Case Study on Compression of Vibration Data for Distributed Wireless Condition Monitoring Systems

Abstract

To build robust condition monitoring solutions, it is important to identify signals that capture relevant information. However, how a degradation affects a given part of machinery might not be clear at the beginning. As a result, exploration measurement campaigns collecting large amounts of data are needed for initial evaluation. Vibration signals are typical examples of such data. Although, for explorative measurement campaigns, the battery-powered wireless node brings extra flexibility in terms of positioning the sensor at the desired location and facilitates retrofitting, the limited energy posed by them is the major downside. Sending high-sampled data over wireless channels is costly energy-wise if all samples are to be sent. When multiple sensor nodes transmit real-time measurement data concurrently over a wireless channel, the risk of channel saturation increases significantly. Avoiding this requires identifying an optimal balance between sampling time, transmission duration, and payload size. This can be done by processing and compressing data before transmission, on the sensor node close to the data acquisition and later reconstructing the received samples on the central node. In this paper, we analyze two compression mechanisms to ensure a good compression ratio and still allow good signal reconstruction for later analysis. We study two approaches, one based on the Fast Fourier Transform and one on Singular Value Decomposition, and discuss the pros and cons of each variant.

1. Introduction

To enable effective predictive maintenance solutions, data are essential. Several solutions exist to capture high-resolution vibration data, which is commonly used for condition monitoring (CM) and predictive maintenance (PM) applications such as bearing health estimation [1]. For initial measurement campaigns, wireless sensor nodes that can be freely positioned and attached to machinery would be ideal, as they allow data acquisition at different locations and thus exploration of the given environment. Wireless sensors are also becoming increasingly popular in the automation industry [2]. If a position is not as relevant, the sensors can be repositioned without the need to change any wiring. This scenario poses several system design challenges and bottlenecks like bandwidth [3], data handling [4] and signal processing [5] in typical Internet of Things (IoT) solutions. Firstly, exploration measurements with vibration result in a large amount of data quite quickly. A detailed study on vibration sensors in the field of predictive maintenance [6] informs that, depending on the sensor type and the usage in the application, the sampling frequency of the vibration signals in industrial predictive maintenance scenarios can range from 2 $\mathrm{k}$$\mathrm{Hz}$ to approx 50 $\mathrm{k}$$\mathrm{Hz}$. Even at a sampling frequency of 2 $\mathrm{k}$$\mathrm{Hz}$, a wireless sensor node with a triaxial accelerometer on it, will have to transfer 12 $\mathrm{kB}$ of raw samples every second. If there are multiple such nodes, then each node handles this amount of data and transfers it to a central node for storing or further processing. The complexity of data handling, transmission, and processing only increases with an increase in sampling frequency. Secondly, if the environment is unknown and the data shall be analyzed later, some common approaches to limit the data amount, such as sampling in certain intervals only or calculating easy metrics such as the signal Root Mean Square (RMS) value, are not suitable, as they incorporate a rather high information loss compared to collecting all vibration measurements. As a result, other options are needed to ensure that the devices can handle the highly sampled data and ideally run on battery power.

In this paper, we explore two options to compress the data at the sensor node instead of transferring the raw signals in order to reduce the communication load. After transmission, the compressed data are reconstructed at the destination device/equipment. The compression ensures a low reconstruction error in comparison to the vibration signals acquired originally. The reconstructed signal thus enables later analysis of the acquired data and builds specific condition monitoring systems. Compression is a common technique to reduce the amount of data and is characterized as lossless or lossy. Lossy compression causes some degree of information loss to enable better compression ratios, typically by removing less relevant information. In contrast to this, lossless compression ensures that the signal can be reconstructed without information loss. Ideally, lossless compression can also be used for the condition monitoring case. However, the cost of lossless compression is lower compression performance, resulting in a comparatively high amount of data to be handled. Therefore, we aim for two lossy approaches in this paper and study the trade-off between the parameters for each approach, the resulting compression performance, and the reconstruction error.

