where Sn is the flux density at the earth from some astronomical source, A is the area of the antenna and Dn is the frequency interval or bandwidth of the measured radiation.

So, the larger antennas collect more power. The antenna also has the capability of discriminating the signals coming from different directions in space.
 

6.2.2 Diffraction and reciprocity

The operation of antennas, and telescopes in general, are governed by electromagnetic theory and diffraction theory plays an important role. In order to understand this, one first needs to know the reciprocity theorem. This theorem states that the telescope operates the same way whether it is receiving or transmitting radiation. So the response pattern of an antenna that is receiving radiation is the same as the pattern produced when the same antenna is transmitting. A schematic of a response pattern of an antenna is given in Figure 6.1.
 

Figure 6.1
 

A telescope’s response pattern is then the same as the far-field diffraction pattern produced by the aperture. In general, when radiation of wavelength l passes through an aperture of diameter D, the radiation diffracts into a beam with angular size q = l /D. At large distances (or the far-field response), the pattern is given by Fraunhofer diffraction theory and the pattern looks like that in Figure 6.1, where the beamwidth is the full width at half power of the main beam. The beamwidth q is also a measure of the directivity of the antenna. A more precise statement that can be made (and will not be derived here) is that the angular pattern of the electric field in the far-field is the Fourier transform of the electric field distribution across the aperture.
 

6.2.3 Parabolic Antennas

Parabolic antennas (or reflectors) are common to both radio and optical astronomy. The reflector focuses plane waves to a single point, or in other words, converts plane waves into converging spherical waves. In a radio telescope these spherical waves are then coupled to a transmission line using a feed horn, which is a horn antenna. The feed horn can be placed at the prime focus or at a secondary focus using a Cassegrain design (Figure 6.2).

Figure 6.2
 

Most radio telescopes have a Cassegrain design since placing the feed horn at the prime focus will block more of the surface. Small telescopes (such as the SRT) have a prime-focus arrangement in which the reflector is illuminated by the feed placed at the focal point on the axis of the parabola. The geometry of a parabola is given by

y=x²/(4F)

where y is the distance from plane, x is the distance from the vertex, and F is the focal length as shown in Figure 6.3.

Figure 6.3
 

The reflector surface must follow a parabola to within a small fraction of a wavelength. An imperfect surface scatters some signal away from the focus and produces a loss known as the Ruze loss after John Ruze, who first derived the expression

L = exp(-(4p d/l )²)

where L is the loss factor, d is the root-mean-square (rms) deviation from a parabola,and is l the wavelength.

For most random distributions, the rms is about one quarter of the peak-to-valley variations. Figure 6.4 shows the Ruze loss in dB as a function of surface quality.

Figure 6.4
 

The angle subtended by the reflector, as seen by the feed, is determined by the ratio of focal length to diameter or F/D. Most dishes have a F/D ratio close to 0.4. A Radio Shack 9-foot satellite TV dish has an F/D of 0.38. For this F/D, the edge of the dish is about 64° out as seen by the feed. The feed should ideally be an antenna with a uniform beam that illuminates only the reflector surface. The efficiency in this case would be close to 100%. In practice, a good feed provides about 60 to 70% efficiency, so that the gain of this 9-foot antenna is 39.6 dB at 4.1 GHz. A very popular feed design is a scalar feed, which consists of a probe in a circular waveguide surrounded by choke rings as illustrated in Figure 4. The beam of the feed, which usually tapers down by about 10 dB at the edge of the dish, can be adjusted to some extent by the choice of opening size and location of the choke rings. Figure 5 shows the effect on efficiency of varying this taper. The beamwidth of a dish illuminated with such a scalar feed is approximately

q = 1.22 l/D

6.2.4 Gain

Radio and radar engineers normally speak about antennas in terms of their gain in dB referred to a half-wave dipole (dBd) or referred to an ideal isotropic antenna (dBi). A half-wave dipole has a gain of 2.15 dBi. Radio astronomers prefer to talk of size and efficiency or effective collecting area The gain, G, of an antenna relative to isotropic is related to its effective collecting area, A, by

G = 4p A/l²

where is the l wavelength

The gain is also related to the directivity of the antenna: An antenna with a smaller beam will have a higher gain. If we think of the antenna as a transmitter, as we can do owing to reciprocity, then if the transmitted energy is confined to a narrow angle, the power in this direction must be higher than average in order for the total power radiated in all directions to add up to the total power transmitted.

