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2009/08/24-28 2009/08/24-28

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2009/08/24-28 Fundamentals of Antenna Theory Electromagnetic potentials Greens 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

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2009/08/24-28 Electromagnetic Potentials Maxwells equations

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2009/08/24-28 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.

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2009/08/24-28 Greens Function for the Wave Equation The form of wave equation Helmholtz wave equation Greens function

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2009/08/24-28 Solutions Solution of the wave equation Solution of the Maxwell equation

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2009/08/24-28 The Hertz Dipole An infinitesimal dipole with a length Δl and a cross-section q

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2009/08/24-28 Rotor or curl of vector in cylindrical coordinates

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2009/08/24-28 Rotor or curl of vector in spherical coordinates

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2009/08/24-28 Electric and Magnetic Fields

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2009/08/24-28 Radiation Field

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2009/08/24-28 The Reciprocity Theorem The antenna parameters for receiving and transmitting are the same

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2009/08/24-28 Descriptive Antenna Parameters --The Power Pattern The power pattern Normalized power pattern Directivity The Hertz dipole

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2009/08/24-28 The Definition of the Main Beam Main beam efficiency

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2009/08/24-28 Beamwidth

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2009/08/24-28 The Effective Aperture

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2009/08/24-28 The Concept of Antenna Temperature

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2009/08/24-28 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 isθ1.02λ/D. (a) Determine the value of θ, in arc min, for the human eye where D=0.3cm, at λ=5×10 -5 cm. (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.

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2009/08/24-28 Contd Hertz usedλ26cm 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) Hertzs transmitter was a spark gap. Suppose the current in the spark was 0.5A, what was the average radiated power from equation (5.41)?

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2009/08/24-28 Contd For the Hertz dipole, P(θ)=P 0 sin 2 θ. Use equation (5.51), (5.53) and (5.59) to obtain Ω A, Ω MB, η B and A e. (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 <

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2009/08/24-28 (b) Suppose that a Gaussian-shaped source has an actual angular size θ s and actual peak temperature T 0. This source is measured with a Gaussian-shaped telescope beam size θ B. The resulting peak temperature is T B. The flux density S ν, 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 T B Show that when the source is small compared to the beam, the main beam brightness temperature and further the antenna temperature.

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2009/08/24-28 Suppose that a source has T 0 =600K, θ 0 =40, θ B =8, and η B =0.6, what is T A ?

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2009/08/24-28 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

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2009/08/24-28 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

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2009/08/24-28 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)

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2009/08/24-28 Expectation Values Of the random variable x Of a function f(x) Of the transformation y=f(x)

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2009/08/24-28 Mean Value and Time Average Mean value Dispersion Time average

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2009/08/24-28 Autocorrelation and Power Spectrum Fourier transform Mean-squared expected value

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2009/08/24-28 Dirichlets Theorem

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2009/08/24-28 Contd ACF: Auto Correlation Function PSD: Power Spectral Density Wiener-Khinchin theorem –ACF and SPD of an ergodic random process are Fourier transform pairs

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2009/08/24-28 Linear Systems

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2009/08/24-28 ACF and PSD

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2009/08/24-28 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

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2009/08/24-28 Square-law Detector

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2009/08/24-28 Probability Density P y (y)

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2009/08/24-28 For a Gaussian Variable x(t)

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2009/08/24-28 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

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2009/08/24-28 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 10 4 K

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2009/08/24-28 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

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2009/08/24-28 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.

