# IEEE 802.11 CTF

To address IEEE 802.11 standard applications, we applied the SDR-based measurement setup as presented in the section concerning CTFM Measurement Systems. The software is based on an OFDM modulation and provides 64-frequency bin wide raw CTF measurement data. Next, we present pre-processing approaches, our measurement environments, as well as the evaluation results.

## Contents

## Pre-Processing Approaches for Secrecy Extraction

The aim of the measurement series presented in this section is to predict the channel correlation behavior versus the spatial distance between both receivers B and C. To perform this calculation, we implemented and evaluated several pre-possessing techniques of the 64 binCTF. The following subsection provides an overview of different approaches. We introduce the notation that describes a notation. Figure 4.13 provides a graphical overview of each evaluation approach.

Please note that most results are presented using box plots showing the statistic of the channel correlation versus the distance between both receivers (cf. Section 2.5.9). In our evaluations, we typically choose to parametrize the box plot in a way that each box represents a statistical evaluation of a spatial point, e.g., one box for each millimetre or centimetre.

### Full-CTF / FULL-CIR - CTF64/CIR64

The first approach presented has the purpose of determining the correlation based on a single CTF or CIR. Since both are defined by a sixty-four value long vector, the correlation can be calculated based on a single message that is validly received by both receivers B and C. The calculation can be performed in both the frequency domain and time domain. Furthermore, since the input data is based on complex numbers, the input data is first transformed into either the magnitude or the angle. That also makes sense from a physics point of view since the correlation is then determined by the signal’s magnitude or the signal’s angle with respect to time.

The different approaches for evaluating the correlation can be described by the following equations

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{i}(H_{B}(f),H_{C}(f)) = corr(abs(H_{B,i}(f)), abs(H_{C,i}(f))}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{i}(H_{B}(f),H_{C}(f)) = corr(angle(H_{B,i}(f)),angle(H_{C,i}(f))}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{i}(h_{B}(t),h_{C}(t)) = corr(abs(h_{B,i}(t)), abs(h_{C,i}(t))}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_{i}(h_{B}(t),h_{C}(t)) = corr(angle(h_{B,i}(t)), angle(h_{C,i}(t)) }**

where abs is the operation to calculate the complex number’s magnitude, angle is the operation to calculate the complex number’s *angle*, corr is the operation to calculate the Pearson Correlation of to data sets, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle H_{B,i}(F)}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle H_{C,i}(F)}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle h_{B,i}(t)}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle h_{C,i}(t)}**
are the CTF or CIR of one of the receivers with respect to a specific message **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle i}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho_i}**
is the correlation for a single message.

### Only Non-Zero Sub-Carriers

In IEEE 802.11 only 52 frequency bins are useful for channel measurements. The other 12 subcarriers include pilot carriers which are consistently set to one or zero and useless for channel measurements. The pilot carriers are at position 1−6, 32 and 60−64. We drop those values and, therefore, extract large non-channel dependend signal effects. This approach can be described by the following equations:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho _{i}(H_{B}(f),H_{C}(f) = \text{corr}(\text{abs}(H_{B,i,(7-32,34-59)}(f)), \text{abs}(H_{C,i,(7-32,34-59)}(f)) }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \rho _{i}(H_{B}(f),H_{C}(f) = \text{corr}(\text{angle}(H_{B,i,(7-32,34-59)}(f)), \text{abs}(H_{C,i,(7-32,34-59)}(f))}**

### Max-Peak over a Time-Series

A different approach to the 52/64 CTF/CIR correlation is to determine the correlation over a series of messages. That also enables several variants, e.g., which values of a CTF or CIR are selected. The approach presented in this subsection is based on the work of Mathur et al. ^{[1]}. Here, the CIR of received WLAN-messages is used to extract the maximum peak. After receiving several messages and extracting the corresponding CIR, a series of subsequent peaks are merged to a vector that is used to calculate the correlation. The vector size of our implementation is 1000.