To achieve an effective area or aperture of many square wavelengths (gains of more than, say, 26 dBd), a parabolic reflector is the simplest and best approach. For long wavelengths, for which an antenna with more than 26 dBd would have enormous dimensions, other approaches are more appropriate. As radio amateurs doing Moon-bounce know, it is hard to beat an array of Yagi antennas for simplicity and minimum wind loading. A single 20-wavelength-long Yagi can give a gain of 20 dBd. Stacking 2 Yagis adds 3 dB and another 3 dB for every doubling. The effective aperture of a 20 dBd Yagi, however, is only 13 square wavelengths, so that stacking 50 Yagis to get 37 dBd of gain at, say, 4 GHz doesn’t make much sense when you can do as well with a 9-foot-diameter dish of 60% efficiency. At UHF frequencies around 400 MHz, the choice between Yagis and a dish is not so clear.

With the availability of excellent LNAs, optimizing the antenna efficiency is less important than optimizing the ratio of efficiency to system noise temperature or gain over system temperature, G/T_s. This means that using a feed with low sidelobes and slightly under-illuminating the dish may reduce T_s by more than it reduces G and so improve sensitivity.
 

6.2.5 Spillover

With advent of superb Low Noise Amplifiers (LNAs), the antenna noise is also a very important performance parameter along with the gain or equivalent effective aperture. Antenna noise originates from the sky background, ohmic losses, and ground pickup or spillover from sidelobes. While the sky noise is fundamental, the losses and sidelobes can be made small by a good design. Sky noise is frequency dependent but never gets any lower than the cosmic 3-K background. The minimum is near 1.4 GHz, where Galactic noise has declined and atmospheric attenuation, due primarily to the water line at 22 GHz, is still low. The lowest system noise achievable is about 18 K. At the NASA deep space network (DSN), where every fraction of a dB of performance is worth $millions, the S-band (2.3 GHz) system budget is approximately 3 K from cosmic background plus 7 K spillover plus 3 K atmospheric plus 5 K LNA. At 408 MHz, galactic noise will dominate, and system noise will be at least 50 K increasing to over 100 K toward the Galactic center. Figure 6.5 shows the relative system-noise contributions as a function of frequency.

Figure 6.5
 

where T_sys is the system temperature in Kelvins, b is the noise bandwidth,

which is only approximately equal to the resolution in spectroscopy or the total usable bandwidth in continuum, and t is the total integration time (on plus off) for normal switched observing. Set a to 2 for ordinary single switching where 2 is the product of two Ö 2, one for spending half the time on source, another for differencing two equally noisy measurements. The correlation quantization correction g is approximately 1.16 for Haystack’s spectrometer with its modified 3×3 multiplication table. Set g to 1 for continuum. Then DT is the rms fluctuation in the corresponding measurement. The bt in the denominator of this equation is, in effect, the number of samples averaged to make the measurement.

B. Why heterodyne?

Most receivers used in radio astronomy (all receivers used for spectroscopy) employ so-called superheterodyne schemes. The goal is to transform the frequency of the signal (SF) down to a lower frequency, called the intermediate frequency (IF) that is easier to process but without losing any of the information to be measured. This is accomplished by mixing the SF from the LNA with a local oscillator (LO) and filtering out any unwanted sidebands in the IF. A bonus is that the SF can be shifted around in the IF, or alternatively, the IF for a given SF can be shifted around by shifting the LO.

C. Why square-law detectors?

Inside radio-astronomy receivers, a signal is usually represented by a voltage proportional to the electric field (as collected by the antenna). But we normally want to measure power or power density. So, at least for continuum measurements and for calibration, we need a device that produces an output proportional to the square of the voltage, a so-called square-law detector, and also averages over at least a few cycles of the waveform.
 

6.3.2 Extracting weak signals from noise

As mentioned above, radio-astronomy systems usually operate close to the theoretical noise limits. With a few exceptions, signals are usually extremely weak. One such exception is the Sun. Depending on frequency, Solar cycle, antenna size, and system noise temperature, pointing an antenna at the Sun normally increases the received power several fold. Toward other sources, it is not unusual to detect and measure signals that are less than 0.1% of the system noise. The increase in power, measured in K, due to the presence of a radio source in the beam is given by

T_a = AF/(2k)

where A is the effective aperture (m²) or aperture efficiency times physical aperture, F is the radio flux density in watts/m²/Hz, and k is Boltzmann’s constant, 1.38 x 10^-23 w/Hz/K.

