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The 1=f effect in the PLANCK LFI radiometers
N. Mandolesi,y M. Seiffert,z M. Bersanelli,x and C. Buriganay
yIstituto TESRE, Consiglio Nazionale delle Ricerche, Via Gobetti 101, 40129 Bologna, Italy
zPhysics Department, University of California -- Santa Barbara, Santa Barbara, CA 93106, USA
xIFCTR, Consiglio Nazionale delle Ricerche, Via Bassini 15, 20133 Milano, Italy
We use an analytic approach to examine the susceptibility of the PLANCK LFI correla­
tion radiometers to various systematic effects. We examine the effects of amplifiers gain
fluctuations, of fluctuations in the amplifiers noise temperatures, in the reference load tem­
perature and in the DC gain ratio. We find that 1=f gain fluctuations are suppressed by
proper gain modulation, leaving fluctuations in the amplifiers noise temperatures as the
main source of 1=f type noise. We estimate that the overall 1=f knee frequency will be
relatively small (¸ 0:1 Hz) and can be reduced further by efficient cooling of the reference
loads.
1 Introduction
The PLANCK Surveyor is a European Space Agency (ESA) satellite mission to map spatial anisotropy
in the Cosmic Microwave Background (CMB) over a wide range of frequencies with an unprecedented
combination of sensitivity, angular resolution, and sky coverage (Bersanelli et al. (1996)). The data
gathered by this mission will revolutionise modern cosmology by shedding light on fundamental cos­
mological questions such as the age and present expansion rate of the universe, the average density
of the universe, the amount of dark matter, and other questions. As with any CMB experiment,
achieving the desired performance requires careful attention to the control of systematic effects. In
this paper we examine some of the systematic effects pertaining to the Low Frequency Instrument
(LFI) radiometers. A more complete analysis, which takes also into account the effect of imperfect
matching of amplifier gains and noise temperatures and of imperfect isolation in the two legs of the
radiometer, can be found in a forthcoming paper (Seiffert et al. (1997)).
A simplified version of the LFI radiometer design is depicted in Figure 1 (see Bersanelli et al.
(1995)). We have made the simplifying assumption that there is no 1=f contribution from the detector
diodes, and we have not considered the effects of the phase shifting that is designed to control this
contribution. The radiometer is a modified Blum correlation receiver (Blum (1959), Colvin (1961)).
The modification is that the temperature of the reference load is quite different from the sky temper­
ature. To compensate, differing DC gains are applied after the two detector diodes. Adjusting the
ratio of DC gains, r, allows one to null the output signal, minimise sensitivity to RF gain fluctuations,
and achieve the lowest white noise in the output. Although it may not be immediately apparent, the
fact that the reference load is not at the same temperature as the sky does not increase the white
noise level compared to a standard correlation receiver (Seiffert et al. (1997)). There are several
potential concerns for the current radiometer scheme. Amplifier noise temperature variations, for
example, might be confused with real changes in the radiometer input. Fluctuations in the reference
load temperature can also mimic input changes. In what follows, we will see that gain fluctuations
and noise temperature fluctuations have the characteristics of 1=f noise. This leads to 1=f­type noise
in the radiometer output. We will define the ``knee'' frequency as the post­detection frequency, f k at
which the 1=f contribution and the ideal white noise contribution are equal.
1=f­type noise in the radiometer output is a concern because it may lead to striping in the final
sky maps and increase the noise level. In general, a value of f k that is significantly greater than the
spacecraft rotation frequency will lead to some degradation in sensitivity (Janssen et al. (1996)).
In the following sections we will examine the underlying assumptions (section 2), and calculate the
radiometer's sensitivity to amplifier gain fluctuations, fluctuations in the amplifier noise temperature,
in the reference load temperature and in the DC gain ratio (section 3). We will use the results of these
calculations to estimate the 1=f knee frequency of the radiometer output. A summary of our results
and a brief discussion of their implications for PLANCK observations are presented in section 4.
