**2.2 General tester structure**

Fig. 6 presents general tester structure. The tester generates excitation signal and makes decision about CUT state (fault) based on analysis of measured CUT responses.

According to different goals of performed fault diagnosis (detection, location or identification) structure of a diagnostic system is shown on fig. 7. The D–Tester (fault detector) returns on of the following decisions:


Fig. 4. "Step-shape" (0th-order polynomial) approximation of input excitation

Fig. 5. Piece-wise linear (1st-order polynomial) approximation of input excitation

If fault detection is the only performed diagnosis type, the "unknown" decision can be replaced by NO GO decision (the worst case). This obviously reduces test yield, but does not deteriorates diagnosis trust level.

The L–Tester (fault location) points which circuit element is faulty or decision "?", if proper classification cannot be performed.

The deepest level: fault identification (information about faulty element value or at least its shift – represented by I–Tester) has not been analysed in this work.

Fig. 6. General tester structure

200 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

Fig. 6 presents general tester structure. The tester generates excitation signal and makes

According to different goals of performed fault diagnosis (detection, location or identification) structure of a diagnostic system is shown on fig. 7. The D–Tester (fault

*x4,t4*

*t6*

*x5,t5*

*n*

*x8,t8*

*n*

*x7,t7*

decision about CUT state (fault) based on analysis of measured CUT responses.

"unknown" if, for any reason, classification cannot be performed.

*x(n)*

Fig. 4. "Step-shape" (0th-order polynomial) approximation of input excitation

*x(n)*

*t2 t3*

Fig. 5. Piece-wise linear (1st-order polynomial) approximation of input excitation

If fault detection is the only performed diagnosis type, the "unknown" decision can be replaced by NO GO decision (the worst case). This obviously reduces test yield, but does not

**2.2 General tester structure** 

detector) returns on of the following decisions: GO – meaning "non-faulty - healthy circuit",

*x1,t1*

*x6*

1 


*x2*

*x3*

NO GO – means "faulty circuit" or

deteriorates diagnosis trust level.

Fig. 7. Fault diagnosis levels

$$F = \{F\_0, F\_1, F\_2, \dots, F\_{NF}\} \tag{4}$$

$$\mathbb{S} = \{\mathbf{S}\_0, \mathbf{S}\_1, \dots, \mathbf{S}\_{NF}\} = \begin{bmatrix} \mathbf{S}\_0 \\ \mathbf{S}\_1 \\ \dots \\ \mathbf{S}\_{NF} \end{bmatrix} \tag{5}$$

$$\mathbb{S} = \begin{bmatrix} \mathbb{S}\_0 \\ \mathbb{S}\_1 \\ \dots \\ \mathbb{S}\_{NF} \end{bmatrix} = \begin{bmatrix} \mathcal{V}^{\mathcal{S}\_0}(n) \\ \mathcal{V}^{\mathcal{S}\_1}(n) \\ \dots \\ \mathcal{V}^{\mathcal{S}\_{NF}}(n) \end{bmatrix} = \begin{bmatrix} \mathcal{V}\_1^{\mathcal{S}\_0} & \mathcal{V}\_2^{\mathcal{S}\_0} & \dots & \mathcal{V}\_{N\_P}^{\mathcal{S}\_0} \\ \mathcal{V}\_1^{\mathcal{S}\_1} & \mathcal{V}\_2^{\mathcal{S}\_1} & \dots & \mathcal{V}\_{N\_P}^{\mathcal{S}\_1} \\ \dots & \dots & \dots & \dots \\ \mathcal{V}\_1^{\mathcal{S}\_{NF}} & \mathcal{V}\_2^{\mathcal{S}\_{NF}} & \dots & \mathcal{V}\_{N\_P}^{\mathcal{S}\_{NF}} \end{bmatrix} \tag{6}$$

