#### 3.1. Sparse Representation Mathematical Model of the BTT Measurement

Suppose there are

L imaginary probes installed uniformly around the casing as shown in

Figure 1. For each rotation,

L data are sampled from vibration signal. Thus, the sampling rate of this system equals to

Lf_{r} (

f_{r} = 1/T_{n} ). According to Nyquist sampling theorem [

19], if the sampling rate of this system satisfies

Lf_{r}/2 >

f_{blade} these calculated deflection data contain all information about the frequency of vibration signals. Considering the engine constraints, however, only a limited number of probes can be mounted. Therefore, as shown in

Figure 1, only

I probes are embedded in certain angular positions selected from these

L ones.

The sampling stream for an arbitrary blade vibration signal could be then viewed as discarding all but I actual samples in every rotation of L imaginary samples periodically. An integer set $\mathsf{\Lambda}=\left\{{\lambda}_{i}|i=1,2,\cdots ,I\right\}$ describes the position of I probes. The ith probe placement is then written as ${\alpha}_{i}=2\pi {\lambda}_{i}/L$. According to Equation (4), for an arbitrary blade, the vibration signals recorded by I probes can be represented as a vector of acquired samples $y\left[j\right]\text{}\left(1\le j\le IN\right)$, where the sampling sequence is under-sampled.

Suppose the vibration signal of an arbitrary blade can be recorded by

L uniformly displaced probes, then the angular positions of

L probes can be formulated as

$2\pi l/L\text{}\left(1\le l\le L\right)$ respectively. Based on Equation (4), the imaginary measurement vector can be written as:

Obviously, $x\left[\tilde{j}\right]$ includes the frequency information of blade vibration signal, which can be solved through Fourier transform according to the celebrated Shannon sampling theorem.

Signals sampled in two periods from a vibration waveform in vector

$y\left[j\right]$ and

$x\left[\tilde{j}\right]$ are presented in

Figure 2. Therefore, the relationship between

$y\left[j\right]$ and

$x\left[\tilde{j}\right]$ can be converted into a matrix equation.

where matrix

$\Phi ={\left[\begin{array}{cccccc}1& 0& \cdots & \cdots & \cdots & 0\\ 0& \cdots & 1& 0& \cdots & 0\\ \vdots & \cdots & \cdots & \cdots & \cdots & \vdots \\ 0& \cdots & \cdots & 1& \cdots & 0\end{array}\right]}_{p\times q}$ is determined by the sampling time sequence of

I probes in each rotation. The number of row

p and column

q of the matrix

**Φ**, which determined by the scale of

$y\left[j\right]$ and

$x\left[\tilde{j}\right]$, satisfy

p =

IN and

q =

LN respectively. Matrix

**Ф** has entries

$\phi \left(\u2022,k\right)=1$ in each row. where

$\frac{k}{L}{\mathrm{T}}_{\mathrm{n}}$ equals to the arriving time of the blade in each rotation, i.e.,

$k={\lambda}_{i}+nL$ (

i = 1, 2, ...,

I,

n = 1,2, ...,

N).

The frequency of the blade vibration can be derived in the frequency-domain. Hence the Fourier transform of

$x\left[\tilde{j}\right]$ can be formulated as follows:

where

${\Psi}_{FFT}\in {C}^{q\times q}$ is the Fourier basis matrix. Apparently,

$\theta \left(f\right)\in {C}^{q}$ is the frequency spectrum of the vibration signal obtained by Fourier transform. As the blade vibration signal has a finite number of modes (harmonic features),

$\theta \left(f\right)$ has a sparse nature. So the non-zero coefficients of vector

$\theta \left(f\right)$ are constrained by number of modes. According to Equation (8), Equation (7) can be transformed into:

Up to now, the sparse representation model of BTT measurement has been built. In addition, the recovery of vibration frequency spectrum

$\theta \left(f\right)$ from Equation (9) has been transformed into a classic Single-Measurement Vector problem [

20]. As

$p<q$, Equation (9) is referred to be an overcomplete transform. Thus, further information is needed to uniquely extract the solution of

$\theta \left(f\right)$.

#### 3.2. Uniqueness of Solution to SR Model

#### 3.2.1. Restricted Isometry Property and Orthogonalization Preprocessing

The Restricted Isometry Property (RIP) is a useful notion for robust recovery of a sparse signal from an under-sampled measurement vector [

21]. As for the single measurement vector model in Equation (9), RIP can be equivalently transformed into the goal that the columns of matrix

$\Phi {\Psi}_{FFT}^{T}$ should be nearly orthogonal [

17]. Referring to [

22], the following orthogonalization preprocessing procedure can satisfy such property.

Define a sensing matrix

**R** as

$\mathbf{R}=\Phi {\Psi}_{FFT}^{T}$. The orthogonalization process is done on both sides of Equation (9).

where

**Q** = [orth(

**R**^{T})]

^{T}, an orthogonal basis for the range of matrix

${\mathbf{R}}^{\mathbf{T}}$, and

${\mathbf{R}}^{\u2020}$ is a pseudoinverse of matrix

**R**.

It has been proved [

22] that

$\mathbf{Q}{\mathbf{R}}^{\u2020}\mathbf{R}=\mathbf{Q}$. Therefore, the vibration frequency problem formulated in Equation (7) can be reformulated as:

where

$z=\mathbf{Q}{\mathbf{R}}^{\u2020}y\left[j\right]$.

