### Model of the output signal from the particle analysis structure

The capacitance changes when the flexible capacitive plate is collided by particles. Because a fixed DC voltage is applied between two plates, an alternating current will produce a change in capacitance, the amplitude of which is the superposition of all weak AC signals caused by collisions between particles (including smoke particles and interfering particles) and capacitive cells, and the signal will be output by the signal stacker between two plates. The mathematical model has been developed by assuming sinusoidal currents.

$$I_{sum} = U_{0} \cdot \frac{{dC_{sum} }}{dt} = U_{0} \cdot \left[ {\frac{{d\left( {C_{{\vartriangle L_{1} }} } \right)}}{dt} + \frac{{d\left( {C_{{\vartriangle L_{2} }} } \right)}}{dt}} \right]$$

(7)

where \(I_{sum}\) is the total alternating current signal synthesized by the signal stacker and, \(U_{0}\) is the constant voltage between capacitor's terminal,\(C_{sum}\) is the superposition of changes in the capacitance of the capacitor. The AC voltage signal is generated on the precision resistor in series between two signal stacks.

$$U_{sum} = I_{sum} *R_{samp}$$

(8)

where \(R_{samp}\) is the electrical resistance of the precision sampling resistor, and it has a resistance value of 10MΩ, \(U_{sum}\) is the AC voltage applied to the precision sampling resistor. A superposition of sinusoidal voltages with different frequencies and amplitudes will be formed after filtering and amplification by the signal processing circuit (as shown in Fig.1).

$$U\left( t \right) = \sum\limits_{{R_{i} = R_{s} ,R_{{N_{1} }} ,R_{{N_{2} }} \cdots }} {A_{{R_{i} }} \cdot \sin \left[ {\omega_{{R_{i} }} \cdot t + \varphi } \right]}$$

(9)

where \(R_{i}\) is the diameter of different particles, \(R_{s}\) is the diameter of smoke particles to be detected, \(R_{{N_{1} }}\), \(R_{{N_{2} }}\), et al. are the diameters of interfe ring particles, \(\omega_{{R_{i} }}\) is the frequency of the signal produced by particles with a diameter \(R_{i}\), \(A_{{R_{i} }}\) is the amplitude of the signal produced by particles with a diameter \(R_{i}\), \(\varphi\) is the offset angle of the signal, and \(t\) is the time (Fig.4).

### Smoke concentration detection algorithm

#### Overall design of the multiscale smoke particle concentration detection algorithm

The signal output of the detector is in this form the superposition of signals generated by particles at different times. The weak signal needs to be amplified with the signal enhancement technique because the size of the smoke particles is tiny. These drawbacks prevent the use of a single method for signal processing from meeting the demand for smoke concentration detection. The multiscale smoke concentration detection algorithm is a combinatorial algorithm for continuous wavelet transform, smooth wavelet transform, sensitization of smoke signals, and single-frequency point concentration calculations. Therefore, the multiscale smoke concentration detection algorithm—a combination of multiple signal analysis methods—will be used for this detection, and its main steps can be divided as follows:

- a.
First, determine the time position of the smoke particle signal in the detector output signal.

- b.
After that, the smoke particle signal needs to be extracted.

- c.
Subsequently, the signal after extraction is sensitized and amplified.

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Finally, the smoke concentration is calculated via single-frequency analysis.

#### Time‒frequency analysis of signals

First, a time-spectrum analysis of the detector output signal is performed via a continuous wavelet transform along the time axis, and the moment at which the smoke particle signal appears is determined. The continuous wavelet transform of the continuous signal \(f\left( t \right)\) can be expressed as follows:

$$WT_{f} \left( {a,b} \right) = \left\langle {f(t),\psi_{a,b} \left( t \right)} \right\rangle \frac{1}{\sqrt a }\int_{ - \infty }^{ + \infty } {f\left( t \right)} \psi^{*} \left( {\frac{t - b}{a}} \right)dt$$

(10)

where \(a\) is the scale parameter of the wavelet function, \(b\) is the translation parameter of the wavelet function, \(\psi_{a,b} \left( t \right)\) is the wavelet basis function for parameters \(a\) and \(b\), \(\psi^{*} \left( t \right)\) is the conjugate function of the wavelet basis function, and \(f\left( t \right)\) is the source signal function.

The relationship between the wavelet decomposition scale and signal frequency after transformation can be expressed as follows:

$$f_{a} = \frac{{f_{c} f_{s} }}{a}$$

(11)

where \(f_{a}\) is the actual signal frequency after decomposition, \(f_{c}\) is the center frequency of the wavelet basis function, and \(f_{s}\) is the sampling frequency of the signal. According to the sampling theorem, the value ranges of the scale parameter satisfy \(a \in \left[ {2f_{s} ,\infty } \right]\) so that the value ranges of the frequency of the wavelet basis function can satisfy \(f_{c} \in \left[ {0,f_{s} /2} \right]\).