$$\text{Fig. 9.}\text{ Schema of fault dictionary creation}$$

$$\mathbf{S} = \begin{bmatrix} \mathbf{S}(\boldsymbol{\nu}\_{1}(n)) \\ \mathbf{S}\_{\mathcal{Y}\_{1}(n),1} \\ \vdots \\ \mathbf{S}\_{\mathcal{Y}\_{1}(n),NF} \\ \mathbf{S}\_{\mathcal{Y}\_{2}(n),0} \\ \mathbf{S}\_{\mathcal{Y}\_{2}(n),1} \\ \cdots \\ \cdots \end{bmatrix} = \begin{bmatrix} \boldsymbol{\mathcal{Y}}\_{1}^{\circ\_{0}}(n) \\ \boldsymbol{\nu}\_{1}^{\circ\_{1}}(n) \\ \cdots \\ \boldsymbol{\nu}\_{1}^{\circ\_{N}}(n) \\ \boldsymbol{\nu}\_{2}^{\circ\_{N}}(n) \\ \boldsymbol{\nu}\_{2}^{\circ\_{0}}(n) \\ \cdots \\ \cdots \\ \boldsymbol{\nu}\_{2}^{\circ\_{N}}(n) \end{bmatrix} \tag{8}$$

$$\mathbb{S} = \begin{bmatrix} \mathbb{S}\{\boldsymbol{\nu}\_1(n)\} \\ \mathbb{S}\{\boldsymbol{\nu}\_2(n)\} \end{bmatrix} \xrightarrow{\mathcal{MC}\text{ analysis}} \mathbb{S} \, ^\prime = \begin{bmatrix} \mathbb{S}\{\boldsymbol{\nu}\_1(n)\}\_{\mathcal{MC}=\text{"norm"}} \\ \mathbb{S}\{\boldsymbol{\nu}\_1(n)\}\_{\mathcal{MC}=1} \\ \mathbb{S}\{\boldsymbol{\nu}\_1(n)\}\_{\mathcal{MC}=2} \\ \mathbb{S}\{\boldsymbol{\nu}\_2(n)\}\_{\mathcal{MC}=\text{"norm"}} \\ \mathbb{S}\{\boldsymbol{\nu}\_2(n)\}\_{\mathcal{MC}=1} \\ \mathbb{S}\{\boldsymbol{\nu}\_2(n)\}\_{\mathcal{MC}=2} \end{bmatrix} \tag{9}$$

$$\mathbf{S}\_{t,k} = \begin{bmatrix} \mathbf{S}\_{k,MC="nom"} \\ \mathbf{S}\_{k,MC=1} \\ \mathbf{S}\_{k,MC=2} \end{bmatrix}; \quad k = 0, 1, \ldots, N\_F \tag{10}$$

$$d^l(k,m) = \sqrt{\sum\_{n=1}^{N\_p} \left[ \mathbf{y}\_n^l - \mathbf{s}\_{l,k,n}^m \right]^2} \qquad \begin{array}{l} k = 0, 1, \dots, N\_F \\ m = 0, 1, \dots, N\_{\text{MC}} \end{array} \tag{11}$$

$$r^l(k,m) = \begin{cases} 1 & \text{if } \quad \left| y\_n^l - s\_{l,k,n}^m \right| > U\_{mln} \\ 0 & \text{elsewhere} \end{cases}, \quad \begin{array}{l} k = 0, 1, \dots, N\_F \\ m = 0, 1, \dots, N\_{\text{MC}} \end{array} \tag{12}$$

$$d^{\dot{l}}(k,m) = \sum\_{m=0}^{N\_{MC}} \sum\_{k=0}^{N\_F} r^{\dot{l}}(k,m) \tag{13}$$


$$X(a,b) = \frac{1}{\sqrt{|a|}} \int\_{-\infty}^{\infty} x(t) \, \psi\left(\frac{t-b}{a}\right) dt \; ; \qquad a \neq 0; \quad a, b \in \mathbb{R} \tag{14}$$

$$
\psi\_{a,b}(t) = \psi\left(\frac{t-b}{a}\right); \qquad a \neq 0; \quad a, b \in \mathbb{R} \tag{15}
$$

$$\text{tr}\{\mathbf{t}\} = \int\_0^\infty \int\_{-\infty}^\infty \frac{1}{a^2} X(a, b) \frac{1}{\sqrt{|a|}} \varphi\left(\frac{t - b}{a}\right) \, db \, da \tag{16}$$

$$X(m,n) = a\_0^{-\frac{m}{2}} \int\_{-\infty}^{\infty} x(t) \, \psi(a\_0^{-m}t - n \, b\_0) \, dt \tag{18}$$

$$a = a\_0^m; \quad b = nb\_0 a\_0^m; \quad \quad m, n \in \mathbb{C}; \; a\_0 > 1; \; b\_0 > 0 \tag{19}$$

