Implementing extreme upsampling filters with a multiplyless architecture
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Laffely, A. et al. “Implementing extreme upsampling filters with a
multiply-less architecture.” Military Communications Conference,
2009. MILCOM 2009. IEEE. 2009. 1-5. © 2010 Institute of
Electrical and Electronics Engineers.
As Published
http://dx.doi.org/10.1109/MILCOM.2009.5379811
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Institute of Electrical and Electronics Engineers
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Wed May 25 18:00:48 EDT 2016
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http://hdl.handle.net/1721.1/59522
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Paper ID #901053
IMPLEMENTING EXTREME UPSAMPLING FILTERS
WITH A MUL TIPLY-LESS ARCHITECTURE
Andrew Laffely
United States Air Force
Academy
Colorado Springs, CO
Balasubramanian Ramakrishnan
The MITRE Corporation
Bedford, MA
ABSTRACT
For ease of implementation, communications
systems have been steadily converted to digital
implementations. FPGA technologies and high-quality,
high-speed DACs have enabled this trend. While this is
for
modern
high
bit-rate
commonly
done
communications systems, legacy systems like the MILSTD-188-165A modem are not often considered. One
issue is the need to up-sample these slower standards by
factors of tens of thousands in order to interface them
with the modulation system.
This paper presents an architectural case study on
the implementation of a direct digital synthesis MILSTD-188-165A modem. It briefly describes a multiplyless single stage filter architecture with unlimited upsampling capabilities. The filter implements a Farrow
type design. By selecting the appropriate filter
coefficients from a set of look-up-tables (LUT) the filter
can be designed to suppress harmonic distortion below
the required filter mask. Mathematical evaluation of
these properties proves that a reasonable size LUT of
1024x14 bits is sufficient to suppress harmonics below 60dB. A full analysis of harmonic suppression vs. LUT
size is included to extend this work beyond the MILSTD-188-165A case study. 1
INTRODUCTION
For ease of implementation, communications
systems have been steadily converted to digital
implementations. FPGA technologies and high-quality,
high-speed DACs have enabled this trend. [1, 2] To
make these digital implementations competitive they
must have limited footprints in both size and power
consumption. While much attention has been placed on
high performance digital signal processing of high bit
rate communications systems, little has been done to
modernize legacy communication systems.
This is
1
This work is sponsored by the United States Air Force under Air
Force Contract FA8721-05-C-0002. Opinions, interpretations,
conclusions and recommendations are those of the authors and are
not necessarily endorsed by the United States Government
978-1-4244-5239-2/09/$26.00 ©2009 IEEE
Chayil Timmerman, Huan Yao,
and Jason Hillger
MIT Lincoln Laboratory
Lexington, MA
especially problematic in fielded military systems where
existing combat capabilities depend on these legacy
communications systems. Interoperability regulations
demand that new systems be constructed so they can
communicate with the existing military infrastructure.
[3] For new systems this creates the need to implement
many different communications systems on their
platform. For small mobile systems this can create
unwanted overhead in size, weight, and power
consumption.
To resolve this it would be ideal to implement the
legacy communications in the same architecture as the
new high data rate systems. The disparity of data rates
can make this end difficult to obtain as legacy
communications systems would require extreme
upsampling in order to match the bit rates of modem
approaches. Conventional upsampling approaches fall
short due to the large footprints of the upsampling filters
on FPGA hardware.
This paper presents an architectural case study on
the implementation of a direct digital synthesis of a
legacy communications system. A multiply-less single
stage Farrow style filter is shown which can implement
virtually unlimited upsampling in a single stage. [4, 5]
The MIL-STD-188-165A, an update of a legacy
government standard for SHF Satellite Communications,
is used to illustrate the requirements of for this
upsampling. [6] An arbitrary direct synthesis L-Band
system is targeted as the modem system.
MIL-STD-188-165A WAVEFORM FOR
SATELLITE COMMUNICATION
The Type A modem has the minimum data bit rates
shown in Table 1. In addition, the exact bit rate must be
selectable in increments of lkbps. As the desired
implementation is for satellite communications, the
A
carrier will be selected in L-Band, ~1-2GHz.
minimum DAC to accomplish this modulation runs at
~2.3Gsps. To use this DAC with these input bit rates
requires up sample ratios shown in Table 2. The spectral
mask for the MIL STD is shown in Figure 1.