The main contributions of this work are the following:

• Development of a real-time data collection and streaming approach for distributed wireless sensor nodes for Condition Monitoring.
• Parallel data streaming capability enabled through the signal processing based compression techniques. Since the proposed compression techniques do not rely on prior knowledge of the signal characteristics, they can be readily applied across diverse application domains.
• The reduction of payload achieved due to compression enhances the battery life of the sensor nodes, making them energy efficient.

2. State-of-the-Art

Compressing data to reduce communication or storage cost is nothing new, and various algorithms have been proposed for different applications. All data compression techniques map input data into a different representation and try to reduce the amount of data significantly. Compression algorithms can be classified into lossy and lossless types [7]. Lossy compression denotes a set of algorithms that reduce the amount of data by either calculating simple metrics or performing other transformations that result in a certain information loss. Any lost information cannot be reconstructed at the receiver’s side. If no information shall be lost, only lossless approaches are possible. However, they come at the cost of a less effective reduction in the amount of data.

In [8], the authors evaluated whether the root mean square (RMS) value of the vibration data can be used as a simple metric. They compared the energy required for the calculation of the RMS by the MCU on the sensor node to the energy saved due to the reduced payload. The total energy was significantly reduced from 3600 $\mu$$\mathrm{J}$ to 172 $\mu$$\mathrm{J}$ by transmitting the RMS values compared to the raw data. However, this approach is only suitable if the RMS value contains the relevant information for further analysis. Especially, with deep learning approaches for condition monitoring, the RMS value transfer approach is not appropriate. If more information is needed, different approaches are required, that retain relevant information while still reducing the data.

Other approaches recently discussed focus on advanced algorithms such as compressed sensing [9], specific codecs [10] based on voice compression, or general signal coding and quantization [11]. Although these approaches achieve good compression results, they either require previous knowledge about the expected signals in order to tune the approach or are not designed for later implementation on MCUs.

An approach similar to ours is explored in [12], where the authors explore a CMSIS implementation of Discrete Cosine Transformation (DCT). The performance reported there is an average of 59% for the best case in a real world experiment. Although this allows for a reduction in required energy and transmission time, streaming the data from multiple nodes in parallel is not possible with this scheme. A complete data collection and processing framework is presented in [4] to address the high data rate and bandwidth challenge. The authors have mainly focused on the fragmentation technique of the data packets and the protocols efficient for transmission to the edge node. Another approach using Fast Fourier Transform (FFT) for compression is described in [13]. In this case, FFT is combined with Markov models and downsampling to transform time domain information to state information. This focuses only on specifically on bearing signals and expects the vibration impulses to be cyclostationary. In [14], the authors describe a detection system, that uses BLE and FFT-based data analysis on the node to detect faulty conditions of bearings. However, this is similar to the initially described idea to only send results or metrics like RMS values of the observed signal and does not allow a detailed analysis of a given data stream at the server side. A customized data collection technique is used in [15], where data are collected when a certain threshold criterion is satisfied. The data are only collected in a storage device connected to the MCU first and then later transmitted to the server. Continuous monitoring by data collection and streaming is not possible using this technique.

Table 1. Comparison among the state-of-the-art works and the current work.

SOTA	Data	Energy	Data Compression/	Comments
	Collection	Saving	Feature Extraction
[8,14]	✗	✓	✓	Feature is transmitted, information
				loss restricts later analysis
[9,11]	✓	✓	✓	Previous knowledge about expected
				signal is required
[10]	✓	✗	✓	Previous knowledge about expected
				signal is required
[12]	✓	✓	✓	Parallel streaming is not possible
[4]	✓	✗	✗	Focuses only on data collection
				framework and communication protocols
[13]	✓	✓	✓	Specific to bearing data and expects
				vibration impulses to be cyclostationary
[15]	✓	✓	✗	Collection and transmission of
				data are not simultaneous

can be used to find the missing link between the current work and the state-of-the-art. The ✓ in Table 1 implies that the implementation is part of the SOTA and ✗ implies that the implementation is not included in the SOTA. The studies are either suitable for inspection of a particular machinery part or presuppose familiarity with the characteristics of the signals being analyzed, indicating that relevant data collection or exploratory campaigns have already been conducted. If the feature extraction is implemented, the information loss is such that reconstruction of data and implementation of other algorithms are not possible in the later stages. Hence, the cited studies are not designed for data collection or exploration purposes and lack emphasis on parallel and continuous data streaming and monitoring.

