Utilising convolutions of random functions to realise function calculation via a physical channel
Predrag Jakimovski
Stephan Sigg
Karlsruhe Institute of Technology
Karlsruhe, Germany
[email protected]
TU Braunschweig
Braunschweig, Germany
[email protected]
Convolution of random distributions for function computation
We discuss the utilisation of an algebra of random functions for the calculation of mathematical operations
on a physical communication channel for actual implementation with resource restricted nodes. In
particular, we present a transmission scheme for the computation of functions on the wireless channel and
discuss various properties from combinations of random functions as well as requirements and restrictions
of resource restricted hardware.
Convolution of independent random variables
If
is a probability distribution with mean and variance
tion with mean
and variance
, the convolution
is a probability distribu-
From this, we can realise the four basic mathematical operations on the wireless channel
Yusheng Ji
Michael Beigl
National Institute of Informatics
Tokyo, Japan
[email protected]
Karlsruhe Institute of Technology
Karlsruhe, Germany
[email protected]
Convolution of Poisson distributions
We divide a burst sequence of length t into κ sub-sequences of length . Each of these subsequences contains with
probability κ one or more of a finite number of bursts.
The Poisson distribution then defines the probability to find k bursts in this sequence as
To transmit a value we create a Poisson-distributed burstsequence
with mean such that each of a set of sub-intervals has a probability
to contain exactly bursts.
The receiver extracts the count
of sub-sequences with exactly
bursts as well as the total number of bursts
.
If
is large, we expect that
Estimation of errors due to collisions
Therefore
Stochastic values of combinations of RVs.
(Exponential distribution)
If
and
are exponential random variables
with mean
and
respectively,
is an exponential random variable with mean
and
001001100000010110010000010100100100100011010 1010100010010011
...
110010000010100
Superimposition
of sequences
0 1 001 .
Transmitted burst sequence at a receiver
...
burst
...
time
...
...
...
Experimental setting
...
1 0001 .
. .0
1 1 001 .
. .1
...
...
Superimposed burst sequence
K
+
...
...
...
...
...
...
...
...
...
Linear combinations of independent random variables
For a linear combination
of random variables, the variance is defined as
...
. .1
...
t
Results (Simulation and Case study)
We created a WSN platform with 15 nodes and a central
receiver gathering incoming signals. The nodes use simple
ON-OFF-Keying to transmit burst sequences.
The simple WSN nodes estimate the mean temperature
with high accuracy (conditioned on the length of the burst
sequence). Our transmission scheme is feasible with
unsynchronised nodes and lowest complexity hardware.
We used a dataset from the Intel Berkeley Research lab as a
benchmark. The dataset consists of several days readings from
sensory data such as air temperature, humidity, light and voltage.
Mean temperature
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Temperature
Stochastic values of combinations of RVs
(Geometric distribution)
If
and
are geometric random variables
with failure probabilities
and , then the
function
is a geometric random
variable with failure probability
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We calculate the average temperature curve obtained during the
day via superimpositions on the channel.
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1 Day
System architecture of a highly
resource restricted IoT node for
calculation on the channel via
convolutions of random distributions.
Mean temperature
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Temperature
For a 1-day recording from 15 nodes we mapped each time series
to one of our nodes.
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大学共同利用機関法人 情報・システム研究機構
国立情報学研究所
National Institute of Informatics
1 Day