We must observe also that this kind of shaping is

Info icon This preview shows pages 269–271. Sign up to view the full content.

We must observe also that this kind of shaping is nonrealistic, due to the bilateral infinite width. Nevertheless, the results of the “optimum filtering” theory are very useful because they represent the limit to be approached by the practical filters. A discussion about practical filter is given in Section “Practical Signal Processing”. Practical Signal Processing A practical signal processor can only “approach” the optimum ENC obtained with the ideal optimum filters described in Section “Optimum Signal Process- ing” (p. 245). for any filter, characterized by the shape factors A 1 , A 2 , and A 3 , it can be seen from (4.25) and (4.30) that its optimum shaping time τ opt is related to the noise corner time constant τ c (which is the optimum shaping time for the ideal cusp-shaped filter) by τ opt τ c = A 1 A 3 . (4.34) By comparing (4.26) and (4.32) it is possible to see how worse is the ENC obtainable with the considered filter with respect to the one obtainable with an ideal filter. In particular, if only white noise is present, it turns out that ENC 2 w ( τ opt ) ENC 2 w opt = A 1 A 3 . (4.35) If also 1 /f noise is present, the worsening of the ENC with respect to the ideal case is less direct to see. It can be shown that the ENC obtainable with the considered filter satisfies the following inequality ENC 2 ( τ opt ) ENC 2 w opt (1 + K ) A 1 A 3 , (4.36) where the factor K , which is related to the amount of the 1 /f noise present, has been defined in (4.33). In the following the shape factors of some practical filters are given. Nowa- days, the most used filtering amplifier is the “semi-Gaussian” one based on a constellation of complex poles. Semi-Gaussian filters based on real poles and CR–RC filters can still be found in many laboratories. An important filter is the trapezoidal one, which can nowadays be also implemented as a time-variant parameter circuit. The triangular filter is important for didacti- cal reasons (it is easy to make mathematics and determine the shape factors with this filter as well as with CR–RC filter).
Image of page 269

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

X-Ray Detectors and XRF Detection Channels 249 It should be noted that nowadays digital signal processing (DSP) tech- niques allow practically the implementation of any weighting function, with any kind of shape constraint (e.g. finite width of the impulse response, flat top, etc.). See 4.2.11 Appendix 3 (pp. 259–262) for a short introduction to the digital filtering techniques in X-ray spectroscopy. 4.2.7 Shape Factors of some Filtering Amplifiers The ENC can be easily evaluated in the case of some practical signal proces- sors, once the “shape factors” are known. We consider a few typical cases and the values of the shape factors for each filter are given in Table 4.2. CR–RC Shaping The CR–RC filter is the simplest among the shaping amplifiers. Nowadays, it is very seldom used because other higher order filters giving a better ENC can be easily implemented. The output pulse of this shaper, fed by a unitary amplitude step-like pulse, is v uso ( t ) = t τ c exp t τ c , (4.37) where τ c = RC . The peak value of the output signal is obtained at
Image of page 270
Image of page 271
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '14
  • MichaelDudley

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern