Data from all 14 receivers after subtraction of running-median baselines. Sources appear as spikes in both polarization channels (R and L) of one or two beams. Interference is usually visible in all 14 receivers simultaneously.
The radio emission from celestial
sources is essentially random noise, nearly indistinguishable from the
noise generated by a warm resistor. Normally this noise is also stationary;
that is, its time-averaged power does not change systematically during
an observation, even though the instantaneous power produced in the
receiver by
the source varies erratically on time scales as short as the inverse of
the receiver bandwidth.
The purpose of the simplest
total-power radiometer is to
measure the timed-averaged power of this noise in some well-defined
radio frequency (RF)
range
$$\nu_{\rm RF} - {\Delta \nu_{\rm RF} \over 2} {\rm ~~to~~}
\nu_{\rm RF} + {\Delta \nu_{\rm RF} \over 2} ,$$
where $\Delta \nu$ is the receiver bandwidth.
For example, the receivers used on
the 300-foot telescope to make the $\lambda \approx
6$ cm continuum survey of the northern
sky had a center radio frequency
$\nu_{\rm
RF}\approx 4.85 \times 10^9$ Hz, bandwidth $\Delta \nu_{\rm RF}
\approx 6
\times 10^8$ Hz, and output averaging time $\tau \approx 0.1$
second.
It is convenient to describe the noise
power generated by a radio source in units of antenna temperature
at the radiometer input. Since the noise power per unit bandwidth
generated by a resistor of temperature $T$ is $P_\nu = k T$, we can
define the
noise temperature
of any noiselike source in
terms of its power per unit bandwidth
$P_\nu$:
$$\bbox[border:3px blue solid,7pt]{T_{\rm N} \equiv {P_\nu
\over k}}\rlap{\quad \rm {(3F1)}}$$
where $k \approx 1.38 \times 10^{-23}$ Joule K$^{-1}$ is
the Boltzmann constant.
The conceptually simplest radiometer consists of three stages in series:
(1) an ideal (lossless) bandpass
filter that passes input noise only
in the desired
frequency range,
(2) an ideal square-law
detector
whose output
voltage is
proportional to the square of its input voltage; that is, its output
voltage is proportional to its input power,
(3) and a signal averager or integrator
that smoothes out the
rapidly fluctuating detector output.
The temperature equivalent to the
total noise power from all
sources referenced to the input of an ideal
receiver input is called the
system
noise temperature:
$$\bbox[border:3px blue solid,7pt]{T_{\rm sys} = T_{\rm cmb} + \Delta
T_{\rm source} + T_{\rm atm} +
T_{\rm spillover} + T_{\rm rcvr} + \dots}\rlap{\quad \rm
{(3F2)}}$$
The contributions listed explicitly
in Equation 3F2 are $T_{\rm cmb} \approx 2.73$ K
from the cosmic microwave background, $\Delta T_{\rm source}$ from the
astronomical source being observed, $T_{\rm atm}$ from atmospheric
emission in the telescope beam, $T_{\rm spillover}$ to account for
radiation that the feed picks up in directions beyond the edge of the
reflector, and $T_{\rm rcvr}$ to represent the noise power generated by
the receiver itself, referenced to the receiver input. The desired
signal $\Delta T_{\rm source}$ was written with a $\Delta$ to emphasize
that it is usually much smaller than the total system noise: $\Delta
T_{\rm source} \ll T_{\rm sys}$. For example, in the $\nu_{\rm RF}
\approx 4.85$ GHz sky survey made with the 300-foot telescope, the
system
noise was $T_{\rm sys} \approx 60$ K, but the faintest sources detected
contributed only $\Delta T_{\rm source} \approx 0.01$ K.
After passing through an input filter
of width $\Delta \nu_{\rm RF} <
\nu_{\rm RF}$ the noise signal is no longer completely random; it
becomes
more like a sine wave of frequency $\approx \nu_{\rm RF}$ whose
amplitude envelope is modulated
(varies) on time scales $\Delta t
\approx (\Delta \nu_{\rm RF})^{-1} > \nu_{\rm RF}^{-1}$. The
positive
and negative envelopes are similar so long as $\Delta \nu_{\rm RF} \ll
\nu_{\rm RF}$.
