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Silvia Zorzetti

USPAS 2024

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Time and Frequency Domain Beam Signal

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Time and Frequency Domain

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Each musical note corresponds to a frequency

FT

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Time and Frequency Domain

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Delta Dirac δ(x)

  • The Delta Dirac is a generalized function used as unit impulse function
  • A sine wave in the time domain has infinite energy since if it continues over an infinite amount of time.
  • In the Frequency domain all this energy is concentrated on a single frequency.
  • The Delta’s Dirac can be seen as the limit of the normal distribution

 

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Fourier Synthesis

A periodic function f(x) can be expressed as a series of harmonics, weighted by Fourier coefficients cn

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Particle beam with gaussian longitudinal distribution

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A0, pick amplitude

σt

bunch length

σf

 

Time domain

 

Frequency domain

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Train of gaussian bunches (1)

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Time domain

 

Frequency domain

 

DC, beam intensity

f0

2f0

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Train of gaussian bunches (2)

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Discrete Fourier Transform (DFT)

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  • T: signal length
  • N: number of samples

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Discrete Fourier Transform (DFT)

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DFT properties (3)

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DC error

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DFT properties (1)

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Longer signal in time, better frequency resolution

For a wider spectrum increase the signal resolution

 

 

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DFT properties (4)

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DC error

 

 

 

 

 

 

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DFT properties (2)

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Odd harmonics

 

 

Sampling rate

 

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Exercise 1

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Sampling of a continuous signal

In many applications there is a significant advantage in converting a signal in discrete-time and digitize for further signal processing.

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Convolution in frequency

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Analogue to Digital Conversion

  • Analogue to Digital Converter (ADC)
    • From continuous to discrete time
    • Sampling rate: number of output samples per unit time
    • Resolution: minimum change of input voltage that can be resolved by the ADC
      • Related to the number of bits in the digital output

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Quantization

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Quantization Error (1)

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Rounding the function to the closest integer number

 

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Quantization Error (2)

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Histogram of N samples of quantization error.

The statistics are approximately uniform.

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Quantization Error (2)

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Histogram of N samples of quantization error.

The statistics are approximately uniform.

 

 

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Least significant bit (LSB)

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Q or less significant bit is the analog input corresponding to the threshold of LSB

 

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Keep this in mind ... Will come back!

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Signal to Noise Ratio (SNR) (1)

Signal + Noise

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Signal + More Noise

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Signal to Noise Ratio (SNR) (2)

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The temperature of the system determines the noise floor.

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Thermal Noise Power:

N = kTB

K: Boltzmann constant

T: temperature

B: noise bandwidth

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Ratio between the desired signal and the background

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Signal to Quantization Noise Ratio

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The SNR of quantization increases by >6dB for every bit added to the quantizer

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14-16 bit ADC Technology (2018)

  • Dual Channel
    • I-Q sampling with separate ADCs
  • Pipeline architecture
    • Continuous CLK
    • Data latency
  • .
  • Signal post-processing
    • Mixers, NCO, CIC, etc

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Type

Res.�[bit]

Ch.

Power

[W]

fs (max)

[GSPS]

BW

[GHz]

SNR @ fin

[dB @ GHz]

AD

AD9208

14

2

3.3

3

9

59.5 @ 2.6

TI

ADC32RF45

14

2

6.4

3

3.2

56.8 @ 2.6

TI

ADS54J60

16

2

2.7

1

1.2

67.5 @ 0.35

TI AD9208 Simplified Block Diagram

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January 28th – February 1st, 2019, USPAS Knoxville (TN) – Beam Position Measurements – M. Wendt

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Analogue to Digital Conversion

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Uniform error distribution

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For a uniform error distribution

Clock jitter: deviation of the sampling edge

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Exercise 2

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Signal sampling

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s(t)

s[n]

GASIOR, MAREK. IMPROVING FREQUENCY RESOLUTION OF DISCRETE SPECTRA: ALGORITHMS OF THREENODE INTERPOLATION.

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Sampling and DFT

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s(t)

s[n]

DFT

N

 

 

 

 

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Windowing technique (1)

  • Windowing technique for the DFT
  • Multiply the input signal s[n] by the ‘window’ w[n]
  • Convolution in frequency
  • Smooth the boundaries
  • Window function
    • Zero at the bounders
    • Tapering from the middle
    • Symmetric

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s(t)

s[n]

DFT

N

 

w[n]

 

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Windowing technique (2)

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s(t)

s[n]

DFT

N

 

w[n]

 

By smoothing the boundaries the FFT is no more affected by the spectral leakage

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Floor of the quantization noise

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Windowing techniques (3)

Windows shapes

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Sigma scale

 

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Windowing technique (4)

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Effect of noise on the DFT

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Perturbing effects can reduce the precision of the measurement

  • Noise in the pick-up
  • ADC resolution
  • Clock jitter effect

The SNR measures the level of the ‘desired’ signal with reference to the background noise

  • Measure of the quality of the acquisition

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Exercise 3

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Betatron oscillation and tune

  • Characteristic oscillation
  • Number of oscillations in the accelerator in a single revolution period
  • Tune Q+q
    • Q: integer number
    • q: fractional part
  • Control the fractional part to avoid beam instability
  • Study of the betatron tune for accelerator operation

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Tune measurements

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Observation

Feedback

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Tune control system

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Pick up

Kicker

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Analogue conditioning

Peak detector

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PLL

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Excitation

Observation

 

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LPF

 

VCO

 

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Illustration of PLL tune tracking

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A

q

Φ

q

Single carrier PLL locks on 900 point of BTF;

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Beam Transfer Function (BTF)

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Pick up

Kicker

Excitation

Observation

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FFT

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Frequency sweep

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FFT of the frequency chirp with sweep modulation [-fm +fm]

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Beam Transfer Function

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BTF simulation

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Simple example: FFT analysis

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G(ω) == flat

(i.e. excite all frequencies)

Made with random noise kicks

Measure beam position over many consecutives turns

apply FFT → H(ω)

BTF = H(ω)

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The Average Phase Advance (APA) method

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Interpolation

Average of the N samples

Δϕ

Phase advance

BARTOLINI, R., MASSIMO GIOVANNOZZI, A. BAZZANI, WALTER SCANDALE, AND EZIO TODESCO. ALGORITHMS FOR A PRECISE DETERMINATION OF THE BETATRON TUNE. NO. CERN-SL-96-048. 1996

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APA measurement of the tune

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APA measurement and FFT comparison

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Number of iterations

 

Accuracy

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APA measurement and FFT comparison w/ NOISE

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Different perturbing effects can reduce the precision of the measurement

  • Pick-up noise
  • ADC resolution

The effect of the Hanning window is almost lost as the SNR decrease

 

Accuracy

Number of iterations

 

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Conclusions

  • Beam signal in frequency and time domain
  • ADC
    • Sampling error
    • Quantization noise floor
    • Optimal design
  • DFT
    • Proprieties
    • Spectrum
    • Windowing
  • Tune measurements
    • PLL
    • Beam transfer Function
    • Average Phase Advance
    • Measurements in presence of noise

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