### Fixed Bin over a Time-Series

This approach is very similar to the previous described max-peak approach. Using a single time tap or a frequency bin, a series of time or frequency taps/bins at a particular position is merged to a vector. For instance, in one evaluation only the first tap of the CIR is used. This approach enables one to observe the correlation’s behaviour over single parts of the signal, e.g., on a particular frequency range.

### Sum-Bins

For this approach all taps or bins of a single measurement are summed up. The corresponding scalar is then concatenated with scalars of further measurements. The corresponding time vector is then used as the input for the correlation calculation.

## The Environments of Interest

In this section, we present an overview of the different environments in which we performed channel measurements. The inner yard and the anechoic chamber are also used as a template for reconstructing an environment for the simulation experiments using ray-tracing.

### Anechoic-Chamber

For evaluating effects of single scatterers and obstacles in a clearly defined environment, we chose to use anechoic chambers that are typically used to measure characteristics of antennas in different frequency ranges. The used chamber has a length of 5.5m, a width of 4.4m and a height of 5m. The whole absorber-chamber can be filled with foam-pyramids. In combination with the ferrite-absorber-plates, the chamber has an absorption factor of nearly −25 dB for signals in the frequency spectrum of 2.4 GHz. During the measurements, the transmitter was placed on a foam-pyramid based table at one side of the anechoic chamber. The receiver including the small antenna-positioning-system was placed on the other side. The whole floor was then filled up with foam-pyramids. In several measurements, we added self-build aluminum reflectors as obstacles. Figure 4.14 presents a schematic and panorama view of the anechoic chamber including the measurement setup and aluminum reflectors.

### Indoor - Floor

For indoor measurements, a floor was chosen that has few obstacles or complex structures, e.g., tube-systems under the ceiling. The floor has a length of 20m and a width of 2.6m. The floor and the walls are build of thick concrete walls. The surface of the floor is covered by a thin layer. Figure 4.15 (a) presents a picture of the line-of-sight measurement arrangement where we arranged the receiver setup at the one end of the floor at a distance of 15m. Figure 4.15 (b) presents the receiver setup located in the neighbor floor. This arrangement was used to simulate a non-line-of-sight scenario whereby both setups are about 6m apart from each other according to the beeline.

### Outdoor - Inner Yard

For outdoor measurements, we choose one of the inner yards of the Ruhr-University Bochum. Here, the inner yards can be compared to a colossal box without a cover. Hence, the area is predestinated for multipath propagation. The inner yard measures a length of approximately 50m and a width of 28m. The building has a height of about 20m whereby the surface partly comprises of concrete and alloy because of the air-conditioning system. Please note that the chosen area is only partially optimal for our experiments. This is because the corresponding spatial resolution for a bandwidth of B = 5MHz is about d = 60m which is longer than the inner yard. For our experiments, the measurement setups are placed at various positions in the inner yard.

Figure 4.16 (a) provides a view of the inside of the inner yard. Figure 4.16 (b) presents a schematic overview of the inner yard. The figure also includes markers in form of RX[1-8]/TX[1-8] that tag the different spots for the receiver and transmitter setups during a measurement. Here, RX1 always belongs to TX1, RX2 to TX2 and so on. For instance, RX1/TX1 defines the spots for the first measurement.

## Channel Evaluation and Results

We append the suffixes -complex, -abs or -phase to the results when either the complex representation or the magnitude or the phase is used for channel correlation. For instance, CTF52-abs signifies for the CTF52 method using the magnitude of the complex representation.

### Evaluation with Focus on Bandwidth

We compare several measurements that are conducted using different signal bandwidths (corresponding to the sampling frequencies) of the transmitter’s and receivers’ setups. The leading question is, how the evaluation of channel correlation behaves when wireless setups operate with different bandwidths. As a reference scenario for this evaluation, we chose several measurement series conducted in the inner yard having the transmitter and receiver setups aligned orthogonally at a fix distance of 49m. The arrangement refers to the TX1/RX1 placement according to Figure 4.16 (b). In each measurement, we adjusted the sampling frequency of both transmitter and receiver to values between 2MHz and 5 MHz.

