IEEE Aerospace and Electronic Systems Magazine - September 2020 - 44

Real-Time Signal Generator for Noise Radar
2
distributed random qvariables,
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i.e., x0 $ Nð0; s Þ and
x1 $ Nð0; s 2 Þ, r ¼ ðx20 þ x21 Þ where r $ RayleighðsÞ.

Noise samples with Rayleigh distribution are generated by
using (6), which can be obtained by using the Box-Muller
equations for Normal variables
r¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
À2 ln u0 :

(6)

The pdf of a Ricean distributed random variable is as
follows:
x Àðx2 þm2 Þ xm
fðxÞ ¼ pﬃﬃﬃﬃﬃ e 2s2 I0 2 0
s
s2

x < 1:

(7)

By definition, Ricean random variable is equal to the
square root of the sum of squares of two independent Normal
distributed random variables, i.e., x0 $ Nðm cos ðuÞ; s 2 Þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
and x1 $ Nðm sin ðuÞ; s 2 Þ, r ¼ ðx20 þ x21 Þ, where
r $ Riceðjmj; sÞ and u are any real number.
By definition, if u is uniformly distributed random
number, À ln ðuÞ= is a random number with exponential
distribution. The pdf of an exponentially distributed x random variable is as follows:

fðxÞ ¼

eÀx ;
0;

if x ! 0
if x < 0:

(8)

Up to this point, pseudo-random samples are generated by only considering the probability distribution. In
addition, there are other methods that focus on generating
samples with specified probability distribution and power
spectral density (PSD). In [18], a method is proposed to
generate realizations from a wide-sense stationary random
process with a given continuous PSD and with a given
first-order marginal pdf. This method is based on the addition of cosine signals whose amplitude, phase, and frequency parameters are random variables.
The method uses the following equation:
M
1 X
X½n ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Ai cos ð2pFi n þ Fi Þ
M=2 i¼1

(9)

where Ai , Fi , Fi are random variables with pdfs pA ðaÞ,
pF ðfÞ, pF ðfÞ and represent amplitude, frequency, and
phase, respectively. By changing these pdfs, the power
spectrum density and statistical distribution of the noise
samples can be controlled independently and samples
with desired characteristics can be obtained much more
easily. The derivation of the pdfs, i.e., pA ðaÞ, pF ðfÞ,
pF ðfÞ required to generate random samples with Gaussian, Gamma, and K distributions are given in [18].
We investigated the method in [18] in terms of
usability of real-time generation of noise samples on
FPGA [19]. In [20], we compared the colored noise generator method with the traditional method. The traditional
44

Figure 4.
Usage of IFFT for colored noise generation.

method of generating narrowband colored noise is to pass
an independently generated white noise through a bandpass linear time invariant (LTI) filter. The CPRN generator provides extensive control in noise signal design. As
opposed to the traditional technique, where the transfer
function of the LTI filter determines the PSD of the signal at the output, it is possible to generate colored noise
with a given continuous PSD and with a given first-order
marginal pdf.
Usage of inverse fast Fourier Transform (IFFT) for
generating noise samples ([19]) can be implemented by
supplying the frequency, phase, and amplitude values to
an IFFT algorithm to obtain the signal that is composed of
parallel cosine signals with different amplitude and phase
values. Figure 4 shows the idea of using IFFT for generating noise samples.

FPGA IMPLEMENTATION
Our real-time signal generator is created by combining
PRN and CPRN generators. The main block diagram is
given in Figure 5. The PRN generator is able to generate
samples with uniform, Gaussian, expontial, and Rayleigh
distributions. The CPRN generator block generates colored pseudo-random samples with the IFFT method. Outputs of these two generators are switched through a
multiplexer and routed to the pulse generator block, which
PRN generates continuous and pulsed signals. Also the
pulse repetition frequency of the generated signals can be
adjusted as constant, staggered, and random values.
Finally, controller block provides peripheral component
interconnect express (PCIe) based control signals from
host PC for controlling and selection of distribution

Figure 5.
FPGA block diagram.

IEEE A&E SYSTEMS MAGAZINE

SEPTEMBER 2020

IEEE Aerospace and Electronic Systems Magazine - September 2020

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