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2009/08/24-28 Bandwidth and Time

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2009/08/24-28 Some Filters

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2009/08/24-28 Receiver Stability Large gains of the receiver system of the order of 10 14 Variations of the power gain enter directly into the determination of limiting sensitivity

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2009/08/24-28 The Dicke Switch

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2009/08/24-28 Generalized System Sensitivity

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2009/08/24-28 Best Integration Time

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2009/08/24-28 Receiver Calibration

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2009/08/24-28 Incoherent Radiometers Coherent radiometers –The phase of the received wave field is preserved –A full reconstruction of the wave field is allowed –In the sub-mm and far-infrared region, coherent radiometers with high sensitivities and large bandwidths become difficult to build and operate Incoherent radiometers –Phase is not preserved –Bolometers Offer a wider bandwidth and sufficient sensitivity Sensitive to all polarizations A wide range of effects that depend on the intensity of the radiation is used for such detector For astronomical purposes, thermal detectors made of semiconductor dominate the field of continuum measurements in the mm and sub-mm regions

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2009/08/24-28 Bolometer Radiometers Theory –The resistance R of a material varies with the temperature. When radiation is absorbed by the bolometer material, the temperature varies and this temperature change is a measure of the intensity of the incident radiation Characteristics –Intrinsically broadband devices

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2009/08/24-28 Device The power from an astronomical source, P 0, raises the temperature of the bolometer elements by ΔT, which is much smaller than the temperature T 0 of the heat sink. Heat capacity, C, is analogous to capacitance, and the quantity conductance, g, is analogous to electrical conductance, G, which is 1/R. The noise performance of bolometers depends critically on the thermodynamic temperature T 0 and on the conductance G. The temperature change causes a change in the voltage drop across the bolometer.

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2009/08/24-28 What is a bolometer and how does it work

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2009/08/24-28 Requirements for a Bolometer in astronomy Energy balance equation Thermal time constant Amplitude of the temperature variation Requirements –Respond with maximum temperature step to a given power input –Have a short thermal time constant –Produce a detector noise as close to the theoretical minimum as possible

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2009/08/24-28 The Noise Equivalent Power of a Bolometer Noise sources –Johnson noise in the bolometer –Thermal fluctuations, or phonon noise –Background photon noise –Noise from the amplifier and load resistor Cooling will reduce all of these noise contributions For ground bolometers the background photon noise will determine the noise of the system

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2009/08/24-28 Currently Used Bolometer Systems Large bolometer arrays –MAMBO2 117 element array, the IRAM 30m telescope, 1.3mm, angular resolution of 11, FOV of 240 –SCUBA 37 pixel array at 0.87mm and 91 element array at 0.45mm, the JCMT 15m telescope Types of bolometers –Superconducting bolometers –Polarization measurements –Spectral line measurements

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2009/08/24-28 Coherent Receivers Basic components Semiconductor junctions Front end Back end

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2009/08/24-28 Basic Components Thermal noise of an attenuator –Noise temperature of an attenuator T N Cascading of amplifiers

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2009/08/24-28 Mixers The unit that is performing the actual frequency conversion –In principle any device with a nonlinear relation between input voltage and output current can be used as a mixer Local Oscillator (LO) Intermediate Frequency

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2009/08/24-28 The thick arrows are the given values. There are two methods to specify the system: (a) when two input frequenciesν L and ν S are given, (b) ν L and ν IF are specified. In this case, signals from both the upper sideband (ν L +ν IF ) and lower sideband (ν L –ν IF ) contribute to the IF signal

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2009/08/24-28 Semiconductor Junctions

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2009/08/24-28 Low-Noise Front Ends and IF amplifiers The mixer converts the RF frequency to the fixed IF frequency where the signal is amplified by the IF amplifier. The main part of the amplification is done in the IF. The IF should only contribute a negligible part to the system noise temperature. But because usually some losses are associated with frequency conversion, the first mixer may be a major source for the system noise. Two ways exist to decrease this contribution –Choosing a low loss nonlinear element to mix the signal to a lower frequency –Placing a low-noise amplifier before the mixer Receiver frontends in historical order –Uncooled mixers –Maser amplifiers –Parametric amplifiers –HEMT: High Electron Mobility Transistors –SIS: Superconducting mixers –HEB: Hot Electron Bolometers