Figure 4.17 presents the evaluation results for both bandwidths. Both measurements were conducted subsequently with a time difference of about an hour and hence, underlie the same outdoor situation. When comparing both evaluations, we observe that the correlation graph for a sampling frequency of 2MHz follows a more flat progression compared to 5 MHz. Here, the correlation varies between 0.60 and 0.87. Especially at an antenna displacement of 10 cm, it is hard to predict whether the communication partners are very close or more than half a wavelength apart from each other.

To prevent from drawing false conclusions, Figure 4.18 provides an evaluation of the very same measurement scenario using a different evaluation approach. Here, the channel correlation is calculated using the Sum-Bins-abs approach. Using a lower sampling frequency must not result in a flat progression but instead can provide another view. Here, the correlation progression for the lower frequency shows peaks around +/-19 cm and two smaller and broader peaks directly around the zero point whereby the zero point itself only has a correlation coefficient 0.35 and 0.45. This example shows us that before interpreting and comparing correlation progressions of different measurement, we may first have to agree on a particular evaluation approach.

### Evaluation with Focus on Different Pre-Processing Approaches

We start comparing the results of measurements based on the complex representation of the CTF, the magnitude as well as the angle. The leading question is, how the channel correlation behaves.

As a reference scenario for this evaluation, we chose the TX1/RX1 arrangement in the inner yard according to Figure 4.16 (b). The following observations are based on this measurement using the CTF52-complex, CTF52-abs, and CTF52-angle evaluation approach. In the following, Figure 4.19 (a)-(b) presents the CTF52-abs and CTF52-angle results for the referenced measurement.

Our results show that the phase provides zero correlation all over the 60 cm measurement. We assume that the consistent, low channel correlation results from the non-clock-synchronized daughter boards of the USRP X300. The results of the magnitude show an almost wavelength dependent decorrelation to zero (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle \lambda = 12.5 cm}**
). Then the correlation increases again up to a level of 0.5.

Figure 4.20 illustrates the results of Mathur et al.’s approach ^{[1]} of extracting the maximum peak of the CIR. The results show very low correlation. The approach seems to be not optimal
for key extraction.

Finally, we compare several measurements based on the Single-Bin evaluation method. We chose the TX1/RX1 arrangement in the inner yard according to Figure 4.16 (b). In the following, we focus on the Single-Bin-abs results as presented in Figure 4.21. Here, the Single-Bin-abs results for the frequency bins 7, 17, 32, 37, 47 and 57 are presented in increasing order. Based on that results, we like to point out that the channel correlation varies when analyzing the frequency bins at different positions over an OFDM symbol.

In a first step, we focus on the zero point where both the static and the dynamic antenna meet during a measurement. Here, the channel correlation for frequency bin 32 is very bulky and stretches over a distance of 7cm before falling below a value of 0.75. We note that our OFDM symbols consist of sixty-four bins and hence, frequency bin 32 is located in the mid of the frequency spectrum. Compared to frequency bin 32, all other frequency bins have a more spiky peak at the zero point whereby the actual channel correlation value fluctuates mostly consistent between 0.90 and 0.95.

Further larger differences can be identified when observing the channel correlation between both receivers at a distance of more than 25cm. Here, the channel correlation for frequency bin 7 is compared to frequency bin 57 very high. Here, the channel correlation for frequency bin 7 moves between 0.60 and 0.83 whereas the channel correlation for frequency bin 57 shows a decorrelation around 0.10 and 0.34.

Both observations present only an example for the varying channel correlation behaviour on
single frequency bins. Analyzing that information in detail, we may learn lots of the individual
channel characteristics and frequency selective fading effects. The channel correlation on
different frequency bins can also be influenced by the electronic layout of both transmitter and
receiver. For instance, an inaccurate filter may treat the mid of the desired frequency spectrum
differently than the edges of the spectrum, e.g., when using a bandpass filter.

