Signal Processing

Simulation Parameters

General
Parameter	Value
Bunches per train	150
Bunch separation/ns	0.667
Multiple bunch charge/nC/bunch	0.05
Single bunch charge/nC	0.6
Bunch length/μm	225
Calibration
Number of steps	5
Half-range/μm	20

Raw waveform signals at -20 μm offset for one train (left) and one single bunch (right):

Signal Processing

• Digital down conversion with 45 MHz gaussian filter:

• Digital down conversion before deconvolution:
  • Calculate the inverse frequency response of a digitally down-converted single bunch signal and multiply the fourier transform of the multiple bunch down-converted waveform.

• Digital down conversion after deconvolution:
  • Use inverse frequency response (left) of raw single bunch waveform then down-convert deconvolved waveform (right) with a DDC filter bandwidth of 64 MHz.

The result of these two deconvolution algorithms are shown below.

• Principle component analysis: The digitised waveform was split into slices of length 30 samples and a basis was found for the full calibration set, of which the first two components correspond to position variation along the train.

Calibration

The horizontal channel was used for calibration with no jitter and a ±20 μm position scan. 100 trains were simulated and there were 20 trains per step in the calibration.

Resolution

The calibration was then used on the y channel where the jitter was set to 100 nm around a constant offset of 1 μm:

The spatial resolution was determined from the horizontal data which had no position jitter. The deconvolution performs worse in terms of resolution since the noise from the single bunch data is propagated. The correlation between position measurements along the train in the y direction was used as a measure for the temporal resolution.

The time resolution was investigated from the correlation of the measurements along the train. The deconvolution has a noticeable effect, quickly reducing the correlation to the random level of .

The matrices of absolute correlations also shows improvement from the deconvolution:

In the above plots the time within the window for which the beam was present was determined from the down-converted reference signal. This worked alright since the beam started, in all cases, at the beginning of the window. When this is not the case, the timing for each algorithm must be determined individually since the deconvolution, when performed before the digital down conversion, changes the timing of the beam within the sample window depending on the timings of both the single and multiple bunch data. In the bottom two plots (DDC left, deconvolution right), there was no difference between the beam timing.

Reason for Swapping Order of Deconvolution and DDC

The deconvolution algorithm is very sensitive to the timing of the single bunch data. Preferably, the single bunch waveform should start at time, t=0. It is not so sensitive to the timing of the multibunch data. In each case, the average resolution of the digitiser samples numbered 20-200:

The solution to this turned out to be performing the deconvolution before the digital down conversion instead of after. There is also trade-off between resolution at the beginning of the bunch train for resolution towards the end of the bunch train when the order of operations is swapped.

To deal with high frequencies and improve resolution of the deconvolution algorithm, some roll-off can be added to the inverse frequency response IR of the single bunch waveform which is used for the deconvolution. This is done by varying the response over the range of frequencies such that

for ,

for ,

elsewhere

where R is the frequency response given by the fourier transform. This effect must be applied to both positive and negative frequency components. It can then be seen that the best choice of (with GHz) is 200 MHz which is the frequency of the baseband signal. The choice of (with MHz) has an effect on the resolution but also on single bunch calibration which is no longer valid when is below about 300 MHz.

Factors Affecting the Resolution

• non-linearity of the electronics
• noise
• digitiser noise

The simulation was run three times with a calibration half-range of 100μm so that the effect of the non-linearities across the intended dynamic range could be studied. The simulation was run three times, once with all contributions present, once without the non-linearities in the electronics and once with no electronics noise or non-linearities. For this, and were set to 200 MHz and 400 MHz respectively.