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2009/08/24-28 Summary of Presently Used Front Ends Single pixel receiver systems –λ>3mm, HEMT –mm and sub-mm, SIS –λ <0.3mm, superconducting HEB Multibeam systems –HEMTs front ends are rather simple systems, there has been a trend to build many receivers in the focal plane Effelsberg, Parkes, Lovell, FCRAO –SIS IRAM, CSO, Onsala –Bolometers: cooled GeGa bolometers, TES (Transition Edge Sensors)

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2009/08/24-28 Correlation Receivers and Polarimeters Cross-correlation function for two ergodic processes

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2009/08/24-28 Contd Mixing two partially coherent signals with an LO preserves the correlation between the two signals at the intermediate frequencies

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2009/08/24-28 Block Diagram of Correlation Receiver

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2009/08/24-28 Correlation Receiver Components –The signals from the antenna and from the reference are input to a 3dB hybrid –The two outputs of the hybrid are amplified by two independent radiometer receivers which share a local oscillator –The IF signals are correlated Advantage –The stability is the same as that of a Dicke receiver Limiting sensitivity

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2009/08/24-28 Polarimeter

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2009/08/24-28 Circular polarized wave Linearly polarized wave

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2009/08/24-28 Spectrometers Spectral information contained in the radiation field –The receivers must be single sideband –The frequency resolution is usually small –In the kHz range, the time stability must be high –Local oscillator frequency should be stable Types –Multichannel Filter Spectrometer –Fourier and Autocorrelation Spectrometer –Acousto-Optical Spectrometer –Chirp Transfer Spectrometer

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2009/08/24-28 Multichannel Filter Spectrometer Multi separate channels simultaneously measure different parts of the spectrum Design aims –The shape of the bandpass for the individual channels must be identical –The square-law detectors must have identical characteristics –Thermal drifts should be as identical as possible Problems –Stability requirement is very high –The flexibility in varying the spectral resolution is low

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2009/08/24-28 Fourier and Autocorrelation Spectrometer

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2009/08/24-28 Autocorrelation Spectrometer Wiener-Khinchin theorem –Autocorrelation function and SPD are FT pairs Digital autocorrelator Advantages –Flexibility Various spectral resolution or bandwidth –Stability long-time integration possible Sampling and quantization

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2009/08/24-28 One-bit Quantization

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2009/08/24-28 Autocorrelation Function arcsin law van Vleck clipping correction

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2009/08/24-28 Basic Hardware A filter limits the IF input frequency band from 0 to B A clipper transforms the signal x(t) into clipped signal y(t) y is then sampled at equidistant time intervals by the sampler, which multiplies y with a very short pulse at the clock frequency 2B The samples are shift at the clock frequency into a shift register The shift register content with time delay is compared with the un-delayed sample The autocorrelation value R y is calculated R y is converted to R x

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2009/08/24-28 FT of the measured ACF Lag window Measure PSD Frequency resolution

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2009/08/24-28 Hanning Window Lag window Frequency resolution

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2009/08/24-28 Sensitivity

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2009/08/24-28 Improvements Multi-bit digitization –For a 2bit or 4 level autocorrelator, the sensitivity factor will be replaced by 1.14 Large bandwidth

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2009/08/24-28 Acousto-Optical Spectrometers Theory –A sound wave causes periodic density variation in the medium which it passes. These density variations in turn cause variations in the bulk constantsεand n of the medium, so that a plane EM wave passing through this medium will be affected. Such a medium will cause a plane monochromatic EM wave to be dispersed. –Acousto-Optical SpectrometerAcousto-Optical Spectrometer

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2009/08/24-28 Characteristics Dispersion angle is proportional to the frequency of the acoustic wave Intensity of the diffracted light is proportional to the acoustic power Resolution Bandwidth

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2009/08/24-28 Chirp Transform Spectrometer In contrast to AOS, the CTS is a device which makes exclusively use of radio technology, instead of both radio and optical technologies. For not too wide bandwidth, CTS is an alternative to AOS.

