IEEE - Aerospace and Electronic Systems - February 2020 - 32

GnuRadar: An Open-Source Software-Deﬁned Radio Receiver Platform for Radar Applications

Figure 3.
Illustration of a typical under-sampled signal showing Nyquist
zones and the effects of aliasing of the RF/IF signal.

bandwidth is less than the Nyquist rate, effectively spreading
the sampled signal's noise power across a broader range
of frequencies. By reducing the Nyquist rate through decimation, an increase in the signal-to-noise ratio (SNR).
Applications meeting this criteria can utilize a class of decimating low-pass filters, known as Cascading Integrator
Comb (CIC) filters [15], which efficiently perform both
operations without multipliers, making these filters desirable in resource-limited hardware designs. An overview of
the CIC filter's operation and optimizations will be provided in the following section.

CIC FILTER
sampled at a maximum rate of 64 MSPS, producing
images in adjacent Nyquist zones, as shown in Figure 3.
Accessing information from the signal requires that one of
the replicated images be shifted to the origin in a process
commonly referred to as down conversion. In the past, this
process was commonly performed in the analog domain
[12], [13], using one or more mixing stages to bring the signal down to its baseband frequency for sampling. In more
recent years, down conversion in the digital domain has
become prevalent, providing perfectly balanced amplitude
and phase in the case of complex sampling. Complex digital down conversion in the USRP is produced using the digital computer (CORDIC) algorithm [14]. This algorithm,
originally developed in 1959, has since been expanded to
efficiently compute trigonometric, hyperbolic, logarithmic,
and exponential functions. In this application, the CORDIC
algorithm is used in rotation mode, for which a rotation
matrix, given by the original USRP implementation, a
12-stage pipelined CORDIC algorithm was applied using
16-b input and output precisions with a fixed-width performance variable and two guard bits for internal stages. In
our application, a sizable reduction in hardware resources
for the CORDIC module was achieved by the following.
1) Creating a submodule for the shift adder stage,
resulting in inferred addsub primitives available in
the FPGA.
2) 1-b reduction in each subsequent stage's performance variable (i.e., each stage reduces the error by
at least a factor of 2).
As in the original implementation, an output bit was
trimmed from the left- and right-most bits, effectively,
reducing the gain incurred by a factor of 2, resulting in an
overall gain of Av ¼ 1:6468=2:0 ¼ 0:82338.

FILTERING
After down conversion, the incoming signal is filtered, isolating desired information from extraneous noise using a
low-pass filter with the extent of the passband determined
by the bandwidth of the signal. In many cases, the signal's
32

The CIC filter [15] is an efficient low-pass decimating filter
suitable for resource-limited hardware implementations.
The basis of the filter is provided by a moving average (MA)
filter with unity coefficients (i.e., boxcar function), given by
HðzÞ ¼

RÀ1
RÀ1
X

Y ðzÞ X
bðkÞzÀk
)
zÀk :
¼
XðzÞ k¼0
bðkÞ¼1
k¼0

(1)

Equation (1) can be rewritten in geometric form
HðzÞ ¼

1 À zÀN
1 À zÀ1

(2)

which can be further decomposed into the following two
elementary operations:
1) Integration : HI ðzÞ ¼ 1Àz1À1 and
2) Differentiation : HC ðzÞ ¼ 1 À zÀN .
An Nth order CIC filter can be synthesized by cascading
N pairs of these stages, resulting in the general form
HðzÞ ¼ ðHI ðzÞHC ðzÞÞN :

(3)

Using

P ðfÞ ¼ jHðzÞj

z¼eÀjv

#2N
"

1 À eÀjvRN
sin
pf

¼
1 À eÀjvN
sin pf
R
(4)

the filter's magnitude and overall frequency response can
be evaluated directly. Inspection of the equation reveals the
following two potential problems in the implementation:
1) a large dc gain; and
2) a nonflat passband response.
In hardware, large filter gains require a proportional increase
in register widths, which can be prohibitive, depending on
available resources. The filter's maximum gain, given by
"
lim

f!0

sin pf
sin

IEEE A&E SYSTEMS MAGAZINE

pf
R

#N
) RN

(5)

FEBRUARY 2020

IEEE - Aerospace and Electronic Systems - February 2020

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