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
LO/Mixer/
IF/Filter
(Br)
Antenna
Integrator
 = 1/2Bi
∫
RF
Amplifier
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,
V  Vac  Vdc

 kGF Tsky  Tsys


2 Br Bi  kGFBr Tsky  Tsys

(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),


dVG  kFB r Tsky  Tsys dG
(2)
dVT  kGFBr dT
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,

kGFBr dT  kGF Tsky  Tsys

2Br Bi
Rearranging we get the well-recognised radiometer equation for temperature sensitivity,

dT  Tsky  Tsys


2 Br Bi / Br
Tsky  Tsys
(3)
Br
Dividing the two equations in (2) gives the temperature sensitivity to gain variations.

dTG  Tsky  Tsys
 dG
G
(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,
dG

G
1
(5)
Br
Temperature and gain errors combine as root sum of squares with the estimate,

dT  Tsky  Tsys

2
1
 dG 

 
Br
 G 
(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
Antenna
LO/IQ Mixer/
Averager
IF/Filter
Buffer
N-FFT’s
(Br)
Memory
I
ADC
FFT
Q
P
RF
2P
bins
Amplifier
samples
P-point
Spectrum
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 ( R 2  I 2 ) 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,
V p  Vac  Vdc

 kGp FB p Tsky  Tsys
p  kGp FB p Tsky  Tsys p
(7)
where, Gp is 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,

dV pG  kFB p Tsky  Tsys
 p dG p
(8)
dV pT  kGp FB p dT p
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,

kGp FB p dTpT  kGp B p F Tsky  Tsys
p
Rearranging we get the digital FFT radiometer equation,

dTpT  Tsky  Tsys
p
(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,

dT pG  Tsky  Tsys
 p dG
G
(10)
The combined temperature uncertainty is therefore (no correlation),

dTp  Tsky  Tsys

 dG p

 Gp

2

 1


(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,
dTp
Tsky  Tsys 

N
 dG p

 Gp

2

 1


(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
Antenna
LO/Mixer/
IF/Filter
(Br)
Integrator
 = 1/2Bi
-
Tcal
RF
Amplifier
+
∫
Square-law
Detector
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,
V 1  Vac  Vdc

 2Br Bi  kGFBr Tsky  Tsys 
V 2  kGF Tcal  Tsys  2 Br Bi  kGFBr Tcal  Tsys 
 kGF Tsky  Tsys
(13)
The voltage output sensitivity to RF gain and temperature variations can again be estimated by
differentiating the DC components in equation (13),
1
V 1  V 2
2
1
dVoG  kFBr Tsky  Tcal dG
2
Vo 


(14)

The factor ½ appears assuming equal interval switching. Also,
dVo 2T
 
 

1
1
kGFBr dT1  kGF Tsky  Tsys 4 Br Bi
2
2
1
1
 kGFBr dT 2  kGF Tcal  Tsys 4 Br Bi
2
2
dVo1T 



(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,

dT1  Tsky  Tsys

4 B r Bi / B r

4 B r Bi / B r
Tsky  Tsys

Br / 2
dT 2  Tcal  Tsys


(16)
Tcal  Tsys
Br / 2
From equations (13) and (14), the temperature uncertainty due to RF gain variations is,

dTG  Tsky  Tcal
 dG
G
(17)
On synchronous subtraction, these uncorrelated components combine as the root sum of squares
(sum by power) to give,
dT  dTG 2  dT12  dT 2 2


Tsky  Tcal 2  dG

G


2
2
Tsky  Tsys 2  Tcal  Tsys 2
(18)
Br
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
Antenna
LO/IQMixer/
IF/Filter
(Br)
Averager
N-FFT’s
Buffer
Memory
/
FFT
P-point
Spectrum
P
2P
bins
samples
RF
Amplifier
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,
R
Tsky  Tsys
Tcal  Tsys
(19)
Knowing Tsys and Tcal, the wanted temperature Tsky and the calibration for R is simply obtained.
Tsky  RTcal  1  RTsys
(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,
V 1  Vac  Vdc

 p  kG1 p FB p Tsky  Tsys  p
V 2  kG2 p FB p Tcal  Tsys  p  kG2 p FB p Tcal  Tsys  p
 kG1 p FB p Tsky  Tsys
(21)
Again, Gp is 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),
dV1 p
dG1 p
 kFB p Tsky  Tsys

p
dV1 p

p
dV 2 p
dV 2 p
dG2 p
 kFB p Tcal  Tsys
dT p
dT p
 kG1 p FB p
(22)
 kG2 p FB p
The voltage output sensitivity to RF gain and temperature variations can again be estimated by
differentiating defining ratio,
V1
V2
V 1  dV1 dV 2 
dR 



