Tetsuo Sasao and Andr´eB. Fletcher Introduction to VLBI Systems Chapter 4 Lecture Notes for KVN Students Partly based on Ajou University Lecture Notes Version 1. Issued on November 14, 2006. Revised on June 18, 2010. Revised on February 10, 2011. (to be further edited) NOT COMPLETE YET. Very Long Baseline Interferometry This chapter will focus on technological bases, actual implementation, and data processing of modern VLBI systems. Contents 1 Technologies Which Made VLBI Possible 6 1.1 Basics of Digital Data Processing . 7 1.1.1 Analog Processing Versus Digital Processing . 7 1.1.2 SamplingandClipping . 8 1.1.3 Discrete–Time Random Process . 9 1.1.4 Stationary Discrete–Time Random Process . 10 1.1.5 Sampling ......................... 11 1.1.6 CombFunction ...................... 13 1.1.7 Fourier Transform of Comb Function Is Comb Function 15 1.1.8 Spectra of Discrete–Time Processes . 16 1.1.9 SpectraofSampledData. 19 1.1.10 Inverse Relations for Spectra of Sampled Data . 20 1.1.11 SamplingTheorem . 21 1.1.12 Optimum Sampling Interval . 25 1.1.13 SamplingFunction . 26 1.1.14 Correlations of Nyquist Sampled Data with Rectangu- larSpectra......................... 30 1.1.15 S/N Ratio of Correlator Output of Sampled Data . 34 1.1.16 Nyquist Theorem and Nyquist Interval . 37 1.1.17 Higher–Order Sampling in VLBI Receiving Systems . 38 1 1.1.18 Clipping (or Quantization) of Analog Data . 40 1.1.19 Probability Distribution of Clipped Data . 43 1.1.20 Cross–Correlation of 1–Bit Quantized Data: van Vleck Relationship . 47 1.1.21 van Vleck Relationship in Autocorrelation . 53 1.1.22 Spectra of 1–Bit Quantized Data . 53 1.1.23 Price’s Theorem . 57 1.1.24 Cross–Correlation of 2-Bit Quantized Data . 61 1.1.25 Cross–Correlation Coefficient of 2–Bit Quantized Data 63 1.1.26 Power and Cross–Power Spectra of 2-Bit Quantized Data 65 1.1.27 Dispersion of Digital Correlator Output . 68 1.1.28 S/N Ratio of Digital Correlator Output . 70 1.1.29 Optimum Parameters v0/σ and n in 2–Bit Quantization 72 1.1.30 Effect of “Oversampling” in S/N Ratio of Clipped Data 73 1.1.31 Coherence Factor and Sensitivity with Given Bit Rate 76 1.2 FrequencyStandard. 79 1.2.1 VLBI Requires “Absolute” Frequency Stability . 79 1.2.2 How to Describe Frequency Stability? . 81 1.2.3 Types of Phase and Frequency Noises . 84 1.2.4 Time Domain Measurements . 84 1.2.5 “True Variance” and “Allan Variance” of Fractional Frequency Deviation . 86 1.2.6 True Variance and Allan Variance through Power Spec- trum of Fractional Frequency Deviation . 88 1.2.7 Time–Interval Dependence of Allan Variance . 90 1.2.8 True Variance and Allan Variance through Autocorre- lationofPhaseNoise . 91 1.2.9 Coherence Function . 92 1.2.10 Approximate Estimation of Coherence Time . 95 1.2.11 Estimation of Time–Averaged Phase Noise . 96 1.3 TimeSynchronization . 98 1.4 RecordingSystem......................... 99 2 Overview of VLBI Systems 103 2.1 MK–3 As Prototype of Modern VLBI Systems . 103 2.1.1 Dual–Frequency Reception . 104 2.1.2 First Frequency Conversion . 104 2.1.3 Transmission of Observed Data to Control Building . 105 2.1.4 Intermediate Frequency Distributer . 105 2.1.5 Baseband Conversion . 106 2.1.6 Formatter .........................107 2 2.1.7 DataRecorder. .108 2.1.8 Phase and Delay Calibration . 108 2.1.9 Hydrogen Maser Frequency Standard . 110 2.1.10 Automated Operation by Control Computer . 110 2.2 ModernVLBISystems . .111 2.2.1 New Recording and Fiber–Link Systems . 111 2.2.2 High–Speed Samplers and Digital Baseband Converters 114 2.2.3 e–VLBI ..........................118 2.2.4 VLBI Standard Interface (VSI) . 121 2.2.5 Future Geodetic VLBI System Proposed in “VLBI2010”122 3 Effects of Clock Offset and Atmospheric Delay 123 3.1 ClockOffset. ...........................123 3.1.1 Measuring Time with Clocks . 123 3.1.2 Clock Time Deviation Due to Fluctuations in Frequency Standard..........................124 3.1.3 LO Phase Deviation Due to Fluctuations of Frequency Standard..........................125 3.1.4 Clock and LO–Phase Offsets between Two VLBI Stations126 3.1.5 How Do Fluctuations in Frequency Standards Affect VLBIData?........................127 3.2 Atmospheric Propagation Delay . 129 3.2.1 Three Major Components of Atmosphere That Cause Propagation Deley . 130 3.2.2 Cross–Power Spectrum of IF Signals with Atmospheric Delay ...........................132 3.3 Expected Correlator Output with Clock and Atmospheric Effects133 3.3.1 Expected Correlator Output, General Expression . 133 3.3.2 Correlator Output for Continuum Spectrum Source . 134 3.3.3 Simplest Case of Rectangular Filters . 135 3.4 Limitations Imposed by Atmosphere and Clock Effects . 135 3.4.1 Example of VLBI Fringe Phase . 136 3.4.2 Difficulty in Direct Use of VLBI Phase . 