That defines a further point for future work. For our current work approach, we can summarize
that when analyzing channel correlation characteristics, we have to take care of our input
and evaluation approach before interpreting the results. Even when only investigating a single
measurement, the channel correlation can vary when laying the focus on different aspects of the
input.

## Evaluation of Low-reflection Environments (Anechoic Chamber)

Now we compare several measurement series that we conducted within an anechoic chamber. The leading question is, how the channel correlation behaves in anechoic environments where both large and small scale fading effects have a comparative low influence.

First, we focus on measurements conducted in the anechoic chambers using the 2m antennapositioning system, self-developed aluminum reflectors and the LEGO®-Train. Both the transmitter and receiver setups were placed at a distance of about 4m apart from each other. During this measurement series, the large antenna-positioning-system was aligned both orthogonal and inline on the transmitter and using the following helper hardware. Table 4.4 presents a comprehensive overview over the different constellations. Figure 4.22 shows the measurement setup arranged in the anechoic chamber.

When first reviewing the CTF52-abs measurement results for measurement 1 and 5 having no reflectors arranged and no LEGO®-Train involved, we observe that over the full distance a very high channel correlation of > 0.95 is present using the CTF52-abs evaluation method (see Figure 4.23 (a), (c)). At several positions minimal variations of about +/ − 0.02 can be observed but no drops. Both measurements show that in anechoic environments with absent fading effects the channel correlation remains in most cases equal even over longer distances.

Following this, measurement 2 and 6 implement seven reflectors arranged at the corners and sides of the anechoic chambers. Compared to the aforementioned results, we can see in Figure 4.23 (b), (d) a more vivid channel correlation progression. Over the full distance, the channel correlation holds a value of > 0.90. However, at several positions this progression drops to values between 0.70 and 0.85. Here, the up and downs occur nearly in the scale of the radio signals wavelength. Though a consistent periodicity is not directly visible. Based on these measurements, we can see that adding obstacles to an anechoic environment introduces fading effects leading to channel distortion and hence, to a dissolution of the spacious channel correlation.

In a third approach, we let the LEGO®-Train drive around the transmitter’s antenna to introduce further movement to the anechoic environment. The results are presented in Figure 4.24 the very same way as the aforementioned results. Compared to these, the additional movement and reflection source do not change the channel correlation progression at all. Instead, the train causes more distributed correlation values. We can verify that at the number of red crosses in the box plot that presents the statistical outliers.

When analyzing the measurements of the anechoic chambers using other evaluation methods, we can observe another characteristic. Figure 4.25 shows the Sum-Bins results for the measurements 1) and 5) as described in Table 4.4. This evaluation method calculates the channel correlation using the summarized energy over the full CTF. At the zero point, we can observe that peaks exist that state a higher channel correlation. Especially, in Figure 4.25, we can identify the highest correlation peak at the zero point whereby over the full distance no second peak with the same height exists. Compared to the CTF52-abs method, the Sum-Bin method puts the received energy over subsequently received CTFs in focus. In highly anechoic environments this approach seems more suitable than our standard CTF52-abs method. We assume that this phenomenon is caused by the receiver that mainly receives a single line-of-sight signal. When adding more reflectors leading to multipath propagation, the effectiveness of this evaluation method decreases.

A further observation is shown by the red bar of outliers that we can observe in each plot around the value −0.2, especially in the results that are based on measurements conducted in the anechoic chamber. We assume that this statistical collection is caused by statistical characteristics of the Pearson Correlation. However, during this work no evidence could be provided that confirms this assumption. That states a point for future work.

## Evaluation with Focus on Different Positions

According to ^{[2]} and ^{[3]}, the auto-correlation over the distance of a received radio signal behaves like a Bessel function of the zeros grade. In the following, we provide an overview of different measurements and compare these with the so-called Jake’s Theorem.

