Principles of Interferometry Hans-Rainer Klöckner IMPRS Black Board Lectures 2014 acknowledgement § Mike Garrett lectures § Uli Klein lectures § Adam Deller NRAO Summer School lectures § WIKI – for technical stuff Lecture 3 § radio astronomical system § heterodyne receivers § low-noise amplifiers § system noise performance § data sampling/representation § Fourier transformation a basic system relate the voltages measured at the 2k receiver system to the antenna temperature Sν = TA Aeff

alternating current (AC) detector input power ~ 10-5 W

Tsys = 20 K, Δν 50 MHz P = 1.4 10-14 W direct current (DC) ~108 amplification / gain heterodyne receiver after all its just listening to radio the most used setup

T1 needs cooling low noise amplifier

High Electron Mobility Transistor

need to stay in the linear regime mixer – frequencyFrequency (down) conversiondown conversion vRF vIF

A typical receiver tries to down-convert the “sky signal” or “Radio vLO Frequency” (or RF) to a lower, “Intermediate Frequency” (or IF) signal.

The reasons for doing this include: (i) signal losses (e.g. in cables) typically go as frequency2; (ii) it is much easier to mainpulate the signal (e.g. amplify, filter, delay, sample/process/digitise it) at lower frequencies.

We use so-called “heterodyne” systems to mix the RF signal with a pure, monochromatic frequency tone, known as a Local Oscillator (or LO).

Consider an RF signal in a band centred on frequency vRF, and an LO with frequency vLO, these can be represented as two sine waves with angular frequencies w and wo:

-- Difference frequency -- -- Sum frequency --

vRF-vLO vRF+vLO Output freq

Inputs vLO vRF LSB - regime USB - regime mixer – frequency down conversion

The higher frequency component (“sum frequency” vRF+vLO) is usually removed by a The higher frequency component (“sum The higher frequency component (“sum filter that is included in the LO electronics. frequency” vRF+vLO) is usually removed by a frequency” vRF+vLO) is usually removed by a filter thatHence is included the process in the LO of electronics. down-conversion, takes USB = upper sidefilter band that is included in the LO electronics. Filter Hence athe band process with of centredown-conversion, frequency takes vRF and converts Hence the process of down-conversion, takes LSB = lowerFilter side band Filter a band with centre frequency vRF and converts a band itwith to centrea lower frequency (difference) vRF and frequency,converts vRF-vLO. it to a lower (difference) frequency, vRF-vLO. it to a lower (difference) frequency, vRF-vLO. TheThe mixermixer signal signal products products preserves preserves the the The mixer signal products preserves the noise characteristics of the input RF (sky) noise characteristics of the input RF (sky) USB noise characteristics of the input RF (sky) signal,signal, but theythey contain contain an arbitraryan arbitrary phase- phase- signal, but they contain an arbitrary phase- shiftshift duedue toto the the unknown unknown phase phase of the of LO. the LO. shift due to the unknown phase of the LO.

Usually there will be several mixers and Usually there will be several mixers and Usuallyfrequency there conversions will be in several a receiver mixers and frequency conversions in a receiver frequencysystem. Eventually conversions one edge in of a thereceiver system. Eventually one edge of the system.frequency Eventually band reaches one 0 Hz,edge known of the as a frequency band reaches 0 Hz, known as a frequency“base-band” band or “video” reaches signal. 0 Hz, known as a “base-band” or “video” signal. “base-band” At high frequencies or “video” (e.g. millimetre signal. At high frequencies (e.g. millimetre wavelengths), down-conversion occurs wavelengths), down-conversion occurs Atbefore high amplification. frequencies (e.g. millimetre before amplification. wavelengths), down-conversion occurs before amplification. low noise amplifier we have covered that already bandpass filter low noise amplifier we have covered that already

4 5

4 5 detector

Since radio astronomy signals have the characteristics of white noise, the voltage induced in the receiver output alternates positively and negatively about zero volts. Any measurement of the Voltage expectation value or time average will read zero (e.g. hooking up a receiver to a DC voltmeter will not measure any signal).

