Study of Sallen–Key digital filters in nuclear pulse signal processing

NUCLEAR ELECTRONICS AND INSTRUMENTATION

Study of Sallen–Key digital filters in nuclear pulse signal processing

Huai-Qiang Zhang，

Bin Tang，

He-Xi Wu，

Zhuo-Dai Li

Nuclear Science and Techniques

Vol.30, No.10

Article number 145

Published in print 01 Oct 2019

Available online 27 Sep 2019

DOI：10.1007/s41365-019-0679-y

45000

The Sallen–Key filter (S–K) is widely used in nuclear pulse signal processing because of its simple working principle and good performance. Related research has only reviewed the recursive numerical model of digital S–K using idealized parameters. The use of digital S–K thus has limitations under these circumstances. This paper comprehensively deduces a recursive numerical model of digital S–K and discusses the effects of resistance and capacitance on the filter quality factor, cutoff frequency and amplitude-frequency response. The numerical recursive function, transfer function and amplitude-frequency response are analyzed using different parameters. From a comparative analysis of the shaper in a simulation and an actual nuclear signal, an optimal parameter selection principle is obtained. Using different forming parameters, the energy resolution and pulse counting rate of the ⁵⁵Fe energy spectrum are compared and analyzed based on a Si-PIN detector. Capacitance has a stronger influence on the Gaussian shape, whereas the influence of resistance is stronger on the shaping amplitude.

Digital Sallen–KeyAmplitude-frequency responseGaussian shapingEnergy resolution

1 Introduction

As a second-order integration circuit, S–K is widely used in the filtering process of nuclear pulse signals due to its simple shaping and high signal-to-noise ratio. Especially in nuclear energy spectrum measurement systems, the negative nuclear signal is shaped into a quasi-Gaussian shaper [1-9]. In recent years, with the development of high-speed and high-resolution analog-to-digital converters (ADC) and high-performance digital signal processors, digital nuclear energy spectrum measurement systems have been an area of research focus. The recursive function of S–K in the time domain has been studied [10]. The shaping algorithm based on digital S–K is widely used in the Gaussian shaping process of nuclear pulse signals [11-15], but the values of the resistor and capacitor are idealized in the specific derivation process. For example, the values of the six resistors and two capacitors have been made exactly equal. The digital S–K recursive function model has been improved, and is actively used in Gaussian shaping and filtering of nuclear pulse signals [16, 17]. The improvement, however, is limited in that the gain adjustment of resistors are set to different values, and the values of the four resistors and two capacitors are also equal.

The above two digital S–K methods can effectively realize Gaussian shaping of the nuclear pulse signal and smoothing of the energy spectrum signal. However, the quality factor and cutoff frequency of the filter are not studied using different resistance and capacitance parameters. No studies have been done on the recursive function, amplitude-frequency response, and application of nuclear pulse processing when all the resistors and capacitors are not equal in the S–K filter. Based on this, the digital S–K shaping recursive numerical general model has been deduced, and its application to nuclear pulse signal processing is further examined.

2 Shaping principle of Digital S–K

The S–K filter is a second-order integration filter circuit. A schematic diagram is shown in Fig. 1.

Fig. 1.

Scheme of Sallen–Key filter circuit.

As shown in Fig. 1, V_i is the input voltage of Sallen–Key filter circuit, V_R is the voltage between R₁ and R₂, V_n is the negative input voltage of operational amplifier, V_p is the positive input voltage of operational amplifier, and V_o is the output voltage of Sallen–Key filter circuit. According to Kirchhoff’s current law, the differential equation of the established S–K filter is as shown in Eq. (1).

A \cdot \frac{d^{2} V_{o}}{d t^{2}} + B \cdot \frac{d V_{o}}{d t} + V_{o} = D \cdot V_{i}

(1)

As shown in Eq. (1), A=R₁R₂C₁C₂, B=R₁C₂+R₂C₂+R₁C₁(1-D), D=(R₃+R₄)/R₃. The transfer function, cutoff frequency and quality factor of S–K filter are as shown in Eq. (2), Eq. (3) and Eq. (4) respectively.

H (s) = \frac{D}{A \cdot s^{2} + B \cdot s + 1}

(2)

f_{c} = \frac{1}{2 π \sqrt{R_{1} R_{2} C_{1} C_{2}}}

(3)

Q = \frac{\sqrt{R_{1} R_{2} C_{1} C_{2}}}{R_{1} C_{2} + R_{2} C_{2} + R_{1} C_{1} (1 - D)}

(4)

When V_i is converted into a sequence of numbers of xi, V_o is converted into a sequence of numbers of yi, further derivation of Eq. (1) can be obtained.

