New methods to remove baseline drift in trapezoidal pulse shaping

New methods to remove baseline drift in trapezoidal pulse shaping 1 first-author 1 corresp Corresponding author, myj@cdut.edu.cn Corresponding author, myj@cdut.edu.cn myj@cdut.edu.cn 1 2 1 1 1 College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu 610059, China College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu 610059, China Chengdu Gaotong Isotope CO., LTD. (CNNC), Chengdu 610000, China Chengdu Gaotong Isotope CO., LTD. (CNNC), Chengdu 610000, China Trapezoidal pulse shaping algorithm is widely applied to improve signal-to-noise ratio (SNR), throughput and energy resolution with the properties of noise suppression, pile-up pulse separation and ballistic deficit correction. The algorithm can be acquired by z transform method which is easier for derivation. However, the baseline drift of trapezoidal pulse appears because the noise superimposes on the input signal. In this paper, two new methods based on convergence analysis and noise suppression are proposed to remove the baseline drift resulting from trapezoidal pulse shaping. Simulations and experimental tests are carried out to verify the methods. The results demonstrate that the proposed methods can remove baseline drift in trapezoidal pulse shaping. Baseline drift removal Trapezoidal pulse shaping z transform method I. INTRODUCTION 1 Digital pulse shaping algorithm is a main contents of digital spectrometer. They are injected to realize pulse shaping in field programmable gate array (FPGA). Trapezoidal pulse, with its rise time being equal to the fall time, provides a near optimum signal-to-noise ratio (SNR). The flat top of trapezoidal pulse can be flexibly set to adapt different measurement conditions. Therefore, trapezoidal pulse shaping algorithm is widely applied to improve SNR, throughout and energy resolution. The theory of trapezoidal pulse shaping has been well developed and published. Radeka designed a trapezoidal filter based on a gated filter system, which achieved good resolution with large germanium detectors at high counting rates and higher energies [ 1 1 ]. Jordanov et al . developed the recursive algorithm of trapezoidal pulse shaping by using a convolution method [ 2 2 , 3 3 ]. The algorithm based on convolution method made trapezoidal pulse shaping possible by digital means. Stein et al . used moving window deconvolution technique to realize trapezoidal and triangular shaping in digital signal processor (DSP) [ 4 4 ]. In addition, a real time digital pulse shaper, which was synthesized by a concave and a convex pulse shapes, was also implemented in programmable logic [ 5 5 ]. Also, z transform method was described to obtain the recursive algorithm of trapezoidal pulse shaping by Cosimo Imperiale [ 6 6 ]. Some simulations of the algorithm based on z transform method were discussed [ 7 7 - 10 10 7-10 ]. Recently, Trapezoidal shaper was employed in digital spectrometer for ballistic deficit correction, neutron-gamma discrimination and pile up correction [ 11 11 , 12 12 ]. Furthermore, typical shapers, including triangular, trapezoidal and cusp-like ones, were generated in a new adaptive digital shaper which enabled automatic adjustment of coefficients for shaping an input signal [ 13 13 ]. However, when the input signal is associated with noise, the baseline drift of trapezoidal pulse appears. It is especially obvious in the process of consecutive pulses. In this paper, we propose two methods to remove the baseline drift. The feasibility and accuracy of the methods are verified by simulations and experiments. The results show that the methods can remove the baseline drift in trapezoidal pulse shaping. II. TRAPEZOIDAL PULSE SHAPING 1 The trapezoidal pulse shaping algorithm was raised by z transform method [ 6 6 ]. As shown in Fig. 1 Fig. 1 , a trapezoidal pulse can be directly synthesized by Eq. (1): <math display="block" id="M1"><mrow><msub><mi>v</mi><mtext>o</mtext></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>4</mn></munderover></mstyle><msub><mi>v</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where, v 1 ( t )=( V max / t a )· tu ( t ), v 2 ( t )=- v 1 ( t-t a ) u ( t-t a ), v 3 ( t )=- v 1 ( t-t b ) u ( t-t b ), and v 4 ( t )= v 1 ( t-t c ) u ( t-t c ); t a is the rise time of trapezoidal pulse, t b - t a is the duration of flat top, t c is the total width of pulse, and V max is the height. Equation (1) can be described as Eq. (2) according to z transform <math display="block" id="M2"><mrow><msub><mi>V</mi><mtext>o</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mrow><mtext>max</mtext></mrow></msub><mi>z</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub></mrow></msup><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>c</mtext></msub></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><msub><mi>n</mi><mtext>a</mtext></msub><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where, n a = t a / T s , n b = t b / T s , n c = t c / T s , and T s is the sampling time of ADC. The input signal is defined as <math display="block" id="M3"><mrow><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>t</mi><mo>/</mo><mi>τ</mi></mrow></msup><mi>u</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>t</mi><mo>≥</mo><mn>0,</mn></mrow></math> where A is the height and τ is the time constant. The function of the input signal in z transform notation is <math display="block" id="M4"><mrow><msub><mi>V</mi><mtext>i</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo>⋅</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><mi>d</mi><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M5"><mrow><mi>d</mi><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>T</mi><mtext>s</mtext></msub><mo>/</mo><mi>τ</mi></mrow></msup></mrow></math> . The transfer function of Fig. 1 Fig. 1 can be represented as <math display="block" id="M6"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mtext>o</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>V</mi><mtext>i</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>d</mi><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub></mrow></msup><mo stretchy="false">)</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><msub><mi>n</mi><mtext>a</mtext></msub><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> then <math display="block" id="M7"><mrow><msub><mi>V</mi><mtext>o</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><msub><mi>V</mi><mtext>i</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>d</mi><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub></mrow></msup><mo stretchy="false">)</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><msub><mi>n</mi><mtext>a</mtext></msub></mrow></mfrac><mn>.