1

Ref Load
Antenna
Hybrid
Coupler
Hybrid
Coupler
Amplifiers
DC gain
Modulation
Detector
Diodes
Output
Figure 1: Shown is a simplified version of the PLANCK LFI radiometer design. The reference load
has a temperature in the range of 4--100 K, depending on the cooling option, and the ratio of DC
gains, r, after the diodes is adjusted to produce a null output.
2 Basic assumption
Referring to Figure 1, we denote the voltage gain of amplifier 1 as g 1 and the voltage gain of amplifier
2 as g 2 . The output, p(t) is then given by:
p(t) = a
2
Ÿ`
x+y
p
2
+n 1
'
g 1 +
`
x \Gamma y
p
2
+n 2
'
g 2
– 2
\Gamma r
a
2
Ÿ`
x+y
p
2
+n 1
'
g 1 \Gamma
`
x \Gamma y
p
2
+n 2
'
g 2
– 2
: (1)
Here, x(t) is the noise voltage at the sky horn, y(t) is the noise voltage of the reference load, n 1 (t)
and n 2 (t) are the noise voltages contributed by the amplifiers, r is the ratio of the DC gains after the
two detector diodes, and the diodes are assumed to be perfect square law detectors with a constant
of proportionality of a.
We will now denote the power gain of amplifier 1 as G 1 (= g 2
1 ) and of amplifier 2 as G 2 (= g 2
2 )
and calculate the time average of p(t), denoted by p. Terms like n 1 x vanish because the signal and
amplifier noise are uncorrelated. Under the assumption that the noise signals in the two amplifiers
are uncorrelated, the n 1 n 2 terms also vanish. The term x 2 is equal to kfiT x , where fi is the effective
bandwidth, T x is the noise temperature of the signal entering horn 1, and k is Boltzmann's constant.
We have not specified where the bandpass of the signal gets defined, and we have assumed that it is
the same in both legs of the radiometer. Using similar formulae for y 2 , n 2
1 , and n 2
2 , p becomes:
p = akfi
4 (1 \Gamma r) [G 1 (T x +T y +2Tn1 ) +G 2 (T x +T y +2Tn2 )] + akfi
4 2(1 + r)
p
G 1 G 2 (T x \Gamma T y ) : (2)
We now see that in order to null the output (i.e. make p = 0), we must adjust r to the proper value.
In the simple case that G 1 =G 2 and Tn1 = Tn2 , this value is
r = T x +Tn
T y +Tn : (3)
Equation 2 identifies some potential systematic effects. If the reference load temperature, T y , changed
slightly, or if the noise temperature, Tn , of one the amplifiers fluctuated for example, there is the
potential that these changes could be confused with the sky signal variations that we are interested
in measuring. In the following section of this paper, we will calculate the magnitude of these effects
under the assumption that the fluctuations in the various parameters are uncorrelated.
