1. The RF Phase Determination for MICE Alexander Dick For the MICE RF Team

2. Relative Phase and Amplitude error of RF cavities are know to 0.5º and 1% Muons can arrive in the cavity at any point. –so it could arrive at a phase that was not optimal for acceleration Phase of the cavity on each muon transit will have to be measured –This will allow the RF gradient experienced by each muon to be determined –Will allow verification of sustained cooling by selecting correct particles for analysis The Timing Problem

3. Do we need to know the Absolute phase of each Muon? –Relationship between cavity centre and detector point. –Cable delays from all detectors and the recorders –Scintillation delay in TOF slabs Is it enough to be able to ‘bucket’ the muons by the acceleration gradient that they experience? –Dependence on track reconstruction –Use variation in axial momentum from exit to entrance vs RF phase to calibrate ToF-RF phase detector Absolute Phase vs. Phase Bunching

4. Frequency of RF – 201.25 MHz –1 Period of RF ~ 5ns –Nyquist limit implies ~1GSa/s for 1 ms –1MB/s –Capture, transfer and storage in 1 sec? –Undersampling preferable. Possible but signal can be reconstructed from undersampled data. –Bandwidth of signal is ~ 10’s kHz –Nyquist comfortably satisfied with ~ 100’s kSa/s –Potential to reduce raw data handling by large factor Accuracy requirements –Require better than <20ps ideally (especially stability) –Does not compromise accuracy of ToF system Sampling

5. Subsampling Amplitude 1 0.5 0 Time/ms -100 0 100 201.25MHz, 1ms, Sine Wave with ~100us ramps used for analysis

6. Subsampling Amplitude 1.25 MHz Alias 18.75 MHz ‘Image’ Alias Frequency MHz 200 10 0 600 1200 1800 FT of 201.25MHz pulse digitised at 20MSa/sec

7. Subsampling Amplitude 341 Time/us 339337 335 -100 0 +100 Comparison of initial sub sampled record with IFT of its FT

8. How to ‘rebuild’ the signal from Undersampled data In Subsampling – Can we rebuild a wave with required accuracy? Subsampled Signals Express a different (alias) frequency compared to a signal sampled at Nyquist or above. –Nyquist rate is 2-3 x carrier signal frequency –In our case ~ 400-600 MSa/s at the bare minimum

9. Fourier plot shows distinct peaks at aliases of the sample/carrier frequency However, off these peaks – the energy is very close to zero. The FT peaks of a sub-sampled signal can be ‘mapped’ to the correct frequency –Will the IFT of the mapped data reproduce the original input data with adequate accuracy? How to ‘rebuild’ the signal from Under sampled data

10. Take peaks from Subsampled FT Signal ‘Pad’ out the data (with zeros) to ‘mimic’ higher sampled FT –Essentially this moves the alias signals to the real frequency Perform IFT on this ‘padded’ signal Compare this to a Higher sampled signal –Shown next with a 20MSa/s signal ‘mapped’ to a 2GSa/s signal How to ‘rebuild’ the signal from Under sampled data

11. FT Mapping Amplitude Time/ms 0.51940.5190.5186 0.5182 -100 100 0 Comparison between Sine waves sampled at 2G.Sa/sec and 20M.Sa/sec

12. Frequency Domain Padding 201.25 MHz Signal 1.7988 GHz Image Amplitude Frequency GHz 20 1 0 6000 12000 18000 Fourier Transform of signal sampled at 2GSa/sec

13. Frequency Domain padding 201.25 MHz Signal 1.7988 GHz Image Amplitude Frequency GHz 20 1 0 6000 12000 18000 Comparison between FT of 2G.Sa/sec and padded 20M.Sa/sec data

14. FT Mapping Amplitude Frequency GHz 2.0125 0 6000 12000 18000 Comparison between FT of 2G.Sa/sec and padded 20M.Sa/sec data

15. Time Domain Comparison Amplitude Time/ms 0.402470.40246 0.40245 0.40244 -100 100 0 Comparison of 2GSa/sec Sine wave with IFT of padded 20MSa/sec data 10ns

16. Previous Graphs have been at computer precision In reality the data will be acquired on digitisers –Bit depth likely to be 8 or 10 bit Will finite resolution affect accuracy of this approach? –Repeat analysis at 20M.Sa/sec –Impose finite vertical resolution on data Impact of Bit Depth

17. Pulse sampled with 8-Bit vertical resolution Amplitude 1 0.5 0 Time/ms -100 0 100

18. Comparison of 8-Bit digitised 2G.Sa/sec sine wave with IFT of padded 20M.Sa/sec data Amplitude 0 100 0.203260.2035 Time/ms 0.20327 -100

19. 8-Bit – Zero-Crossing 1 ps 2 ps 0 2 -2 Amplitude Time ms 0.5044

20. Proposed System Layout Sketch illustrates relationships of key components in STEP V configuration Work in progress: Mathematical tests of digitiser interpolation Test sensitivity to vertical resolution, temporal sample rate, noise Work in progress: Understand cable stability Work to be undertaken: Test TDC/Discriminators in 201.25 MHz environment If necessary test alternative hardware ToF 1 Cavities 1&2 RF Amp 1 LLRF Beamline HPRF RF Drive LLRF Feedback TDC’s (ToF) TDC’s (RF) Digitisers Datarecorders RF Clock Trigger Discriminators (RF) Discriminators (ToF) ToF Signals RG213 201.25 MHz LLRF MO MO Signal (RG213) Computers RF Amp 2 HPRF Cavities 3&4 RF Drive Cavities 1&2 (RG213) Cavities 3&4 (RG213)

21. RF Cabling –RG213 used in ToF system has moderate RF performance In ToF detectors, cables laid in parallel Similar environments- helps compensate for thermal sensitivity At 200 MHz VNA measurements show 20 deg phase shift from 14 o C to 30 o C for 20m- corresponds to ~14ps/m sensitivity –High Performance RF Cables exist e.g.: Czuba A. et al, Acta. Phys. Pol. A., 119, p333, also EuCARD-PUB-2011-001 LCF38-50J — 3/8” cable at 200MHz <0.1ps/m thermal sensitivity over 14- 34 o C Hardware: Cables

22. Hardware: Instruments

23. Test different sample rates for reconstruction –How low can the Sample Rate get before reconstruction is too inaccurate? Investigate limits on bit resolution –Can we achieve the required precision with realistic digitisers –Role of vertical resolution/precision, clock precision? Obtain real data to allow comparison to undersampled ‘padded’ signal –Both from Cavity tests Tests on High speed oscilloscopes –Understand the toleration to noise (real measurements and further mathematical tests) Work ongoing

24. Future Work Obtain and test Hardware –ADC’s, TDC’s, Discriminators, cabling etc. Oscilloscope testing – Bit depth, Sample rates etc. How will equipment cope in Hall Environment ? Phase relation from measurement point to centre line of cavity etc.