The amount of signal power received in each 100 kHz frequency range (or ‘bin’ of the histogram plot) is represented in terms of an ‘effective temperature’, 
. This isn't the actual physical temperature of the molecular cloud since the amount of power received will also depend upon how many molecules the cloud contains. A ‘thin’ or almost transparent cloud won't radiate very much power even if the molecules inside it are hot. Despite this it's often convenient to represent the power level in terms of a temperature since this relates more simply to the way most receivers of this type are calibrated.
In order to obtain the kind of information illustrated in figure 3.2 we need to perform a spectral analysis on the IF output from the mixer. This can be done in various ways. Here we can imagine using a heterodyne system whose output section has been modified as illustrated in figure 3.3.
Here the single bandpass filter, power detector, and post-detection filter shown the receiver considered in the last lecture has been replaced with a multi-channel Bank of filters and detectors. Each of the filters, 
, passes a narrow frequency range, 
to 
, where B is the frequency resolution we require, and 
. By using J filters we can cover a total spectral range, JB, with a resolution, B. To obtain the spectrum illustrated in figure 3.2 we would need 64 channels like this, each passing a 100 kHz bandwidth.
The bank of filters and detectors provide a set of output voltages, 
, whose mean levels are proportional to the average power in each of the frequency ranges, 
to . So far as each of these output voltages are concerned the existence of all the other filters and detectors is irrelevant. As a result we can imagine the system as being equivalent to a ‘parallel’ set of separate heterodyne receivers, each having an IF bandwidth, B. By observing each output voltage for a period of time, t, we can therefore measure the apparent input source temperature as seen in that band with an accuracy
Using J filters we can collect information over a total signal bandwidth, 
, with a resolution, B.
To measure the spectrum with improved frequency resolution we can reduce the chosen value of B. Of course, if we reduce B we also have to increase the number of filter-detector channels if we still want to collect information covering the same total signal frequency range. This means that there tends to be a practical limit (usually appearing in terms of the cost of the receiver!) to the resolution and signal bandwidth we can obtain.
In principle, we can make B a small as we wish, and J as large as we wish to collect spectra with arbitrarily high frequency resolution. Since the amount of information reaching us in a given time is finite we must expect this improvement in frequency resolution to cost us either a reduction in the accuracy of the related power/temperature measurements or an increase in the measurement time required. For example, from expression 3.2 we can expect that halving B to increase the frequency resolution means we have to double the measurement time, t, if we wish to maintain the same accuracy for measurements of the power or temperature level.
The ‘filter bank’ arrangement isn't necessarily the most convenient way to measure the signal spectrum. Modern systems tend to employ various forms of Fourier analyser or correlator to process signal information. These are more convenient in use, but from the point of view of information theory the relationship between the frequency resolution, power level accuracy, and measurement time remains the same irrespective of the way the information is processed.
The main practical problems which arise when using and calibrating a millimetre-wave heterodyne receiver can be illustrated by considering figure 3.4. This represents the ‘front end’ of a typical receiver system. Since the signal and local oscillator frequencies are much higher than is usual for conventional electronics or microwave systems they are processed and sent from place to place in the form of ‘optical’ beams just as we would visible light.
Signals coming from the astronomical source are collected with a telescope and beamed into the mixer via a suitable set of optics. The local oscillator required to drive the mixing process is also transmitted to the mixer as another optical beam. In order for the mixer to work it must simultaneously see both the signal and the local oscillator power. The two beams must therefore be ‘overlaid’ in some way. This action is performed by a device called a Diplexer. In conventional radios this diplexer is usually a couple of tuned circuits or resistors.
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.
