射电天文基础

1. 射电天文基础 姜碧沩 北京师范大学天文系 2009/08/24-28日，贵州大学

2. Fundamentals of Antenna Theory • Electromagnetic potentials • Green’s function for the wave equation • The Hertz Dipole • The reciprocity theorem • Descriptive antenna paramters • The power pattern • The definition of the main beam • The effective aperture • The concept of antenna temperature 射电天文暑期学校

3. Electromagnetic Potentials Maxwell’s equations 射电天文暑期学校

4. The Lorentz Gauge Neither A nor Φ are completely determined by the definitions. An arbitrary vector can be added to A without changing the resulting B. While E will be affected, unless Φ is also changed. The induced parameter Λ is free. 射电天文暑期学校

5. Green’s Function for the Wave Equation The form of wave equation Helmholtz wave equation Green’s function 射电天文暑期学校

6. Solutions Solution of the wave equation Solution of the Maxwell equation 射电天文暑期学校

7. The Hertz Dipole • An infinitesimal dipole with a length Δl and a cross-section q 射电天文暑期学校

8. Rotor or curl of vector in cylindrical coordinates 射电天文暑期学校

9. Rotor or curl of vector in spherical coordinates 射电天文暑期学校

10. Electric and Magnetic Fields 射电天文暑期学校

11. Radiation Field 射电天文暑期学校

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13. The Reciprocity Theorem • The antenna parameters for receiving and transmitting are the same 射电天文暑期学校

14. Descriptive Antenna Parameters--The Power Pattern The power pattern Normalized power pattern Directivity The Hertz dipole 射电天文暑期学校

15. The Definition of the Main Beam Main beam efficiency 射电天文暑期学校

16. Beamwidth 射电天文暑期学校

17. The Effective Aperture 射电天文暑期学校

18. The Concept of Antenna Temperature 射电天文暑期学校

19. Examples • The full width half power (FWHP) angular size,θin radians, of the main beam of a diffraction pattern from an aperture of diameter D isθ≈1.02λ/D . (a) Determine the value of θ, in arc min, for the human eye where D=0.3cm, at λ=5×10-5cm. (b) Repeat for a filled aperture radio telescope, with D=100m, atλ=2cm, and for the very large array interferometer (VLA), D=27km andλ=2cm. 射电天文暑期学校

20. Cont’d • Hertz usedλ≈26cm for the shortest wavelength in his experiments. (a) If Hertz employed a parabolic reflector of diameter D ≈2m, what was the FWHP beam size? (b) If the Δl ≈0.3cm, what was the radiation resistance from equation (5.42)? (c) Hertz’s transmitter was a spark gap. Suppose the current in the spark was 0.5A, what was the average radiated power from equation (5.41)? 射电天文暑期学校

21. Cont’d • For the Hertz dipole, P(θ)=P0sin2 θ. Use equation (5.51), (5.53) and (5.59) to obtain ΩA, ΩMB, ηB and Ae. • (a) Use the equations in previous problem, together with equations (5.51), (5.60), and (5.63) to show that for a source with an angular size <<the telescope beam, TA=SνAe/2k. Use the relations above and equation (5.64) to show that TA=ηBTBwhere TB is the observed brightness temperature. 射电天文暑期学校

22. (b) Suppose that a Gaussian-shaped source has an actual angular size θs and actual peak temperature T0. This source is measured with a Gaussian-shaped telescope beam size θB. The resulting peak temperature is TB. The flux density Sν, integrated over the entire source, must be a fixed quantity, no matter what the size of the telescope beam. Use this argument to obtain a relation between temperature integrated over the telescope beam TB Show that when the source is small compared to the beam, the main beam brightness temperature and further the antenna temperature . 射电天文暑期学校

23. Suppose that a source has T0=600K, θ0=40”, θB=8’, and ηB=0.6, what is TA? 射电天文暑期学校

24. Signal Processing and Receivers • Signal processing and stationary stochastic processes • Limiting receiver sensitivity • Incoherent radiometers • Coherent receivers • Low-noise front ends and IF amplifiers • Summary of presently used front ends • Back ends: correlation receivers, polarimeters and spectrometers 射电天文暑期学校

25. Signal Processing and Stationary Stochastic Processes • Stationary stochastic processes x(t) • Probability density, expectation values and ergodicy • Autocorrelation and power spectrum • Linear systems • Gaussian random variables • Square-law detectors 射电天文暑期学校

26. Probability Density Function p(x) • Definition • The probability that at any arbitrary moment of time the value of the process x(t) falls within an interval (x-½dx, x+ ½ dx) 射电天文暑期学校

27. Expectation Values • Of the random variable x • Of a function f(x) • Of the transformation y=f(x) 射电天文暑期学校

28. Mean Value and Time Average • Mean value • Dispersion • Time average 射电天文暑期学校

29. Autocorrelation and Power Spectrum • Fourier transform • Mean-squared expected value 射电天文暑期学校

30. Dirichlet’s Theorem 射电天文暑期学校

31. Cont’d • ACF: Auto Correlation Function • PSD: Power Spectral Density • Wiener-Khinchin theorem • ACF and SPD of an ergodic random process are Fourier transform pairs 射电天文暑期学校

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33. Linear Systems 射电天文暑期学校

34. ACF and PSD 射电天文暑期学校

35. Gaussian Random Variables • Normally distributed random variables or Gaussian noise • Probability density distribution is a Gaussian function with the mean μ=0 • FT of a Gaussian is also a Gaussian 射电天文暑期学校

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37. Square-law Detector 射电天文暑期学校

38. Probability Density Py(y) 射电天文暑期学校

39. For a Gaussian Variable x(t) 射电天文暑期学校

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41. Limiting Receiver Sensitivity • Radio receivers are devices that measure the spectral power density • Basic units • The reception filter with the power gain transfer function G(ν) defining the spectral range to which the receiver responds • The square-law detector used to produce an output signal that is proportional to the average power in the reception band • The smoothing filter with the power gain transfer function W(ν) that determines the time response of the receiver 射电天文暑期学校

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43. The Minimum Noise Possible with a Coherent System • Heisenberg uncertainty principle • Coherent system • Phase reserved • Minimum noise • For incoherent detectors, this limit does not exist since phase is not reserved • In the cm and even mm wavelength regions, this noise temperature limit is quite small, e.g. at 2.6mm, it is 5.5K, while at optical wavelengths, it is about 104K 射电天文暑期学校

44. The Fundamental Noise Limit • Sources of system noise • Receiver • atmosphere • spillover of the antenna consisting of ground radiation that enters into the system through far side lobes • Noise produced by unavoidable attenuation in wave guides and cables • System temperature • Addition of all the noise temperature 射电天文暑期学校

45. Sensitivity • Nyquist sampling theorem • The highest frequency which can be accurately represented is less than one-half of the sampling rate. If we want a full 20 kHz bandwidth, we must sample at least twice that fast, i.e. over 40 kHz. 射电天文暑期学校

46. Bandwidth and Time 射电天文暑期学校

47. Some Filters 射电天文暑期学校

48. Receiver Stability • Large gains of the receiver system of the order of 1014 • Variations of the power gain enter directly into the determination of limiting sensitivity 射电天文暑期学校

49. The Dicke Switch 射电天文暑期学校

50. Generalized System Sensitivity 射电天文暑期学校