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Digital Signal Processing

This presentation discusses the selection and optimization of a transfer function for signal processing, specifically in the context of digital signal processing. It covers various examples and techniques for improving the extraction of physical quantities from the processed signal.

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Digital Signal Processing

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  1. Digital Signal Processing J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  2. Selection of an appropriate sequence of transfer function for the processing Example: Simulated ADC response ADC gain = 1000 Delta charge injection: Time Value 120 0,34 121 0,33 122 0,33 160 0,2 161 0,2 FADCn n=50,250 Processed: Original: Fn ; n=z,N <= FADCn ; n=z,N transfer function? Optimized to extract physical quantities (charge, etc.) J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  3. Example:The moving window deconvolution transfer function For an arbitrary window of L samples : F[n] = ai * FADC[n-i] i=0,N L a0 = 1 ai = 1/TAUpreamp i = 1, L-1 (TAUpreamp in units of the sampling period) aL = -1 + 1/TAUpreamp Properties Transforms an exponential into a rectangular function of L points. J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  4. Hardware implementation: Accumulator += a * FADC[n-1] a * FADC[n-L] Data In Dual Port Memory Data Out A - B A - B Write address Sampling Clock Sampling Clock Counter Read Address Constant N-1 G[n] Simplified implementation in favorable cases In the previous example, ai = 1/TAUpreamp i = 1, L-1 (equal weight factors) The term with identical ai’s,: G[n] = ai * FADC[n-i] i=1,L-1 Add the new element at the head Reduces to : G[n] = G[n-1] + a * (FADC[n-1] – FADC[n-L] ) Remove the out of range element at the tail Value for the previous point J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  5. Deconvolution in the presence of noise Remark: For series noise, the RMS value of the noise in the resulting function is increased by a factor SQRT(2) Note: It can be demonstrated that the transfer function shown on the next slide will yield the best estimate of the trend of the “flat” portion of the deconvolution J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  6. Floating average (boxcar) filter applied to the deconvolution result Transfer function: G[n] = aj * F[n-j]; aj = 1/K j = 0, K -1 Example with K = 16; Note parameter K =>  Peaking time G[n] J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  7. Some interesting properties of the filter 1- For an input step function, the resulting shape is a symetrical trapeze with a peaking time of K and a flat-top equal to L - K 2- As long as the charge collection in the detector is shorter than L - K, the pulse shape will reach its full amplitude. => NO ballistic deficit 3- The S/N ratio is slighly better than that of an analog CR-(RC)n or pseudo gaussian filter of the same FWHM. K L J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  8. Performance summary of the « trapezoidal » filter - The S/N of the trapezoidal signal is a few % better than that of a pseudo-gaussian analog filter • For signal rise-times shorter than the parameter K, the filtered signal has zero • ballistic deficit. (Same filtered pulse height for all rise-times) • The trapezoidal signal has no « tail » . (Good behaviour for pile-up) Other considerations: As for its analog counterpart with pole-zero suppression, the transfer function is not zero for the DC or low frequency components. It requires the equivalent of a « baseline restorer », or double sampling. J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  9. Time measurement Example: the Constant Fraction Discriminator (CFD) Principle: Compensates for the time walk associated with the pulse height. Tr Threshold set at MAX * Fraction: “Black” Threshold “Blue” Threshold Δt Same for all amplitudes if Tr is constant If Tr is not constant: Use a “delay line clip” ≤ than the shortest rise time Tr1 Not Clipped Tclipped Clipped “Black” Threshold “Blue” Threshold Δt Same again! (in the case of a linear rise time) J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  10. Time measurement, digital CFD implementation example Step 1: Clip the raw data samples: F[n] = ai * FADC[n-i] ; (a0=1, aMinTr=-1) = FADC[n] – FADC[n-MinTr] i=0,N Step 2: Arm the “find Max” process when F[n] goes above a pre defined threshold (leading edge) Step 3: Find the maximum value of F[n] Step 4: Calculate the constant fraction threshold ( F[Max] * Fraction) Step 5: Produce a delayed clipped pulse shape Step 6: Find the two points of F[n] delayed on either side of the threshold level Step 7: Interpolate the value between the two points result: 1) Value of the index “n” at the crossover point 2) Time interpolation value (“vernier”) ( precision << sampling period) => “High resolution Time Stamp” J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  11. Error on the evaluation of the maximum = Nrms Error on the evaluation of the signal amplitude = Nrms Error on the evaluation of the fraction threshold = Nrms * Fraction Timing resolution in the digital CFD Sources of error in the presence of noise: Amplitude Tr fraction threshold S Δt Time J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  12. Resulting error in the evaluation of time: TError_rms = Nrms * (1+Fraction) * Tr/S Timing resolution in the digital CFD (zoom) Tr error S Notes: - Valid for analog or digital CFD - independant of digital sampling rate to first order - Error may be much smaller than the sampling rate for large signal to noise (S/Nrms) ratios fraction threshold Extra source of errors for the discrete sampling: - linear intrapolation of the rise time function Position of the sample with no noise nominal Δt Position of the sample with noise J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  13. The TIG-10 Module Characteristics: Form factor: VXI-C Interface :a) Stand-alone: VME-A24D16 :b) System: 200 MHz source synchronous LVDS Number of channels: 10 Digitizers : 100 MHz 14-bit Signal processing: Raw data - Trigger latency buffer - Data sample buffers Charge Channel: - Preamplifier decay pole deconvolution - Trapezoidal filter - Baseline restorer Timing channel - Hit detector - CFD - Trigger generate / accept logic Data flow/control: - Parameters read/write - Event builder - Communication links J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  14. Example 4: the VF48 card, (Rev 0 shown) 48 Differencial Channels FADCs: - 10 bit, 20-65 MS/sec • Interfaces • Serial LVDS • VME64 • Signal processing: • 7 Altera Cyclone FPGAs • Raw data segments • Hit detection • Charge calculation • Time stamp • Event formatting • Applications: • TPC readout • ILC prototypes • TACTIC detector • PET readout Silicon and scintillation detectors readout ASIC preamp multiplexer readout (ALPHA) J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  15. Properties of the VF48 card • Form Factor : VME 6U • Number of channels : 48 • Number of bits : 10 (12 bits under development) • Max sampling frequency : 65 MS/sec. • Max number of samples/event : 2048 (for each channel) • Interface: : 1) VME64X • 2) Source synchronous serial, 200 mbits/sec, copper (RJ45) • Common system clock : From front panel connector or serial link • Local trigger signalling output : Front panel conector or serial link • Trigger accept input : «  «  «  J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  16. Example 3, TIGRESS DAQ architecture Trigger decision Run control (parameters) System clock Optional logic signals Master Communication links Interface to computers Sub Events,(one clover or more) System concentrators TIG-C Communication links Event fragments, (one crystal) Local Collectors TIG-10 Communication links Trigger requests, Data elements: -pulse shapes - charge - time - other “features” 720+ Channels 720 Signals + Aux. Detectors J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

  17. Example 2, TIG-C serial readout module, PCB, component layer 1 RJ45 master link connector (820 Mbit/sec. Max) VME64 12 RJ45 links connector Altera Stratix FPGA J.P.Martin, Université de Montréal, ILC EndCap Meeting, Paris, Sept 12-14 2006

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