1 Introduction

Services based on precise location, electronic healthcare, and home care apps are just a few examples of the new applications made possible by the proliferation of the Internet of Things (IoT) [1]. The dynamic behavior of the propagation channel is a source of frustration for radio-frequency engineers [2]. Because wireless signals are prone to noise, interference, and other impediments, the main target is improving signal predictability at any location. The received signal strength (RSS) is one of the most common parameters used nowadays for localization [3]. It is preferred due to its inexpensive cost, does not require timing synchronization, and robustness to non-line-of-sight (NLOS) propagation scenarios [2]. Signal propagation within indoor environments has garnered significant interest in the research community. Researchers have been actively developing models to elucidate signal propagation behavior within these environments [4]. Some models are empirical, where they use the best fit for measurements. Numerous empirical models are proposed in [5,6,7,8]; however, these models should be used in similar environments. Others are stochastic, where the channel is random at any time, yet stochastic models describe an indoor channel using its statistical characteristics [9,10,11,12,13]. Furthermore, other models are deterministic [14, 15], which can estimate channel parameters by solving Maxwell’s equations. Generally, the solvers are either ray-tracing method [16] or finite difference time domain FDTD-based methods [17]. There are also semi-deterministic models, which are hybrids of deterministic and empirical or stochastic models. These models improve the deterministic model’s performance [18, 19].

RSS level is widely used in localization; the current mainstream Wi-Fi positioning method is the RSS-based fingerprint localization approach [20]. When it comes to localization performance, there is a lot of bandwidth available; therefore, time of arrival (TOA) is better at short-range estimate than RSS is at long-range; nonetheless, a hybrid system that takes use of both approaches is suggested in [21]. RSS level is affected due to human presence and movement, especially if the person is found close to the transmitter or the receiver; for the case of movement, a similar observation was recorded when a person is passing through the LOS path [22].

For personal communication services, RSS is usually demonstrated as the sum of three mechanisms: path loss attenuation representing signal changes at a macro-level. Over distances ranging from 10λ to 30λ, where λ is the signal wavelength, the signal experiences log-normal fluctuations as a result of shadowing. Due to the incoming rays' rapid phase shift, it also undergoes small-scale fading caused by multipath, resulting in changes across ranges less than a wavelength. Therefore, the RSS level follows rapid change even over a small distance, making using RSS in localization more challenging [23]. A consistent RSS reading is achieved by taking many readings inside the target region and then averaging them to mitigate the fast-fading impact [24]. Local average power is a metric that shows the average signal strength over a small distance or time interval. It helps to describe the fading statistics and channel quality in a multipath environment [25].

This paper aims to investigate the effect of averaging on signal strength smoothing using multi-frequency, multi-antenna heights, and a combination of them (i.e., that's what we call a hybrid). The paper starts with the physical model, plane-earth loss model (PEL), and then results are contrasted by applying ray tracing methods.

Here is the layout of the paper: The relevant prior research is reviewed in Sect. 2. Section 3 investigates the effect of RSS averaging on the PEL model using different frequencies and antenna heights. Section 4 examines the effect of RSS averaging on the ray-tracer results using different frequencies and antenna heights. Finally, conclusions are drawn in Sect. 5.

2 Related previous work

The power sum (PS) technique is one approach to mitigate the fast change caused by multipath that relies solely on the power level of arriving signals and disregards phase fluctuations [26]:

$$\langle {P}_{PS}\rangle =\sum_{M}{P}_{M}$$
(1)

where 〈PPS〉 is the mean power, M is the number of detected signals, and PM is the power of each detected signal. Nevertheless, this method requires multiple antenna elements at the receiver to detect all arrival angles and a massive bandwidth to detect all arrival times. Ray tracing software can track the paths of rays and, hence, measure the characteristics of the channel using the PS method, which can be helpful as a reference. The other approach considers the effect of phase on the average signal power as shown below; this method is called the vector sum (VS) [26]:

$$\langle {P}_{VS}\rangle ={\left|\sum_{M}\sqrt{{P}_{M}}{e}^{-j{\phi }_{M}}\right|}^{2}$$
(2)

where 〈PVS〉 is the mean power and \({\phi }_{M}\) is the phase (rad) of the Mth ray. To consider the effect of the shadowing effect only (no phase effect on signal strength), researchers tend to calculate the local mean RSS at a particular point by getting the average of many RSS measurements around that point [27]. In contrast, some researchers consider taking the mean value, while others take the median value to suppress the odd results from affecting the result [27]. Usually, there are three parameters to consider: the window length (the distance/area considered for averaging), the number of samples, and the distance between samples [28]. Lee criteria [29] was one of the early attempts to estimate local RSS value; the averaging window was set to be 20λ-40λ over the ultra-high frequency (UHF) band; though, this distance was found to be too large for indoor measurements [26, 30].

In [31], the mean power was estimated over a range of 5λ-15λ. At the same time, in [32], averaging was performed over a 10λ window with 0.25λ spacing between samples, and 2-dimensional (2D) sample distribution for averaging was conducted over (3λ)2 region [33]. Similarly, local mean RSS was calculated over (3λ)3 region [34]. The optimal size of configuration and spacing between samples was examined by [35], which showed that the best result was obtained using a 2D configuration with a 1.25λ spacing. Authors in [36] modified Lee's criteria; they found that the averaging window for indoor propagation measurements should be within 5λ-10λ at UHF frequencies and that the decorrelation distance is around 2–4 m. Recent published works estimate mean RSS based on principal component analysis (PCA) [37]. In [38], researchers found that averaging window for factory environments is performed on squares of dimensions 86.7λ × 86.7λ. Authors in [39] examines how the local average power for LOS and NLOS paths changes with the averaging length in an indoor corridor. The study considers 3, 6, 10 and 17 GHz frequencies, the averaging interval was 2λ, 4λ, 6λ, 8λ and 10λ. As the averaging window increased, the number of measurements increases, therefore, the accuracy of the measurement results enhanced.

In [40], the authors modified the previous work by Lee [29] and developed new expressions for the variance of both types of window averaging. It also shows how the correlation between samples varies with the sample spacing and the window width. Generally, the stability of the averaged signal is predicted to vary as the window averaging increases.

3 RSS averaging: the PEL model case

The fact that the RSS level varies wildly with distance, leading to uncertainty in position prediction, is a major challenge in RSS localization. This is reflected in the reliability of the estimation based on the RSS-distance relationship, therefore, with the help of averaging techniques, we may bring this connection as near to a one-to-one relationship as possible, so that it decreases monotonically.

The out-of-phase multipath signals cause the RSS level to fluctuate rapidly. Considering Fig. 1, in the absence of reflection, refraction, scattering, and diffraction, the RSS for a single ray propagating in space (the free-space path loss model, or FSPL) will decrease linearly. On the other hand, in the presence of two rays (the PEL model), the RSS level will change as a result of constructive and destructive interference between the two rays.

Fig. 1
figure 1

PEL & FSPL model

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3.1 RSS averaging over frequency

Using the PEL, the phase difference (\(\Delta \phi \)) between the two rays is a function of the access point (AP) and the mobile antenna heights (hb and hm), wavelength (\(\lambda \)), and distance between the two antennas (d). Hence, one can expect different variations if one or more of these parameters changed.

$$\Delta \phi =\frac{{4\pi h}_{b}{h}_{m}}{\lambda d}$$
(3)

It is anticipated that varying distances would result in distinct variances when the operating frequency is changed. Averaging RSS values of different frequencies will suppress the deep fading, as shown in Fig. 2 as the PEL model was adopted (where (\(f\)) is the operating frequency). Given that (\({h}_{b}=2m, {h}_{m}=1m\), f = 175:5:220 MHz), the average variations taken for these frequencies are represented by the red line.