Modem
Type
A
Bit Rate Range (kbps)
I QPSKlOQPSK
BPSK
64.0 to 6,300.0 I 64.0 to 8,472.0
18 tables
,
0 -
Table 1: Data Bit Rates [6]
Modem
Type
A
-
-
,
-0 -
-
Up Sample Ratio
T QPSKlOQPSK
BPSK
72k to 543
36k to 365.8 I
-
-
• • •
,
----,
LUT
512 Coel
Table 2: Up Sample Ratios
Coel
MIL-STD-188-165A Spectral Mask
• • •
!
0
'\ \
-1 0
\
-2 0
iii'
~
~
i
•
- 30
inner mask = [0 .00 -0.25:
0.10-0.40:
0.20 -0.40:
0.40 -1.50:
0.45 -4.00:
0.50 -12.0:
~
:;: - 40
- 50
outer mask = [0
<.
-----------
- 60
Sign
I
e
0.8 -36.5:
0.83 -36.5:
1.6 -50.5:
3.0 -61.5]: .. .
~
,-,
1
0.25:
0.1 0.25:
0.2 0.25:
0.4 0.25:
0.45 -0.50:
0.5 -2.00:
0.56 -8.50:
0.59-16.0:
0.7 -34.5:
0.501 65.0):
[h
ata
e
DAC
Sample
Rate
Top 9 bits
-.:.::
Lowest p::>if':t is 61.5dB
-70
o
1.5
0 .5
,
2 .5
e
Normal ized Frequency . (Rs/pO' Up Sample Rate (= 1)
Figure 1: Spectral Mask
FILTER ARCHITECTURE
Figure 2 shows a single channel modulator. For
QPSK or OQPSK a second identical filter would be
required. The parallel input nature of the DAC requires a
further duplication of this architecture by a factor of 8.
The look up tables (LUTs) could be shared amongst
these parallel structures. A key element is the reduction
of multiplies required. For BPSK and OQPSK each
symbol contains sign information only . The symbol
magnitude can be adjusted for in the LUTs
The access of the LUTs via a Gardner loop makes
this filter a Farrow structure. This structure eliminates
harmonics by providing intermediate estimates of the
samples. Harmonics could be created if the rate of
upsampling is greater than the number of intermediate
sample estimates captured in the LUTs. Upsampling the
MIL-STD-188-165A waveform without harmonics
could force the LUTs to 70k coefficients per tap. The
remainder of this paper shows a how these LUTs can be
dramatically reduced with no impact on system
performance.
Figure 2: Simplified Filter Architecture
MATHEMATICAL EVALUATION OF THE UP
SAMPLED SIGNAL
The difficulty in upsampling is the fact that
repeating samples creates harmonics in the spectrum
which could violate the spectral mask of Figure 1 and
thus interfere with other signals in the system. The filter
architecture can suppress these harmonics by using
multiple coefficients per filter tap . The upper bound of
the harmonics for our upsampling pulse shape filter
depends on the number of coefficients used for each tap.
To prove this, start with the definition of the DFT in
equation 1.
N-1
Eq 1:
Y(k) = Ly(n)e-j2nknIN
n=O
Where y(n) is the filter output. When repeating values
y(n) = y(n+ I) = ... = y(n+L-1), where L is the number of
repeat values. Using this, the DFT in equation 1 can be
separated into 2 summations as shown in equation 2.
(N I L)-l L-I
Eq2:
Ly(m)e- j2iTk(mL+/) IN
Y(k) = L
m=O
1=0
Rearranging the order of summation and separating the
exponent yields equation 3.
2 in the number of coefficients per tap. This is
highlighted in each graph by the star at the maximum
product of the ideal filter and the harmonic mask. By
1024 coefficients the first harmonic should be no greater
than -62dB. This is highly consistent with the trials with
this filter architecture in both C and Matlab.
0
Eq 3:
-10
(N IL)-l
L-I
Y(k) = Le-j2iTlk lN
1=0
Ly(m)e- j2iTk(mL) 1N
-20
m=O
-30
co
"0
.1;
Which is really the OFT of the non-repeated spectrum,
Ym(k) multiplied by a term which causes harmonics.
'"
I~
-40
"0
;:J
I:
0)
-50
'"
::;; -60
L-I
E4:
Y(k) = Le-j211ikIN .Ym(k)
1=0
The result is that Y(k) can be no larger that the nonrepeated spectrum times the harmonics term. In a sense
this first term becomes a harmonic mask. Figure 3
shows this mask when L = 100. Because the harmonic
mask steadily decreases it is most important to worry
about the first harmonic. If the filter response to the first
harmonic is small enough the results of Figure 3 show
that all other harmonics will be smaller.