A \cdot {y^{'}}^{'} + B \cdot y^{'} + y = D \cdot x

(5)

According to the differential numerical analysis, Eq. (5) can be changed into Eq. (6) and Eq. (7).

A \cdot \frac{\frac{y_{i} - y_{i - 1}}{Δ t} - \frac{y_{i - 1} - y_{i - 2}}{Δ t}}{Δ t} + B \cdot \frac{y_{i} - y_{i - 1}}{Δ t} + y_{i} = D \cdot x_{i}

(6)

y_{i} = \frac{(\frac{2 A}{Δ^{2} t} + \frac{B}{Δ t}) y_{i - 1} - \frac{A}{Δ^{2} t} y_{i - 2} + D x_{i}}{\frac{A}{Δ^{2} t} + \frac{B}{Δ t} + 1}

(7)

Where R₁=mR, R₂=R, C₁=nC, C₂=C, $k = \frac{R C}{Δ t}$ (Δt is sampling interval of ADC), $Q = \frac{\sqrt{m n}}{m + 1 + m n (1 - D)}$ , $f_{c} = \frac{1}{2 π R C \sqrt{m n}}$ . According to A=R₁R₂C₁C₂, B=R₁C₂ +R₂C₂ + R₁C₁ (1-D), D = (R₃R₄)/R₃, Eq. (7) can be changed into Table. 1. The numerical recursive model, transfer function and amplitude-frequency response of S–K filter are shown in Table. 1 at different resistance and capacitance values.

The recursive model, transfer function and amplitude-frequency response of S–K filter at different resistance and capacitance