#### Smoke particle signal separation

In addition, the smoke particle signal is extracted from the detector output signal by a stationary wavelet transform.

In the stationary wavelet transform, the scale parameter \(a\) needs to be discretized, and the translation parameter \(b\) remains constant so that the signal after the transform has the same length as the original signal \(f\left( t \right)\). The stationary wavelet transform can be obtained through discrete sampling to the scale parameter \(a\) within the binary sequence \(\{ 2^{j} \}\) (where \(j \in Z\)).

$$SWT_{f} \left( {j,b} \right) = \left\langle {f\left( t \right),\psi_{a,b} \left( t \right)} \right\rangle = \frac{1}{{\sqrt {2^{j} } }}\int_{ - \infty }^{ + \infty } {f\left( t \right)} \psi^{*} \left( {\frac{t - b}{{2^{j} }}} \right)dt,j \in Z$$

(12)

Equation(12) shows that only the scale parameter \(a\) is discretized by the stationary wavelet transform, and the translation parameter \(b\) remains constant. In this way, the wavelet coefficients are all retained, and the length of the wavelet coefficients remains constant after each transform.

There are two ways of upsampling and downsampling at the same time so that the lengths of the signal between the original signal and the high- and low-frequency coefficients after the transform remain constant when the original signal is disintegrated by the stationary wavelet transform. This sampling mode is achieved by interpolating \(2^{j}\) zeros between the two coefficients of the high-pass and low-pass filters; the high-pass and low-pass filter coefficients are stripped in this way; and the high-pass and low-pass filters in the transformation can be expressed as follows:

$$g\left( k \right) = \left\{ {\begin{array}{*{20}l} {g\left( {\frac{k}{{2^{j} }}} \right),} \hfill & {k = 2^{j} m} \hfill \\ {0,} \hfill & {others} \hfill \\ \end{array} } \right.$$

(13)

$$h\left( k \right) = \left\{ {\begin{array}{*{20}l} {h\left( {\frac{k}{{2^{j} }}} \right),} \hfill & {k = 2^{j} m} \hfill \\ {0,} \hfill & {others} \hfill \\ \end{array} } \right.$$

(14)

where \(j,k,m \in Z\), \(g\left( k \right)\) and \(h\left( k \right)\) denote the unit response functions of the high-pass and low-pass filters, respectively.

Furthermore, the decomposition based on the Mallat algorithm can be obtained as follows:

$$\left\{ {\begin{array}{*{20}c} {S_{j + 1} \left( n \right) = \sum\limits_{k = 1}^{M} {S_{j} \left( k \right)g^{ * } \left( {k - 2n} \right)} } \\ {d_{j + 1} \left( n \right) = \sum\limits_{k = 1}^{M} {d_{j} \left( k \right)h^{ * } \left( {k - 2n} \right)} } \\ \end{array} ,\;j = 0,1, \cdots J} \right.$$

(15)

where \(j\) is the decomposition depth of the Mallat algorithm, \(J\) is the number of decompositions of the signal, \(n\) is the degree of decomposition of the signal, \(k\) is the order number of the decomposed sequence, \(M\) is the sampling point upper limit of the decomposed sequence, and \(S_{j} \left( k \right)\) and \(d_{j} \left( k \right)\) denote the coefficients of the high-pass and low-pass filters, respectively, at the *j*th signal decomposition.

The detector output signal, which includes the smoke particle signal period, is decomposed by the stationary wavelet transform based on the Mallat algorithm. Suppose that the eigenfrequency of the awaiting detection smoke particle signal is \(\omega_{{R_{S} }}\) and that the eigenfrequency of the interfering particle signal is \(\omega_{{R_{i} }}\). The signal that contains only smoke particles can be acquired after \(i\) the step of stationary wavelet decomposition.

In Fig.5, 2-s2-step decomposition is shown as an example. First, the original signal \(f\left( t \right)\) is decomposed by high-pass and low-pass filters with coefficients \(g_{{R_{{N_{1} }} }}\) and \({h}_{{R}_{N1}}\), respectively, and the signal \(S_{1}\) filters the interference caused by interference particles of size \(R_{{N_{1} }}\) and the interference signal \(d_{{R_{{N_{1} }} }}\) generated by particles of this size. Subsequently, the signal \(S_{1}\) is decomposed again by another high-pass and low-pass filter with coefficients \({g}_{{R}_{s}}\) and \({h}_{{R}_{s}}\), respectively, and the signal \(S_{{R_{S} }}\) contains only the signal generated by smoke particles and the signal \(d_{{R_{{N_{2} }} }}\) generated by interference particles of size \(R_{{N_{1} }}\).

