MAXIMIZING SIGNAL-TO-NOISE RATIO (SNR) IN 3-D LARGE BANDGAP SEMICONDUCTOR PIXELATED DETECTORS IN OPTIMUM AND NON-OPTIMAL FILTERING CONDITIONS

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1 9 International Nuclear Atlantic Conference - INAC 9 Rio de Janeiro,RJ, Brazil, September7 to October, 9 ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN ISBN: MAXIMIZING SIGNAL-TO-NOISE RATIO (SNR) IN 3-D LARGE BANDGAP SEMICONDUCTOR PIXELATED DETECTORS IN OPTIMUM AND NON-OPTIMAL FILTERING CONDITIONS Miesher L. Rodrigues, Zhong He, Yuefeng Zhu, André da S. Serra Centro Tecnológico da Marinha em São Paulo, CTMSP Fuel Cycle Division Av. Professor Lineu Prestes São Paulo, SP miesher@ctmsp.mar.mil.br, aserra@ctmsp.mar.mil.br University of Michigan Nuclear Engineering and Radiological Sciences 9 Cooley 355 Bonisteel Boulevard Ann Arbor, MI hezhong@umich.edu, zhuyuef@umich.edu ABSTRACT 3-D pixelated semiconductor detectors are used in radiation detection applications requiring spectroscopic and imaging information from radiation sources. Reconstruction algorithms used to determine direction and energy of incoming gamma rays can be improved by reducing electronic noise and using optimum filtering techniques. Position information can be improved by achieving sub-pixel resolution. Electronic noise is the limiting factor. Achieving sub-pixel resolution position of the interaction better than one pixel pitch in 3-D pixelated semiconductor detectors is a challenging task due to the fast transient characteristics of these signals. This work addresses two fundamental questions: the first is to determine the optimum filter, while the second is to estimate the achievable sub-pixel resolution using this filter. It is shown that the matched filter is the optimum filter when applying the signal-to-noise ratio criteria. Also, non-optimum filters are studied. The framework of 3-D waveform simulation using the Shockley-Ramo Theorem and the Hecht Equation for electron and hole trapping is presented in this work. This waveform simulator can be used to analyze current detectors as well as explore new ideas and concepts in future work. Numerical simulations show that assuming an electronic noise of 3.3 kev it is possible to subdivide the pixel region into 5x5 sub-pixels. After analyzing these results, it is suggested that sub-pixel information can also improve energy resolution. Current noise levels present the major drawback to both achieve sub-pixel resolution as well as improve energy resolution below the current limits.

2 . INTRODUCTION Reference [] shows that 3-D large band gap semiconductor pixelated detectors are being used as radiation detection sensors in applications that require both spectroscopic and imaging information from radiation sources. Reconstruction algorithms use Compton scattering information from multiple interaction events to determine the direction of incoming gamma rays. Neighboring pixel signals can be used to improve position information resolution better than one pixel pitch. Electronic noise is the most important parameter that limits both energy and position resolution. Optimum filtering techniques improve signal-to-noise ratio, especially in applications having small signal relative to high electronic noise. This works simulates both induced signals and electronic noise in 3-D pixelated semiconductor detectors. Filtering techniques in time domain and the theory of maximization of signal-to-noise ratio, see reference [] for detailed explanations, is used in this work. The matched filter is the optimum filter. Since it is usually difficult to use the matched filter in real applications, this work also shows how to calculate the best filter in non-optimum conditions. The framework of 3-D waveform simulation in semiconductor radiation detectors was used. A program in MATLAB was written applying the Shockley-Ramo Theorem and the Hecht Equation for electron and hole trapping. This waveform simulator can be used to analyze current detectors as well as explore new ideas and concepts. One of the main advantages is the fact that fewer counts will be necessary to both locate and identify unknown sources of radiation. Then, achieving subpixel resolution fewer events will be needed to deconvolve the location of the source ameliorating the current disadvantage of semiconductor pixelated detectors. In the future, if these devices can be built in larger volumes faster and more efficient algorithms will be able to be used to take more advantage of this potential technology.. FILTERING IN TIME DOMAIN Linear systems have proved to be useful for many physical processes. This is fortunate because the principle of superposition applies to these systems, thus allowing general response simulations. Many filtering devices, independent of hardware realization, are accurately modeled by linear systems. An important member of this class is the system described by ordinary differential equations. When the system is time-invariant in addition to being linear, the coefficients of the differential equations describing it are constants. These systems are useful as models for many portions of real-life systems. The time-domain approach is useful when the problem is stated in terms of differential equations, or alternatively, when the system impulse response is defined. Time is the primary variable, thus the physical description of the filtering system should always be expressed in the time domain. After all, the input and output signals are defined in this domain. Today, with current computer capabilities, it is possible to calculate in a reasonable amount of time the response of any linear system, knowing its response to an impulse excitation, using the convolution integral. When no definite advantage is gained by the use of Laplace and Fourier transformations, the time variable should be retained, since then a physical