Shaping parameters(m, n, D, k)	Recursive model (R)	Transfer function (T)	Amplitude-frequency response (A)
R₁ = mR,R₂ = R	R: $y_{i} = \frac{(2 m n k^{2} + (m + 1 + m n - m n D) k) y_{i - 1} - m n k^{2} y_{i - 2} + D x_{i}}{m n k^{2} + (m + 1 + m n - m n D) k + 1}$
C₁ = nC,C₂ = C	T: $H (z) = \frac{D}{m n k^{2} {(1 - z^{- 1})}^{2} + (m + 1 + m n - m n D) k (1 - z^{- 1}) + 1}$
D = (R₃ + R₄)/R₃	A: $\| H (e^{j w}) \| = D \sqrt{\frac{1}{1 + 6 M + N - (8 M + N + 2 k^{2} m n) \cos (w) + (2 M + 2 k^{2} m n) \cos (2 w)}}$
k = RC/Δt	M = k⁴m²n² + (((1 - D)n² + n)m² + mn)k³
	N = 2k((1 + ((1 + (1 - D)n)m)²k + (2 + (1 + 2(1 - D))n)m)k + 1 + (1 + (1 - D)n)m)
(r)1-1 R₁ = R₂ = R	R: $y_{i} = \frac{(2 n k^{2} + (2 + n - n D) k) y_{i - 1} - n k^{2} y_{i - 2} + D x_{i}}{n k^{2} + (2 + n - n D) k + 1}$
C₁ = nC,C₂ = C	T: $H (z) = \frac{D}{n k^{2} {(1 - z^{- 1})}^{2} + (2 + n - n D) k (1 - z^{- 1}) + 1}$
D = (R₃ + R₄)/R₃	A: $\| H (e^{j w}) \| = D \sqrt{\frac{1}{1 + 6 M + N - (8 M + N + 2 k^{2} n) \cos (w) + (2 M + 2 k^{2} n) \cos (2 w)}}$
k = RC/Δ t	M = k⁴n² + ((1 - D)n)² + 2n)k³
	N = 2k((4 + ((1 - D)n)²k + (1 + 4(1 - D))nk + (2 + (1 - D)n))
R₁ = mR,R₂ = R	3lR: $y_{i} = \frac{(2 m k^{2} + (2 m + 1 - m D) k) y_{i - 1} - m k^{2} y_{i - 2} + D x_{i}}{m k^{2} + (2 m + 1 - m D) k + 1}$
C₁ = C₂ = C	T: $H (z) = \frac{D}{m k^{2} {(1 - z^{- 1})}^{2} + (2 m + 1 - m D) k (1 - z^{- 1}) + 1}$
D = (R₃ + R₄)/R₃	A: $\| H (e^{j w}) \| = D \sqrt{\frac{1}{1 + 6 M + N - (8 M + N + 2 k^{2} m) \cos (w) + (2 M + 2 k^{2} m) \cos (2 w)}}$
k = RC/Δ t	M = k⁴m² + ((2 - D)m² + m)k³
	N = 2k((1 + ((2 - D)m)²k + (1 + 2(2 - D))mk + (1 + (2 - D)m))
R₁ = mR,R₂ = R	R: $y_{i} = \frac{(2 m n k^{2} + (m + 1) k) y_{i - 1} - m n k^{2} y_{i - 2} + x_{i}}{m n k^{2} + (m + 1) k + 1}$
C₁ = nC,C₂ = C	T: $H (z) = \frac{1}{m n k^{2} {(1 - z^{- 1})}^{2} + (m + 1) k (1 - z^{- 1}) + 1}$
D = 1	A: $\| H (e^{j w}) \| = \sqrt{\frac{1}{1 + 6 M + N - (8 M + N + 2 k^{2} m n) \cos (w) + (2 M + 2 k^{2} m n) \cos (2 w)}}$
k = RC/Δ t	M = k⁴m²n² + (mn + m²n)k³
	N = 2k(((1 + m² + (2 + n)m)k + (1 + m))
R₁ = R₂ = R	3lR: $y_{i} = \frac{(2 k^{2} + (3 - D) k) y_{i - 1} - k^{2} y_{i - 2} + D x_{i}}{k^{2} + (3 - D) k + 1}$
C₁ = C₂ = C	T: $H (z) = \frac{D}{k^{2} {(1 - z^{- 1})}^{2} + (3 - D) k (1 - z^{- 1}) + 1}$
D = (R₃ + R₄)/R₃	A: $\| H (e^{j w}) \| = D \sqrt{\frac{1}{1 + 6 M + N - (8 M + N + 2 k^{2}) \cos (w) + (2 M + 2 k^{2}) \cos (2 w)}}$
k = RC/Δ t	M = k⁴ + (3 - D)k³
	N = 2k((1 + (3 - D)²)k + (3 - D))
R₁ = R₂ = R	R: $y_{i} = \frac{(2 k^{2} + k) y_{i - 1} - k^{2} y_{i - 2} + 2 x_{i}}{k^{2} + k + 1}$
C₁ = C₂ = C	T: $H (z) = \frac{2}{k^{2} {(1 - z^{- 1})}^{2} + k (1 - z^{- 1}) + 1}$
R₃ = R₄(D = 2)	A: $\| H (e^{j w}) \| = 2 \sqrt{\frac{1}{1 + 6 M + N - (8 M + N + 2 k^{2}) \cos (w) + (2 M + 2 k^{2}) \cos (2 w)}}$
k = RC/Δ t	M = k⁴ + k³
	N = 2k(2k + 1)

(1) R₁ =mR, R₂=R, C₁=nC, C₂=C, D=1, and $Q = \frac{\sqrt{m n}}{m + 1}$ , $f_{c} = \frac{1}{2 π R C \sqrt{m n}}$ .

(2) R₁ =mR, R₂=R, C₁=C₂=C(n=1), and $Q = \frac{\sqrt{m}}{2 m + 1 - m D}$ , $f_{c} = \frac{1}{2 π R C \sqrt{m}}$ .

(3) R₁=R₂=R(m=1), C₁=nC, C₂=C, and $Q = \frac{\sqrt{n}}{2 + n (1 - D)}$ , $f_{c} = \frac{1}{2 π R C \sqrt{n}}$ .

(4) R₁=R₂=R, C₁=C₂=C(m=n=1), and $Q = \frac{1}{3 - D}$ , $f_{c} = \frac{1}{2 π R C}$ .

(5) R₁=R₂=R₃=R₄, C₁=C₂=C(m=n=1, D=2), and Q=1, $f_{c} = \frac{1}{2 π R C}$ .

As shown in Table. 1, the values of the resistor and capacitor parameters are divided into five cases.

3 Amplitude-frequency response of S–K

After the input signal is processed by the S–K numerical recursive model function, the amplitude-frequency response curves under different parameters are analyzed in order to obtain the output variation of the filter shaping output pulse amplitude at different frequencies. The amplitude-frequency response curve of digital S–K with different parameters is shown in Fig. 2.

Fig. 2.

Frequency response curve at different parameter.

Fig. 2 shows a low-pass filter. When the value of D is increased, the amplitude response increases but the cutoff frequency of the low-pass filter is basically the same. When the value of k is increased, the amplitude response decreases, the width of shaping output is wider and the cutoff frequency is decreased.

