Deconvolution

Since future high energy colliders are designed for very frequent collisions, the response time of the amplifiers must be fast enough to distinguish particles coming from different bunch crossings. For charge-sensitive amplifiers this implies that the peaking time should be in the order of the bunch crossing period, which is challenging in terms of noise performance. An alternative solution has been developed which works with a larger peaking time. The principal target of the deconvolution method [32,33] is to restore the original detector current pulse by processing the shaper output signal with a special digital filter.

The shaper output is sampled with the bunch crossing clock frequency and stored in a pipeline. Three consecutive values are added with individual weights to numerically compensate the shaping process. With the sampled shaper output values $p_i$ and the weights $w_i$, the deconvoluted output $d_k$ becomes

$$d_{k}=w_3\,p_{k-2}+w_2\,p_{k-1}+w_1\,p_{k}\quad.$$ (2.33)

It has been shown that this procedure is exact for an integrating preamplifier followed by a CR-RC shaper using the weights

$$w_1 = A\,\frac{e^{x-1}}{x}\quad,$$ (2.34)

$$w_2 = A\,\frac{-2 e^{-1}}{x}\quad{\rm and}$$

$$w_3 = A\,\frac{e^{-x-1}}{x}$$

with the ratio between sampling time and peaking time $x=T/T_p$ and a normalization factor $A$.

The APV chip (see section , p. ) of the CMS Silicon Strip Tracker has a peaking time $T_p=50\,\rm ns$ and is clocked with the bunch crossing period of $T=25\,\rm ns$. It includes an analog pipeline and an analog pulse shape processor (APSP), which performs the deconvolution using switched capacitors.

The deconvolution method has been included in the silicon detector simulation discussed in section , p. . Fig.  shows the APV shaper output (``peak mode'') and the processed signal (``deconvolution mode''). Although the real output consists of sampled values in steps of the bunch crossing time, continuous waveforms have been calculated for easier comparison.

Figure: APV output in peak and deconvolution modes. In reality, the output is sampled in steps of $T=25\,\rm ns$ in both cases, but continuous curves have been calculated for better visualization.

In fact, the deconvoluted output, when properly timed with respect to the sampling points, is approximately zero except for one sample. Two particles, producing a signals in consecutive bunch crossings, cannot be recognized in the shaper output, but easily after deconvolution. Such a clear separation could not be obtained by simply shaping with a shorter peaking time of $T_p=25\,\rm ns$, because the long tail would result in a few non-zero samples after the peak. Thus, the deconvolution method is a powerful tool for unambiguous bunch crossing separation. This advantage has to be paid off by an increased noise figure, as discussed in the following section.