3 interpretation is attached to the mathematics. Optimum system performance is often only achievable mathematically by time-domain approaches... Linear Systems A system is linear if the input c f( + c f( produces an output c u ( + cu( for all f ( and f (, when it is known that an input f ( produces and output u ( and an input f ( produces an output u (. The constants c and c are arbitrary and may be complex numbers. The property of superposition is characteristic of linear systems. An ordinary differential equation with constant coefficients describes a linear time-invariant system. For this system, an input delay of T seconds leads to an output delay of T seconds. Thus if an input f ( produces an output u (, then f( t T ) produces an output u( t T ). A system is nonanticipative if the output value at any instant is independent of the input values at all later instants. This system, also referred as a causal or physically realizable system, and has the property that if the input is zero for t < t, then the output is also zero for t < t. The differential equation describing the system implicitly relates its input and output functions. However, it fails to define a unique response for each input function unless additional information about the system is given. This includes the initial system state (initial conditions), the anticipatory nature of the system (ordering of the time scale), and the domain for which the differential equation description is valid (the time interval). In general, we consider a single-input, sinle-output, linear system whose input f ( and output u ( are implicitly related on the interval t < t < t by the n th -order ordinary differential equation n n d u d u a( + a( a ( u f ( n n n = dt dt () The a i ( are assumed to be defined and continuous throughout the interval t < t < t and we assume a ( on this interval. The initial conditions for this system are expressed by the first ( n ) derivatives of u ( evaluated at t, u( t du dt M d dt n ) = t = t u n = u'( t t = t = u ) = n ( t ) = () where t is a particular instant of time within the interval t < t < t. Since the system is assumed to be nonanticipative, the initial conditions given, referred to as the zero initial conditions, imply that the system is in a relaxed state immediately prior to t =, that is, there is no output if there is no input. t

4 In general, the system initial conditions are not the zero initial conditions. However, from the theory of differential equations we can form u, the general solution of the differential (3) equation, as u = u p + u h where u p is any particular solution and u h is the sum of n linearly independent homogeneous solutions. For the particular solution we choose the one that satisfies the zero initial conditions; thus all initial conditions are satisfied by appropriately scaling the homogeneous solutions. The existence theorem for a linear differential equation guarantees that there is a unique function u ( that identically satisfies the above system of differential equations and the specified initial conditions on the interval t < t < t. This is a powerful theorem, since it tells us that not only a unique solution exists but also no matter how we find this solution, it is the correct solution... The Convolution Integral The convolution integral is of fundamental importance in the analysis and design of a linear time-invariant system, for it is an explicit representation of the system response. Let p ( and q ( each be a piecewise continuous function defined for < t <. The convolution of p and q, denoted by p q, is defined to be the third function v ( = p q = p( t τ ) q( τ ) dτ (4) and the integral on the right side is called the convolution integral. Furthermore, the convolution of p and q, p q, is identical to q p. An important special case occurs when both p ( and q ( are zero for t <, then it becomes t = p( t v( = p * q τ ) q( τ ) dτ (5) It can be shown that the solution of the differential equation that describes a linear system, as stated above, can be expressed as the convolution of its kernel function and the driving function f (. In the case of filtering in time domain, it can be shown that the kernel function is directly related to the response of the filter to an impulse function. 3. FILTER OPTIMIZATION A primary function of the electronic readout system is to convert the electronic signal generated after the interaction of the radiation within the detector into a meaningful signal.

5 Usually, a radiation induced signal is readout superimposed in a noisy electronic background. Then, its desirable to be able to distinguish the true signal from noise in an optimum fashion. Although it seems obvious the function of the electronic readout system, it is not so obvious to define optimality under many different possible scenarios. Thus, the basic signal processing problem is to define the necessary operations needed to optimally convert the input signal, information and noise, into useful information. This problem can be divided into signal detection, the determination of whether a message signal is present in the noisy environment, and signal extraction, the measurement or recovery of the important parameters contained in the message signal corrupted by noise. In many different criteria when the input noise is additive, white and Gaussian, the optimum filter is the matched filter. Although, it is only realizable in special cases, it remains useful as a standard against which other realizations can be compared. 3.. Optimum Conditions The matched is the optimum filter when the signal is corrupted by additive white Gaussian noise under a wide variety of criteria. It is the optimum filter in respect to signal-to-noise criterion. It works essentially like a decoder, recognizing a specific signal shape in the presence of noise and accordingly yields a higher output peak signal-to-mean-noise power ratio (SNR) for this signal shape than for any other signal shape with the same energy. The matched filter when the noise power density spectrum is constant, white noise, has a very simple impulse response h ( = Kf ( t (6) where f ( is the input signal and t is a constant delay. Here, h ( t ) has the same shape as the input signal f ( but inverted and delayed in time to be physically realizable. From Fig., SNR can be calculated as SNR = u( t ) max u n (7) where u ( is the voltage across a one-ohm load resistor and noise power dissipated in this same resistor. u n is the total mean output After substituting the correct terms in Equation 7, one can write Equation 8 to compute SNR. SNR = f ( t τ ) h( τ ) d τ N h ( dt max (8)