By choosing Ym(k) to meet the filter mask in Figure
I an engineer can derive the LUT size required to
prevent harmonic distortion.
Figures 4, 5 and 6 show the harmonic mask
multiplied by ideal FIR filters using 2 to 4096
coefficients per LUT. Recall the number of coefficients
per table represents the maximum up sampling without
harmonics. As such the ideal filter with cutoff at I must
be scaled in frequency by these numbers. So the filter
with 4 coefficients has a cutoff of .25. The 8 coefficient
filter has a cutoff of .125 and so on. When we up
sample by repeating the filter output once, we must scale
the filter pass band by another factor of 2. This is why
you see the first cut off at .125 for the 4 coefficient filter.
This causes a harmonic, a flipped version starting at I
and moving backwards. This harmonic is multiplied by
the harmonic mask which slopes to a zero at I. The
results are more clearly represented in Figures 4 and 6,
where the axis is expanded to view the first harmonic.
The key here is that the filter cutoff is small bringing it
close to the null in the harmonic mask. The higher the
number of coefficients in each filter table the more
closely the cutoff is to the null. The result is that the
first harmonic rolls off by about 6 dB for every factor of
-70
-80
-90
-100
0
0.05
0. 1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency Normalized to the Symbol Rate (Rs)
Figure 3: Harmonic Mask with 100 Repeated Values
Imprint of Ideal Filter w ith X Coel/Tap on First Harmonic of Repeat Sample
0.1
0.2
03
04
O~
OB
O~
03
0.9
Frequency Normalize d to the Symbol Rate (Rs)
Figure 4: Harmonic Mask Showing First Harmonic and
the Frequency Scaled Up Sample Filters
Table 3 shows the maximum harmonic size for each
case. To confirm this I used the Simulink model to test
the case where the filter had x coef/tap and the up
sample rate was 2x. These graphs are shown below.
They correlate well with the data in Table 3.
Imprint of Ideal Filter with X C oef/Tap on First Harmonic of Repeat Sa mple
-------
-10
-20
.
0
<, -,
\
-30
'.~1J
~'"
'e
MIL-STD- I 88- 16SA F~ter Tests With 18 Tapsand 2 CoeflTap Up Sampled by 4
~
- 10
\
\
~4 0
-50
g'
:2 -60
' 128 Coef/Tap
-70
- 5 12
Coef/Tap
-80
" --
'-"""
.
o
0.1
-60
0.3
0.2
0.4
0.5
0.6
0.7
0.8
Frequency Normalized to the Sy mbol Rate (Rs)
-70
Imprint of Ideal Filter with X Coef/T ap on First Harmonic of Repeat Sample
.........; r- ·-- c---- .;.---- -'-. - - - .:,..- -
lI ii
0
:l
;
1
.!
;.........
I ·················I . ! .
······•··
····:·...... ;::
j
0.15
0.2
0.25
0.3
0.35
0.4
I
L-....-.
"'"\'
0.45
0.5
Rolioff
f\.
\
1\
c
<h
,
.....
i!'\ Ii
-50
-60
-70
0.10
•
•
::
• "
0.1
-2 0
: i
;
0.05
- 14.2dB oredieted
I I
•
o
- I0
:!
~---t~-ff3------;--.J~JL~
r• • • ! ················!·l·
~.
,I
I·
------ -------
~
MIL-STD-188-165A Filter Tests with 18 Taps and 4 CoefiTap, Up Sampled by 8
1 I
·t
--- ~.
\,
Figure 7: 2 Coef/Tap with 4x Up Sampling
-_...:._._-_:~---,-' - 1.~
-
.......
r ··•· · ·····.· : ·;····· ···•··· ···.· ··
r · ···•··· ······.·· j····· •····· ···;··· · ·
~
\.
0.9
Figure 5: Harmonic Mask and First Harmonic Zoomed
to Show First Harmonic
- .. .. .
~~
I...
-50
,- u o " I ap
-90
-100
~~
Hollotf _ 0 .10
rI ""
I "'-riP- oredlcted
-20
I
o
0.05
~
<,
"
0.1
\
~
....... .--------- I-----.
