To generate compressive sensing measurements, a sparse signal and a random sensing matrix Φ are required [107]. This matrix Φ should satisfy the RIP shown in [113]. In [125] Candes and Tao define the properties required for random sensing matrices to be used for signal recovery. For instance, they show that both Bernoulli and Gaussian matrices satisfy the properties required for compressive sensing with high probability [107]. From the matrices that satisfy the RIP with overwhelming probability [107], the Bernoulli matrix was chosen. This matrix was selected, generated and imple- mented in this thesis as part of the wireless vibration sensing strategy for signal encoding at the sensor nodes. This matrix was selected and used as the sensing matrix Φ. It was selected over other matrices because it uses less memory when embedded in a wireless sensor node, the computation is reduced and the production of compressed measurements is faster than using floating-point matrices. This sensing matrix Φ was formed by sampling independent identically distributed binary entries from a symmetric Bernoulli distribution with probability P=1/2, this matrix Φ is then stored in the sensor node. The product of this matrix Φ with a sparse signal
x produces a vector of measurements y. The condition for that is M = O(S log
(N /S))≪ N as mentioned in [107]. Hence, as the sparsity increases the number of measurements M grows but only logarithmically in N, the signal length. An example of a test vector, corresponding to a single row in the sensing matrix Φ is shown in Figure 3.7.
Figure 3.7: Example of a single binary test vector of length N = 125.
A Bernoulli binary generator in Simulink was used to generate all the test vectors for the sensing matrix Φ as illustrated in Figure 3.8, this function generates Bernoulli random binary numbers. The following parameters were used: probability of a zero=0.5, initial seed= from 1 to N (125), output data type: Boolean.
Figure 3.8: Examples of generated test vectors that form the Bernoulli Random Sensing Matrix Φ. Each binary test vector was embedded in the sensor node as binary entries as shown in Figure 3.9.
After the generation of the N × N sensing matrix Φ, it was then embedded in the microcontroller’s flash memory of the sensor node to allow on-board compressive sensing. An example of a portion of the sensing matrix stored in the sensor node is shown in Figure 3.9.
1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 1 0 0 1 0 Ǥ Ǥ Ǥ Φ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 Ǥ Ǥ Ǥ 0 Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ Ǥ
Figure 3.9: A section of the implemented binary matrix in the sensor node.
This section presented the selected and generated random sensing matrix which was embedded in the sensor node as part of the compressed sensing procedure. This same matrix is also stored in the base station for signal recovery. To generate compressive sensing measurements, a sparse signal is required (e.g. the output signal XS of
the proposed Algorithm 3.1 in Section 3.5) and a sensing matrix Φ which can be generated as described in this section.
The next section presents the use of compressive sensing for signal encoding in the sensor node including how to generate a vector of compressive sensing measurements and the number of measurements required for signal recovery. Compressive sensing generates this vector of measurements y using a sparse signal (Section 3.5) and a sensing matrix (Section 3.6). This vector contains the encoded vibration signal which is transmitted by the wireless sensor node to the base station.
3.7 Chapter Summary
The framework proposed in this research work mitigates the effect of random packet loss and performs data compression by using local signal processing at the wireless sensor node through frequency domain analysis, adaptive thresholding and compress- ive sensing. More importantly, after wireless data transmission, the performance during signal recovery is increased. More details in Section 4. This thesis may be divided into two main sections: vibration data encoding which was presented in this chapter, and vibration data decoding procedures. The vibration data is encoded at the wireless sensor node (TX) through local signal processing and compressive sensing. To encode the vibration data at the sensor node, a spectral representation of the vibration signal through the Fast Fourier transform was used as the basis for signal compression using compressive sensing. Subsequently, the vibration signal dimension was reduced via an adaptive thresholding algorithm that induced sparsity while maintaining the main spectral components. To produce compressive sensing measurements, a sparse signal and a suitable measurement matrix are required. Hence, a Bernoulli matrix was generated and stored in the sensor node as part of the wireless vibration sensing strategy for signal encoding. For signal decoding, the vibration data received at the base station (RX) is decoded using an enhanced signal recovery method for compressive sensing measurements which is presented and described in the next chapter.
Signal Recovery with
Frequency Support
4.1 Introduction
Mitigating the effect of random packet loss during wireless transmissions is chal- lenging especially if the data transmission occurs in noisy environments such as in a gas turbine engine. Chapter 3 presented the encoding procedure. The data packets are encoded at the sensor node through on-board frequency domain analysis, dimensionality reduction and compressive sensing prior to wireless vibration data transfer. These procedure aims to help deal with the packet loss problem through local signal processing and data compression. At the receiver, the data needs to be decoded to reconstruct the original signal sent by the transmitter.
The present chapter shows the set of steps for frequency domain sparse signal es- timation and signal recovery at the receiver. The number of compressed sensing measurements required for signal recovery is reduced by exploiting prior information from the application. The methodology to achieve that is as follows:
1. Demonstrate that the selected hardware is able to collect, transmit and recover vibration data within a Gas Turbine Engine (Section 4.3).
2. Observe and analyse the collected vibration data to help identify patterns and zones of higher energy within the frequency spectrum (Section 4.3).
3. Propose a method to capture the signal characteristics from the application or frequency components where most of the energy is concentrated. The idea is to then use that information to recover the vibration signal accurately and reduce the number of samples required (Section 4.4).
4. Generate synthetic vibration signals to validate the proposed strategy (Section 4.5).
5. After validation, incorporate the prior knowledge obtained in Step 3 into the proposed signal recovery algorithm (Section 4.6).
In summary, this chapter presents the signal decoding procedure to recover the original vibration signal sent by the wireless sensor node is described. More spe- cifically, the received vector of compressed sensing measurements is decoded using a proposed signal recovery algorithm. This algorithm is a novel contribution in this research work. It considers information from the real application and is used as prior knowledge to enhance signal recovery performance. This prior information refers to vibration signals collected from wireless sensor nodes deployed on a Trent1000 aeroengine during a running engine test. The structure of these frequency domain signals is extracted using a novel algorithm which outputs a probability density function which was used as frequency support for the proposed signal recovery algorithm.
This chapter is structured as follows:
• Section 4.2 presents the experimental methodology of the wireless vibration sensing system.
• Section 4.3 describes the deployment of wireless sensor nodes in the active Gas Turbine Engine and presents the collected vibration data.
• Section 4.4 introduces the properties of a probability density function followed by a proposed novel algorithm to estimate the Probability Density Function (PDF) for the frequency spectrum.
• Section 4.5 presents the procedure to generate synthetic vibration signals based on the same probabilistic structure from the collected data in the aeroengine.
• Section 4.6 presents a novel algorithm, the Enhanced Orthogonal Matching Pursuit which increases signal recovery performance including the PDF estim- ated in Section 4.4.
• Section 4.7 presents final remarks and summarises the impact of the proposed algorithm to enhance the performance during signal recovery of vibration sig- nals from compressive sensing measurements.