By comparing Fig. 2(a) with Fig. 2(b), when m=1, as the value of n increases, the cutoff frequency of the filter decreases, and the amplitude response increases. By comparing Fig. 2(a) with Fig. 2(c), when n=1, as the value of m increases, the cutoff frequency of the filter is less, and the amplitude response is the same. By comparing Fig. 2(b) with Fig. 2(d), when n=2, as the value of m increases, the cutoff frequency of the filter is less, and the amplitude response is greater.

From Fig. 2, it can be seen from the amplitude-frequency response curves under different conditions that the shape output from digital S–K is affected by the parameters m, n, D, and k. The effect of the parameter n on the filter shaping output pulse is greater than the parameter m. The amplitude of the shaping output pulse is mainly determined by the value of D, and the width of the shaped output pulse is mainly determined by the value of k. The best method for selecting the above parameters in order to obtain the optimal filter forming performance will be discussed in the following sections.

4 Selection principle of shaping parameters

For the digital S–K filter, when R₁ = m R, R₂=R, C₁ = nC, C₂=C, D=(R₃+R₄)/R₃, the quality factor $Q = \frac{\sqrt{m n}}{m + 1 + m n (1 - D)}$ , and the cutoff frequency $f_{c} = \frac{1}{2 π R C \sqrt{m n}}$ .

4.1 Relationship between quality factor and parameters

If Q becomes negative, the poles move into the right half of the s-plane, which leads to oscillation in the circuit. When $D < 1 + \frac{m + 1}{m n}$ , Q will be a positive number. For example, let m=n=1, 1 le; D le;3, m=2, n=1, 1 le; D le;2.5, m=1, n=2, 1 le; D le;2, m=n=2, 1 le; D le;1.75. When m and n are between 1 and 2 and D is between 1 and 1.75, the relationship between Q and m, n is as shown in Fig. 3(a) and Fig. 3(b) respectively.

Fig. 3.

Relation diagram of m, n and Q.

As can be seen from Fig. 3(a), as m increases, Q increases continuously, and a larger value of n results in a faster increase of Q. As can be seen from Fig. 3(b), as n increases, initially Q remains substantially constant, followed by a sharp increase at higher values of n, and larger values of m result in a faster increase in Q.

4.2 Relationship between cutoff frequency and parameters

The cutoff frequency decreases as m and n increase. If R=10 kΩ, C=10 nF, then RC=10 μs, and the cutoff frequencies at different m and n are as shown in Fig. 4.

Fig. 4.

Relation diagram of m and f_c.

As shown in Fig. 4, as m increases, f_c decreases continuously. With larger n, the downward trend is more obvious. The relationship between f_c and n is consistent with Fig. 4.

5 Digital S–K shaping of nuclear signals

To verify the effect of digital S–K shaping, the S–K shaping under different parameters is taken offline and the optimal shaping parameters are obtained for the simulated and the actual sampled nuclear signal. The specific shaping process is as shown in Table 1. R is the recursive model, xi is the digitized nuclear pulse signal, and yi is the shaped signal, and the shaping parameters are m, n, D, k, respectively.

5.1 Digital S–K shaping of simulated nuclear signals

In digital S–K shaping, the simulated nuclear signal with different parameters is processed by the digital S–K. When m=1, n=1, the shaping output of a simulated nuclear signal with different D and k is as shown in Fig. 5.

Fig. 5.

Shaper of simulated signal at different parameters.

As can be seen from Fig. 5(a), when m=1, n=1, a larger D results in a higher shaping output pulse amplitude. With a larger k, the shaping output pulse is wider and the amplitude is lower; the shaping output is closer to the Gaussian shape.

As can be seen from Fig. 5(b), the shaping rule is described in Fig. 5(a), and when n=2, the shaping output approaches the Gaussian shape. In other words, when n is larger, the shaping output is closer to the Gaussian shape.

As can be seen from Fig. 5(c), the shaping rule is described in Fig. 5(a). As m increases, the pulse of the shaped output is wider and the amplitude is lower, which is not conducive to subsequent pulse pile-up identification and amplitude extraction.

As can be seen from Fig. 5(d), when m=2, n=2, the shaped output satisfies the shaping law shown in Fig. 5(a). As the m and n increase, the shaping output is closer to the Gaussian shape, but at this time, with the increase of D, the shaping output oscillates. This affects subsequent pulse pile-up identification and amplitude extraction.