</mn></mrow></math> Applying inverse z transform, we have the time domain output of v o : <math display="block" id="M8"><mrow><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi><mtext>o</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mn>2</mn><msub><mi>v</mi><mtext>o</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>−</mo><msub><mi>v</mi><mtext>o</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mo>{</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>+</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>c</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>d</mi><mo>⋅</mo><mo stretchy="false">[</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>c</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>}</mo><mo>⋅</mo><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mtext>a</mtext></msub></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi><mtext>o</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mtext> </mtext><mtext> </mtext><mi>n</mi><mo>≤</mo><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></mrow><mn>.</mn></mrow></math> The trapezoidal pulse shaping is implemented by Eq. (7). It is suitable for processing ideal input signal without noise. Generally, the input signal is superimposed by Gaussian white noise. An input signal with SNR=30 dB is simulated by Eq. (3) with A =2000 and τ =100. The result of trapezoidal pulse shaping with n a =150, n b =300, n c =450, T s =1 is shown in Fig. 2 Fig. 2 . The baseline drift of trapezoidal pulse appears when the input signal is associated with Gaussian white noise. The simulations results indicate that the lower SNR, the more serious baseline drift will be. III. SIMULATIONS AND EXPERIMENTAL TESTS 1 A. Convergence conditions 2 Figure 2 Figure 2 shows that the algorithm based on z transform method is not convergent. To remove the baseline drift, a new method based on convergence analysis is used. Letting y [ n ]= v o [ n ] and x [ n ]= v i [ n ], Eq. (7) can be written as <math display="block" id="M9"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mn>2</mn><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where <math display="block" id="M10"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mtext>a</mtext></msub></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mo>⋅</mo><mo>{</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>c</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>d</mi><mo>⋅</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>c</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>}</mo><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> The recursive equations of Eq. (8) are as follows: <math display="block" id="M11"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo stretchy="false">]</mo><mo>+</mo><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo stretchy="false">]</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>4</mn><mo stretchy="false">]</mo><mo>+</mo><mi>X</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mtext> </mtext><mo>⋮</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>y</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>+</mo><mi>X</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>y</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>+</mo><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> An equation only about y [ n ], y [ n -1], y [ 1 ], y [ 0 ] and X [ n ] is acquired by accumulating the equations above <math display="block" id="M12"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mi>y</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover></mstyle><mi>X</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mn>.</mn></mrow></math> The general formula of y [ n ] is <math display="block" id="M13"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mi>n</mi></mrow></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>y</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>+</mo><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mn>3</mn></munderover></mstyle><mi>X</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mo>⋯</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover></mstyle><mi>X</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover></mstyle><mi>X</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> Assuming that y [0]= y [ 1 1 ]=0, one can simplify Eq. (9) into Eq. (10): <math display="block" id="M14"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mn>3</mn></munderover></mstyle><mi>X</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover></mstyle><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover></mstyle><mi>X</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mn>.</mn></mrow></math> v i [ n ] is an equal ratio progression, with an equal ratio of q = e -1/ τ . It can be proved that X [ n ] is also an equal ratio progression and the equal ratio is q . Equation (10) is expressed as <math display="block" id="M15"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>+</mo><mfrac><mrow><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mrow><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mfrac><mrow><mi>X</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></mfrac><mn>.