Before calculating these effects we briefly examine the expected magnitude of gain and noise
temperature fluctuations. We can infer that cryogenic HEMT amplifiers have noise temperature
fluctuations with a 1=f type spectrum because we know that the amplifiers have 1=f type gain
fluctuations (Wollak (1995), Jarosik (1996), Seiffert et al. (1996)). We can estimate the magnitude of
noise temperature fluctuations from the following argument. Assuming that each stage of the amplifier
2

has the same level of fluctuation, we can conclude that the transconductance of an individual HEMT
device also fluctuates according to:
\Deltag m
g m
= 1
2 p
N s
\DeltaG
G
; (4)
where N s is the number of stages of the amplifier, typically ¸ 5. An optimal low noise amplifier design
will have equal noise contributions from the gate and drain of the HEMT, which mean the changes in
gm will lead to changes in Tn (Pospieszalsky (1989)). This can be expressed as
\DeltaT n
Tn
'
\Deltag m
gm
: (5)
We can write the 1=f spectrum of the gain fluctuations as:
\DeltaG
G
= C
p
f
: (6)
Putting this together we get:
\DeltaT n
Tn
'
1
2 p
N s
C
p
f
: (7)
We can therefore write the noise temperature fluctuations as
\DeltaT n
Tn
= A
p
f
(8)
with A=C=(2
p
N s ); a normalisation of A' 1:8 \Theta 10 \Gamma5 (relying on the references above) is appropriate
for the 30 and 45 GHz radiometers. Throughout, we will use units of K= p
Hz for \DeltaT so that we will
not need to refer to the sampling frequency of the radiometer. In these units then, \DeltaT =T has units of
Hz \Gamma1=2 and A is dimensionless. We also note that the value of A will generally depend on the physical
temperature of the amplifier. The values for A given here should be regarded as estimates rather than
precise values. For the radiometers at higher frequencies, it will be necessary to use HEMT devices
with a smaller gate width to achieve the lowest amplifier noise figure. We expect that the gate widths
will be roughly 1=2 that of the devices used for the lower frequency radiometers and this will lead
to gm fluctuations that are roughly a factor of
p
2 higher (Gaier (1997), Weinreb (1997)). We will
therefore adopt a normalisation of A= 2:5 \Theta 10 \Gamma5 for the 70 and 100 GHz radiometers.
3 Susceptibility to various systematic effects
Small fluctuations in each of the terms Tn1 , Tn2 , G 1 , G 2 , T y and r (generically denoted as w i )
appearing in equation 2 will lead to a change in the observed signal which can mimic a true sky
fluctuation \DeltaT equiv according to:
@¯p
@w i
\Deltaw i = @¯p
@T x
\DeltaT equiv : (9)
The results of section 2 can be used to calculate the susceptibility of the radiometer to each source of
spurious fluctuation.
3.1 Sensitivity to amplifier noise temperature variations
In this section, we will calculate the change in the output signal for a small change in the noise
temperature of one of the amplifiers. We start with equation 2, let G 1 = G 2 = G, and then consider
the derivative of the output with respect to Tn1 :
@p
@Tn1
= aGkfi
Ÿ
1 \Gamma r
2
–
: (10)
Note that:
@p
@T x
= aGkfi : (11)
3

Putting these together, one finds that a change in amplifier noise temperature, \DeltaT n can mimic a
change in input signal, \DeltaT equiv . Incorporating the fact that both amplifiers (which have uncorrelated
noise) can contribute increases this effect by p
2 (this is equivalent to adding the two contributions in
quadrature). The result is:
\DeltaT equiv =
p
2\DeltaT n
Ÿ
1 \Gamma r
2
–
: (12)
If the reference load is at the same temperature as the sky, then r = 1 (no DC gain difference) and
there is no effect to worry about. As r departs from 1 the effect is of more concern.
Given this, we must now calculate the post­detection frequency, f k , at which the contributions
from gain fluctuations are equal to the white noise from an ideal radiometer:
\DeltaT equiv = \DeltaT white : (13)
The ideal sensitivity of the radiometer is (see the Seiffert et al. (1997) for a detailed discussion):
\DeltaT white =
p
2(Tn +T x )
p
fiÜ
: (14)
Substituting for the two sides of equation (13) one gets
p
2\DeltaT n (f k )
`
1 \Gamma r
2
'
=
p
2(Tn +T x )
p
fiÜ
: (15)
Dividing each side by Tn and rearranging yields:
\DeltaT n
Tn
=
`
2
1 \Gamma r
'
1
p
fiÜ
`
Tn +T x
Tn
'
: (16)
We will use Ü = 1=(2\Deltaf ) and \Deltaf = 1 Hz. Finally, by using equation 8 for the noise temperature
fluctuations, we have the knee frequency
f k = A 2 fi
8 (1 \Gamma r) 2
`
Tn
Tn +T x
' 2
: (17)
Assuming a 20% bandwidth for our frequency channels and an antenna temperature T x = 3K, we
tabulate the resulting knee frequencies in column 4 of Table 1 for several choices of T y , Tn and the
corresponding values of r.
