The document characterizes an FPGA-based spectrometer prototype designed for use in heterodyne receivers at the Large Millimeter Telescope (LMT). Testing showed the spectrometer could successfully calibrate for offset, gain, and phase issues and reduce spurious tones in data. It was also found to be linear in nature and sufficiently follow the radiometer equation, with an Allan time of at least 400 seconds. The spectrometer uses a CASPER-designed ROACH2 board with a Xilinx FPGA and two analog to digital converter boards to process signals at sample rates up to 2.5 Giga samples per second across two channels.

1. Characterizing
an
FPGA
Based
Spectrometer
Prototype
for
Heterodyne
Receivers
at
the
LMT
Tim
Costa1
1.
University
of
Massachuse7s
Amherst
Abstract
We
have
characterized
a
field
programmable
gate
array
(FPGA)
based
spectrometer
prototype.
We
tested
it
for
offset,
gain,
and
phase
(OGP)
calibraIon,
linearity,
and
Allan
variance.
We
have
found
that
we
are
able
to
calibrate
the
OGP
and
reduce
spurious
tones
in
the
data.
We
have
also
found
that
the
spectrometer
is
linear
in
nature,
passing
both
the
homogeneity
and
addiIvity
tests
and
sufficiently
follows
the
radiometer
equaIon,
having
an
Allan
Ime
of
at
least
400
seconds.
IntroducCon
The
Five
College
Astronomy
Department
(FCAD)
operated
the
Five
College
Radio
Astronomy
Observatory
(FCRAO)
from
1969
unIl
it
was
closed
in
2011.
During
that
Ime,
the
FCRAO
hosted
the
Second
Quabin
Observatory
Imaging
Array
(SEQUOIA)
which
was
the
fastest
imaging
array
at
3
mm
wavelengths.
The
FCRAO
closed
in
favor
of
the
Large
Millimeter
Telescope
(LMT).
With
an
acIve
surface
diameter
of
50
meters,
the
LMT
is
the
largest
single
dish
telescope
observing
in
the
0.85
to
4
mm
wavelength
range.
These
radio
wavelengths
easily
pass
through
the
dust
and
gas
of
the
interstellar
medium
(ISM)
and
can
carry
informaIon
on
star-­‐formaIon,
planetesimals,
and
extra-­‐solar
protoplanetary
disks.
Astronomers
have
also
proposed
to
use
the
LMT
to
study
fluctuaIons
in
the
cosmic
microwave
background
(CMB)
and
acIve
galacIc
nuclei
(AGN).
The
LMT
currently
lacks
a
focal
plane
array
spectrometer.
Inclusion
of
such
a
spectrometer
would
allow
for
the
mapping
of
external
galaxies
and
the
mapping
of
several
spectral
lines
simultaneously.
MulIple
spectral
lines
are
needed
to
derive
a
complete
picture
of
the
ISM
and
being
able
to
map
mulIple
lines
simultaneously
would
increase
mapping
speed,
thus
freeing
up
expensive
telescope
Ime.
The
Spectrometer
The
spectrometer
is
based
on
a
field
programmable
gate
array
(FPGA)
designed
by
the
CollaboraIon
for
Astronomy
Signal
Processing
and
Electronics
Research
(CASPER).
FPGAs
can
be
repeatedly
reprogrammed
by
a
consumer
and
tend
to
have
many
logic
and
RAM
blocks.
This
specific
design
uses
the
second
generaIon
reconfigurable
open
architecture
compuIng
hardware
(ROACH2)
board.
The
ROACH2
consists
of
a
Xilinx
Virtex-­‐6
series
FPGA,
a
PowerPC
running
Linux,
and
two
docks
that
support
analog
to
digital
converters
(ADCs).
The
ADC
boards
each
have
four
cores
with
max
sample
rates
of
1.25
Gs/s.
The
cores
are
interleaved
as
two
pairs
giving
two
channels
with
max
sample
rates
of
2.5
Gs/s.
Methods
• Offset/Gain/Phase
CalibraCon
Aligning
the
cores
of
the
ADCs
is
necessary
for
reducing
the
noise
of
the
system
which
can
obscure
scienIfic
signals.
