Research on a reference signal optimisation algorithm for indoor Bluetooth positioning

The distance in Bluetooth positioning system can be obtained, depending on the free space propagation model, by using the following formula: (1) d=10−RSSI+A+ξ10n d = {10^{{{- RSSI + A + \xi} \over {10n}}}} where RSSI, A and n denote the measured signal strength, reference signal strength and attenuation coefficient, respectively, and ξ represents the Gaussian random variable with zero mean.

Even the measured signal strength at the same point at different time periods is quite different due to the non-linear time-varying characteristics of wireless channels. One way to alleviate the noise, in practice, is to pass the sampling RSSI values via a Gaussian filter, after which RSSI measurements are averaged or further processed.

Assuming that RSSIs obey the Gaussian distribution (μ, σ²) at any point in a relatively small space, the probability density function at d₀ is given as follows: (2) P(RSSId0)=12πk−1∑i=1k(RSSId0,i−1k∑i=1kRSSId0,i)2e−(RSSId0−1k∑i=1kRSSId0,i)22k−1∑i=1k(RSSId0,i−1k∑i=1kRSSId0,i)2 P({RSSI}_{{d_0}}) = {1 \over {\sqrt {{{2\pi} \over {k - 1}}\sum\limits_{i = 1}^k {{({RSSI}_{{d_0},i} - {1 \over k}\sum\limits_{i = 1}^k {RSSI}_{{d_0},i})}^2}}}}{e^{- {{{{({RSSI}_{{d_0}} - {1 \over k}\sum\limits_{i = 1}^k {RSSI}_{{d_0},i})}^2}} \over {{2 \over {k - 1}}\sum\limits_{i = 1}^k {{({RSSI}_{{d_0},i} - {1 \over k}\sum\limits_{i = 1}^k {RSSI}_{{d_0},i})}^2}}}}} where RSSI_d0,i denotes the ith measured signal strength of the total k signal strength at d₀.

Let (3) φ(RSSId0)=12π∫−∞RSSId0e−ξ22dξ \varphi ({RSSI}_{{d_0}}) = {1 \over {\sqrt {2\pi}}}\int_{- \infty}^{{RSSI}_{{d_0}}} {e^{- {{{\xi ^2}} \over 2}}}d\xi (4) F(RSSId0)=∫−∞RSSId01σ2πe−(ξ−μ)22σ2dξ F({RSSI}_{{d_0}}) = \int\limits_{- \infty}^{{RSSI}_{{d_0}}} {1 \over {\sigma \sqrt {2\pi}}}{e^{- {{{{(\xi - \mu)}^2}} \over {2{\sigma ^2}}}}}d\xi and the probability in the interval μ − σ ≤ RSSI_d0 ≤ μ + σ equals (5) P(μ−σ≤RSSId0<μ+σ)=F(μ+σ)−F(μ−σ)=2φ(1)=0.6828 P(\mu - \sigma \le {RSSI}_{{d_0}} < \mu + \sigma) = F(\mu + \sigma) - F(\mu - \sigma) = 2\varphi (1) = 0.6828

As mentioned above, the mathematic average method may lead to large errors, even after the Gaussian filter alone, when the sampling space is not big enough due to disturbance brought by measurements with large probability. Therefore, we insist that those RSSI measurements after the Gaussian filter be further processed via the supervised learning algorithm, relying on the deep learning method rather than simply averaging.

Assuming that the number of the probability density function of RSSI equals k and all the samples obey the Gaussian distributions, we surmise that the probability density function of RSSI is given by the following equation: (6) P(RSSId;μ,σ)=∑i=1kαiϕ(RSSId;μi,σi) P({RSSI}_d;\mu,\sigma) = \sum\limits_{i = 1}^k {\alpha _i}\phi ({RSSI}_d;{\mu _{i,}}{\sigma _i}) where μ_i and σ_i denote the μ and σ of the i^th sample of the k measurements, respectively, and α_i represents the weight of the i^th Gaussian component.

