Radiometer Detection Analysis
PW East
Abstract
The performance of various types of radiometer receiver algorithms is analysed for both analogue and digital FFT receiver processes.
1. Total Power Radiometer
a)Square-law detector
Figure 1. Total Power Radiometer (Square-law Detector)
With sky noise temperature Tsky and system temperature Tsys (made up of feeder loss and input RF amplifier noise), the output signal voltage V from a square-law detector followed by an integrator contains both AC, Vac and DC Vdc components, given by,
(1)
where, k is Boltzmann’s constant
G is the total receiver gain
F is the overall noise figure (equivalent temperature = (F-1)Tambient)
Br is the detector input RF bandwidth and Bi the post-detector integrator bandwidth
The DC output provides a measure of the sky + system temperature whilst the AC component adds uncertainty to the measure. Estimation of the system temperature from component parameters allows computation of the wanted sky noise source. This is a simplification as the temperature measured from the antenna may, in addition to the wanted source, comprise background components and ground noise entering antenna side and back lobes.
The voltage output sensitivity to RF gain and temperature variations can be estimated by differentiating the DC component in equation (1),
(2)
The second part of equation (2) represents the temperature uncertainty which is equal to the AC component observed at the output of the integrator, or,
Rearranging we get the well-recognised radiometer equation for temperature sensitivity,
(3)
Dividing the two equations in (2) gives the temperature sensitivity to gain variations.
(4)
This shows that receiver gain variations directly affect the wanted signal output and to ensure the output uncertainty is only due to the integrator, the gain uncertainty should be much less than the desired integrator residual noise, or,
(5)
Temperature and gain errors combine as root sum of squares with the estimate,
(6)
This equation shows that RF gain variations are directly reflected in the output temperature estimate. It is important therefore that the RF chain gain is stabilised in some way; by means of a temperature controlled oven for example.
b)Fast Fourier Transform Detector
Figure 2. Total Power Radiometer (Digital FFT)
Figure 2 shows a schematic of a digital FFT receiver configuration for radiometry. The analogue receiver part includes a quadrature mixer producing two IF outputs of 90 relative phase, denoted I, in-phase and Q, in quadrature. They are digitised by twin analogue-to-digital converters (ADC) and groups of P samples are collected in a buffer memory.
The FFT algorithm is then performed on this block of data, so generating the spectrum of the RF signals present in the sample. The buffer data window duration is the ratio of the FFT buffer length P to the clock frequency C, whereas the spectrum resolution is equal to the reciprocal of this data block duration (ie, C/P). For real input signals, the spectral cover is up to half the clock frequency (Nyquist frequency), but for complex (I/Q) input data, band cover up is to the full clock frequency. Ideally, to ensure continuous operation, the FFT processing should be completed before the next buffer fills; typically requiring some Plog2P multiplications for binary number block samples using efficient fast discrete algorithms. For the analysis of noise, it is not necessary to apply an aperture-weighting function to the FFT input data.
The FFT spectral output comprises real, R and imaginary, I component measures of the signal frequency components in the P FFT bins. These need to be combined vectorially () to produce the spectral amplitudes. The radiometer integration process is accomplished by averaging these amplitudes over many sample data blocks.
The FFT output bin voltage amplitudes again contains DC and AC parts (now nominally equal, a property of the FFT bin vector noise statistics), but since there is no integration, for a single block and the pth spectrum bin, become,
(7)
where, Gpis the total gain effective within the pth FFT bin, so allowing for nominal gain variation across the possible wider RF band Br.
Bp is the FFT bin bandwidth = Br/P. Similarly, (Tsky + Tsys)p is the equivalent noise temperature falling in the frequency bin p.
Again, the voltage output sensitivity to RF gain and temperature variations can be estimated by differentiating equation (6) with respect to G and T,
(8)
The second part of equation (2) represents the temperature uncertainty which is equal to the AC component observed at the output of the integrator, or,
Rearranging we get the digital FFT radiometer equation,
(9)
Because of equal RF and post detection bandwidths, this shows that there is no AC noise reduction from integration.
Dividing the two equations of equation (7), we find the effective temperature uncertainty due to gain variations as,
(10)
The combined temperature uncertainty is therefore (no correlation),
(11)
Smoothing only occurs in a digital FFT radiometer by averaging successive data blocks. In this case the radiometer equation for an N-block average, becomes,
(12)
This appears to show the FFT radiometer performs better than the analogue integrating radiometer. But, by choosing the value of N equal to Brτ, over the same integration period, the analogue receiver averages out the same gain variations. When considering continuous observation, any gain drift is reflected in the apparent output temperature variation equally. However, the fact that the FFT receiver can be configured over a broad bandwidth and the number of points/bins/sub-bands chosen as required to produce a broad-band spectrum increases the operational value.
2. Dicke Radiometer
Figure 3. Dicke Switch Radiometer
In the conventional Dicke Switched receiver, The two switches in Figure 3 are switched synchronously so that the difference between the Antenna path signal and the Calibration reference path signal after square-law detection is integrated. The aim of this architecture is to minimise the effect of RF gain variations on the detected total power output.
The analysis approach is similar to that above. For the two paths,
(13)
The voltage output sensitivity to RF gain and temperature variations can again be estimated by differentiating the DC components in equation (13),
(14)
The factor ½ appears assuming equal interval switching. Also,
(15)
The extra factor 2 under the square root signs accounts for the halving of integration time, again subject to equal switching periods.