Since the input filter does not pass
signals with zero frequency (DC),
the time-averaged voltage is nearly zero. However, the average power at
the filter output is not zero; it is
$$P \approx P_\nu \Delta \nu_{\rm
RF}$$
The filter output is sent to a
square-law detector, a device whose output
voltage is proportional to the square of the input voltage, which in
turn is proportional to the input power. The detector output waveform
looks like:
The oscillations under the envelope
approach zero every $\Delta t
\approx (2 \nu_{\rm RF})^{-1}$. Thus the oscillating component of the
detector output is centered near the frequency $2 \nu_{\rm RF}$. The
detector output also has frequency components near zero (DC) since the
mean output voltage is clearly nonzero. The output frequency spectrum
of the detector looks like:
Both the rapidly varying component with frequencies near $2 \nu_{\rm RF}$ and its envelope vary on time scales that are normally much shorter than the time scales on which the average signal power $\Delta T$ varies. Thus it is possible to filter out the unwanted rapid variations and smooth the detected envelope with some time constant $\tau \gg (\Delta \nu_{\rm RF})^{-1}$ by integrating or averaging the detector output. This integration might be done electronically by smoothing with an RC (resistance plus capacitance) filter whose time constant is $\tau = RC$ or by sampling and digitizing the detector output voltage and integrating it numerically (e.g., by computing its running mean).
Integration greatly reduces the
receiver output
fluctuations. In the time interval $\tau$ there are $ N \approx \Delta
\nu_{\rm RF} \times \tau$ independent samples of the total noise power
$T_{\rm sys}$, each of which has an rms error $\sigma_{\rm T} \approx
T_{\rm sys}$. The rms error in the average of $N \gg 1$ independent
samples is reduced by the factor $\sqrt{N}$, so the receiver output
fluctuations are only
$$\bbox[border:3px blue solid,7pt]{\sigma_{\rm T} \approx {T_{\rm sys}
\over
\sqrt{\Delta \nu_{\rm RF} \tau}}}\rlap{\quad \rm {(3F3)}}$$
after smoothing, and the central limit theorem of statistics implies
that they have a Gaussian amplitude
distribution. This important equation is called the ideal
radiometer
equation for a total-power receiver. The weakest detectable
signals
$\Delta T$ only have to be several (typically five) times the output
rms $\sigma_{\rm
T}$ given by the radiometer equation, not several times the total
system noise $T_{\rm sys}$. The number $N \approx \Delta \nu_{\rm RF}
\tau$ may
be quite large in practice (e.g., $ N > 10^8$ is not
uncommon), so signals as
faint as $\Delta T \sim 10^{-4} T_{\rm sys}$ may be detectable.
Example: The $\nu \approx 4.85$ GHz ($\lambda \approx 6$ cm) northern sky survey made with the 300-foot (91 m) telescope.
This survey used total-power
radiometers very similar to the radiometer
described above, but with multistage RF amplifiers that simultaneously
amplified and
filtered the input signals. The telescope was driven up and down in
elevation at its slew rate $\pm 10^\circ$ per minute = 10 arcmin per
second of time. The beamwidth was
$$\theta_{\rm HPBW} \approx
{1.2 \lambda \over D} = {1.2 c \over \nu_{\rm RF} D}$$
$$
\theta_{\rm HPBW} \approx { 1.2 \times 3 \times 10^8 {\rm ~m~s}^{-1}
\over 4.85 \times 10^9 {\rm ~Hz~} \times 91 {\rm ~m}} \approx 8.2
\times 10^{-4} {\rm rad} \approx 2.8 {\rm ~arcmin}$$
The scanning time
between half-power points was thus $\approx 0.3$ s.
The data were integrated and sampled
every $\tau = 0.1$ s, so there
were $\approx 3$ samples per half-power beamwidth. A subset of the
samples taken from one receiver during one scan covering the
declination range $\delta \approx -2^\circ$ to $\delta \approx
+73^\circ$ is shown.
The intensity scale has been
calibrated in Kelvins, and the large
mean $T_{\rm sys} \approx 60$ K has been subtracted. By far the biggest
time-dependent signal (spanning a range of about 1 K) is caused by
ground
radiation entering the prime-focus feed via leakage through the
reflector mesh and spillover. Fortunately, this unwanted ground signal
varies smoothly with telescope elevation, so subtracting a
short (about 40 arcmin long) running-median baseline takes out the
spillover signal without removing compact radio sources. The
outputs from all 14 receiver channels (7 beams $\times$ 2
polarizations/beam) after baseline subtraction are shown in the next
viewgraph. Only now are the faint radio sources visible above the noise
fluctuations.
Data from all 14 receivers after
subtraction of running-median baselines. Sources appear as spikes in
both polarization channels (R and L) of one or two beams.
Interference is usually visible in all 14 receivers simultaneously.