$$\text{supp}[\,\ulcorner\chi(n)] \in [0; t\_{\max}] \tag{20}$$

$$a = 1, 2, \dots, a\_{\max} \tag{21}$$

$$b = 0, 1, \dots, N\_P - 1 \tag{2}$$

$$X(a,b) = \frac{1}{\sqrt{a}} \Sigma\_{n=-\infty}^{\infty} \int\_{n}^{n+1} x(t) \, \psi\left(\frac{t-b}{a}\right) dt\tag{23}$$

$$X(a,b) = \frac{1}{\sqrt{a}} \Sigma\_{n=-\infty}^{\infty} x(n) \int\_{n}^{n+1} \psi\left(\frac{t-b}{a}\right) dt\tag{24}$$

$$X(a,b) = \frac{1}{\sqrt{a}} \Sigma\_{n=1}^{N\_P} \ge (n) \left( \int\_{-\infty}^{n+1} \psi\left(\frac{t-b}{a}\right) dt - \int\_{-\infty}^{n} \psi\left(\frac{t-b}{a}\right) dt \right) \tag{25}$$

$$\int\_{-\infty}^{n} \psi(t)dt\tag{26}$$

$$X(a,b) = \frac{1}{\sqrt{a}} \sum\_{n=1}^{N\_P} x(n) \,\psi\left(\frac{n-b}{a}\right); \quad \begin{array}{c} a = 1,2,\ldots,a\_{\max} \\ b = 0,1,\ldots,N\_P - 1 \end{array} \tag{27}$$

$$Y\_l(a,b) = TF[\mathbf{y}\_l(n)]\tag{28}$$

$$\mathbf{Y} = \begin{bmatrix} Y\_{11} & Y\_{12} & \dots & Y\_{1,N\_P - 1} \\ Y\_{21} & Y\_{22} & \dots & Y\_{2,N\_P - 1} \\ \dots & \dots & \dots & \dots \\ Y\_{a\_{\max 1}1} & Y\_{a\_{\max 2}2} & \dots & Y\_{a\_{\max N\_P - 1}} \end{bmatrix}\_{a\_{\max N\_P - 1}} \tag{29}$$

$$d^l(j,k) = \sqrt{\Sigma\_{a=1}^{a\_{\max}} \sum\_{b=0}^{N\_p - 1} \left[ Y^l(a,b) - S\_k^{F,l,j}(a,b) \right]^2} \qquad \begin{array}{l} k = 0, 1, \dots, N\_F \\ j = 1, 2, \dots, N\_{\text{MC}} \end{array} \tag{30}$$

$$d\_P^l = \frac{\frac{\sum\_{a=1}^{a\_{max}} \sum\_{b=0}^{N\_P - 1} \left[ Y^l(a, b) - \overline{Y}^l \right] \left[ S\_k^{F, l, j}(a, b) - S\_k^{F, l, j} \right]}{\sqrt{\sum\_{a=1}^{a\_{max}} \sum\_{b=0}^{N\_P - 1} [Y^l(a, b) - \overline{Y}^l]^2} \sqrt{\sum\_{a=1}^{a\_{max}} \sum\_{b=0}^{N\_P - 1} \left[ S\_k^{F, l, j}(a, b) - \overline{S\_k^{F, l, j}} \right]^2}} \tag{31}$$

$$\overline{Y}^{l} = \frac{1}{a\_{maks} \cdot (N\_P - 1)} \Sigma\_{a=1}^{a\_{maks}} \sum\_{b=0}^{N\_P - 1} Y^l(a, b) \tag{32}$$

$$\overline{S\_k^{F,l,j}} = \frac{1}{a\_{\text{maks}}!(N\_P - 1)} \Sigma\_{a=1}^{a\_{\text{maks}}} \Sigma\_{b=0}^{N\_P - 1} S\_k^{F,l,j}(a, b) \tag{33}$$

$$k = 0, 1, \dots, N\_F \qquad \quad j = 1, 2, \dots, N\_{\mathcal{MC}}$$

$$d^l(j,k) = \sqrt{\Sigma\_{a=1}^{a\_{\max}} \sum\_{b=0}^{N\_P - 1} \left[ Y^l(a,b) - S\_k^{F,l,l}(a,b) \right]^2} \qquad \begin{array}{l} k = 1,2,\ldots,N\_F \\ j = 1,2,\ldots,N\_{\text{MC}} \end{array} \tag{34}$$