3. Experimental Scenario

3.1. VibDemo and Derived Dataset

We use the motor fault test bench VibDemo [16], which provides a practical example to analyze different CM scenarios. It contains an electric motor with a connected primary shaft and an optional secondary shaft, which can be connected via belts or gear wheels. A frequency converter allows seamless adjustment of the rotation speed. Figure 1 shows the setup.

For data acquisition purposes, the test bench is equipped with an LIS2DS12 Micro-Electro-Mechanical System (MEMS) vibration sensor by STMicroelectronics mounted on a bearing of the primary shaft (cf. green box in Figure 1). The sensing element is configured to sample with a frequency of 3.2 kHz for each of the three axes and to measure an acceleration range of ±8 g. It provides data in a 16 bit format with a 12 bit resolution per sample [17].

The measured vibration data are sent via I2C to a microcontroller (MCU) STM32L433, which forwards the data directly to UART. A UART to USB converter transmits the data to a connected laptop, where values are stored in a comma-separated values file. Additionally, the acceleration due to gravity is subtracted on the Z axis.

Using this setup, we recorded the following scenarios:

• Failure-free operation on one shaft only, without a belt (no belt);
• Failure-free operation with belt between the shafts (good belt);
• Broken belt between the shafts (broken belt);
• Broken bearing on the secondary shaft (broken bearing);
• Static imbalance on the primary shaft (imbalance static);
• Dynamic imbalance on the primary shaft (imbalance dynamic).

According to authors in [6], the vibration response is useful in detecting bearing faults, misalignments, imbalances, and looseness. Different datasets were used to investigate compression effects on different signal characteristics.

Each case was recorded two times for 1 $\min$ with a rotation speed of 25 rotations per second (RPS) and 40 RPS, respectively. Hence, one measurement of 1 $\min$ delivers 192,000 acceleration values for each axis with a $3.2$ $\mathrm{k}$$\mathrm{Hz}$ of sampling frequency. As a result, we have a total of 24 $\min$ of vibration data for further analysis [18].

Figure 2a,b show an example of the recorded data for the case of imbalance static for the time span of 0.03 s.

3.2. Node Design

Since we target a wireless sensor system, we use a standard sensor node setup consisting of a vibration sensor, an MCU, a radio transceiver, and a battery pack as energy supply as the base for our evaluation of compression performance. The sensor used here is the same as the sensor used for the VibDemo setup, to ensure similar capturing performance. This sensor can be connected to any MCU if needed via I2C or SPI. We chose a Nordic Semiconductor nRF52840 [19] as MCU since this System on Chip (SoC) controller provides an Arm M4 core and a Bluetooth Low Energy (BLE) capable radio in one package and, thus, allows small designs. This can be advantageous for retrofit diagnosis systems.

We use BLE communication since it is low in energy consumption with in case of the nRF around 6 $\mathrm{m}$$\mathrm{A}$ for both receiving (RX) and sending (TX) at 3 $\mathrm{V}$ and the availability of a 1 Mbps channel, which enables higher data amounts than other low power wireless technologies. This makes BLE more suitable for high-data-rate applications while ensuring a battery powered operation [20]. To ensure the best channel utilization, we use a connection oriented communication scheme, where the sensor node acts as peripheral and sends its data via notifications to a receiving central, which could also be called a gateway or edge device.

Since the sampling results in a high amount of data in a short time period, we do not assume a traditional duty cycle where a node repeats a loop of sampling, sending the data, and then sleeping for an extended period of time. Instead, we analyze whether the node is able to send all available data while sampling the next batch. This is not a very energy efficient approach but is valuable for short measurement campaigns to explore unknown machinery. Once the patterns of failures are known, the sensor node can be optimized to do a frequent monitoring task in a more energy conserving way.

In such a scenario, multiple measurements from different locations around a piece of machinery simultaneously would, however, be ideal. This is only possible if the used communication channels are not occupied by other traffic. Building an application supporting multiple high-rate sensor nodes is therefore especially challenging. Hence, we study in addition how compression enables reduced communication for a single node and how this can be exploited to enable multiple measurement points.

4. Compression Concept and Experiments

In order to reduce the amount of data sent to the edge device, we apply a processing chain on the sensor node that comprises several steps between data acquisition and the actual transmission of the acquired samples. The edge device can either store or analyze the recorded data. We assume that reconstruction takes place on the edge device. Since we use a 12 bit ADC, which, however, returns 16 bit values with zero-padding, the first step is to remove the padding by applying bit packing. We use the result of this step as a baseline data amount for later comparisons and as a base for compression ratio calculations. As next step, we perform the compression of the packed data, using two options: Fast Fourier Transform (FFT) and Singular Value Decomposition (SVD). These will be described in detail below. Afterward, the data are sent via BLE to a receiving central device, which in a real scenario would collect data from multiple connected nodes. The reconstruction and evaluation of the reconstruction error take place at the server side.

FFT is a widely used post processing technique and can be easily implemented on arm MCU processors using Discrete Fourier Transform (DFT) functionality within the CMSIS-DSP library [21], which internally implements FFT. SVD, on the other hand, is an advanced signal processing technique and is recently gathering attention for CM applications [22]. Authors in [23] proposed the use of SVD for unsupervised on-device training to determine the degradation of machinery parts. This shows that SVD can be later extended for complex on-device analysis. Implementation of SVD on embedded systems is not straightforward and can only be made possible by works like [24] and the Eigen library in C++ [25]. Hence, the choice of SVD gives flexibility for further advanced algorithmic scalability with some implementation challenges, while FFT offers the simplicity in implementation for the initial data collection and exploration phase. For the mentioned reasons, FFT and SVD are investigated in this work as compression methods and are intended to be an initial step in system design of a distributed wireless condition monitoring system.

4.1. FFT-Based Compression

We use a DFT and its inverse transformation (IDFT) as base for the first compression variant [18]. The DFT and the IDFT are defined as shown in Equations (1) and (2).

$$X\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)e^{-j{\displaystyle \frac{2\pi }{N}}kn}$$

(1)

$$x\left(n\right)={\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}X\left(k\right)e^{+j{\displaystyle \frac{2\pi }{N}}kn}$$

(2)

We use a block size of 1024. Furthermore, we apply a Hamming window due to advantage of non-zero values to perform a division after inverse transformation in the course of reconstructing the original data. According to the Nyquist frequency, the amplitude spectrum will provide frequencies in the range of 0 $\mathrm{Hz}$ to 1600 $\mathrm{Hz}$ with their individual phases.

In order to compress the data, we select only the most important frequencies after the transformation and remove the others. To select these frequency bins, a threshold T is defined as the arithmetic mean amplitude according to Equation (3). The threshold T is chosen for simplicity as the application is aimed at generalization of data acquisition and streaming for a wide range of CM scenarios. Based on special cases, an $\alpha$ factor can be used to scale the threshold T.

$$T=mean\left(\right|X\left(k\right)\left|\right)={\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}\left|X\left(k\right)\right|$$

(3)

with

$$\left|X\left(k\right)\right|=\sqrt{{\Re \left|X\left(k\right)\right|}^2+\Im {\left|X\left(k\right)\right|}^2}$$

(4)

In $X\left(k\right)$, the values with smaller amplitudes than the threshold T will be removed. The threshold can be tuned if needed.

In addition to the actual values, a bit mask will be sent to identify the transmitted values for reconstruction purposes. In the frequency domain, the complex values that were transmitted are identified via the bitmask to assign them back to their frequencies. The removed complex values are set to 0 to create a complete spectrum of the signal, which can be inverse transformed to reconstruct the signal back to the time domain.