The following evaluation presents the results of several outdoor measurements in the inner yard setup. Here, both the transmitter and receiver setups were located in the inner yard using different arrangements. Figure 4.26 includes the results for the following scenarios on the given enumeration. The following listing also uses the TX/RX notation for the transmitter’s and receiver’s arrangement as shown in Figure 4.16 (b).

- TX1/RX1 - Transmitter and receiver are located at a distance of 49m at the north and south end of the inner yard using an orthogonal alignment.
- TX5/RX5 - The transmitter is located in the center of the inner yard, the receiver is located in the south-west corner using an orthogonal alignment.
- TX6/RX6 - The receiver is located in the center of the inner yard, the transmitter is located in the south-west corner using an orthogonal alignment.
- TX8/RX8 - The transmitter is located at the center of the inner yard, the receiver is located at the west side of the inner yard using an inline alignment. All scenarios approximate the precondition of Jake’s Theorem. Here, it is assumed that the receiver is located in a uniform scattering environment continuously receiving radio signal with equal power over the full angle.

As our evaluation approach, we use the CTF52-abs method. Here, we extract the CTF for each message, drop all pilots and use the shrunk CTF to estimate the channel correlation.

Though we conducted all measurements presented in this chapter in the very same environment, all measurement results show a different progression of the correlation. From this, we can derive that the environment has a direct influence on the individual channel correlation between two receivers considering transmitter and receiver are placed arbitrarily in the area of evaluation.

Reviewing Figure 4.26 (a), we see the CTF52-abs results of a measurement that was taken over the full inner yards length. Based on the correlation progression’s shape, we observe similarities to a zeros grade Bessel function. At the zero point, we have the maximum correlation having a value of 0.90. With increasing antenna displacement, the correlation decreases until it reaches the minimum turning point and then increases again. According to Jake’s Theorem, a full decorrelation should be reached at a distance of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0.4 \cdot \lambda = 0.4 \cdot 12, 5cm = 5cm}**
. Based on this specific measurement, the correlation has a value of 0.58 and hence, can not be defined as fully decorrelated. The first time, the correlation reaches a value of 0.00 is at a distance of 9cm. This point is also very close to the observed minimum turning point around 10-11cm which is almost the wavelength of the radio signal itself. The next higher correlation point having a value of 0.48 point is reached at 25cm which is double the wavelength. Based on Figure 4.26 (a), we can perceive similarities to a zeros grade Bessel function though the theoretical Jake’s Theorem does not fully hold.

Figure 4.26 (b) provides a higher correlated progression with more peaks. First, we can observe a high correlation of 0.75 at the zero point (the point both antennas meet). Though even higher correlation points exist, the zero point distinguishes from other higher correlation points, e.g., at 11cm by a slight amount of statistical outliers. The nearest arrangement of both receivers can be determined even though higher correlation points appear at other, more apartdistances. When reviewing the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0.4 \cdot \lambda}**
point of decorrelation, we can observe that compared to Figure 4.26 (a) the correlation only has a value of about 0.31. Over the next two centimetres, the correlation decreases under a value of 0.00. With an increasing antenna displacement, the correlation in- and decreases periodically. Here, the period does not seem to follow a fixed length. Concerning this measurement, only the periodically in- and decreasing correlation equals Jake’s Theorem.

Concerning Figure 4.26 (a), the measurement results presented by Figure 4.26 (c) provide a further example that the channel correlation’s progression bases on the structure of a zero grade Bessel function.

The measurement presented in Figure 4.26 (d) provides a good example of the channel’s decorrelation after an antenna displacement of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0.4 \cdot \lambda}**
. Here, the initial correlation at the zero point having a value of 0.90 decreases rapidly to values below 0.2. Afterwards, first a smaller peak and later a higher peak of a recurring channel correlation can be observed.

