Histogram-Based Probability Density Function
Estimation on FPGAs
Suhaib A. Fahmy
School of Computer Engineering
Nanyang Technological University
Nanyang Avenue, Singapore
sfahmy@ntu.edu.sg
Abstract—Probability density functions (PDFs) have a wide
range of uses across an array of application domains. Since
computing the PDF of real-time data is typically expensive, var-
ious estimations have been devised that attempt to approximate
the real PDFs based on fitting data to an expected underlying
distribution. As we move to more adaptive systems, real-time
monitoring of signal statistics increases in importance. In this
paper, we present a technique that leverages the heterogeneous
resources on modern FPGAs to enable real time computation
of PDFs of sampled data at speeds of over 200 Msamples per
second. We detail a flexible architecture that can be used to
extract statistical information in real time while consuming a
moderate amount of area, allowing it to be incorporated into
existing FPGA-based applications.
I. INTRODUCTION
Information on the probability density function (PDF) of
sampled data has a wide range of uses across a variety of
application domains. A variety of techniques have been pro-
posed over the years [1], however computational complexity
often means these cannot be implemented in real time. The
preference is thus to rely on techniques that use a reduced
data set, attempting to fit to a predefined parametric model.
This can be inaccurate, especially if the PDF of the data is
unknown or changing.
In most references to applications of PDF estimation in the
literature, one-time statistics are computed on a block of data.
In image processing applications, for example, the statistics
are typically required for a single frame. In this paper, we are
more interested in facilitating real-time monitoring of signals
that may change over time. Specifically, adaptive systems
may need to monitor changes in certain environmental factors,
before making changes based on those statistics. An example
might be a cognitive radio that reacts to channel occupancy
statistics, or an adaptive networking node that may modify its
routing behaviour based on network queue length statistics. In
order for such applications to be feasible, we must be able to
compute PDFs in real-time on streaming data.
In this paper, we present a PDF calculation architecture that
can be integrated within FPGA-based applications, which is
both flexible and moderate in terms of area usage. The aim is
to allow designers of applications that can make use of PDF
estimation to leverage this capability within their hardware
designs. The primary focus is for adaptive applications, that
we feel are becoming more practical on FPGAs.
In Section II, PDF estimation is discussed along with exist-
ing hardware approaches. Section III introduces the proposed
architecture used for PDF calculations and how it has been
tailored for FPGA implementation. Section V discusses some
extensions to the architecture that allow it to compute more
complex functions. In Section IV we present synthesis results
and discuss accuracy issues. Finally, Section VI concludes by
discussing further development of this architecture.
II. RELATED WORK
PDF estimation techniques fall into two categories: paramet-
ric and non-parametric. [1] Parametric techniques try to fit a
known model to the data and deduce values for the model
parameters based on the data. Non-parametric techniques
use the samples themselves to construct the PDF. The most
common non-parametric technique is the use of a histogram,
which when normalised, gives the PDF. Given a good model,
parametric techniques can give more accurate results with less
data than a non-parametric model. However, the requirement
for a good model means that where the PDF’s nature is
unknown or changing, parametric approaches can result in
poor accuracy.
PDF estimation is of value in a number of different ap-
plications areas, including image processing [2], [3], machine
learning [4], computer security [5], and medical imaging [6]
among others.
Existing work on hardware architectures for PDF estimation
is minimal. In [7] a histogram of an image is constructed
for histogram equalisation. However, since this is done for
one image at a time, they process accumulation at the end of
histogram construction, during the blanking period between
frames in a video, using a sequential approach. This would
not be suitable for a constantly updating window. In [8], a
novel technique for constructing histograms is shown, but with
an application to median computation. In this paper, we use
a similar approach to construct the histogram, but tailored to
PDF calculation. In [3] a histogram is constructed within a
hardware architecture but using a sequential approach, again
on a per-image basis.
Elsewhere, in [9], a Parzen window PDF estimator is used as
an example application for their performance migration tool.
However it processes a single block of data loaded from a
host PC, and the performance comparison is made between

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