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DDC Signal Processing Applied to Beam Phase & Cavity Signals Alfred Blas Angela Salom Sarasqueta 30 March 2004

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 2 Input Signal: Analysis of an extreme case I beam Input Signal: Very narrow single bunch Spectrum -> ∞

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 3 Analogue Filtering Stage The aim is to analyze one particular Harmonic of the revolution: h A F REV without being affected by the aliases Practical limitation for the analogue Filter: 52 dB/oct (Ref: MiniCircuits) Practical value for the Dynamic Range = 80 dB

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 4 Sampling Frequency Evaluation MiniCircuits Reference Data: ,6 MHz ,6 MHz F s – F C = 3 F C F S = 4 F C

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 5 Mixing Stage F S has been evaluated for a spectrum Є [0; F NOISE ] but the mixing product Є [0; F Noise + h A F REV ] will be above the Nyquist Limit. F Noise + 1*F REV will fold back to F C – F REV F Noise + h A F REV will fold back to F C – h A F REV = 0 Hz but with an amplitude lost in Noise. In this example with F S = 4 F C all the spectrum is distorted, but with a negligible effect at very low frequency

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 6 Digital Low Pass Filter As the spectrum is only expected to have components at the Harmonics of the Revolution (Beam Phase and cavity return), a Notch Filter can be used to get rid of them All the F REV harmonics are filtered even the 0 th harmonic which we are interested in. We might have a solution by multiplying 0 by ∞ (integrator) at DC

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 7 Digital Low Pass Filter (II) Integrator + Notch Filter = CIC Filter!

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 8 Summary Typical Values for LEIR: F REV Є [360 KHz; 1.43 MHz] h A max =4 (taking into account the dual harmonic mode) F C,LEIR =4*1.43 MHz = 5.72 MHz F S = h ck *F REV ≥ 4*F c = 4 * 5.72 MHz = 22.9 MHz h ck ≥ F s /F REV h ck could remain = 64 if 91.5 MHz can be handled by the circuit.

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 9 Digital Filter Simulation 2 nd Order CIC 1 st Order CIC CIC2 Phase response Relative Frequency [ ] Phase [rad] Relative frequency [ ] Gain [ dB] CIC2 amplitude response Relative Frequency [ ] Gain [ ] CIC1 amplitude response CIC1 Phase response RelativeFrequency [ ] Phase [rad] |CIC 1 (0)| = 36 dB |CIC 1 (F REV -2kHz)/F S | = - 9 dB Group Delay = 1.4 µs |CIC 2 (0)| = 72 dB |CIC 2 (F REV -2kHz)/F S | = - 18 dB Group Delay = 2.8 µs

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 10 Digital Filter Simulation II 3 rd Order Butterworth Filter f co := Normalized to the sampling frequency Filter Coefficients |But (F REV ) | = -89 dB |But (0) | = 0 dB Group Delay ζ = 8 µs

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DDC Signal ProcessingAlfred Blas Angela Salom Sarasqueta 11 Digital Filter Simulation III CIC 2 versus Butterworth Filter Conclusion: For the same attenuation at F REV ± ε, the Butterworth is more complicated and the group delay is almost 3 times higher.

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