4.2. SVD-Based Compression

SVD is a matrix factorization technique used in many different applications [26] and can also be used to compress and reconstruct data [27]. The authors in [22] provide extensive research on SVD and its applications in fault diagnosis. The work focuses on many different aspects of SVD, but in this work we are only concerned with the advantages of the compression aspect of SVD for resource-constrained devices. For time series data as presented in Section 3.1, SVD requires an additional preprocessing step. As SVD operates on two-dimensional matrices, the acquired samples should be formulated as a 2D data matrix $\mathbf{X}\in {\mathbb{R}}^{s\times n}$.

This can be done as follows. The data are acquired at 3200 $\mathrm{Hz}$ per axis. The acquired data $\mathbf{X}$ can be considered as n machine observations, with each observation being s dimensional, where s can be chosen as the number of samples acquired during each rotation cycle of the machine. SVD can be applied on $\mathbf{X}$ using Equation (5).

$$\mathbf{U},\mathrm{\Sigma },{\mathbf{V}}^H=SVD\left(\mathbf{X}\right)$$

(5)

In Equation (5), ${\mathbf{V}}^H$ represents the conjugate transpose of $\mathbf{V}$. As we are not dealing with complex numbers in this work ${\mathbf{V}}^H={\mathbf{V}}^T$. Both $\mathbf{U}\in {\mathbb{R}}^{s\times s}$ and $\mathbf{V}\in {\mathbb{R}}^{n\times n}$ are orthogonal matrices with orthonormal vectors in their columns. The diagonal matrix $\mathrm{\Sigma }\in {\mathbb{R}}^{s\times n}$ has diagonal elements in descending order, which implies ${\sigma }_1\ge {\sigma }_2\cdots \ge {\sigma }_n$. The diagonal elements of $\mathrm{\Sigma }$ are called the singular values and are stable compared to the eigenvalues for the same operation [28]. This is an important criterion to apply in the predictive maintenance of dynamic industrial systems to avoid singularities.

The data matrix $\mathbf{X}$ is decomposed using Equation (5), and the elements of this decomposition can be used for $rank-K$ representation of $\mathbf{X}$. The K used here indicates the number of singular values used from the matrix $\mathrm{\Sigma }$.

This implies the transfer of K singular values along with the right and left singular vectors corresponding to them. The selection of K singular values for transmission corresponds to tuning of the compression ratio here. The higher the value of K, the lesser is the compression, but the reconstruction error is also less. With a lower value of K, a higher compression can be reached, but there can also be a considerable amount of reconstruction error. In order to find the optimum value of K, the total variance to be included must be decided in advance by setting the percentage of total variance to be included in the reconstructed dataset. This decision is based on the application.

4.3. Experiments and Evaluation Metrics

We perform the following experiments to evaluate different approaches.

SVD Parameter Tuning: Since SVD is capable of tuning to signal properties, it is important to select the appropriate parameters despite an unknown setting. In a first evaluation, we analyze how the number of singular values can be selected best.

Compression Performance: Next, we analyze the compression performance of both approaches with respect to the trade-off between compression ratio ($\rho$) and reconstruction error. The $\rho$ is defined as in Equation (6). It calculates the percentage of actually transmitted bytes $nu{m}_{b_{comp}}$ after compression with respect to the original data after bitpacking $nu{m}_{b_{raw}}$.

$$\rho ={\displaystyle \frac{nu{m}_{b_{comp}}}{nu{m}_{b_{raw}}}}$$

(6)

The RMSE according to Equation (7) is used to calculate the reconstruction error.

$$e_{rmse}=\sqrt{{\displaystyle \frac{{\sum }_{s=1}^N{(x_s-{\widehat{x}}_s)}^2}{N}}}$$

(7)

where, $x_s$ represent the original vibration signal acquired with the accelerometer and ${\widehat{x}}_s$ is the reconstructed signal.

Impact on parallel measurement data transmission: Since we target a wireless system with limited bandwidth available and at the same time nodes streaming their acquired data constantly to a gateway for later analysis during a measurement campaign, we also compare the impact of the data compression at an individual node on the spectrum availability. To do this, we measure the required transmission time to send the compressed data for each approach and compare that to the time required to send the base raw and bit-packed data. For better comparison, we assume that the data transmission has to be finished, before the node is done sampling the next input data and thus estimate how many nodes can send their compressed data during the time it takes to sample the raw input for each approach.