$$d\_P^l = \frac{\Sigma\_{a=1}^{a\_{\max}} \sum\_{b=0}^{N\_P - 1} [\mathbf{Y}^l(a, b) - \overline{\mathbf{Y}^l}] \Big[ \mathbf{S}\_k^{F, l, j}(a, b) - \overline{\mathbf{S}\_k^{F, l, j}} \Big]}{\sqrt{\Sigma\_{a=1}^{a\_{\max}} \sum\_{b=0}^{N\_P - 1} [\mathbf{Y}^l(a, b) - \overline{\mathbf{Y}^l}]^2 \sqrt{\Sigma\_{a=1}^{a\_{\max}} \sum\_{b=0}^{N\_P - 1} \left[ \mathbf{S}\_k^{F, l, j}(a, b) - \overline{\mathbf{S}\_k^{F, l, j}} \right]^2}} \tag{35}$$

$$\mathcal{F}^{l} = \frac{1}{a\_{\max} \cdot (N\_P - 1)} \Sigma\_{a=1}^{a\_{\max}} \Sigma\_{b=0}^{N\_P - 1} Y^l(a, b) \tag{36}$$

$$\overline{\mathcal{S}\_{\mathbf{k}}^{F,l,j}} = \frac{1}{a\_{\max} \cdot (\mathbf{N}\_P - 1)} \Sigma\_{a=1}^{a\_{\max}} \sum\_{b=0}^{\mathbf{N}\_P - 1} \mathcal{S}\_{\mathbf{k}}^{F,l,j}(a, b) \tag{37}$$

$$k = \mathbf{1}, \mathbf{2}, \dots, N\_F \qquad \qquad j = \mathbf{1}, \mathbf{2}, \dots, N\_{MG}$$


$$F(m) = \Sigma\_{n=0}^{N\_P - 1} \varkappa(n) \cdot e^{-j2\pi m \frac{n}{N}} \qquad m = 0, 1, 2, \dots, N\_P - 1 \tag{38}$$

$$E\_1 = \sum\_{m=0}^{\frac{N\_P}{2}-1} |F(m)|^2 \qquad \text{or} \qquad E\_2 = \sum\_{m=\frac{N\_P}{2}}^{N\_P-1} |F(m)|^2 \tag{39}$$

$$\begin{aligned} \mathit{jeśli \, E\_1 > E\_2 \, to \, fit \to 2 \cdot fit} \\ \mathit{jeśli \, E\_1 \le E\_2 \, to \, fit \to \frac{1}{2} \cdot fit} \end{aligned} \tag{40}$$

Wavelet Transform in Fault Diagnosis of Analogue Electronic Circuits 213

Fig. 16. Found specialised excitation x2(n)

Fig. 17. Normalised frequency (amplitude) spectrum of found excitation x2(n)


Table 1. Fault detection efficiency (ad. 1)


Table 2. Fault location efficiency for specialised excitation (ad. 1)


Table 3. Fault location efficiency for step excitation (ad. 1)

It can observed that found specialised excitation x1(n) increased test yield in case of fault detection (tab. 1) and efficiency proper fault location was 1.5 ÷ 5 times greater (tab. 2 and 3 main diagonals, better values marked red).

#### **Ad. 2**

Figure 16 presents found excitation in time domain and its normalised amplitude spectrum in figure 17. Table 4 shows efficiency of fault detection for step and specialised excitation (probabilities of a healthy circuit correct detection - true positive HH; healthy circuit incorrect detection – false negative HF; faulty circuit correct detection - true negative FF and faulty circuit incorrect detection - false positive FH). Similar data, but for case of fault location (probabilities of fault Fx classified as Dx, with correct decisions in main diagonal) can be found in table 5 for designed excitation and in table 6 for diagnosis with step excitation.