We define binary hidden variable γ_ji, which denotes whether the j^th RSSI at d comes from the i^th Gaussian component. The expectation of γ_ji, termed expectation-step (E-step), can be obtained as follows: (7) E(γji;RSSId,μi,σi)=P(γji=1;RSSId,μi,σi)=αi𝒩(RSSId,j;μi,σi)∑i=1kαi𝒩(RSSId,j;μi,σi) E({\gamma _{ji}};{RSSI}_d,{\mu _i},{\sigma _i}) = P({\gamma _{ji}} = 1;{RSSI}_d,{\mu _i},{\sigma _i}) = {{{\alpha _i}N({RSSI}_{d,j};{\mu _i},{\sigma _i})} \over {\sum\limits_{i = 1}^k {\alpha _i}N({RSSI}_{d,j};{\mu _i},{\sigma _i})}} where 𝒩(RSSId,j;μi,σi)=12πσie−(RSSId,j−μi)22σi2 N({RSSI}_{d,j};{\mu _i},{\sigma _i}) = {1 \over {\sqrt {2\pi} {\sigma _i}}}{e^{- {{{{({RSSI}_{d,j} - {\mu _i})}^2}} \over {2\sigma _i^2}}}} is the posterior probability of RSSI_d,j. The log-likelihood function of all RSSI_d,j is given by the following equation: (8) ∑j=1N∑i=1k(γjilogαi+γjilog𝒩(RSSId,j;μi,σi) \sum\limits_{j = 1}^N \sum\limits_{i = 1}^k ({\gamma _{ji}}log {\alpha _i} + {\gamma _{ji}}log N({RSSI}_{d,j};{\mu _i},{\sigma _i}) where log𝒩(RSSId,j;μi,σi)=−12log(2π)−logσi−(RSSId,j−μi)22σi2 log N({RSSI}_{d,j};{\mu _i},{\sigma _i}) = - {1 \over 2}log (2\pi) - log {\sigma _i} - {{{{({RSSI}_{d,j} - {\mu _i})}^2}} \over {2\sigma _i^2}} .

We take the partial derivatives of μ_i and σ_i of Eq. (8), respectively, and let them both be equal to 0; then, we have (9) ∂∂μi(∑j=1N∑i=1k(γjilogαi+γjilog𝒩(RSSId,j;μi,σi))=∑j=1N(−γji(RSSId,j−μi)σi2)=0 {\partial \over {\partial {\mu _i}}}(\sum\limits_{j = 1}^N \sum\limits_{i = 1}^k ({\gamma _{ji}}log {\alpha _i} + {\gamma _{ji}}log N({RSSI}_{d,j};{\mu _i},{\sigma _i})) = \sum\limits_{j = 1}^N (- {\gamma _{ji}}{{({RSSI}_{d,j} - {\mu _i})} \over {\sigma _i^2}}) = 0 (10) ∂∂σi(∑j=1N∑i=1k(γjilogαi+γjilog𝒩(RSSId,j;μi,σi))=∑j=1N(−γji1σi+γji(RSSId,j−μi)2σi3)=0 {\partial \over {\partial {\sigma _i}}}(\sum\limits_{j = 1}^N \sum\limits_{i = 1}^k ({\gamma _{ji}}log {\alpha _i} + {\gamma _{ji}}log N({RSSI}_{d,j};{\mu _i},{\sigma _i})) = \sum\limits_{j = 1}^N (- {\gamma _{ji}}{1 \over {{\sigma _i}}} + {\gamma _{ji}}{{{{({RSSI}_{d,j} - {\mu _i})}^2}} \over {\sigma _i^3}}) = 0

Assuming γ_ji to be a value already known, then we obtain the following: (11) μi=∑j=1NγjiRSSIj∑j=1Nγji {\mu _i} = {{\sum\limits_{j = 1}^N {\gamma _{ji}}{RSSI}_j} \over {\sum\limits_{j = 1}^N {\gamma _{ji}}}} (12) σi2=∑j=1Nγji(RSSId,j−μi)2∑j=1Nγji \sigma _i^2 = {{\sum\limits_{j = 1}^N {\gamma _{ji}}{{({RSSI}_{d,j} - {\mu _i})}^2}} \over {\sum\limits_{j = 1}^N {\gamma _{ji}}}}

Taking the partial derivative of α_i of Eq. (9) and using the Lagrange multiplier method along with the constraint condition of α_i, we obtain the following equation: (13) {∑i=1kαi=1αi=(∑j=1Nγji)/(−λ) \left\{{\matrix{{\sum\limits_{i = 1}^k {\alpha _i} = 1} \hfill \cr {{\alpha _i} = \left({\sum\limits_{j = 1}^N {\gamma _{ji}}} \right)/\left({- \lambda} \right)} \hfill \cr}} \right.