From equation (13), the temperature uncertainties are,
(16)
From equations (13) and (14), the temperature uncertainty due to RF gain variations is,
(17)
On synchronous subtraction, these uncorrelated components combine as the root sum of squares (sum by power) to give,
(18)
Equation (60) shows an advantage of the Dicke switch receiver, for if Tcal is constrained close to the antenna temperature Tsky, the effect of gain variations is minimised.
3. Ratiometric Dicke Radiometer
Figure 4. Ratiometric Dicke Switch Radiometer
An alternative Dicke switch processing scheme2 is to take the ratio of the standard Dicke switch states (Figure 4) which ideally, removes the gain sensitivity completely, as the output ratio R, becomes,
(19)
Knowing Tsys and Tcal, the wanted temperature Tsky and the calibration for R is simply obtained.
(20)
This scheme has the advantage that it compensates for RF gain variations with frequency when applied to wideband digital FFT systems. The factor 2 loss in sensitivity due to switching is still apparent. However the loss can be avoided if the receiver switch is operated manually and the calibration done off-line, averaging for a period considerably longer than when measuring Tsky so that measurement uncertainty is only significant when switched to the antenna. There may be some gain drift between switch states and this needs to be defined.
For a single block and the pth spectrum bin, the outputs for the two paths become,
(21)
Again, Gpis the total gain effective within the pth FFT bin, so allowing for nominal gain variation across the possible wider RF band Br.
Bp is the FFT bin bandwidth = Br/P. Similarly, (Tsky + Tsys)p is the equivalent noise temperature falling in the frequency bin p.
The voltage output sensitivity to RF gain and temperature variations can be estimated by differentiating the DC component in equation (21),
(22)
The voltage output sensitivity to RF gain and temperature variations can again be estimated by differentiating defining ratio,
(23)
Equating the second parts of equation (22) to the AC parts of V1 and V2, after averaging N data blocks and dividing by V1,V2,
(24)
The characteristic of RF gain variation is largely that of drift so we can assume that there is no gain change within a measurement block. However the temperature noise is significant and uncorrelated, hence,
(25)
and from (19),
The total uncertainty in temperature measurement after averaging N blocks for each data sample (antenna and reference load) is now obtained,
(26)
Gain variations for even a well insulated amplifier section can be subject to a slow drift which may cause significant differences between sky and reference switched data causing the hidden gain elements in equation (19) not to be completely cancelled out. For a gain difference G, the corresponding ratio change, R = G/G causing a baseline temperature error of,
(27)
4. Meech Ratiometric Radiometer
A ratiometric Dicke switch type of radio telescope radiometer has been described by Meech1 that compensates for system gain variations. It periodically switches a calibrating noise source in parallel with the signal path and calculates the ratio of the received power with and without the calibration source so cancelling any receiver slow gain variations and obviating any switch loss.
The output ratio is R, where,
(28)
Clearly, with a knowledge of Tsys and Tcal, the wanted temperature Tsky can be calculated, from
(29)
To calculate the residual variation due to gain variations between switching states, we need to look at the detected noise signals. For a square law detector with RF bandwidth Br much greater than the integration bandwidth Bi = 1/2, where is the integrator time constant. The AC and DC (noise) outputs are,
(30)
The voltage output sensitivity to RF gain and temperature variations can be estimated by differentiating the DC component in equation (30),
(31)
Assume that there is no gain change within a measurement block, but the temperature noise is significant and uncorrelated,
(32)
and from (29), dT = TcaldR/(1-R)2, or,
(33)
The total uncertainty in temperature measurement after averaging N blocks for each data sample (antenna and reference load) is now obtained,
(33)
For a gain drift G between switch states, the corresponding ratio change, R = G/G causing a baseline temperature error of,
(34)
Appendix: Noise in the Square-law Detector
Band-limited Gaussian noise can be represented by a large set of sinusoidal RF components with random amplitudes and phases,
where f is within the band B1 to B2, and B2 - B1 = B
and the total power is given by,
where, P is the noise power density in the band B Figure A1 Input noise representation
and for receivers, P = kTGF.
The detector output is,
The low frequency detector output is,
The DC component is the sum of B frequencies with total output power Pdc = P2B2. The AC termsshow that there are two possible components at the same difference frequency, one above fp and one below fp.( or at positive and negatibe frequencies).
For any difference frequency fp-fq = f, there are B-f instances, so the output power spectrum exhibits a triangular shape as shown in Figure A2.
Figure A2 Output noise spectrum
Following the detector with a low-pass video filter, Bi the video AC power from integrating under the curve is
If Bi is much smaller than B, i.e. long video integration time, the AC noise output voltage approximates to. .
In the case of an FFT bin, the vector amplitude digital squaring process is equivalent to the square-law function, but in this case, there is no bandwidth restriction in post squaring and the bin amplitude noise voltage is simply Pac = Pdc = PBbin
References
- Simulations of a Ratiometric Switching Radiometer for Radio Astronomy. Marcus Leech, VE3MDL.
- Cheap Hydrogen Line Radio Telescope for £160 using the RTL SDR. PW East.
PWE 18/11/2013