**Excitation HH HF FF FH** x1(n) 0.25 0.75 0.93 0.07 Step 0.03 0.97 0.96 0.04

 **D1 D2 D3 D4 D5 D6 D7 D8 F1 0.32** 0.02 0.03 0.29 0.03 0 0.27 0 **F2** 0.04 **0.36** 0.10 0.01 0 0.05 0 0.17 **F3** 0.05 0.23 **0.47** 0 0.04 0 0 0.14 **F4** 0.14 0.04 0 **0.43** 0 0.18 0.08 0 **F5** 0.01 0 0.14 0 **0.73** 0 0.12 0 **F6** 0 0.01 0 0.08 0 **0.71** 0 0.18 **F7** 0.17 0 0 0.03 0 0 **0.80** 0 **F8** 0 0.19 0.02 0 0 0.05 0 **0.72** 

 **D1 D2 D3 D4 D5 D6 D7 D8 F1 0.14** 0.01 0 0.19 0.29 0.01 0.12 0.22 **F2** 0 **0.13** 0.03 0.03 0.37 0.02 0 0.34 **F3** 0 0.11 **0.12** 0.04 0.35 0 0 0.34 **F4** 0.15 0.02 0 **0.28** 0.20 0.03 0.03 0.21 **F5** 0 0 0.06 0.14 **0.41** 0 0.02 0.37 **F6** 0.04 0.06 0 0.05 0.21 **0.15** 0 0.37 **F7** 0.10 0 0 0.20 0.27 0 **0.27** 0.16 **F8** 0 0.21 0.04 0 0.23 0.01 0 **0.49** 

It can observed that found specialised excitation x1(n) increased test yield in case of fault detection (tab. 1) and efficiency proper fault location was 1.5 ÷ 5 times greater (tab. 2 and 3

Figure 16 presents found excitation in time domain and its normalised amplitude spectrum in figure 17. Table 4 shows efficiency of fault detection for step and specialised excitation (probabilities of a healthy circuit correct detection - true positive HH; healthy circuit incorrect detection – false negative HF; faulty circuit correct detection - true negative FF and faulty circuit incorrect detection - false positive FH). Similar data, but for case of fault location (probabilities of fault Fx classified as Dx, with correct decisions in main diagonal) can be found in table 5 for

Table 1. Fault detection efficiency (ad. 1)

Table 2. Fault location efficiency for specialised excitation (ad. 1)

Table 3. Fault location efficiency for step excitation (ad. 1)

designed excitation and in table 6 for diagnosis with step excitation.

main diagonals, better values marked red).

**Ad. 2** 

Fig. 16. Found specialised excitation x2(n)

Fig. 17. Normalised frequency (amplitude) spectrum of found excitation x2(n)

Wavelet Transform in Fault Diagnosis of Analogue Electronic Circuits 215

Fig. 18. Found specialised excitation x3(n)

Fig. 19. Normalised frequency (amplitude) spectrum of found excitation x3(n)


Table 4. Fault detection efficiency (ad. 2)


Table 5. Fault location efficiency for specialised excitation (ad. 2)


Table 6. Fault location efficiency for step excitation (ad. 2)

Found specialised excitation x2(n) increased test yield in case of fault detection (tab. 4) and, in most cases, increased efficiency of a proper fault location (tab. 5 and 6).

## **Ad. 3**

Figure 18 presents found excitation in time domain and its normalised amplitude spectrum in figure 19. Table 7 shows efficiency of fault detection for step and specialised excitation (probabilities of a healthy circuit correct detection - true positive HH; healthy circuit incorrect detection – false negative HF; faulty circuit correct detection - true negative FF and faulty circuit incorrect detection - false positive FH). Similar data, but for case of fault location (probabilities of fault Fx classified as Dx, with correct decisions in main diagonal) can be found in tab. 8 for found excitation and tab. 9 for diagnosis with step excitation.

**Excitation HH HF FF FH** x1(n) 0.25 0.75 0.93 0.07 Step 0.03 0.97 0.96 0.04

 **D1 D2 D3 D4 D5 D6 D7 D8 F1 0.32** 0.02 0.03 0.29 0.03 0 0.27 0 **F2** 0.04 **0.36** 0.10 0.01 0 0.05 0 0.17 **F3** 0.05 0.23 **0.47** 0 0.04 0 0 0.14 **F4** 0.14 0.04 0 **0.43** 0 0.18 0.08 0 **F5** 0.01 0 0.14 0 **0.73** 0 0.12 0 **F6** 0 0.01 0 0.08 0 **0.71** 0 0.18 **F7** 0.17 0 0 0.03 0 0 **0.80** 0 **F8** 0 0.19 0.02 0 0 0.05 0 **0.72** 