Aligning
the
cores
decreases
the
effect
of
spurs,
which
are
direct
results
of
misalignment
in
the
offset,
gain,
and
phase.
We
followed
many
of
the
methods
laid
out
in
Patel
et
al.
(2014),
with
some
slight
modificaIons.
We
only
had
to
align
two
cores,
as
opposed
to
the
four
they
aligned.
• Linearity
Reliable
spectrometers
demonstrate
a
linear
nature.
Linearity
implies
that
a
change
to
the
input
signal’s
strength
results
in
a
corresponding
change
in
the
output
signal’s
strength.
• Allan
Variance
Spectrometers
should
follow
the
radiometer
equaIon,
σ
α
1/√[b*t],
where
σ
is
the
noise
level,
b
is
the
bandwidth,
and
t
is
the
integraIon
Ime.
When
the
noise
level
is
plo7ed
against
integraIon
Ime
in
a
log-­‐log
scale,
the
radiometer
equaIon
will
says
that
the
noise
should
decrease
linearly
with
respect
to
Ime,
indefinitely.
In
reality,
spectrometers
will
follow
this
trend
unIl
a
certain
point
where
the
noise
will
level
off
and
even
increase
a
bit.
This
point
is
called
the
Allan
Ime.
Acknowledgments:
This
work
was
supported
by
the
University
of
Massachuse7s
at
Amherst
Department
of
Astronomy,
The
Commonwealth
Honors
College,
Massachuse7s
Space
Grant.
I
would
especially
like
to
thank
my
thesis
advisor
Dr.
Gopal
Narayanan,
graduate
student
Aleksandar
Popstefanija
for
helping
me
write
and
understand
python
scripts,
and
Rurik
Primiani
of
SAO
for
aspects
of
the
ASIAA
ADC.
References:
Patel,
N.
A.
et
al,
2014;
Results
OGP
CalibraCon
CalibraIng
the
offset,
gain,
and
phase
(OGP)
of
the
cores
in
the
ADCs
serves
to
smooth
the
spectra
they
produce.
Without
calibraIng
the
OGP,
the
spectra
tend
to
have
spurious
noise
which
can
result
in
poorer
signal
to
noise.
This
figure
demonstrates
the
noisy
tendencies,
uncalibrated
ADC
boards
have.
The
calibrated
plots
clearly
show
a
linear
trend
when
the
input
power
level
is
plo7ed
against
the
output
power
level.
The
uncalibrated
plots
show
the
general
trend
but
are
not
nearly
as
consistent.
Linearity
These
eight
plots
show
four
different
background
noise
levels,
quanIzed
and
unquanIzed
data
and
several
different
input
frequencies.
The
data
clearly
follows
a
linear
trend
unIl
a
threshold
is
reached.
That
threshold
represents
the
point
at
which
amplifiers
in
the
ADCs
being
to
saturate.
The
following
table
has
the
parameters
of
a
few
of
these
lines.
Frequency
QuanCzed
Noise
Level
Slope
100
MHz
Yes
-­‐20
dB
1.02
±
0.03
200
MHz
No
-­‐23
dB
0.96
±
0.01
400
MHz
Yes
-­‐26
dB
0.96
±
0.07
799
MHz
No
-­‐29
dB
0.980
±
0.
006
Allan
Variance
The
Spectrometer
clearly
demonstrates
that
it
follows
the
radiometer
equaIon
unIl
about
400
seconds
of
integraIon.
Results
Cont.
Conclusions
Once
calibrated,
the
spectrometer
is
linear
in
nature
and
has
an
Allan
Ime
on
the
order
of
400
seconds.
These
results
are
promising
but
more
work
is
necessary
before
it
can
be
uIlized.
Currently,
only
one
mode
(800
MHz
bandwidth)
is
complete
with
two
more
modes
(200
MHz,
and
100
MHz)
in
the
works.
Once
completed,
the
spectrometer
will
be
used
with
a
new
one-­‐
millimeter
wavelength
focal
plane
array
receiver
called
OMAR
(One
Millimeter
Array
Receiver),
which
will
eventually
be
integrated
into
the
LMT
by
UMass.