Fig. 2
figure 2

Averaged RSS (red) for RSS (blue) over frequencies range (175 MHz: 5 MHz: 220 MHz) (Color figure online)

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The frequency-averaged PEL demonstrates superior performance when compared to other frequencies. While the number of nulls remains unchanged, the severity of these nulls is reduced, as an illustration, there would be a minimum of twenty-one matching places for each frequency if SS was (-40 dBW); nevertheless, when their numbers were averaged, their number was decreased to one.

In Fig. 3, averaging over different sets of frequencies is presented. It is clear from the graph that the averaged PEL tends to be smoother as frequency separation increases since the nulls of these frequencies are contiguous rather than intersecting.

Fig. 3
figure 3

Comparison between different sets of averaged frequencies RSS

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Wireless local-area network (WLAN) is extensively used in indoor localization systems, that offers low-cost technology with satisfactory accuracy at the free license spectrum in the ranges (2.4–2.5 GHz, 5.15–5.35 GHz, 5.470–5.725 GHz, and 5.735–5.825 GHz). Wi-Fi 6E, which operates at 5.925–7.125 GHz, offers 59 channels, each with a bandwidth of 20 MHz, that boosts capacity and throughput.

The PEL model was adopted and averaged over five WLAN frequencies; it was noticed that averaging over high frequencies has lower efficiency compared to lower frequencies due to the many signal level variations, as illustrated in Fig. 4 that assumes \({h}_{b}=2m, {h}_{m}=1m,\) f = 2.4 GHz: 20 MHz: 2.48 GHz. If RSS was − 70 dBW, using averaged frequencies, the number of possible distances reduces from 33 to 7; this upgrade will make localization better, especially when many APs are used.

Fig. 4
figure 4

Averaged RSS (red) for RSS (blue) over frequencies range (2.4–2.48 GHz) (Color figure online)

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It was observed that using multiple frequencies does not improve the averaging process. This is because all frequencies have the same nulls as the lowest frequencies, as depicted in Fig. 5.

Fig. 5
figure 5

The averaged RSS of doubled frequencies

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3.2 RSS averaging over heights

In this section, we have changed the antenna heights to minimize phase variation. We adopted the maximum ratio combining (MRC) as it shows the best combining technique performance.

In the previous section, it was found that averaging over doubled frequencies will not improve RSS smoothing. Similarly, the impact of doubling the antenna heights was examined, and it was found that the averaged PEL model exhibits an equal number of nulls as the shortest antenna, as shown in Fig. 6, where (f = 2.4 GHz). Once again, this indicates that the double-height antenna will not improve smoothness, hence it is not suggested to use it.

Fig. 6
figure 6

The averaged RSS of doubled antenna heights

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In Fig. 7, APs are operating at (f = 2.4 GHz), where their heights are at (2.1, 2.3, 2.5, 2.7, 2.9 m), and the results of averaging show a better performance. As seen, at (− 60 dBW), there are around 25 possible distances away from the AP given that the AP height is 2.9 m; however, using averaging heights, this value is reduced to two only.

Fig. 7
figure 7

Averaged of multiple antenna heights localization

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3.3 Hybrid RSS-averaging using frequencies and heights

In earlier sections, we saw that averaging over multiple frequencies and heights reduced the fluctuations; here, we apply a hybrid combination to further minimize the variations. It was observed that using the higher frequency with the higher antenna gives better results, as shown in Fig. 8. Three antennae are used, and their corresponding heights are: (1, 1.3, 1.6 m). In “group 1,” the lower frequency was paired with, the lower antenna height and the upper frequency was assigned to the higher antenna, i.e. (hb = 1 m \(-\) f = 400MHz, hb = 1.3 m \(-\) f = 900 MHz, and hb = 1.6m \(-\) f = 2.4 GHz). While in “group 2,” the higher frequency was assigned to the lower antenna (hb = 1 m \(-\) f = 2.4 GHz, hb = 1.3m \(-\) f = 900 MHz, and hb = 1.6 m \(-\) f = 400MHz). Group 1 performs better than Group 2 due to the fewer ripples. Consequently, the configuration of group 1 is adopted.