5. Results and Discussion

5.1. Parameter Tuning for SVD

When looking at SVD-based compression, selecting the optimal value of K for our dataset is important. K offers the right trade off between compression and reconstruction error.

As each singular value is the square root of the eigenvalue of the data covariance matrix, the sum of all singular values corresponds to the total variance present in the signal. The higher the singular value, the larger is the captured variance by that singular value [29]. As the singular values ${\sigma }_1\ge {\sigma }_2\cdots \ge {\sigma }_n$ are always arranged in the descending order of their magnitude, it implies, that the first singular value and the singular vectors corresponding to it have the highest significance. The amount of variance captured by a singular value ${\sigma }_i$ can be calculated using Equation (8) in $va{r}_{{\sigma }_i}$. For k singular values, the total variance captured can be calculated using Equation (8) in $va{r}_{{\sigma }_k}$, where $k\le n$.

$$\begin{array}{c}\hfill va{r}_{{\sigma }_i}={\displaystyle \frac{{\sigma }_i^2}{{\sum }_{i=0}^n{\sigma }_i^2}}\\ \hfill va{r}_{{\sigma }_k}={\displaystyle \frac{{\sum }_{i=0}^k{\sigma }_i^2}{{\sum }_{i=0}^n{\sigma }_i^2}}\end{array}$$

(8)

The variance captured by the singular values can be analyzed by using their cumulative sum The cumulative sum can be denoted as $C{S}_{{\sigma }_k}={\sum }_{i=0}^k{\sigma }_i^2$. Figure 3 shows the cumulative sum of all singular values plotted on the ordinate, and the abscissa denotes the total number of singular values. Considering $50\%$ of the total cumulative sum implies considering $50\%$ of the variance in the data. To capture $50\%$ of the variance in the data, one should select $K=8$, 5, and 6 singular values for axes X, Y, and Z, shown by the dotted green, dotted blue and dotted red lines respectively, as shown in Figure 3. The figure also shows that few initial singular values are sufficient to include most of the variance present in the signal.

Instead of transmitting the original signal, we now send K singular values, K left singular vectors ($U_K\in {\mathbb{R}}^{s\times K}$), and K right singular vectors ($V_K\in {\mathbb{R}}^{n\times K}$). The $rank-K$ reconstruction can be done using Equation (9), where $\widehat{\mathbf{X}}$ is the reconstructed signal.

$$\begin{array}{c}\hfill \widehat{X}={\mathbf{U}}_k·{\mathrm{\Sigma }}_k·{\mathbf{V}}_k^T\\ \hfill {\mathbf{U}}_{\mathbf{k}}\in {\mathbb{R}}^{s\times k}\\ \hfill {\mathbf{V}}_{\mathbf{k}}\in {\mathbb{R}}^{n\times k}\\ \hfill {\mathrm{\Sigma }}_{\mathbf{k}}\in {\mathbb{R}}^{k\times k}\end{array}$$

(9)

In order to verify our choice for K, we evaluate compression performance by including different variances discussed in Equation (8) and calculate the resulting reconstruction error. Figure 4 shows the average compression and RMSE achieved for the different RPS signals. The average RMSE is normalized and plotted to find the intersection point where compression and reconstruction error meet. The results shown here are the average of all different recorded situations discussed in Section 3.1. There is a trade-off between the included information and the achievable data reduction or compression ratio. The more variance is included, the higher the compressed data amount to send. When analyzing that trade-off, we found that a good point for signals at 25 RPS is $\rho =$ 37.7%, which is achieved when including 44% of variance. This is found by determining the included variance and compression ratio at the intersection of normalized RMSE with $\rho$ for 25 RPS shown with the blue dotted line. The percentage of compression was later calculated by multiplying the compression ratio ($\rho$) with 100. In case of 40 RPS, we achieve a similar performance for compression with $\rho =$ 34.2% when including 45%. Similar to above, for the case of 40 RPS, the included variance and compression ratio is determined at the intersection of normalized RMSE with $\rho$ for 40 RPS shown with the green dotted line. The values differ depending on the RPS number since SVD is sensitive to the included information, resulting in slightly different curves.