Considering the presented results, good predictions whether two communication partners are close or apart from each other can be made. Here, the actual environment has to be put into consideration because different environments or alignments of transmitter and receiver units can result in various correlation progressions.

### Evaluation with Focus on Distances

Now, we compare several measurement campaigns based on different distances between transmitter and receiver. As a reference to evaluate the channel correlations behaviour with respect to different distances, we chose the inner yard to conduct several measurements. Here, the receiver was placed at a fix position in the south of the inner yard. The transmitter setup was first places in the north of the inner yard with a distance of about 49m to the receiver setup. In each iteration, we decreased the distance between both setups by about the half (49m, 25m, 12.5m, 4m). In each case, the receiver setup’s alignment was orthogonal to the transmitter. According to the TX/RX notation as described in Figure 4.16 (b) the positions during the here mentioned measurements are RX1-4/TX1-4.

When reviewing Figure 4.27 (a), we can first note that at a distance of 49m the channel correlation between both receivers is clearly visible. When both antennas meet at the zero point, the correlation reaches a value of about 0.9. After **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0.4 \cdot \lambda}**
, the correlation decreases to 0.5. At a distance of 10cm, the correlation reaches its minimum turning point between 0.08 and 0.21 depending on the graph’s side. Later the correlation graph reaches further peaks at a distance of about **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 2 \cdot \lambda = 25cm }**
that reaching a value between 0.50 and 0.58. The main peak at the zero point defines clearly the highest correlation with a margin to further peaks.

Nearly the same observation holds for Figure 4.27 (b) that presents the evaluation results for a distance of 25m. Here, the zero point also defines the main channel correlation peak reaching a value of 0.93. Furthermore, we observe a decorrelation to a range between 0.60 and 0.65
after a distance of a **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0.4 \cdot \lambda = 5cm}**
. However, the correlation at this point is slightly higher when compared to the results for a distance of 49m. At higher antenna displacements, the re-correlation peaks become taller. At distances of 17-20cm and 30cm, the correlation reaches values between 0.50 and 0.65. In contrast to the results for 49m, the re-correlation point at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 2 \cdot \lambda}**
decreases to values between 0.35 and 0.40. Hence, those points transform more to a minimum turning point than to a peak. Though the zero point furthermore defines the obvious correlation point, the re-correlation after an antenna displacement of a wavelength increases.

Reviewing Figure 4.27 c) that presents the evaluation results for a distance of 12.5m, we clearly observe a continuation of the aforementioned phenomena. Over the full measurement distance of 30cm, the channel correlation only decreases to a minimum of 0.50. Though the zero point still and apparently defines the point of a maximal channel correlation having a value of 0.97 other peaks after an antenna’s displacement of a wavelength reach high correlation values between 0.75 and 0.80. This observation intensifies when analyzing Figure 4.27 d) showing the results for a distance of only 4m. At first appearance, the channel correlation remains at high values. Here, the channel correlation reaches a value of 0.89 at the zero point. After a minimal decrease to a value of 0.85 after an antenna displacement of 2-3cm, the channel correlation increases to an even higher value compared to the zero point. At

Based on the presented evaluation results, we observe that in ”larger”, multipath affine areas a higher distance between transmitter and receivers leads to a more clear progression when estimating the channel correlation between two receivers. Decreasing the distance between transmitter and receiver leads to visual compression of the correlation graph on the distance between both receivers. Furthermore, we observed the phenomena at close distances between transmitter and receiver where the maximum correlation is located at a distance of

### Evaluation with Focus on Polarization

Now we analyze two measurement series that makes use of the circular antenna-positioning system polarization setup. To conduct the polarization measurement, we chose one outdoor and one indoor scenario to conduct the measurement series. First, we performed a measurement using the 49m inner yard’s reference scenario. As an indoor scenario, we chose a small floor that connects several offices. Both setups are placed on the ground itself at the side of the floor. Both measurements were conducted in default configuration using a sampling frequency of 5MHz. Furthermore, we implemented circular antenna-positioning-system to simulate a **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 180^\circ}**
movement.