Fig. 8
figure 8

Comparison between different antenna-frequency allocations

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Figure 9 displays a comparison of the averaging strategies that were studied. We used three antennas to perform averaging. We refer to averaging over different heights as “height averaging”. The operating frequency is 2.4 GHz, while antenna heights are 3, 2, 1 m. We refer to averaging over frequencies as “frequency averaging”, where the antenna height is 2 m while operating frequencies are 2.4, 2.44, and 2.48 GHz. In the final case, we used a hybrid averaging where the same set of antenna heights and frequencies are used as mentioned above; however, the lower antenna height is associated with the lower frequency, whereas the higher antenna height is associated with the highest frequency, that we indicate this case as “Hybrid FH”.

Fig. 9
figure 9

Comparison between averaging methods using widely spaced antenna over (2.4–2.48 GHz)

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The “height averaging” method has inferior performance compared to other methods. The Fig. 9 shows that it is affected by deep nulls, while the “frequency averaging” method does not have deep nulls but exhibits more ripples. On the other hand, the Hybrid FH method combines both approaches to achieve optimal performance. It has the same number of nulls as the “height averaging” method, but these nulls are not as deep as those in the “frequency averaging” method.

4 RSS averaging: ray-tracing simulations case

4.1 Ray-tracing methodology

A receiver with a dual WLAN band antenna can observe two channels from the same AP for example, one operates at the 2.4 GHz band, and the other operates at the 5 GHz band. In an indoor environment, the typical number of multipath rays will be higher than two, as shown in Fig. 10. Therefore, it is expected to have a rapid variation in RSS-distance variations.

Fig. 10
figure 10

Simulated multipath rays in a room using Wireless InSite

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The averaging principle can be applied here, although the number of channels is not adequate; though, the results can be used to support the concept of averaging. Therefore, we modeled an indoor scenario using Wireless InSite (WI). The AP operates on dual bands in the model, and the receiver collects the RSS from different locations; WI was validated over WLAN and millimeter-wave frequencies [41].

The fields are merged when the multipath rays reach the receiver. With WI settings, the user may save each ray's electric field phase while using the PS approach (described in Sect. 2) to forecast the received power and path loss. The power sum method simulated results are considered the reference as they represent the ideal scenario. In our analysis, we estimated the RSS at both bands using the PS and the VS methods. In this case, the PS method simulations serve as benchmarks, while the VS method result stands in for the outcomes of our measurements.

4.2 Ray-tracing results

We contrasted the vs and reference PS outcomes at each frequency range. The next step was to average over the two frequencies and then compare the respective reference findings. Then, we compared the outcomes when averaging is applied and when it is not. In all scenarios, the receiver height was set to 1m, representing the typical height of a mobile above ground level while being carried by a human. WI settings are shown in Table 1. The adopted values are based on the software manual recommendations and study conducted within indoor environments by [41].

Table 1 Wireless InSite Simulations Settings
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The market offers dual-band WLAN routers; therefore, two frequencies can be detected from the same AP. Similar to Sect. 3.1, the RSS can be averaged over frequencies. Table 2 shows the root-mean-square error (RMSE) between PS and VS results at 2.4 GHz and 5 GHz bands, and the averaged RSS of the two bands. The AP height was set to 2 m, while the receiver height was set to 1 m. We chose different frequencies from the two bands, and the averaged RMSE tends to have better results for all cases. It is worth mentioning that the averaging process was performed over the linear values of the results rather than the dB values.

Table 2 RMSE (in dB) comparison using 2.4 GHz, 5 GHz bands and averaged RSS
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Figure 11 presents the effect of averaging over two frequency bands to estimate the RSS-distance relationship compared to using a single frequency. The averaging performance is enhanced and closer to the ideal case. This is provided as the RMSE between the RSS collected at 2.412 GHz, and 5.805 GHz are 5.4838 dB and 4.8478 dB, respectively; however, while averaging is performed, the RMSE is reduced to 3.2165 dB.