Summing from 1 to k at both sides of Eq. (13), termed maximisation step (M-step), finally, we can get the following: (14) αi=∑j=1NγjiN {\alpha _i} = {{\sum\limits_{j = 1}^N {\gamma _{ji}}} \over N}

For programming, initially we choose μ_i, σ_i and α_i randomly. For every d, i and j, we repeat E-step and M-step until they converge. The unreasonable outliers can thus be filtered out by defining the threshold of the locally weighed logarithmic likelihood. Then, the threshold of the probability determines which distribution the RSSI belongs to, with the value of k based on the reality.

We select l target nodes at the same height and let the distance between the transmitting node and the target nodes be d₁, d₂, d₃, … , d_l, respectively. We assume that the attenuation coefficient n has been determined with experiments or experience. For every RSSI measured at every node, the following equation is obtained: (15) Adk,i=−|RSSIdk,i|+10nlgdk {A_{{d_k},i}} = - |{RSSI}_{{d_k},i}| + 10nlg {d_k} where k = 1, 2, … , l, I = 1, 2, … , m, RSSI_dk,i denotes the i^th RSSI measurement at d_k and A_dk,i represents the reference signal strength affected by noise at every point.

The input vector Γ and the target output vector Ψ are given as follows: (16) Γ=(|RSSId1,1|,|RSSId1,2|,⋯,|RSSId1,m|,|RSSId2,1|,⋯,|RSSIdk,i|,⋯,|RSSIdl,m|)T {\boldsymbol{\Gamma}} = {(|{RSSI}_{{d_1},1}|,|{RSSI}_{{d_1},2}|, \cdots,|{RSSI}_{{d_1},m}|,|{RSSI}_{{d_2},1}|, \cdots,|{RSSI}_{{d_k},i}|, \cdots,|{RSSI}_{{d_l},m}|)}^T (17) Ψ=(Ad1,1,Ad1,2,⋯,Ad1,m,Ad2,1,⋯,Adk,i,⋯,Adl,m)T {\boldsymbol{\Psi}} = {({A_{{d_1},1}},{A_{{d_1},2}}, \cdots,{A_{{d_1},m}},{A_{{d_2},1}}, \cdots,{A_{{d_k},i}}, \cdots,{A_{{d_l},m}})}^T

We feed the input and the target output vector into the deep neural network for training and we can generate the model. The most possible reference signal strength A will be obtained according to the RSSI inputs without the model.

Table 1 illustrates the basic settings in our experiments. As given, the transmission power of the Bluetooth device is 1 mW by default. Samples are collected every 0.1 m, starting from 0.2 m to 10.0 m, and each of them contains 200 RSSI values, among which 100 measurements are dedicated for Gaussian filter and supervised learning, and another 100 samples are designed for testament of the effects of the reference signal optimisation algorithm (RSOA) that are measured at the rate of once a second.

The sampling number can be obtained via (10 − 0.2 m)*200/0.1 m = 19600.

Figure 1 outlines the experiment scenario in a living room. As seen, there are some furniture and plants in the room. One sender and one receiver are also observed.

Fig. 1

Figure 6 shows the descent of the loss function in the training process, where the blue line and the orange line represent the loss of the training set and the validation set, respectively. As seen, the loss converges and remains stable until the end at both figures. Therefore, the proposed RSOA model is regarded to be rational for prediction. Meanwhile, the training process after the Gaussian filter converges much faster and more steadily, obviously so in comparison with those without Gaussian filter that experience several bounces and fluctuations around 40 epochs. Although the loss rate of the Gaussian-filtered training process had dropped to 3% by the 40th period, it rebounded a little to 5% by the 75th period and remained convergent at 2.7%. In conclusion, the training process can be optimised via the Gaussian filter.

Fig. 6