In Figure 4.28, two evaluation approaches are presented in which the channel correlation is visualized using both the CTF52-abs approaches. In contrast to other plots, the x-axis does not scale over the distance but instead scales over a range of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 180^\circ}**
. This range is defined by the dynamic antenna that starts with a shift of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle -90^\circ}**
concerning the fixed antenna. During the measurement, the dynamic antenna moves over the zero point that is defined by both antennas aligned parallel at **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 0^\circ}**
to the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle +90^\circ}**
point. For the outdoor measurement (cf. Figure 4.28 (a)), we observe that the channel correlation around **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle +/- 90^\circ}**
reaches highest values around 0.98 whereby the channel correlation at the zero point varies ”only” between 0.85 and 0.95.

First, we can confirm that a channel correlation between 0.85 and 0.95 at the zero point when both antennas are parallel matches other results of outdoor measurements presented in this chapter. Additionally, we can note that higher correlations can be achieved when the antenna alignment is mostly orthogonal. Here, we assume that this behaviour is valid for multipath affine environments. Due to the orthogonal alignment of both antennas, the antenna’s re-radiation effect has a decreased influence on the second antenna compared to a parallel alignment. Hence, when a high correlation is preferable, the receivers’ antennas might be aligned orthogonal to each other.

Following this, Figure 4.28 (b) presents the results of the indoor measurement. Here, we can observe a consistent correlation coefficient between 0.80 and 0.92 over the complete **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/wikimedia.org/v1/":): {\displaystyle 180^\circ}**
movement. We assume the last graph can be caused due to a higher influence of the line-of-sight component compared to multipath scatters.

### Evaluation of Indoor LOS/NLOS Measurements

Next, we compare two measurements conducted in an indoor environment. We focus on evaluating indoor measurements within a LoS and non-LoS scenario. When first reviewing the CTF52-abs result of the indoor line-of-sight measurement (see Figure 4.29 (a)), we can observe a fluctuating progression of the channel correlation between 0.0 and 0.9. Here, peaks and also plateaus reoccur in arbitrary intervals. Furthermore, we can not resolve the zero point based on the correlation progression because around the 100cm mark no exposed peak is visible that would refer to close antennas. Here, we note that this can be caused by an inaccurate linear guide of the long antenna-positioning-system itself. Compared to the line-of-sight scenario, the non-line-of-sight scenario presented in Figure 4.29 (b) behaves even more fluctuating. Over the full distance, the channel correlation wobbles statistically between 0.9 and -0.6. Another time, we can not determine the zero point for sure because peaks around 0.8-0.9 occur every few centimetres nearly in periods having a multiple of the radio signals wavelength. Based on the presented and further indoor measurements, we can summarize that using the large antenna-positioning system no results could be achieved where a correlation at the zero point is obvious. However, based on the provided results and on the measurement setup, we can conclude that in indoor environments a periodically occurring high channel correlation has to be expected. Even when the zero point is exposed by a higher correlation value, the quantization and decision algorithms to decide whether two communication partners are close to each other have to be accurate.

## References

- ↑
^{1.0}^{1.1}Suhas Mathur, Wade Trappe, Narayan B. Mandayam, Chunxuan Ye, and Alex Reznik. Radio-telepathy: extracting a secret key from an unauthenticated wireless channel. In J. J. Garcia-Luna-Aceves, Raghupathy Sivakumar, and Peter Steenkiste, editors, Proceedings of the 14th Annual International Conference on Mobile Computing and Networking, MOBICOM 2008, San Francisco, California, USA, September 14-19, 2008, pages 128–139. ACM, 2008. formerly known as: mathur2008radio. - ↑ William C. Jakes. Microwave Mobile Communications. John Wiley and Sons Inc., 1994.
- ↑ Andrea Goldsmith. Wireless Communications. Cambridge university press, 2005.