137 3.4.3 Limitation to Coherence Time . 138 4 Correlation Processing in VLBI 139 4.1 Features of Correlation Processing in VLBI . 139 4.2 FirstStep—HardwareProcessing . 142 4.2.1 Accumulation Period . 142 4.2.2 “Classical” XF–Type Correlator . 144 4.2.3 Data Synchronization . 145 3 4.2.4 Detection of Phase Calibration Tones . 146 4.2.5 RingBufferMemory . .147 4.2.6 Delay Tracking by Programmable Delay Circuit . 148 4.2.7 Address Shift Timing in Programmable Delay Circuit . 148 4.2.8 “Fringe Rotator”: Fringe Stopping and Complex Cor- relation ..........................149 4.2.9 Multi–Lag Correlation Unit . 151 4.2.10 Algorithm for 1-Bit Correlation . 153 4.2.11 Autocorrelation Unit . 154 4.2.12 Coherence Loss in Digital Correlation Processing . 154 4.2.13 Output of Multi–Lag Complex Correlator . 155 4.2.14 Expected Output of Multiplier & Integrator . 157 4.2.15 Cross–Correlation Coefficient as Expected Output . 159 4.2.16 Expected Complex Cross–Correlation . 161 4.2.17 Expected Complex Cross–Correlation for White Spec- trumSource........................164 4.2.18 Rectangular Videoband Case . 166 4.2.19 Behavior of Correlator Outputs . 166 4.2.20 PC–Based Correlators . 170 4.2.21 New 16–Station FX–Type Correlator in Seoul . 171 4.3 Second—Step, Software Processing (Fringe Fitting) . 173 4.3.1 Some Words on Terminalogy . 173 4.3.2 Two–Dimensional Search in Software Processing . 174 4.3.3 How to Apply Corrections to Correlator Outputs? . 176 4.3.4 Spectrum of Complex Cross–Correlation . 177 4.3.5 Amplitude and Phase Spectra of Complex Cross–Correlation180 4.3.6 Two–Dimensional Taylor Expansion of Residual Phase 181 4.3.7 Residual Group Delay and Residual Fringe Frequency . 182 4.3.8 Trial Corrections to Residual Phase . 183 4.3.9 Finding Correlation Peak . 185 4.3.10 Fringe Amplitude and Residual Fringe Phase in VLBI . 187 4.3.11 Coherence Factor Due to Imperfect Delay Model . 190 4.3.12 Theoretical Form of Uncalibrated Fringe Amplitude . 191 4.3.13 SearchWindow . .192 4.3.14 Actual Software Processing . 193 4.3.15 Correlation Peak of Continuum Source . 197 4.3.16 Estimation of S/N of Detected Fringe . 199 4.3.17 Improvement of S/N through Integration . 201 4.3.18 Examples of Detected Fringes for Continuum Sources . 203 4.3.19 Software Processing for Spectralline Sources . 206 4.3.20 If We Do Not Find Any Peak ... 208 4 4.3.21 Threshold of Fringe Detection . 209 5 Observables of VLBI 211 5.1 FourBasicObservables. .211 5.1.1 Direct Products of Correlation Processing . 211 5.1.2 Calibration of Fringe Amplitude . 212 5.1.3 Estimates and Expected Forms for Observables . 213 5.2 Accuracies of VLBI Observables . 214 5.2.1 Statistics of Amplitude and Phase . 214 5.2.2 Expectations of Amplitude and Phase . 217 5.2.3 In Limiting Case When S/N isLarge . .218 5.2.4 Accuracy of Fringe Amplitude Observable . 221 5.2.5 Accuracy of Fringe Phase Observable . 222 5.2.6 Group Delay and Fringe Frequency Observables Free from Cycle Ambiguity . 223 5.2.7 Accuracy Estimation for Group Delay and Fringe Fre- quency...........................224 5.2.8 Equivalence of Peak Search Method and Least Squares Estimation.........................225 5.2.9 Dispersions of Fringe Frequency and Group Delay Ob- servables..........................232 5.2.10 Accuracy of Fringe Frequency Observable . 234 5.2.11 Atmospheric Limitation to Usefulness of Fringe Fre- quency...........................236 5.2.12 Accuracy of Group Delay Observable . 238 5.2.13 Atmospheric Effect in Group Delay Observable . 239 5.2.14 Phase Delay and Group Delay . 241 5.3 Bandwidth Synthesis Technique . 242 5.3.1 For Higher Accuracy of Group–Delay Estimation . 242 5.3.2 Model Delay Resolution Function . 246 5.3.3 Examples of Model Delay Resolution Functions . 246 5.3.4 Ambiguity in Bandwidth Synthesis . 249 5.3.5 Actual Bandwidth Synthesis . 252 5.3.6 Group–Delay Accuracy in Bandwidth Synthesis . 257 6 Atmosphere–Related Difficulties and Their Solutions in VLBI258 6.1 Difficulties Associated with mm–Wave VLBI . 259 5 1 Technologies Which Made VLBI Possible 100 ~ 10000 km Low Noise Amplifier Low Noise Amplifier Local Oscillator Local Oscillator IF Frequency Phase and Phase and IF Frequency Distributer Delay Delay Distributer Calibrator Calibrator Video Converter H-Maser Frequency H-Maser Frequency Video Converter Standard Standard Formatter Formatter Data Recorder Data Recorder Correlation Processor Digital Data Processing Figure 1: Schematic view of three major technologies and digital data processing which realized VLBI. VLBI (Very Long Baseline Interferometry) is a kind of radio interfer- ometer with very long baselines often exceeding thousands of kilometers, equipped with independent frequency standards (clocks) at each station.