6 Figure Input-output relationship of a generic filter with response function h(. Using Schwarz inequality and Parseval's theorem, the value of SNR can be proved to be bounded, SNR f ( t τ ) dτ N (9) where the input signal energy, E, is given by E = f ( dt () The maximum achievable SNR, designated SNR M, corresponds to the response of the system when the filter impulse response is given by Equation 6, the matched filter. 3.. Non-Optimum Conditions Since optimum filters are difficult to implement, cause increased design and manufacturing cost and filter complexity, it s useful to compare non-optimum filters with respect to the degradation of SNR. This can be assessed by comparing the ratios of SNR and SNR M. As an example, considering a rectangular pulse, with amplitude A and period T, and a firstorder filter, with frequency ω, the non-optimum condition shown on Fig. is calculated by c SNR SNR M ( e = ω T c ω T c ) ()

7 Figure Results of non-optimum conditions for a rectangular pulse comparing analytical and numerical solutions. 4. RESULTS A typical 37 Cs gamma-ray induced signal on 3-D large band gap semiconductor pixelated detector, Fig.3, measured on a typical pre-amplifier with a digital oscilloscope has an amplitude around mv with an electronic noise contribution of about 4 kev to the overall energy resolution. Numerical simulations show that the signals induced on neighboring pixels, considering non-charge sharing events, have amplitudes around mv, Fig.4. Since noise starts to play a more important role at this low signal levels, it is important to use a filter that optimizes the measured signal-to-noise ratio. One way to achieve sub-pixel resolution is to compare the ratios of the positive peaks of oposite adjacent neighboring pixels. This ratio determines the relative position of the cloud under the pixel. Simulations considering CR RC, CR RC, CR RC 3, CR RC 4, CR RC 5, and CR RC 6 filters were conducted to calculate SNR/SNR M ratios to determine optimum filtering shaping times for non-optimum filtering conditions. A simple CR-RC filter was determined to be the best non-optimal filter. The optimal shaping time varies with the relative position where the electron-hole cloud is generated with respect to the distance from the cathode and anode.

8 Figure 3 3-D large band gap semiconductor pixelated detector modeled showing the anode side view with x anodes and a guard ring. Figure 4 Induced signals when the electron-hole cloud is generated under the central pixel (pixel#33).

9 Results showed that reducing electronic noise to 3.3keV it is possible to subdivide the pixel region into 5x5 sub-pixels which would improve the reconstruction algorithms used to locate radiation sources. Fig.5, as an example, shows simulated results for an electron-hole cloud starting at a position 3% of detector height from the cathode side. After analyzing the results obtained with the simulator, it was suggested that sub-pixel information can also improve the energy resolution achieved in these devices. Current noise levels present the major drawback to both sub-pixel resolution and improvement in energy resolution below the current limits. Ratio Pixel #3/ Pixel #43 Calculated σ Fractional error Figure 5 Using the best non-optimal filter, results for neighboring pixel ratios, calculated σ and fractional errors for an electron-hole cloud generated at a position 3% of detector height from the cathode side. 5. CONCLUSIONS In this work a 3-D large band gap semiconductor pixelated detector was modeled and simulated applying the Shockley-Ramo Theorem and the Hecht Equation for electron and hole trapping. After analyzing the results obtained with the simulator, it was suggested that subpixel information can also improve the energy resolution achieved in these devices. Current noise levels present the major drawback to both subpixel resolution and improvement in energy resolution below the current limits. Future simulations to determine the influence of the gamma-ray direction on induced signals on neighboring pixels can also improve the reconstruction algorithms used to determine the real direction of the incoming gamma-ray. Also, a more detailed study on charge sharing events combining this code with Monte Carlo simulations are necessary to explore the low energy tail problems found in semiconductor detectors which degrade considerably the energy resolution measured currently.

10 ACKNOWLEDGMENTS The authors wish to thanks the Brazilian Navy Technological Center (CTMSP) and the Nuclear Engineering and Radiological Science Department of University of Michigan for their support which greatly contributed to this work. REFERENCES. Glenn F. Knoll, Radiation Detection and Measurement, John Wiley & Sons, New Jersey, U.S.A. (999).. Herman J. Blinchikoff and Anatol I. Zverev, Filtering in the Time and frequency Domains, Wiley, New York & U.S.A. (975).