</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> Let y [ n ]= X [ 2 2 ]+ Y , where <math display="block" id="M16"><mrow><mi>Y</mi><mo>=</mo><mi>C</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>C</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where C = X [ 2 2 ]/(1- q ), and then <math display="block" id="M17"><mrow><mi>Y</mi><mo>=</mo><mi>C</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>−</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>q</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mrow></math> q < 1 as τ > 1. So, y [ n ] can be convergent when Y =0. It can be seen from Eqs. (8) and (12) that Y can be set to 0 by setting x [ 1 ] to 0. Figure 3 Figure 3 illustrates the result of trapezoidal pulse shaping with v i [ 1 1 ]=0, with the same input signal as Fig. 2 Fig. 2 . Output 1 is used as a contrast. Output 2 is the trapezoidal pulse with the first input data being 0. Therefore, the new method can remove the baseline drift resulting from trapezoidal pulse shaping. B. Noise suppression 2 The baseline drift of trapezoidal pulse is caused by noise accumulation. Filtering the original signal before shaping is another way of removing the baseline drift. Digital S-K filter performs well in signal processing with amplitude filtering and frequency filtering factors [ 14 14 ]. The true height of filtered signal can be obtained by adjusting amplitude of the filtering factor properly. The algorithm of digital S-K filter is written as <math display="block" id="M18"><mrow><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mi>k</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>3</mn><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>⋅</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><msup><mi>k</mi><mn>2</mn></msup><mo>⋅</mo><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mi>a</mi><mo>⋅</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mo>⋅</mo><mo stretchy="false">(</mo><mn>3</mn><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mi>x</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mtext> </mtext><mtext> </mtext><mi>n</mi><mo>≤</mo><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></mrow><mn>,</mn></mrow></math> where x [ n ] is the discrete input signal, y [ n ] is the output signal, k is the frequency filtering factor and a is the amplitude filtering factor. The original signal in Fig. 2 Fig. 2 is filtered by the digital S-K filter, and output of the filter is processed by calling Eq. (7) recursively. The baseline drift of trapezoidal pulse is removed with k =5, a =1.15, as shown in Fig. 4 Fig. 4 . Differential operation can also attenuate the noise. Equation (6) can be rewritten as <math display="block" id="M19"><mrow><msub><mi>V</mi><mtext>o</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>i</mtext></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>d</mi><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub></mrow></msup><mo stretchy="false">)</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><msub><mi>n</mi><mtext>a</mtext></msub><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>z</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mrow></math> Assuming V ( z ) = V i ( z )/(1-1/ z ) 2 , an improved recursive algorithm of trapezoidal pulse shaping is obtained: <math display="block" id="M20"><mrow><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mn>2</mn><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><msub><mi>v</mi><mtext>i</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi><mtext>o</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mtext>a</mtext></msub></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mo>⋅</mo><mo>{</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo>+</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>c</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><mi>d</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>a</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>b</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><msub><mi>n</mi><mtext>c</mtext></msub><mo>−</mo><mn>2</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>}</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>v</mi><mtext>o</mtext></msub><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mi>v</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mtext> </mtext><mtext> </mtext><mi>n</mi><mo>≤</mo><mn>0</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></mrow><mn>.</mn></mrow></math> Equation (15) is the improved recursive algorithm of trapezoidal pulse shaping without baseline drift [ 10 10 ]. Trapezoidal pulse can be implemented by calling Eq. (15). Figure 5 Figure 5 shows that the improved algorithm can remove the baseline drift in Fig. 2 Fig. 2 . C. Measured pulses shaping 2 In order to verify feasibility of the methods, pulses tests were carried out. The time constant of pulses is 3.2 μs, detected by silicon drift diode (SDD) detector and digitized by ADC at 20 MSPS, with T s =50 ns and τ =3.2 μs according to Eq. (7). A trapezoidal pulse with t a =4 μs, t b =4 μs and t c =8 μs corresponding to n a =80, n b =80 and n c =160 is implemented by Eqs. (7) and (15). The experimental tests of different methods are given in Fig. 6 Fig. 6 . The results indicate that both the new method and the improved recursive algorithm can remove the baseline drift in trapezoidal pulse shaping. For the purpose of signal filtering, digital S-K filter can also be used in trapezoidal pulse shaping to remove baseline drift. However, it is complex to realize digital S-K filter in FPGA. Besides, it is also difficult to select the values of k and a . Therefore, the new method and the improved recursive algorithm are recommended in practical applications. The correlation between the new method and the improved recursive algorithm is shown in Fig. 7 Fig. 7 . It indicates that the improved method is suitable for trapezoidal pulse shaping without baseline drift. IV. CONCLUSION 1 In this paper, the recursive algorithm of trapezoidal pulse shaping is derived by z transform method and the approaches to baseline drift removal are discussed. Setting the first input data to 0 and digital S-K filter are used to remove the baseline drift due to the accumulation of noise. The results of experimental tests demonstrate that the methods can remove the baseline drift existing in trapezoidal pulse shaping. A comparison between the new method and the improved recursive algorithm is also carried out. 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DOI: 10.1016/j.nima.2013.09.063 http://doi.org/10.1016/j.nima.2013.09.063 Implementation of real-time adaptive digital shaping for nuclear spectroscopy J B Zhou , W Zhou and X Hong . Improvement of digital S-K filter and its application in signal processing . Nucl Sci Tech , 2013 , 24 : 060401 . DOI: 10.13538/j.1001-8042/nst.2013.06.020 http://doi.org/10.13538/j.1001-8042/nst.2013.06.020 Improvement of digital S-K filter and its application in signal processing 10.13538/j.1001-8042/nst.26.050402 Supported by National High Technology Research and Development Program of China (863 Program) (No. 2012AA061804-03) 2014-05-08 2014-08-18 2015-10-20 2015-10-20 2021-04-11 2015 Nuclear Science and Techniques 5 26