suggests that a robust system would use both bands: the higher frequency and rate from the high band (when it is available), and the lower rate but more reliable lower band otherwise.

Figure 18. Propagation gain comparison for pulses in the lower and upper bands of the spectral mask. Receiver moves along line A in Fig. 16, from deep shadow to LOS; Reflector *is* present.

The model can simulate a large number of reflectors as well, and this is illustrated in Fig. 19. The higher frequency band responds to all the reflectors and obtains many overlapping in time paths. The lower band responds to them 'in aggregate' with a large channel response, but no sharp dependence. The multiple paths have implications on detecting the signals, as shown below.

Figure 19. (a) The scenario that produces many overlapping reflected pulses in time, each with a slightly different delay and path gain. (b) Comparison of propagation gain for the various UWB pulses as the receiver moves along line A in part (a).

The simplest expectation for detecting UWB pulses is that the received signal will look like what was transmitted, when viewed in time (excluding antenna effects). This will not be true when frequency-dependent distortion arises such as when the receiver is in a deep shadow. Thus, we use fractional derivative templates, found by multiplying the Fourier transform by (i2πf)^{α} before doing the inverse transform back to time. A value of α = 1, 0, -1 corresponds to derivative, nothing, and integral of the signal, respectively. We take templates made by fractional derivatives of the signal with α over the range 0 -> -1, with the number of values called the number templates. The received signal is compared to each of these in an iterative (over the number of paths or iterations) fashion: the largest is removed from the signal, then others are applied and removed, typically with different delays. We judge success by the fraction of signal pulse energy collected. Note that such a method would also be able to account for frequency dependences of the antennas. Figure 20 shows the result of using this method at a few different points in the simulation, and it suggests a small negative α is best for the template.

Figure 20. Energy capture vs. fractional integral templates of various order α for Gaussian monocycle, tp = 0.25 ns for diffracted paths along line A, Fig. 16.

For the two-path scenario of a ground bounce, we expect the signal capture to take only a few templates. The time dependence of the received compared to the transmit pulse template is shown in Fig. 21 (a), and the capture vs. the number of iterations in part (b).

Figure 21. (a) Example of a received signal with interpulse interference for two overlapping pulses, from the single reflector ground bounce plus line of sight, and the reconstructed template waveform using the iterative receiver. (b) Average energy capture vs. number of paths employed at the receive; 2nd order Gaussian pulse, tp = 2.5ns.

We can simulate the case with many reflections by instead using the scenario in Fig. 19. The results here in Fig. 22 do indicate that a larger number of iterations are needed since there are more paths. The situation is even worse for the higher band pulse, since it resolves the time dependence better.

Figure 22. Average energy capture as a function of the number of paths captured for the case with many reflections, Fig. 19 (a). The capture as a function of the number of iterations (paths, templates used) for a 2nd order Gaussian monocycle pulse (lower band), tp = 2.5 ns.

The Optics Laboratory has a reconfigurable outdoor channel measurement system. The system is based upon a high data rate Tektronix AWG7122B arbitrary waveform generator (24 Gsamples/s, 2 channel, 64M/channel) and a fast (12 GHz, 4 channel, 125M/channel), long memory length Tektronix DPO71254B oscilloscope, which are software controlled to acquire either narrowband data for algorithm testing or channel sounding to characterize the environment. A similar system with 1 GHz bandwidth is also available. A four node software radio system operational in the 2.4 and/or 5.8 GHz bands (Ettus Research) is configured with GPS, 'camera looking at a cone' surround imaging, and windo mounted capability.

Figure 23. The reconfigurable wireless measurement system, right, and the four software radio systems, left. The software radio USRPs are mounted on holders that clamp around a window and elevate the antennas above the roof of the car, while having the USRP boxes inside for connection to the MacBook computers.

More info is in the wireless lab papers.

Even more info is in the optics lab papers.

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