Response from Darrel Emerson to Bill Brundage on the implementation of a continuum
detector anti-aliasing filter and sampler, from the point of view of
scientific requirements.

   Bill's strawman model consists of the square law detector at each
2-GHz-wide IF, followed by an anti-aliasing filter, followed by
a digitizer (not itself incorporating square-box or other integration),
followed by a digital averager.  The digital averager would then
represent a very good approximation to square-box integration,
with a data dump frequency of 2 kHz or greater.  This is not the only
possible implementation.

Bill asked for guidance on the following parameters:

"A Science requirement giving nominal or limit values for 8 parameters:
<Fc>, anti-alias filter <Type> with response <R> at frequency <Fr>,
<Fsample> rate, LSB = <N> * delta Vrms, boxcar integrated data rate <Fd>
and integration efficiency <E> ought to be sufficient for implementation
by BE IPT.  Science IPT may prefer a smaller number of parameters,
but that might result in a less satisfactory implementation that limits
the potential science."

   There is a requirement for the phase response of the anti-aliasing
filter to be fairly linear over the range of frequencies of interest.
(If it isn't, it introduces a distortion into the OTF beam shape, which
is different according to each rectangular grid OTF scan direction, and
becomes extremely bad for providing good short-spacing UV data.)
Based on that, I opt for a Bessel low pass filter, and somewhat
arbitarily (because it matches the IC chip Bill already has in mind)
adopt a 4th-order Bessel filter.

   The maximum frequency of interest, after detection, is 663 Hz.
This is derived from a scanning speed of 0.5 degs/second, and
lambda/(2.D) at 950 GHz.  If we want this frequency to be
attenuated by the anti-aliasing LP filter by not more than 1%,
then the -3 dB point of the filter has to be at
3.74 kHz (see below).  [This is fairly conservative.)

    From my notes below, I derive:

<Fc> (the -3 dB point of the anti-aliasing filter): 3.74 kHz.
<Type> 4th-order Bessel low pass anti-aliasing filter(see equations below).
<Fsample> sample rate 18.3 kHz or faster (see below).
          If these samples are averaged to a 2 kHz data
          dump rate, then 20 kHz raw sample rate may be
          convenient, averaging 10 samples.
<N> The LSB should be 25% to 30% of the rms noise, which implies
          a 14-bit digitizer at this 18 or 20 kHz rate. The
          data dumps at 2 kHz should be 16-bit values.
<Fd> Equivalent boxcar integrated data rate is 2 kHz or faster.
       (See notes from Change Request text).
<E> Efficiency: not more than 1% effective observing time
should be lost, in addition to the compromises mention above.

---------------------------------------------------------


Detailed notes on low-pass filter and sampling:
===============================================

AMPLITUDE RESPONSE:

    Choosing a Bessel design because of its near-linear
phase characteristic, which is particularly important when
on-the-fly single dish data is to be combined with
interferometric data:

   The maximum frequency of interest is 663 Hz. If we
allow (rather conservatively) the amplitude to be attenuated
by 1% at 663 Hz, then for a 4th-order Bessel filter the
-3 dB (0.707) frequency comes out as 3.74 kHz.

PHASE RESPONSE:
   With the -3 dB response set to 3.74 kHz, the phase/frequency
response at 663 Hz is still linear, within less than 1 degree.
(With this filter, even at the -6 dB point, the departure from linear
phase/frequency response is less than 3 degrees, although it
deteriorates rapidly beyond that point.)
   The scientific requirement is only that the departure from
linearity is less than 13 degrees at the highest frequency
of interest (663 Hz), which is easily met.

SAMPLING FREQUENCY:
   If the data are sampled at frequency 2.Fs, then a frequency
(Fs + x) present in the original data is aliased back to a frequency
of (Fs - x).  With the 3 dB frequency at 3.74 kHz, using a
4th-order Bessel LP filter the -40 dB (1%) point is
at 17.65 kHz.  We can allow noise at this -40 dB point to alias
back into 663 Hz.  If we set Fs - x = 663 Hz and Fs + x = 17.65 kHz,
then 2*Fs=17.65 + 0.663 kHz = 18.3 kHz.

   If this Bessel filter, with F(3dB)=3.74 kHz, is sampled at
18.3 kHz, then noise products aliasing back within the passband of
interest (0-663 Hz) are at 1% or less.

DYNAMIC RANGE:
   If the anti-aliasing filter has a 3-dB bandwidth t of
3.74 kHz, its effective integration time is about 1/3.74 =
0.27 milliseconds. With a pre-detector bandwidth B of 2 GHz,
the sqrt(B.t) factor in the radiometer equation is
sqrt(2.e9 * 0.00027) = 735.  In order for <1% degradation
from an inadequate number of bits, the LSB should be
no greater than about 25% (maybe 30%) of the rms noise.
So, if the detected DC voltage is at the maximum digitized
value, 735*4 = 2940 digitizer levels are required.
However, the headroom in the detector should allow
for at least a factor of 4(?) above that, to be able
to cope with clouds and other variations in atmospheric
emission and system noise, giving a requirement for
12000 levels: 16384 corresponds to 14 bits.
So, a 14 bit digitizer is required.  Note that, after
reducing the 18.3 kHz raw data rate to 2 kHz by
averaging (in practice a sample rate of 20 kHz might
be used, averaging 10 samples), higher precision is
required, by roughly the square root of the ratio of
the anti-aliasing filter bandwidth to 2 kHz.  A
16-bit averaged result, at 2 kHz, is adequate.


APPENDIX:

RESPONSE OF A 4th-ORDER BESSEL LOW PASS FILTER:

See for example http://www.crbond.com/papers/bsf.pdf

    For the amplitude response, with w the angular frequency
(w=2.pi.f), I use:

         A=105/sqrt(w^8 + 10.w^6 + 135.w^4 + 1575.w^2 + 11025)

    For the phase response, I use

        phi(radians) = arctan[(105.w - 10.w^3)/(105 - 45.w^2 + w^4)]