V 2  V1 V 2 
R
(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,
dV1 pG
V 1 pG
dV 2 pG
V 2 pG
1  dG1 p 
N  G1 p 
1  dG2 p 

N  G 2 p 

dV1 pT
V 1 pT
dV 2 pT
V 2 pT

1
N

(24)
1
N
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,
 dV1G dV 2 G
dR  R 

V 2G
 V 1G
2
  dV1T
 
  V1
T
 
2
  dV 2 T
  
  V 2T
and from (19), dT  Tcal  Tsys dR



2
(25)
The total uncertainty in temperature measurement after averaging N blocks for each data sample
(antenna and reference load) is now obtained,

2
Tsky  Tsys
N
dT 

(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,

T  Tcal  Tsys
 GG
(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,
R
Tsky  Tsys
(28)
Tsky  Tsys  Tcal
Clearly, with a knowledge of Tsys and Tcal, the wanted temperature Tsky can be calculated, from
Tsky 
R
Tcal  Tsys
1 R
(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,
V 1  V ac  V dc

 2Br Bi  kGFBr Tsky  Tsys 
V 2  kGF Tsky  Tsys  Tcal  2 B r Bi  kGFBr Tsky  Tsys  Tcal 
 kGF Tsky  Tsys
(30)
The voltage output sensitivity to RF gain and temperature variations can be estimated by
differentiating the DC component in equation (30),
dV 1G dG1 p

V 1G
G1 p
dV 1T

V 1T
dV 2 G dG 2 p

V 2G
G2 p
dV 2 T

V 2T
1
Br
(31)
1
Br
Assume that there is no gain change within a measurement block, but the temperature noise is
significant and uncorrelated,
2
 dV1G dV 2 G
dR  R 

V 2G
 V 1G
  dV1T
 
  V1
T
 
2
  dV 2 T
  
  V 2T



2
(32)
and from (29), dT = TcaldR/(1-R)2, or,
dT 
Tsky  Tsys  Tcal 2
Tcal
(33)
dR
The total uncertainty in temperature measurement after averaging N blocks for each data sample
(antenna and reference load) is now obtained,
dT 
Tsky  Tsys  Tcal Tsky  Tsys 
Tcal
2
Br
(33)
For a gain drift G between switch states, the corresponding ratio change, R = G/G causing a
baseline temperature error of,
T 
Tsky  Tsys  Tcal 2
G
G
Tcal
(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,
nt  
a
p

cos 2 f p t   p

voltage
p
where f is within the band B1 to B2, and B2 - B1 = B
and the total power is given by,
n 2 t  
a
1
2
2
p
2P
ap
 PB
B1
p
where, P is the noise power density in the band B
and for receivers, P = kTGF.
The detector output is,
n 2 t  
 a cos 2 f t   
  a a cos2 f t    cos2 f
2
p
p
0
p q
p 0 q 0
p
p
q
B2
Figure A1 Input noise representation
2
p
fp
t  q

frequency
The low frequency detector output is,
n 2 t  

p

a 2p
2

p 0 q 0
a p aq
2
 

 
cos 2 f p  f q t   p   q 
p 0 q0
a p aq
2
 

cos 2 f q  f p t   q   p

The DC component is the sum of B frequencies with total output power Pdc = P2B2. The AC
terms show 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.
power
2P2(B-f)
B2- B1
frequency
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


Pac  2P 2 BBi  Bi2 / 2
If Bi is much smaller than B, i.e. long video integration time, the AC noise output voltage
approximates to. P 2 BBi .
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
1. Simulations of a Ratiometric Switching Radiometer for Radio Astronomy. Marcus Leech,
VE3MDL. www.sbrac.org/documents/ratiometric_radiometer.doc
2. Cheap Hydrogen Line Radio Telescope for £160 using the RTL SDR. PW East.
http://www.y1pwe.co.uk/RAProgs/RadAst.doc
PWE 18/11/2013