 **D1 D2 D3 D4 D5 D6 D7 D8 F1 0.14** 0.01 0 0.19 0.29 0.01 0.12 0.22 **F2** 0 **0.13** 0.03 0.03 0.37 0.02 0 0.34 **F3** 0 0.11 **0.12** 0.04 0.35 0 0 0.34 **F4** 0.15 0.02 0 **0.28** 0.20 0.03 0.03 0.21 **F5** 0 0 0.06 0.14 **0.41** 0 0.02 0.37 **F6** 0.04 0.06 0 0.05 0.21 **0.15** 0 0.37 **F7** 0.10 0 0 0.20 0.27 0 **0.27** 0.16 **F8** 0 0.21 0.04 0 0.23 0.01 0 **0.49** 

Found specialised excitation x2(n) increased test yield in case of fault detection (tab. 4) and,

Figure 18 presents found excitation in time domain and its normalised amplitude spectrum in figure 19. Table 7 shows efficiency of fault detection for step and specialised excitation (probabilities of a healthy circuit correct detection - true positive HH; healthy circuit incorrect detection – false negative HF; faulty circuit correct detection - true negative FF and faulty circuit incorrect detection - false positive FH). Similar data, but for case of fault location (probabilities of fault Fx classified as Dx, with correct decisions in main diagonal) can be found in tab. 8 for found excitation and tab. 9 for diagnosis with

Table 4. Fault detection efficiency (ad. 2)

Table 5. Fault location efficiency for specialised excitation (ad. 2)

Table 6. Fault location efficiency for step excitation (ad. 2)

**Ad. 3** 

step excitation.

in most cases, increased efficiency of a proper fault location (tab. 5 and 6).

Fig. 18. Found specialised excitation x3(n)

Fig. 19. Normalised frequency (amplitude) spectrum of found excitation x3(n)

Wavelet Transform in Fault Diagnosis of Analogue Electronic Circuits 217

There have been analysed four active CUT responses (fig. 1). Assumed observation windows was Tmax = 50 s after last falling edge of the excitation. This value is also equal to

1. D-Tester (without wavelet transform) and one-dimensional Euclidean distance metrics

2. DW-Tester with *Meyer* base wavelet and two-dimensional Euclidean distance metrics

Fig. 21 and 22 present found excitations for case 1 and 2 respectively. Tab. 10 presents

time when circuit reaches steady state after step excitation.

diagnosis efficiency for defined faults and excitations.

Fig. 20. Active low-pass filter (Kaminska et al., 1997)

Fig. 21. Found specialised excitation for case 1

There have been investigated two cases:

(11).

(30).


Table 7. Fault detection efficiency (ad. 3)


Table 8. Fault location efficiency for specialised excitation (ad. 3)


Table 9. Fault location efficiency for step excitation (ad. 3)

Designed specialised excitation x3(n) together with utilisation of wavelet transform has increased efficiency of a proper fault location (tab. 8 and 9), with exception of faults F3 and F4. However, it must be noted that specialised excitation together with wavelet transform enabled proper location of faults F2 and F5 (tab. 9, marked blue), which cannot be localised at all using simple step excitation.

#### **4.2 Example 2: Active low-pass filter**

Figure 20 presents active low-pass filter (Kaminska et al., 1997). Designed excitation x(n) has been approximated by a 0th order polynomial (fig. 4). Amplitude of each sample xn is binary coded by NB = 3 bits. Width tw of each interval is changed in range 1 do 8 s and its resolution is MB = 2 bits coded by Gray code. Non-faulty tolerances were equal 2% for resistors and 5% for capacitors. There were selected 8 parametric (soft) faults of discrete elements (R1, R2, R3 and C): 10% shift above and below nominal values.

There have been analysed four active CUT responses (fig. 1). Assumed observation windows was Tmax = 50 s after last falling edge of the excitation. This value is also equal to time when circuit reaches steady state after step excitation.

There have been investigated two cases:

216 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

**Excitation HH HF FF FH** x3(n) **0.17** 0.83 0.90 0.10 Step **0.16** 0.84 0.91 0.09

 **D1 D2 D3 D4 D5 D6 D7 D8 F1 0.35** 0 0.10 0.14 0.07 0 0.30 0 **F2** 0.01 **0.24** 0.16 0.07 0.01 0.07 0 0.28 **F3** 0.04 0.19 **0.20** 0.14 0.02 0.06 0 0.07 **F4** 0.33 0.02 0.17 **0.15** 0.05 0.01 0.18 0 **F5** 0.32 0 0.18 0.23 **0.11** 0 0.12 0 **F6** 0.01 0.15 0.16 0.13 0.03 **0.06** 0 0.31 **F7** 0.08 0 0 0 0 0 **0.92** 0 **F8** 0 0.17 0 0 0 0.13 0 **0.67** 