Fig. 11
figure 11

The effect of frequency averaging over 2.4 and 5 GHz bands compared to single-frequency at 2.4 GHz band

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For optimal performance in real-time applications, follow the steps outlined above. Although the following discussion is not applicable to present-day WLAN routers, it might be useful for future implementations. The importance of these results is that they show how averaging does not apply only to the PEL model but also to a rich multipath propagation environment.

The averaging over different heights (i.e., height averaging) is applied for 2.4 GHz, 5.2 GHz, and 5.8 GHz in Tables 3, 4, and 5, respectively. Each table considers three heights and a single frequency. The APs heights are 1.8, 2, and 2.2m. All tables show better results as we use averaging over heights. For example, in Table 3, the RMSE is enhanced to be 4.427 dB.

Table 3 RMSE comparison for height averaging method at 2.4 GHz
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Table 4 RMSE comparison for height averaging method at 5.2 GHz band
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Table 5 RMSE comparison for height averaging method at 5.8 GHz band
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Figure 12 presents the performance of the AP while applying averaging over 1.8, 2, and 2.2 m heights compared to its performance at a single height of 2.2 m. The operating frequency is 5.8 GHz. The RSS smoothing is enhanced as the RMSE between the investigated methods and the power sum method was reduced from 7.4956 dB to 4.7193 dB.

Fig. 12
figure 12

The effect of height averaging compared to single height at 2.2 m at 5.8 GHz

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The averaging over different frequencies (i.e., frequency averaging) is applied for 2.4 GHz and 5.8 GHz frequency bands in Tables 6 and 7, respectively. Each table considers different frequency channel. As seen, the results show better performance using averaging over frequencies approach. The more frequencies are used, the better performance, we observe, as seen in Table 7, where the RMSE enhanced to be 3.2866 dB. One can indicate that frequency averaging has better performance than height averaging.

Table 6 RMSE comparison for frequency averaging method at 2.4 GHz band
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Table 7 RMSE comparison for frequency averaging method at 5 GHz band
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Figure 13 presents the effect of averaging over frequencies to estimate the RSS-distance relationship compared to using a single frequency. As seen, the averaging reduces the effect of phase on multipath; hence, the performance gets closer to the ideal case.

Fig. 13
figure 13

The effect of frequency averaging compared to single-frequency at 2.4 GHz band

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Finally, we used the Hybrid FH technique. Here in Table 8, we can observe the method's use within the same band; three 2.4 GHz frequencies were employed at different heights, with the lower frequency allocated to the lower height. The performance was satisfactory, with an RMSE of 3.5 dB. In the 5 GHz range, the Hybrid FH technique yielded similar results with an RMSE of 3.796 dB. We employed four frequencies, which enhances the averaging process, which may explain why this number is lower than the Frequency-only technique (3.2866 dB in Table 7).

Table 8 RMSE for hybrid averaging method at 2.4 GHz and 5 GHz bands
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5 Conclusions

Indoor localization using RSS is common because it is inexpensive, doesn't require time synchronization, and works well even in non-line-of-sight (NLOS) propagation conditions. However, the accuracy of localization is affected by the unstable signal level variation with distance. In order to enhance the accuracy of RSS localization, this publication proposes a novel approach that averages over APs with several frequencies and multiple heights to mitigate the impact of signal level volatility. Extending the work to incorporate a simulated multipath indoor propagation environment, the research was initially carried out using the PEL model. The ray-tracing Wireless InSite program was used to model the interior setting. Averaging across many height access points decreased the RSS-distance variance in the PEL model and the simulated indoor multipath propagation scenario.

Averaging over many frequencies also makes the decline more apparent. By combining various APs' heights with varied frequencies, the suggested hybrid technique allowed for even more improvement. Our research also shows that averaging over APs with two frequencies or two heights does not improve the decrease of RSS-distance variance, thus we do not advise using them.