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2009/08/24-28 Pulsar Backends Pulsars –The pulsar signals change rapidly with time Measurements –Determination of average pulse shapes Short time constant due to the short pulse period Narrow bandwidth due to frequency dispersion Many individual pulses are added together to obtain a pulse profile –Searches for periodic pulses with unknown periods

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2009/08/24-28 Pulse Dispersion and Dispersion Removal Pulse dispersion can be described by a transfer function in the time or frequency domain. A filter can be constructed to remove this dispersion for a limited frequency range either by hardware or by software techniques

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2009/08/24-28 Pulsar Searches Rail filtering –Convolution of the received signal with a matching filter whose impulse response is given by rectangular functions spaced at the assumed period

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2009/08/24-28 Exercises The equivalent noise temperature of a coherent receiver T n which corresponds to the NEP of a bolometer is determined by using the relation NEP=2k T n (Δν) 1/2. For Δν=50GHz, determine T n for NEP=10 -16 W Hz -1/2. A bolometer receiver system can detect a 1mK source in 60s at the 3σlevel with the bandwidth of 100GHz. How long must one integrate to reach this RMS noise level with a coherent receiver with a noise temperature of 50K and bandwidth of 2GHz?

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2009/08/24-28 Exercises The definition of a decibel, db, is If a 30db (i.e. gain 1000) amplifier with a noise temperature of 4K is followed by a mixer with a noise temperature of 1000K, what is the percentage contribution of the mixer to the noise temperature of the total if T sys =T stage1 +T stage2 /Gain stage1 ?

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2009/08/24-28 Exercise (a) When observing with a double-sideband coherent receiver system, an astronomical spectral line might enter from either upper or lower sideband. The upper sideband frequency is 115GHz and the lower sideband frequency is 107GHz. What is the intermediate frequency? What is the local oscillator frequency? (b) to decide whether the line is actually in the upper or lower sideband, the observer increases the local frequency by 100kHz. The signal moves to lower frequency. Is the spectral line from the upper or lower sideband?

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2009/08/24-28 Exercise Given below is the rms noise as a function of time for an acousto-optical spectrometer used in Chile on the 1.2m telescope of the National University of Chile at the Cerro Tololo Interamerican Observatory. There were 172 calibration measurements between July 1993 and August 1994. All were 10min scans. After baseline subtraction these spectra had an rms noise of 0.151±0.026K, where the uncertainty is the rms scattering about the mean. Averaging the spectra in groups of four, there were 43 spectra with 40min integration. The rms noise was 0.086±0.013K. Next, averaging groups of 16 scans, then groups of 64 scans, and lastly all 172 scans (28.5 hrs of integration), one has the following rms(160min)= 0.056±0.007K, rms(640min)= 0.039±0.004K, rms(1720min)= 0.031K. Over what period of time did the noise in this system follow a t -1/2 relation?

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2009/08/24-28 Homework On two days, labelled as 1 and 2, you have taken data which are represented by Gaussian statistics. The mean values are x 1 and x 2, with σ 1 andσ 2. Assume the average is given by x=fx 1 +(1-f)x 2 and the corresponding. Determine the value of f which gives the smallest σ by differentiating the relation for σand setting the result equal to zero. Show that

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2009/08/24-28 Homework For an input voltage signal calculate the Fourier transform, autocorrelation function and power spectrum. Note that this function extends (formally) to negative times. This frequently used generalization allows a simplification of the mathematics. Repeat previous calculation for

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2009/08/24-28 Homework Calculate the power spectrum S ν for the function v(t)=A for –τ/2

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2009/08/24-28 Homework What is the minimum noise possible with a coherent receiver operating at 115GHz? At 1000GHz, at 10 14 Hz? If the bandwidth of a receiver is 100MHz, how long must one integrate to reach an RMS noise which is 0.1% of the system noise with a total power system? Repeat for a Dicke switched system, and for a correlation system. Now assume that the receiver system has an instability described by. For a time dependence we take, and K=1. On what time scale will the gain instabilities dominate uncertainties caused by receiver noise? If one wants to have the noise decrease as what is the lowest frequency at which one must switch the input signal against a comparison?

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Fundamental Antenna Parameters

Fundamental Antenna Parameters

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