In order to map the corresponding variance to the number of singular values, we have to select the closest integer number representing the desired variance, as we can only send a singular value or not. Here, we decided to use the $floor\left(\right)$ function, which returns the smaller number, to achieve better data reduction during the compression phase. As a result, we use only 5 singular values for 25 RPS and 6 for 40 RPS.

5.2. Possible Data Reduction

Using both approaches, we study the impact of each approach on the resulting data.

Table 2. Compression rate results.

Parameter	FFT	SVD
Input block size [samples]	1024	3200
RPS [1/s]	25 and 40	25	40
K	NA	5	6
Sample time [s]	0.32	1	1
Input shape ($s\times n$)	1 × 1024	25 × 128	40 × 80
Raw Payload [Byte]	6144	19,200	19,200
Bitpacked Payload [Byte]	4608	14,400	14,400
Compressed Payload [Byte]	1880	4620	4356
Compression Ratio $\rho$ [%]	40.8	32.1	30.1

gives an overview of the input size and data reduction after compression. The data presented in Table 2 is averaged over all different classes of data acquired, which is already discussed in Section 3.1. One should notice that we target a streaming application, and both approaches are tuned to the given parameters. This results in different data amounts and time bases, as each uses a different block size as input.

Since the shape of matrices in SVD depends on the RPS, we show two variants. Both show a compression ratio around 30% compared to the bitpacked data and thus achieve a slightly better result than the FFT-based approach. The FFT-based approach is, however, independent of the RPS and, thus, does not require a tuning step.

5.3. Impact on Reconstruction Error

Regarding the reconstruction error achieved by each approach, the FFT-based approach achieves an average RMSE of 223 mg, while both SVD-based variants show an average RMSE of 258 mg. The compression ratio was better for SVD, and this also results in a higher RMSE.

However, the pure number does not reflect whether the required characteristics are still available in the reconstructed data. For this, we prepared a direct comparison of the signals. Figure 5 shows corresponding examples of reconstructed and original signals for all three axes and both approaches.

This figure shows that the characteristics of each axis are different, and both approaches are able to reconstruct the general trends. Small errors occur in acceleration information. This does, however, not limit further analysis of the streamed data.

5.4. Impact on Parallel Streaming

Finally, we evaluate the impact on parallel operation of multiple nodes at the same time, simulating a use case, where multiple measurement points on one piece of equipment shall be monitored simultaneously. Based on the respective raw, bitpacked, and compressed data, we first estimate the number of packets required for each variant under test and then measure how long it takes to actually send these data.

Table 3. Achievable parallel streams.

Parameter	FFT			SVD
	Raw	Packed	Comp.	Raw	Packed	Comp.
Payload [Byte]	6144	4608	1880	19,200	14,400	4620	4356
Sample time [ $\mathrm{s}$]	0.32	0.32	0.32	1	1	1	1
$\rho$ [%]	133	100	40.8	133	100	32.1	30.3
$nu{m}_{indications}$	48	36	15	150	113	37	35
tx time [ $\mathrm{s}$]	0.211	0.158	0.060	0.710	0.519	0.167	0.152
${\displaystyle \frac{Sampletime}{txtime}}$	0.95	2.13	5.30	1.40	1.92	5.98	6.67
achievable	1	1	5	1	1	5	6
parallel nodes

shows the resulting transmission times and the potential number of nodes operating and streaming their data in parallel. The basis for this is the available spectrum as well as the assumption, that each node has to be able to send its compressed data while sampling the next input batch. Both approaches use a different block size; thus, the sampling time is different depending on the number of samples in each case. As a result, the transmission time is compared to the sampling time for each case, and actual times are difficult to compare.

