I would like to discuss the digital signal processing (DSP) trapezoidal shaping used in radiation measurement, contrasting it with analog Gaussian shaping.

Fig. 1

In analog systems, radiation interacting with a detector produces charge, which is then converted into voltage by a charge-sensitive amplifier. Subsequently, the voltage signal undergoes Gaussian shaping through an amplifier, often referred to as a spectroscopic or linear amplifier. Connecting the Gaussian-shaped pulse to a Multi-Channel Analyzer (MCA) and converting it to a digital signal yields a radiation spectrum.

Gaussian shaping, known for its favorable Signal-to-Noise (S/N) ratio, is widely used in amplifiers, albeit with slight variations in specifications across manufacturers. Achieving ideal Gaussian shaping, which theoretically requires infinite integrations, is impractical in real-world scenarios. The amplifier used in HPGe semiconductor detectors, specifically designed for symmetrical and sharp responses without tails, is termed a spectroscopic amplifier.

The preamplifier signal initially rises rapidly, resembling an integrated charge, but subsequently decays exponentially due to the discharge resistor. The signal-to-noise ratio for Gaussian shaping of such an exponentially decaying pulse theoretically reaches 1.12 (unrealizable Kapus shaping is 1), but achieving this requires an infinite number of integrations.

In spectroscopy amplifiers, an active filter with an adjusted quality factor (Q) performs four integrations of the first derivative within a practical range, enhancing the signal-to-noise ratio to 1.14. Adjusting the time constant allows for customization of pulse width, optimizing the S/N ratio, counting rate, and operational settings to suit specific requirements. The time from charge generation to the peak value in Gaussian shaping typically ranges from 2.2 to 2.4 times the time constant. On the other hand, digital signal processing (DSP) incorporates analog amplifier and Multi-Channel Analyzer (MCA) functionalities.

Trapezoidal shaping is predominantly utilized within DSP, an advancement over triangular shaping, offering a signal-to-noise ratio (S/N ratio) of 1.08. The rise time indicates how long it takes to ramp up to the peak, typically equating to 2.2 to 2.4 times the analog time constant. Consequently, the time to reach the peak is nearly identical for both analog and DSP systems.

Triangular shaping, characterized by a sharp peak, is more vulnerable to trajectory imperfections compared to the smoother Gaussian shaping. This is particularly noticeable in coaxial HPGe semiconductor detectors where signals rise unevenly and include very slow components. Incorporating a trapezoidal flat-top can mitigate the degradation in resolution. The duration of the flat-top can vary: it ranges from 0.6 to 0.8 microseconds for coaxial HPGe detectors and 0.1 to 0.3 microseconds for low-energy planar types like SDD. In scintillation detectors, the rise time remains largely unchanged, allowing for flat-top durations as short as 0.1 to 0.3 microseconds.

Equation (1) represents a recursive formula where an exponentially decaying pulse input into v(n ) produces a trapezoidal-shaped pulse output in s(n).

This recursive algorithm is compatible with digital signal processing using FPGA technology. By connecting v(n) (A/D converted data) to this logic, trapezoidal shaping s(n) is applied on each clock cycle. For instance, with A/D conversion operating at 100MHz and FPGA running at the same frequency, real-time trapezoidal shaping can be achieved with minimal latency.

It is widely adopted as a straightforward algorithm suitable for implementation on FPGAs.

The derivation of this recurrence formula stems from a convolution integral, where function is shifted in parallel and added to function . Here, represents an exponentially decaying pulse. The convolution of continuous functions and is defined as follows.

Since the target is a discrete signal,

The detailed explanation can be found in the paper titled "Digital Synthesis of Pulse Shapes in Real Time for High Resolution Radiation Spectroscopy," and I would suggest referring to it for further information.

Let's compare the performance of trapezoidal shaping and Gaussian shaping.

Fig. 2 displays the waveforms of the amplifier with a time constant of 6μs (green) and DSP with a risetime of 14μs (purple) for the preamplifier signal (blue). Trapezoidal shaping returns to the baseline quicker than Gaussian shaping. This allows for faster preparation to accurately measure subsequent pulses with trapezoidal shaping, resulting in a higher output count rate.

Fig. 3 Comparison of DSP15μs and AMP6μs

As expected, the counting rate is higher with trapezoidal shaping.

Fig. 4 Change in resolution

The resolution is almost the same at low counting rates. As the counting rate increases, the difference become more significant. From Fig.3 and Fig.4, it can be concluded that DSP is on par with analog systems at low counting rates, but demonstrates its true value even more at high counting rates.

DSP performance is not determined only by the recurrence formula of trapezoidal shaping.

The first factor is A/D conversion. In order to accurately sample the preamplifier's rise time of tens to hundreds of nanoseconds, a conversion frequency of 50M to 100Msps is required. Additionally, when performing spectrum analysis with 16k channels, a resolution of 14 to 16 bits is necessary.

In the qualitative analysis of spectra, the linearity of peak positions is very important, so the integral nonlinearity must be as good as possible. It is also very important to have good differential nonlinearity in order to obtain a smooth spectrum with few statistical errors.

The other factor is BLR (Base Line Restorer), which is performed after trapezoid shaping. Baseline fluctuations must be suppressed to less than 0.01% while applying feedback. It is a robust algorithm that maintains the baseline even when the counting rate increases. This BLR is one of the factors that determines resolution, and can be said to be the most difficult process. Since no definitive algorithm has been published, it is up to the manufacturer to decide what kind of algorithm is used.

Since the recurrence formula is a numerical calculation, no deterioration occurs, so the accuracy of A/D conversion and the BLR algorithm are major factors that determine the performance of the DSP. DSP has various features, and I will continue to introduce them in the future, but that's all for now.

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References
[1] E.Kowalski, Nuclear Electronics, Asakura Publishing Co., Ltd.(1971)
[2] V.T. Jordanov and G.F. Knoll, Nucl Instr. and Meth.A353(1994)261-264