It can be seen from Fig. 5 that as n increases, the shaping output approaches the Gaussian shape, but the shaping output has oscillation. As m increases, the amplitude of the shaping output pulse decreases. As m and n increase, the shaping output peak position moves to the right.

In summary, when using digital S—K, the pulse shaping output amplitude is affected by D and k. For larger D and smaller k, the amplitude of the shaping output pulse is higher. The shape of the pulse shaping output is affected by n and k. For larger n and larger k, the shaping output is closer to the Gaussian shape; m has a smaller influence on the amplitude of the shaping output.

5.2 Digital S–K shaping of actual nuclear signals

The Moxtek company’s Si-PIN detector of XPIN-XT type is used to measure ⁵⁵Fe, and the τ=3.2 μs, the obtained digital nuclear pulse signal is processed using digital S–K. When D=1.5 and k=20, the shaping outputs at different m and n are as shown in Fig. 6.

Fig. 6.

Shaper of actual nuclear signal at different parameters.

As can be seen from Fig. 6(a), when m is 1 or 2, the shape of the shaping output signal is substantially the same at n=1, and when n=2,the shaping output is closer to a Gaussian shape. At this time, when m=2, the shaping output oscillates. Given these results, m=1 and n=2 are the best choices for the parameters.

It can be seen from Fig. 6(b) that with an increase in k, the shaping output amplitude is lower, the shape is closer to the Gaussian shape, the shaping output appears the zero crossing signal, which affects the extraction of the subsequent pulse amplitude, it can be considered as k=40.

It can be seen from Fig. 6(c) that as D increases, the shaping output amplitude is lower, and the shaper is closer to the Gaussian shape, but the oscillation of shaping output is more severe. When D=1.2, there is no oscillation, but when D=1.5, the shaping output oscillates. In order to comprehensively the shaper and amplitude of the output, a shaping output of 1.3 to 1.45 is used for D, and the shaping output is therefore as shown in Fig. 6(d).

As shown in Fig. 6(d), when considering the amplitude, shape and oscillation of the shaping output, the optimum shaping parameters are m=1, n=2, D=1.4,and k=40. For the nuclear pulse signal using different detector systems, which use digital S–K, the above-mentioned optimal shaping parameters can be referred to. The optimal digital S–K shaping parameters can also be obtained by the shaping process shown in Fig. 6.

6 Experimental example

⁵⁵Fe is measured using a Si-PIN detector. For the nuclear signal obtained by actual sampling, when D=1.4, digital S–K Gaussian shaping is performed using different parameter values (m, n, k), and the acquired energy spectrum is shown in Fig. 7. The specific energy resolution and pulse counting rate are shown in Table. 2.

Performance comparison of energy resolution and counts.

Shaping parameters	Energy resolution (eV)	Pulse counting rate
m=1, n=1, k=20	172	1803
m=1, n=2, k=20	168	1836
m=1, n=2, k=40	166	1770

Fig. 7.

Energy spectrum of different parameters.

From Table. 2, when m=1, n=1, k=20, the energy resolution and pulse counting rates are respectively 172 eV and 1803. When m=1, n=2, k=20, the energy resolution and pulse counting rates are respectively 168 eV and 1836. When m=1, n=2, k=40,the energy resolution and pulse counting rates are respectively 166 eV and 1770. It can be seen that when k is higher, the energy resolution is higher, and when n is higher, the pulse counting rate is higher.

When k is higher, the shaped pulse is wider, and the noise suppression performance and the high energy resolution are better. However, due to the pile-up identification, the counting rate is reduced. When n is higher, the shaping output approaches the Gaussian type more quickly, the shaped pulse is narrower, and the counting rate is increased.

7 Conclusion

The numerical recursive function formula of a digital S–K filter is deduced in this paper for the first time, and the quality factor, cutoff frequency and amplitude-frequency response of the filter at different resistance and capacitance settings are analyzed. A simulated nuclear signal and an actual sampled nuclear signal are processed by digital S–K shaping at different parameters. Results shows that when the parameters of n and k are larger, the shaping output is closer to the Gaussian shape, and the parameter of m has little effect on the shaper of the shaping output. When D is larger, the amplitude of the shaping output pulse is higher. However, as D increases, the quality factor (Q) of the filter may become negative, and oscillation occurs. The optimal shaping parameters are shown to be: m=1, n=2, D between 1-1.4, k between 20-40. The specific selection of parameters can balance the advantages and disadvantages between energy resolution and pulse counting rate in energy spectrum measurement.

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