From equation 17 it results that that in the space T y , Tn , T x the curves of equal f k are hyperboles
on any plane parallel to the plane T y , Tn . Figure 2 shows several curves of equal f k for the four
considered frequencies. The knee frequency must be compared to the spin frequency f s ; for the
PLANCK observational strategy f s = 1 r.p.m., i.e. 0.017 Hz.
3.2 Overall effect
This radiometer is not sensitive to gain fluctuations in first order. We have indeed calculated how the
output will change with respect to a small change in the gain of one of the amplifiers.
In the case G 1 =G 2 =G and by using the expression for r, we have obtained that the output change
cancels completely. The conclusion is that, to first order, gain fluctuations in the both amplifiers do
not mimic a sky signal fluctuation. We note that the second order cross terms are not zero, but 1=f
contribution is too small to be of concern here.
By carrying out analogous calculations, we derive the output changes mimicked by reference load
fluctuations and by fluctuations in r; we find respectively: \DeltaT equiv = \Gammar\DeltaT y and \DeltaT equiv = \Gamma\Deltar(T y +
Tn ). Therefore they are equal to the white noise respectively for \DeltaT y =
p
2=fiÜ (T y +Tn ) and \Deltar=r =
p
2=fiÜ . In these cases the fluctuations in r or T y became important.
We have above discussed to what extent fluctuations in the different parts of our radiometer can
mimic true signal variations. A complete treatment of all contributions together is quite difficult. On
the other hand, under the assumption that all fluctuations terms are uncorrelated, an estimate of their
4

Table 1: 1=f knee frequency for PLANCK LFI radiometers.
30 GHz
T y (K) Tn (K) r f k (Hz) Comments
100 15 0.157 1:20 \Theta 10 \Gamma1 Phase A study baseline
50 15 0.277 8:82 \Theta 10 \Gamma2 optimised passive cooling
20 10 0.433 4:62 \Theta 10 \Gamma2 active cooling of loads and amps
4 10 0.929 7:34 \Theta 10 \Gamma4 amps cooled to 10 K, loads to 4 K
45 GHz
T y (K) Tn (K) r f k (Hz) Comments
100 40 0.307 1:51 \Theta 10 \Gamma1 Phase A study baseline
50 25 0.373 1:14 \Theta 10 \Gamma1 optimised passive cooling
20 15 0.514 5:87 \Theta 10 \Gamma2 active cooling of loads and amps
4 15 0.947 7:01 \Theta 10 \Gamma4 amps cooled to 15 K, loads to 4 K
70 GHz
T y (K) Tn (K) r f k (Hz) Comments
100 50 0.353 4:07 \Theta 10 \Gamma1 Phase A study baseline
50 40 0.478 2:58 \Theta 10 \Gamma1 optimised passive cooling
20 25 0.622 1:24 \Theta 10 \Gamma1 active cooling of loads and amps
4 20 0.958 1:44 \Theta 10 \Gamma3 amps cooled to 20 K, loads to 4 K
100 GHz
T y (K) Tn (K) r f k (Hz) Comments
100 60 0.394 5:21 \Theta 10 \Gamma1 Phase A study baseline
50 50 0.530 3:07 \Theta 10 \Gamma1 optimised passive cooling
20 40 0.717 1:09 \Theta 10 \Gamma1 active cooling of loads and amps
4 40 0.977 6:98 \Theta 10 \Gamma4 amps cooled to 40 K, loads to 4 K
5

Figure 2: The curves of equal f k on the plane T y , Tn are plotted; an antenna temperature T x = 3 K is
assumed. Each panel refers to a different frequency channel (30, 45, 70 and 100 GHz). The different
lines refer to: f k (Hz) = 0.3 (solid line), 0.1 (dotted line), 0.03 (dashed line), 0.01 (long dashes),
0.003 (dotted­dashed line), 0.001 (three dots­dashes). For the channels at 30 and 45 GHz the case
f k = 0:3Hz does not occur independently of cooling optimisation and is not reported.
global effect can be derived by comparing the change of ¯
p due to a true sky temperature variation
with the quadrature sum of the signal mimicked by the different instrumental effects.