The factor of 2 in the denominator is because radio astronomers usually define the flux density as that present in both wave polarizations, but a receiver is sensitive to only one polarization. Radio telescopes use linear or circular polarization depending on the type of observations being made, and with two LNAs and two receivers, one can detect two orthogonal polarizations simultaneously. In order to detect and measure signals that are a very small fraction of the power passing through the receiver, signal averaging or integration is used. If the receiver gain were perfectly stable, our ability to measure small changes in signal is given by the noise equation in the previous section. There DT is the one-sigma measurement noise.

If the receiver bandwidth is 1 MHz and T_sys = 100 K, for example, then we can measure down to 0.013 K in one minute. For a sure detection, we need to see a change of 10 sigma or about 0.1 K change. The receiver gain in practice is seldom exactly constant, and the additional spillover noise and atmospheric noise may also be changing, so it will be difficult at this level to distinguish a real signal from a change in gain or atmospheric noise. There are several solutions to this problem, depending on the type of observing, all of which rely on some way of forming a reference. If we are making spectral-line measurements, the reference is often just adjacent frequencies. If we scan the frequency or simultaneously divide the spectrum into many frequency channels, then the gain or atmospheric noise changes will be largely common to all frequencies and will cancel with baseline subtraction in the final spectrum. In making measurements of broadband or continuum radio emission, we usually use a synchronous detection technique known as Dicke switching after its inventor Robert Dicke. An example of Dicke switching is the use of a switch to toggle the input of the LNA between two antenna outputs that provide adjacent beams in the sky. If we switch fast enough in this case and take the difference between the power of the two outputs synchronously with the antenna switch, then receiver gain changes will largely cancel. Furthermore, if the two antenna beams are close together on the sky, then changes in the atmospheric noise will tend to be common to both beams and will also cancel. Since we are taking a difference and spending half the time looking at the reference, the DT given above will have to be doubled.

Another powerful technique for extracting weak signals from noise is correlation. The radio telescope in this case has two or more receivers either connected to the same antenna, or, more often, two or more separate antennas. The signal voltages are multiplied together before averaging instead of multiplying the signal voltage by itself to obtain the power. With separated antennas, the correlation output combines the antenna patterns as an interferometer, which generates lobes on the sky that are separated in angle by the wavelength divided by the projected baseline between the antennas. Correlation techniques are common in radio astronomy, and they are becoming popular also in communications. Correlation is used, for example, to detect and demodulate spread-spectrum signals as in code-division multiple-access (CDMA) digital cellular telephones.
 

6.3.3 An analog-to-digital converter (ADC)

Since all the final processing of a radiometer output is done with a computer, we need to convert analog voltages from the detector to numbers that can be processed in software. A very accurate and effective ADC is a voltage-to-frequency converter followed by a counter. This ADC provides integrated power with as many bits as are needed to represent the count over the integration interval. If reading the output of the counter at perfectly regular intervals is difficult, then another counter can be used simultaneously to count the constant-frequency output of a crystal oscillator. The integrated power is then proportional to the ratio of the counts from the voltage-to-frequency converter to the counts from the crystal oscillator.
 

6.3.4 Interference

Radio astronomy is often limited by interference especially at low frequencies. The spectrum is overcrowded with transmitters: Earth-based TV, satellite TV, FM, cellular phones, radars, and many others. Radio astronomy has some protected frequency bands, but these bands are often contaminated by harmonics accidentally radiated by TV transmitters, intermodulation from poorly designed transmitters, and noise from leaky high-voltage insulators and automobile ignition noise. Some of the worst offenders are poorly designed satellite transmitters, whose signals come from the sky so that they effect even radio telescopes that are well shielded by the local terrain. Radio telescopes and their receivers can be made more immune to interference by:

1. Including a bandpass filter following the LNA to prevent interference from being generated inside the receiver by intermodulation.
2. Placing the telescope in a location with as much shielding as possible from the local terrain. Low spots (e.g., valleys) are good for low-frequency radio telescopes because they reduce the level of interference from ground-based transmitters. (Prefer a dry mountain-top to reduce atmospheric attenuation at millimeter and shorter wavelengths.)
3. Tracking down interference and trying to reduce it at the source.
4. Designing and using an antenna with very low sidelobes.
5. Using an interferometer and correlation processing, which is far more immune to interference.
6. Using data editing to remove data corrupted by interference.