The relationship between the coefficients \(g_{{R_{{N_{1} }} }}\) and \(h_{{R_{{N_{1} }} }}\) of high-pass and low-pass filters in the first decomposition layer and the eigenfrequency \(\omega_{{R_{{N_{1} }} }}\) of the interference signal caused by particles with size \(R_{{N_{1} }}\) can be expressed as follows:

$$g_{{R_{{N_{1} }} }} = \beta_{{R_{{N_{1} }} }} \omega_{{R_{{N_{1} }} }} g\left( {\frac{{k_{{N_{1} }} }}{{2^{{j_{{N_{1} }} }} }}} \right)$$

(16)

$$h_{{R_{{N_{1} }} }} = \beta_{{R_{{N_{1} }} }} \omega_{{R_{{N_{1} }} }} h\left( {\frac{{k_{{N_{1} }} }}{{2^{{j_{{N_{1} }} }} }}} \right)$$

(17)

where \(g\left( {\frac{{k_{{N_{1} }} }}{{2^{{j_{{N_{1} }} }} }}} \right)\) and \(h\left( {\frac{{k_{{N_{1} }} }}{{2^{{j_{{N_{1} }} }} }}} \right)\) are the unit response functions of the high-pass and low-pass filter decomposition depths, respectively \(N_{1}\), and \(\beta_{{R_{{N_{1} }} }}\) is the correction coefficient for the eigenfrequency \(\omega_{{R_{{N_{1} }} }}\).

Similarly, the relationship between the coefficients \(g_{{R_{S} }}\) and \(h_{{R_{S} }}\) of the high-pass and low-pass filters in the second decomposition layer and the eigenfrequency \(\omega_{{R_{s} }}\) of the smoke signal caused by particles of size \(R_{s}\) can be expressed as follows:

$$g_{{R_{S} }} = \beta_{{R_{S} }} \omega_{{R_{S} }} g\left( {\frac{{k_{{N_{S} }} }}{{2^{{j_{{N_{S} }} }} }}} \right)$$

(18)

$$h_{{R_{S} }} = \beta_{{R_{S} }} \omega_{{R_{S} }} h\left( {\frac{{k_{{N_{S} }} }}{{2^{{j_{{N_{S} }} }} }}} \right)$$

(19)

where \(g\left( {\frac{{k_{{N_{S} }} }}{{2^{{j_{{N_{S} }} }} }}} \right)\) and \(h\left( {\frac{{k_{{N_{S} }} }}{{2^{{j_{{N_{S} }} }} }}} \right)\) are the unit response functions of the high-pass and low-pass filter decomposition depths, \(N_{S}\) respectively, and \(\beta_{{R_{S} }}\) is the correction coefficient for the eigenfrequency \(\omega_{{R_{s} }}\).

#### Signal sensitization and smoke concentration calculations

A programmable circuit, as shown in Fig.6, is included in the signal processing circuit in Fig.1. The circuit comprises 2 operational amplifiers (op. amps.) U28A and U29A, and a digital potentiometer U25. The very low-amplitude raw output at the sensitive element is amplified through a two-stage amplifier circuit consisting of U28A and U29A. The gain of the output signal can be adjusted by changing the tap position of the digital potentiometer U25. Finally, the processed analog signal is passed to an analog-to-digital converter (ADC). "ADC5V" is the DC 5V power supply for the analog simulation circuit section. Capacitor C46 is used to filter the signal. Diode D3 is used to provide voltage-limited protection for the ADC signal. When the ADC voltage exceeds 12V, diode D3 conducts to limit the ADC signal to 12V. Electrolytic capacitors C42 and C43 are connected in reverse series to form an unpolarized capacitor of halved capacitance. This is to double the voltage rating of the capacitor.

Where \(S_{{R_{S} }}^{*}\) is the sensitized smoke particle concentration signal and \(Gain\) is the signal magnification.

The fast Fourier transform (FFT) algorithm was used to calculate the modulus of a single frequency point after separation and sensitization. Near the characteristic frequency ω of the smoke particle signal, the characteristic frequency modulus \(M_{{R_{S} }}\) can be obtained.

Finally, the smoke concentration can be calculated by bringing the modulus \(M_{{R_{S} }}\) into the smoke concentration characterization line as follows:

$$Col_{{R_{S} }} = \gamma_{{R_{S} }} \times M_{{R_{S} }} + \rho_{{R_{S} }}$$

(20)

where \(Col_{{R_{S} }}\) is the calculated smoke concentration, \(\gamma_{{R_{S} }}\) is the slope of the smoke concentration characteristic line, and \(\rho_{{R_{S} }}\) is the constant of the smoke concentration characteristic line.