DDC Signal Processing Applied to Beam Phase & Cavity Signals

1. DDC Signal ProcessingApplied to Beam Phase & Cavity Signals Alfred Blas Angela Salom Sarasqueta 30 March 2004

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

3. Analogue Filtering Stage The aim is to analyze one particular Harmonic of the revolution: hAFREV 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 Alfred Blas Angela Salom Sarasqueta

4. Sampling Frequency Evaluation MiniCircuits Reference Data: -20 dB @ 13,6 MHz -40 dB @ 15,6 MHz Fs – FC = 3 FC  FS = 4 FC Alfred Blas Angela Salom Sarasqueta

5. Mixing Stage • FS has been evaluated for a spectrum Є [0; FNOISE] but the mixing product Є[0; FNoise + hA FREV] will be above the Nyquist Limit. • FNoise + 1*FREV will fold back to FC – FREV • FNoise + hA FREV will fold back to FC – hA FREV = 0 Hz but with an amplitude lost in Noise. • In this example with FS = 4FC all the spectrum is distorted, but with a negligible effect at very low frequency Alfred 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 FREV harmonics are filtered even the 0thharmonic which we are interested in. We might have a solution by multiplying 0 by ∞ (integrator) at DC Alfred Blas Angela Salom Sarasqueta

7. Digital Low Pass Filter (II) Integrator + Notch Filter = CIC Filter! Alfred Blas Angela Salom Sarasqueta

8. Typical Values for LEIR: FREVЄ [360 KHz; 1.43 MHz] hA max=4 (taking into account the dual harmonic mode) FC,LEIR=4*1.43 MHz = 5.72 MHz FS = hck*FREV≥ 4*Fc = 4 * 5.72 MHz = 22.9 MHz Summary hck ≥ Fs/FREV • hck could remain = 64 if 91.5 MHz can be handled by the circuit. Alfred Blas Angela Salom Sarasqueta

9. 100 10 CIC1 Phase response CIC2 Phase response Gain [ dB] 100 10 Gain [ ] Phase [rad] Phase [rad] 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0.1 0.1 RelativeFrequency [ ] Relative Frequency [ ] Relative Frequency [ ] Relative frequency [ ] Digital Filter Simulation 2nd Order CIC 1st Order CIC CIC1 amplitude response CIC2 amplitude response |CIC1(0)| = 36 dB |CIC1(FREV-2kHz)/FS| = - 9 dB Group Delay = 1.4 µs |CIC2(0)| = 72 dB |CIC2(FREV-2kHz)/FS| = - 18 dB Group Delay = 2.8 µs Alfred Blas Angela Salom Sarasqueta

10. Digital Filter Simulation II 3rd Order Butterworth Filter Filter Coefficients fco:= 0.005 Normalized to the sampling frequency |But (FREV) | = -89 dB |But (0) | = 0 dB Group Delay ζ = 8 µs Alfred Blas Angela Salom Sarasqueta

11. Digital Filter Simulation III CIC2 versus Butterworth Filter Conclusion: For the same attenuation at FREV± ε, the Butterworth is more complicated and the group delay is almost 3 times higher. Alfred Blas Angela Salom Sarasqueta