 **D1 D2 D3 D4 D5 D6 D7 D8 F1 0.16** 0.02 0.10 0.36 0.04 0.07 0.04 0.10 **F2** 0.10 **0** 0.15 0.09 0.02 0.05 0.05 0.45 **F3** 0.07 0.02 **0.28** 0.08 0 0.04 0.01 0.40 **F4** 0.12 0.01 0.11 **0.37** 0.04 0.02 0.05 0.16 **F5** 0.12 0.06 0.19 0.14 **0** 0.06 0.05 0.26 **F6** 0.11 0.02 0.25 0.08 0.06 **0.06** 0.02 0.30 **F7** 0.24 0.03 0.04 0.52 0.01 0.01 **0.09** 0.01 **F8** 0.05 0.03 0.23 0 0.01 0.03 0 **0.57** 

Designed specialised excitation x3(n) together with utilisation of wavelet transform has increased efficiency of a proper fault location (tab. 8 and 9), with exception of faults F3 and F4. However, it must be noted that specialised excitation together with wavelet transform enabled proper location of faults F2 and F5 (tab. 9, marked blue), which cannot be localised

Figure 20 presents active low-pass filter (Kaminska et al., 1997). Designed excitation x(n) has been approximated by a 0th order polynomial (fig. 4). Amplitude of each sample xn is binary coded by NB = 3 bits. Width tw of each interval is changed in range 1 do 8 s and its resolution is MB = 2 bits coded by Gray code. Non-faulty tolerances were equal 2% for resistors and 5% for capacitors. There were selected 8 parametric (soft) faults of discrete

elements (R1, R2, R3 and C): 10% shift above and below nominal values.

Table 7. Fault detection efficiency (ad. 3)

Table 8. Fault location efficiency for specialised excitation (ad. 3)

Table 9. Fault location efficiency for step excitation (ad. 3)

at all using simple step excitation.

**4.2 Example 2: Active low-pass filter** 


Fig. 21 and 22 present found excitations for case 1 and 2 respectively. Tab. 10 presents diagnosis efficiency for defined faults and excitations.

Fig. 20. Active low-pass filter (Kaminska et al., 1997)

Fig. 21. Found specialised excitation for case 1

Wavelet Transform in Fault Diagnosis of Analogue Electronic Circuits 219

It has been also found that in some cases (example 2) utilisation of wavelet transform allowed 100% location of a selected faults. Merging genetic algorithm and wavelet transform in example 1 allowed design of test excitation which enabled location of faults completely

It must be also added that abovementioned results have been achieved for simple, non-

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hidden for diagnosis using step excitation.

**6. References** 


Table 10. Fault location efficiency

Fig. 22. Found specialised excitation for case 2

Found specialised excitation improved efficiency of analysed single parametric (soft) faults. Utilisation of wavelet transform brought further improvements: i.a. there has been reached 100% proper location of faults F7 and F8. Improvement of fault diagnosis was also obtained in testing using simple step excitation (tab. 10, case 2).

## **5. Conclusions**

Utilisation of a wavelet transform as a feature extraction from CUT responses and in building fault dictionary resulted in general improvement of diagnosis efficiency. There have been investigated single catastrophic (hard) and parametric (soft) faults of passive and active analogue electronic circuits. It must be emphasized that the last faults are much more difficult to diagnose, because their influence on circuit behaviour (e.g. transfer function) is much weaker than catastrophic ones. It must be also noted that fault location is more difficult diagnostic goal than fault detection ("just" a differentiation between healthy and faulty circuits). Wavelet transform has been found useful tool in diagnosis of analogue electronic circuits, both in reference cases of simple excitations (step function, real Dirac pulse, linear function) and in cases when excitation has been designed by genetic algorithm. In every case, combination of specialised excitation and wavelet transform resulted in highest efficiency of fault diagnosis.

It has been also found that in some cases (example 2) utilisation of wavelet transform allowed 100% location of a selected faults. Merging genetic algorithm and wavelet transform in example 1 allowed design of test excitation which enabled location of faults completely hidden for diagnosis using step excitation.

It must be also added that abovementioned results have been achieved for simple, nonoptimised classifiers based on simple, the closest neighbourhood metrics.