2 What is needed is a non-linear device (Vout = AVin ) that will only measure the passage of the signal in one preferred direction (either positive or negative) i.e. we must incorporate a semiconductor diode into our measuring system

alternating current (AC)

direct current (DC) integrator

capacity needs time τ to charge

reads out the capacity signal processing tools 1d Convolution Fourier Transformation

2πiνt Ff (t)= f(ν)e dν ￿ 2πiνt Ff (ν)= f(t)e− dt Convolution theorem ￿ F(f * g) = F(f) F(g) heterodyne receiver Fourier transformation convolution theorem in action

USB The higher frequency component (“sum frequency” vRF+vLO) is usually removed by a filter that is included in the LO electronics. Hence the process of down-conversion, takes broad bandFilter IF signal a band with centre frequency vRF and converts it to a lower (difference) frequency, vRF-vLO. time dependent voltage U(τ) frequency dependent voltage U(ν) The mixer signal products preserves the noise characteristics of the input RF (sky) signal, but they contain an arbitrary phase- shift due to the unknown phase of the LO. Continuum measurement UsuallyPower( thereτ) will~ |U be2 (severalτ)| mixers and frequency conversions in a receiver system.Line Eventuallymeasurement one edge of the frequency band reaches2 0 Hz, known as a “base-band”Power(ν or) ~ “video” |U (ν signal.)|

At high frequencies (e.g. millimetre wavelengths), down-conversion occurs before amplification. The higher frequency component (“sum frequency” vRF+vLO) is usually removed by a filter that is included in the LO electronics. Hence the process of down-conversion, takes Filter a band with centre frequency vRF and converts it to a lower (difference) frequency, vRF-vLO. The mixer signal products preserves the noise characteristics of the input RF (sky) signal, but they contain an arbitrary phase- shift due to the unknown phase of the LO.

Usually there will be several mixers and frequency conversions in a receiver system. Eventually one edge of the frequency band reaches 0 Hz, known as howa to get P(ν) = |U2(ν)| “base-band” or “video” signal. - approach 1 - Hardware

Theoretically At high frequencies (e.g. millimetre

wavelengths), down-conversion occurs filters split signal into channels feed each channel into detector before amplification. FFT spectrometer

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FPGA – Field-Programmable Gate Array recap what we measure in a single dish heterodynereceiver τ ν d d ντ i ντ π i 2 π 2 − Integration time τ e e ) ) τ ν ( ( f f Observing time ￿ ￿ )= )= ν τ ( ( f f F F Fourier Transformation Transformation Fourier

Base Band spectrum in ν per integration time τ

note usually the integration time will be defined as t how to get P(ν) = |U2(ν)| - approach 2 - Convolution Theoretically auto correlation function how to get P(ν) = |U2(ν)| - approach 2 - old style hardware to shift the data signal processing

clipper - quantisation 1 bit (0 or 1) signal processing signal processing signalOnline dataprocessing sampling

Square-law detectors are not used so very often these days. The receiver produces a varying analogue output voltage that is usually digitised and stored for further (offline) processing. How often must be sample the signal ? Consider the following sine wave:

If we sample twice per cycle time If we sample once per cycle time For lossless digitisation we must (period) we get a saw-tooth wave that (period) we would consider the signal sample the signal at least twice per is becoming a good approximation to a to have a constant amplitude. cycle time. sinusoid.

Reconstructed

Signal

Nyquist’s sampling theorem states that for a limited bandwidth signal with maximum frequency fmax, the equally spaced sampling frequency fs must be greater than twice the maximum frequency fmax, i.e. fs > 2·fmax in order for the signal to be uniquely reconstructed without aliasing.

The frequency 2fmax is called the Nyquist sampling rate. e.g. If a reciever system provides a baseband signal of 20 MHz, the signal must be sampled 40E6 times per second. how to get P(ν) = |U2(ν)| auto correlation

digital data auto correlation cross digital data

using the signal from different antennas we build an interferometer Correlator !"##$%&'"#()%&'*"#+(",$#,-$.(

Development effort required

x CPU Correlator Reuse- capacity per ability hardware $$ GPU

FPGA

ASIC

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solid line unresolved dashed line resolved aperture synthesis mix the signal from all the telescope that they are in phase