The rms noise observed is consistent
with the prediction of the
total-power radiometer equation:
$$\sigma_{\rm T} \approx {T_{\rm sys}
\over \sqrt{\Delta \nu_{\rm RF} \tau}} \approx {60\,{\rm K} \over
\sqrt{ 6 \times 10^8\,{\rm Hz} \times 0.1\,{\rm s} } } \approx
0.008\,{\rm K}$$
Gain stability
Note that the output of a total-power
receiver scales in
proportion to the overall gain $G$ of the receiver:
$$P_\nu = G k
T_{\rm sys}$$
If $G$ isn't perfectly constant, the change in output
$$\Delta P_\nu = \Delta G k T_{\rm sys}$$
caused by a gain
fluctuation
$\Delta G$ produces a change
$$\Delta T_{\rm G} = T_{\rm sys} \biggl(
{\Delta G \over G}\biggr)$$ which is indistinguishable from a
comparable change $\Delta T$ in the system noise temperature produced
by
an astronomical source. Since gain fluctuations and input
noise fluctuations are independent random processes, their variances
(the variance is the
square of the rms) add, and the total receiver
output fluctuation becomes:
$$\sigma_{\rm total}^2 = \sigma_{\rm
noise}^2 + \sigma_{\rm G}^2$$ $$\sigma_{\rm total}^2 = T_{\rm sys}^2
\biggl[ {1 \over \Delta \nu_{\rm RF} \tau} + \biggl( {\Delta G \over G}
\biggr)^2 \biggr]$$
The
practical total-power radiometer equation
is thus: $$\bbox[border:3px blue solid,7pt]{\sigma_{\rm T} \approx
T_{\rm sys} \biggl[ {1 \over \Delta
\nu_{\rm RF} \tau} + \biggl( {\Delta G \over G} \biggr)^2
\biggr]^{1/2}}\rlap{\quad \rm {(3F4)}}$$
Clearly, gain fluctuations will
significantly degrade the
sensitivity unless $$\biggl( {\Delta G \over G} \biggr) \ll { 1 \over
\sqrt{\Delta \nu_{\rm RF} \tau}}$$
For example, the 5 GHz receiver used
to make the sky survey with the 300-foot telescope had $\Delta \nu_{\rm
RF} \approx 6 \times 10^8$Hz and $\tau \approx 0.1$ s, so the
fractional
gain fluctuations on time scales up to a few seconds (the time to scan
one baseline length) had to satisfy
$${\Delta G \over G} \ll {1 \over
\sqrt{6 \times 10^8{\rm ~Hz~} \times 0.1{\rm ~s}}} = 1.3 \times
10^{-4}$$
This is difficult to achieve in practice.
Fluctuations in atmospheric emission
also add to the noise in the
output of a simple total-power receiver. Water vapor is the main
culprit because it is not well mixed in the atmosphere, and noise from
water-vapor fluctuations can be a significant
problem at frequencies of $\sim 5$ GHz and up. One way to minimize the
effects of fluctuations in both receiver gain and atmospheric emission
is to make a differential measurement by comparing signals
from two adjacant feeds. The method of switching rapidly between beams
or loads is called
Dicke switching
after Robert Dicke, its
inventor.
If the system temperatures are $T_1$
and $T_2$ in the two positions
of the switch, then the receiver output is proportional to $T_1 - T_2
\ll T_1$ and the effect of gain fluctuations is only
$$\Delta T_{\rm G}
\approx (T_1 - T_2){\Delta G \over G} \ll T_1{\Delta G \over G}~.$$
Likewise, the atmospheric emission in two nearly overlapping beams
through the troposphere is nearly the same, so most of the tropospheric
fluctuations
cancel out. The main drawback with Dicke switching is that the receiver
output fluctuations, relative to the source signal in a single beam,
are doubled, so the radiometer equation for a Dicke switching
receiver is:
$$\bbox[border:3px blue solid,7pt]{\sigma_{\rm T} = {2 T_{\rm sys}
\over \sqrt{\Delta
\nu_{\rm RF} \tau}}}\rlap{\quad \rm {(3F5)}}$$
Few actual radiometers are this
simple. Nearly all practical radiometers are superheterodyne
receivers, in which a mixer multiplies
the RF signal by a sine wave of frequency $\nu_{\rm LO}$ generated by a
local
oscillator. The
product of two sine waves contains the sum and difference frequency
components
$$2 \sin (2 \pi \nu_{\rm LO} t) \times \sin (2 \pi \nu_{\rm RF} t) =
\cos [2 \pi (\nu_{\rm LO} - \nu_{\rm RF}) t] -
\cos [2 \pi (\nu_{\rm LO} + \nu_{\rm RF}) t]~.$$
The difference frequency is called the intermediate
frequency (IF).
The advantages of superheterodyne receivers include doing most of the
amplification at lower frequencies ($\nu_{\rm IF} < \nu_{\rm RF}$),
which is usually easier, and precise control of the $\nu_{\rm RF}$
range covered via tuning only
the local oscillator so that back-end
devices following the untuned IF amplifier, multichannel filter banks
or digital spectrometers for example, can operate over fixed frequency
ranges.