By using the above results, in the case G 1 =G 2 =G, Tn1 = Tn2 = Tn , after algebraic manipulations
we have:
(\DeltaT equiv ) 2 = (T y +Tn ) 2 (\Deltar) 2 + r 2 (\DeltaT y ) 2 + 1
2
`
T y \Gamma T x
T y +Tn
' 2
T 2
n
A 2
f
: (18)
The basic information of the above equation was already implicit in the equation 11 of Bersanelli et
al. (1995), when T x is derived from the condition \Delta / (rT y \Gamma T x ) +Tn (r \Gamma 1) = 0 (with the present
notation for the interesting quantities), and its fluctuations are obtained by the sum in quadrature
of the fluctuations of r, T y and Tn . We note that in equation 18 the two terms related to the two
amplifiers gain fluctuations do not appear, because they are negligible at first order, as previously
discussed.
We can also see from this equation that the effect of white noise fluctuations in r or T y is to
raise the overall white noise level, thereby lowering the knee frequency (but decreasing the overall
sensitivity). On the other hand the limits on r and T y fluctuations given above can be realistically
met with present technology.
More generally, the fluctuations in r and T y may have a complicated spectral shape. In this case,
a single knee frequency and white noise level are an inadequate description of the noise; one must
instead consider the shape of the composite noise spectrum.
4 Discussion and conclusions
We have shown that the dominant source of 1=f noise in the radiometer output is amplifier noise
temperature fluctuations. The contributions to the 1=f noise from other sources, such as amplifier
gain fluctuations, reference load fluctuations and fluctuations in the ratio of DC gains, are either
zero at first order or much less than the amplifier noise temperature fluctuations term under quite
reasonable assumptions.
6

For a total power radiometer, laboratory measurements have found knee frequencies between 10
and 100 Hz; the modified correlation radiometer scheme reduces the knee frequency by more than two
order magnitudes.
The value of the knee frequency depends upon several factors including the radiometer bandwidth,
reference load temperature, and the intrinsic level of fluctuation in the HEMT devices; values of ¸ 0:1
Hz should be reached with only passive cooling of the radiometer to ¸ 50 K. A f k of 0:1 Hz will
not significantly contaminate the PLANCK observations. For example, considering the standard
PLANCK observational strategy for the channel at 30 GHz (sample time of about \Deltat ' 0:09 s per
pixel and about 680 pixels, n pix , for a scan circle with an angle of 70 ffi between spin axis and telescope
direction), and referring to Janssen et al. (1996), we find that the maximum excess noise factor Fmax
for a typical scan circle, with respect to the case of pure white noise, is ' 1:06 For the channel at 100
GHz (\Deltat ' 0:035 s and n pix ' 1700) we find Fmax ' 1:03.
Active cooling to ¸ 20 K of the amplifiers and reference loads would, of course, allow the reduction
of the knee frequency and the total noise to very low values.
A refinement of the present analysis for the determination of f k will be pursued in the future by
including in the analytical formalism the phase shifts between the signals entering the two legs of
the radiometer, by software simulations of the radiometer functions to accurately study the combined
effect of all components and finally by testing it with laboratory bread boarding and precursor sub­
orbital CMB experiments.
Acknowledgements
It is a pleasure to thank S. Weinreb, T. Gaier, P. Meinhold, P. Lubin, G. Smoot, C. Lawrence, S.
Levin, M. Janssen, and J. Delabrouille for useful discussions.
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