Hence the detection of vibration frequency spectrum can be transformed into estimating the solution of

$\theta \left(f\right)$ in Equation (11) with minimal sparsity. It is shown in [

22] that the orthogonalized sensing matrix

**Q** obeys the RIP. If the length of BTT sampling sequence also obeys the corresponding principles, then the vibration frequency spectrum

$\theta \left(f\right)$ has high probability to be solved from

**z** by Basis Pursuit (BP) algorithm that solves the following

l_{1} norm minimization problem [

16]:

Thus, the frequency of the blade vibration can be extracted from the solution of $\widehat{\theta}\left(f\right)$.

#### 3.2.2. Requirement of Probe Number

According to the theory of CS [

17], the vibration frequency spectrum can be well recovered from

**z** with high probability when the number of validate sampling data

${p}_{v}$ obeys the following rule:

where

K_{s} is the sparsity of

$\theta \left(f\right)$, i.e.,

${\Vert \theta \left(f\right)\Vert}_{0}$,

q is the length of spectrum vector

$\theta \left(f\right)$.

However, the definition of

p_{v} various from synchronous modes to asynchronous ones. As for synchronous vibration (engine-ordered) [

3], whose response frequency is an integer multiple of the rotational speed of the assembly, the same phase of the vibration cycle is detected in each rotation by probes. Thus, the amount of data that is available for the estimation of the response frequency is limited. As shown in

Figure 3a, no matter how many measurement results are sampled, the feasible results is equal to the number of probes, i.e.,

${p}_{v}=I$. Asynchronous vibration, however, contains different values of blade displacements in each rotation as shown in

Figure 3b, Thus,

${p}_{v}$ is equal to the length of sampling sequence, i.e.,

${p}_{v}=IN$.

Therefore, the requirement of probe number can be summarized from Equation (13) as follows:

For the most ideal case, which the blade responds at a synchronous mode, at least four probes are needed to get the unique solution as the length of q is in the order of 10^{3} to 10^{4} considering both the computational complexity and the time of engine to rotate in a constant speed.

For the multi-mode case, which usually mixed with asynchronous resonances caused by flutter [

23], three sensors are abundant for the monitoring of multiple mode vibration as long as the sampling time is long enough.

#### 3.2.3. The Coherence of the Sensing Matrix and Choice of Probe Placement Location

According to the theory of SR/CS, the coherence property of matrix **Q** also has effect on the accuracy of the recovery of vibration spectrum $\theta \left(f\right)$.

The two-sided coherence [

24] for matrix

**Q** with columns

${q}_{i}$ is defined as:

For the linear system of equations

**z** =

**Q****θ**(

f), if a nonnegative solution exists such that

${\Vert \theta \left(f\right)\Vert}_{0}<\frac{1}{2}\left(1+\frac{1}{\mu \left(\mathbf{Q}\right)}\right)$, then several efficient algorithms are guaranteed to find it exactly [

24]. As

$\theta \left(f\right)$ is the frequency spectrum, the sparsity of spectrum

${\Vert \theta \left(f\right)\Vert}_{0}$ represents the number of modes that contains in vibration signals. To get the high recovery probability of

$\theta \left(f\right)$,

$\mu \left(\mathbf{Q}\right)$ must be bounded with an certain value.

According to Equation (7) and Equation (9), **Q** is defined by the number of sensors around the casing and their positions. When the imaginary probe number L and probe number I are fixed, $\left(\begin{array}{c}L-1\\ I-1\end{array}\right)$ kinds of probe placements can be selected after locking the initial probe. In an attempt to reconstruct the frequency with high probability, minimal $\mu \left(\mathbf{Q}\right)$ under proper probe placement need to be calculated.

It can be concluded from the simulation results that several arrangements of probe placements, which satisfies the minimal two-sided coherence of matrix

**Q**, can be chosen in BTT measurement. Assuming that

L is set as a constant 25, the relationship between the two-sided coherence of

**Q** and the number of probes is shown in

Table 1.

Table 1 shows that increasing the number of probes can in some extent improve the incoherence of matrix

**Q**. However,

$\left(1+1/\mu \left(\mathbf{Q}\right)\right)/2$ is not more than

$\frac{1}{2}\times \left(1+1/0.4\right)=1.75$, which seems that spectra containing multi-mode frequencies can hard to be recovered. Fortunately, pioneers [

25] have proved that the sparse solution still has high probability to be recovered with a

$\mu \left(\mathbf{Q}\right)$ exceeding the bounded value.

In summary, optimized probe placements can be selected through finding the minimal $\mu \left(\mathbf{Q}\right)$. It must be noted that the uniform placement of probes must be avoided because $\mu \left(\mathbf{Q}\right)$ in that case is larger than any other kinds of placements regardless of the number of probes.