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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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**Input Signal: Analysis of an extreme case**

I beam Input Signal: Very narrow single bunch Spectrum -> ∞ DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**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 DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**Sampling Frequency Evaluation**

MiniCircuits Reference Data: -20 13,6 MHz -40 15,6 MHz Fs – FC = 3 FC FS = 4 FC DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**Angela Salom Sarasqueta**

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 = 4FC all the spectrum is distorted, but with a negligible effect at very low frequency DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**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 0th harmonic which we are interested in. We might have a solution by multiplying 0 by ∞ (integrator) at DC DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**Digital Low Pass Filter (II)**

Integrator + Notch Filter = CIC Filter! DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**Angela Salom Sarasqueta**

Summary Typical Values for LEIR: FREV Є [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 hck ≥ Fs/FREV hck could remain = 64 if 91.5 MHz can be handled by the circuit. DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**Digital Filter Simulation**

1st Order CIC 2nd Order CIC CIC1 amplitude response 0.005 0.01 0.015 CIC1 Phase response RelativeFrequency [ ] Phase [rad] CIC2 amplitude response 0.005 0.01 0.015 CIC2 Phase response Relative Frequency [ ] Phase [rad] 0.005 0.01 0.015 0.1 10 100 Relative Frequency [ ] Gain [ ] 0.005 0.01 0.015 0.1 10 100 Relative frequency [ ] Gain [ dB] |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 DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**Digital Filter Simulation II**

3rd Order Butterworth Filter Filter Coefficients fco:= Normalized to the sampling frequency |But (FREV) | = -89 dB |But (0) | = 0 dB Group Delay ζ = 8 µs DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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**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. DDC Signal Processing Alfred Blas Angela Salom Sarasqueta

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