Comparison of the sampling time is not the right parameter to evaluate the results. The nodes communicate with the central via Bluetooth, and the wireless channel can be accessed only by one node at a time. While one node transmits the acquired and processed data over the wireless channel, new batch of data is being processed by the nodes. If the transmission time exceeds the next sampling time, this leads to saturation and consequently loss of data. Hence, if the increase in sampling time is proportionate to the transmission time, there will be no loss in information, and more nodes can be part of the distributed wireless condition monitoring system. The proportionality of sampling time and transmission time for compressed data for SVD and FFT can be verified using Table 3.

Although, based on our results, it is possible to stream all raw data, it restricts the operation to a single node in that case. Adding bit packing reduces transmission time, but is alone not sufficient to increase the possible number of nodes. Only when compression is added, we achieve 5 or 6 nodes in parallel, depending on the chosen approach.

The compression ($\rho$) achieved in each case of FFT and SVD is averaged over all the different scenarios of signals acquired in Section 3.1. All different scenarios were used in equal proportion. This implies that in some scenarios from Section 3.1 compression was better than the average compression, and in some scenarios the compression was worse. Five or six wireless sensor nodes transmitting real-time signals in parallel is the average when all different scenarios were used in equal proportion.

One should also note that SVD is not working on a fixed input size. Therefore, it could provide better compression results if a larger number of input samples is used with a similar low number of singular values, which are transferred for reconstruction. This would further increase the number of parallel nodes. We want to explore this and a DCT-based approach in the future.

If more nodes are required for a measurement task, one would have to carefully assess further options to relax the timing constraint of the nodes. One option for this could be to use local storage and later transmission of the stored data, if the process or machinery allows for such a scenario. In addition, further compression options exist, which should also be taken into account. Here, the DCT-based approach in [12] could be interesting, if adopted to allow data streaming. In addition, using novelty detection methods on the node could help to reduce the data to send further without losing information.

6. Conclusions and Future Work

The work highlights the missing links in the state-of-the-art and proposes solutions for compressing the vibration data for distributed wireless condition monitoring systems. In order to enable exploratory data acquisition campaigns using wireless sensor nodes for high data-rate applications, this work demonstrates the possibilities of parallel streaming using data compression with one basic (FFT) and one advanced (SVD) signal processing technique. The paper explores two approaches to compress vibration data for machine health prognostics at an early stage of the investigation, where measurements are performed to later analyze the signals and derive more sophisticated diagnostic approaches based on the collected data. Since such a system is used in an unknown environment, other techniques such as calculating metrics or using longer sample intervals do not apply. In both cases, relevant information could be lost for later analysis.

This work deals with high-data rate ($3.2$ $\mathrm{k}$$\mathrm{Hz}$) applications that can lead to saturation of the wireless channels and restrict distributed systems with multiple nodes. The goal is to send all the data from one node while the same node is sampling the next batch. The constraint is, only one node can use a channel at a given point in time for BLE and error-free transfer. In order to enable multiple nodes to do this, the transfer has to be finished before the next sampling phase is complete and ideally leave the channel free for other nodes to send their data within the same time frame (assuming identical sampling/sending schemes) as well.

We use the well known mechanisms FFT and SVD to study how they perform in these scenarios. Our tests show that, one can achieve streaming all data from multiple nodes in parallel by incorporating simple signal processing based compression techniques. Both approaches achieve compression that can increase the number of nodes operated in the system, both approaches are tuneable to tolerate a desired reconstruction error, and can be easily implemented in MCUs. Therefore, we enable up to 5 parallel nodes streaming all captured data to a central. This study represents a preliminary stage of an exploratory measurement campaign, conducted in situations where the specific objectives or signal characteristics are not yet known.

In the future, it is planned to enhance the current work by exploiting larger input sizes for SVD as well as the use of novelty detection mechanisms to reduce the amount of data to send. Adapting the threshold, in the FFT-based compression technique, to be based on the signal energy constraint in the future can provide a robust industrial implementation for exploratory measurements. The dynamic memory allocations required by the Eigen library written in C++, pose severe implementation challenges on embedded systems, which will be improved in future work using libraries designed specifically for arm processors.