I
..}
-------:
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency Normalized to the Symbol Rate (Rs)
Figure 6: Harmonic Mask and First Harmonic Zoomed
to Show First Harmonic for Larger Coefficient Tables
Figure 8: 4 Coe f/Tap with 8x Up Sampl ing
MIL-STD-188-165A Filter Tests with 18 Taps and 64 CoefiTap Up Sampled by 128
LUT Size
2
4
8
16
32
64
128
256
5 12
1024
2048
4096
Peak of First
Harmonic (dB)
-8.34
-14.2
-20 . 17
-26 . 18
-32.20
-38.21
-44.22
-50.25
-56.26
-62.25
-68 .22
-74 . 13
Table 3: Maximum First Harmonic Size
Holloff - 0.10
0
- I0
-20
-30
~
AO
'"
'0
.g
-50
~
:2 -60
~
"-L
-70
Supressed Harmonics
I
'+t : IIITil
I
I
I
0.35
0.4
-80
I
I
-90
-100
o
0.05
0.1
0.15
0.2
0.25
0.3
0.45
Frequency Normalized to the Symbol Rate (Rs), Up Sample Rate = 128
Figure 9: 64 Coef/Tap with 128x Up Sampling
0.5
IMPLEMENTATION
MIL-STD-188-165AFilter Tests with 18 Taps and 1024 Coet/Tap
-
A model of a single filter was constructed in C and
in Matlab Simulink. As a proof of concept these two
models focused on the BPSK implementation of the
MIL-STD modem . Encoding and modulation were not
implemented.
Both models implemented the
architecture of figure 2 with 18 taps/LUTs. Each LUT
had a bit width of 9 bits. The Bernoulli Binary
Generator generated bit streams which were filtered
directly. The Matlab model generated figures 7-9, but
became slow when dealing with larger up-sample rates .
At higher upsample rates the C model produce faster
results and gave the opportunity to take the FFT of
signals with > 106 samples. Many tests where run with
various table sizes and upsample rates. In each test the
harmonics in upsampled filter response were below the
ideal presented in table 3. Figures 10 and 11 show two
views of an I 8 tap filter with 1024 Coefficients per tap
when the upsample rate was 40k samples per bit. Higher
levels of upsampling were difficult to measure
accurately with the current computer power as the FFT
would exceed the allowed memory space.
MIL-STD-188-165A Filter Tests with 18 Taps and 1024 Coet/Tap
-10
-~
FFT with 1677 samples
\
-20
-30
~
,\'" <,
-40
~
\
'"
"0
.~ -50
--
"-.....---.--- -- ---
:if
::> -60
-70
-
-80
-90
-100
o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency,Up Sample Rate = 40000
0.9
Figure 10: 18 Tap 1024 Coef/Tap Filter Design with
40000 Up-Sampling
-,
x 10
,
:
;
:
:
:
~
· · · · · · , · · · · · · · · · · · · : · · · · · · · · · · · · : · · · · · · · · · · · · ;·
,
;
;
:
:
-l
-60 ~ · · · · · · · · · · · ·: · · · · · · · · · · · · ; · · · · · · · · · · · · : · · · · · · · · · · · ..,
,
---:
,
;
:
-l
0.45
0.5
-20 ~ · · · · · · · · · · · · · · · · · · · · · · · · ; · · · · · · ·· · · · · · ; · · · · · · · · · ·
-40 ~ ·
co
:s.
FFT with 1677 samples
'"
"0
;:J
·c
(J)
:i
-80 .
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Normalized Frequency, Up Sample Rate = 40000
Figure 11: 18 Tap 1024 Coef/Tap Filter Design with
40000 Up-Sampling
CONCLUSIONS
This paper presented a methodology and test case
for developing direct synthesis of low bandwidth signals.
An upper bound on harmonic distortion was derived and
experimentally confirmed. This paper confirms that a
single stage low footprint pulse shaping filter can be
used to implement legacy military communications
protocols digitally .
BIOGRAPHY
[1] G. Goslin , "A Guide to Using Field
Programmable Gate Arrays (FPGAs) for
Application-Specific Digital Signal Processing
Performance," XILINX Inc., 1995
[2] R. D. Tumey, C. Dick, A. M. Reza, "Multirate
Filters and Wavelets : From Theory to
Implementation" XILINX Inc.
[3] DoD Directive 5000.01, 2003
[4] H. Meyr, M. Moeneclaey, S. A. Fechtel ,
"Digital
Communication
Receivers:
Synchronization, Channel Estimation and Signal
Processing," John Wiley and Sons, 1998
[5] F. M. Gardner, "Interpolation in Digital Modems
- Part I: Fundamentals," IEEE Trans on Comm.,
vol. 41, pp. 501-507, March 1993.
[6] MIL-STD-188-165A, 2000