This document discusses energy detection of unknown signals in fading environments. It proposes modeling the received signal power distribution under combined slow and fast fading. This allows deriving the distribution of the detector's decision variable in closed form. Specifically: 1) It models the received signal as the sum of the signal and noise, scaled by a complex channel amplitude representing fast and slow fading. 2) It derives an expression for the sufficient statistic at the detector's output and simplifies it under assumptions of high sample numbers and independent samples. 3) It expresses the distribution of the decision variable as an integral of the distribution for a fixed SNR, averaged over the SNR distribution due to fading. 4) It provides the specific

1. Detection of Unknown Signals in a Fading Environment
Yogesh Wankhede1, Abhishek Kumar2, Amutha Jeyakumar3
Department of Electrical Engineering, V.J.T.I
Matunga, Mumbai, Maharashtra, India
1
yogesh17.wankhede@gmail.com
2
abkvjti@gmail.com
3
amuthajaykumar@vjti.org.in
Abstract— In this paper we consider the effect of combined slow Various combinations of fast/slow fading
and fast fading on the energy detection. We model the received
signal power distribution and propose a simplified parameters. In addition, it allows analyzing
approximation to this distribution. The approximation allows us distributed detection schemes where each sensor
to derive the distribution of the decision variable at the detector’s observes different fast/slow fading values.
output in closed-form. By using the suggested distribution, we
find that in the block fading channel the impact of slow fading is
not significant. In the case of multiple independent fast fading
realizations with common slow fading value we can show how the
SYSTEM MODEL
II.
slow fading starts to dominate the detection performance. The energy detection problem is usually treated as
a hypothesis test. The detector has to separate
Keywords— Fading channels, Rayleigh fading, signal detection.
between the noise, hypothesis H0, and the jointly
I. INTRODUCTION presence of signals and noise, hypothesis H1. Under
The energy detection is a common approach to these two hypotheses the received band pass
decide whether unknown signals exist in the waveform has the following form
medium. The first step of the detector design
requires a model for the distribution of the noise r (t ) = R{[ hs (t ) + n(t )]e j 2π fct },..........H1
and the signal. It is reasonable to describe the noise
as a simple white Gaussian process. The signal R { n ( t ) e j 2 π f c t }, .......... H 0 (1)
model has to be more complex to incorporate the where we use a common complex signal
fading effects. representation and the real part R(.) of it. The signal
The energy detection of unknown signals in s(t)=sr(t)+jsi(t) and the noise n(t) = nr(t)+jni(t) are
additive white Gaussian noise (AWGN) Recently, expressed in terms of their equivalent lowpass
the energy detector has been revised to incorporate components. The signal is also scaled with the
decision after fast fading channels. In these studies complex channel amplitude h = αejθ and fc stands
the detector performance is described analytically for the carrier frequency.
in Nakagami and Rayleigh channels. However, the At the input of the detector the received
combined impact of fast and slow fading on the waveform is filtered by the ideal band pass filter of
energy detection lacks attention in the literature. positive bandwidth W. If N0/2 denotes the two-sided
This is partly due to the difficulty to obtain a power spectral density of noise, the noise power PN
closed-form description for the detection at the output of the filter equals PN = N0W. The
performance. One has to resort either to numerical filter output is squared and one complex sample is
integration or to simulations. Both of these collected in every 1/W s. equivalently, for N
approaches makes the system analysis very measured samples the total observation time T
cumbersome. equals T = N/2W s. The N samples are summed
In this paper we propose a model that describes the together and normalized by the noise power. The
signal power distribution in fast/slow fading calculated metric serves as a sufficient statistic L
environment. The proposed model allows us to for the hypothesis test. By using the sampling
obtain in closed-form, the distribution of the energy theorem, the sufficient statistic in case of
detector’s decision variable. This distribution helps hypothesis H1 can be approximated as
us to investigate the detector performance with

2. 1 N 2 power, γ . In the presence of combined fast/slow
L≃
PN
∑ (α (cos(θ )sr ,k − sin(θ )si,k + nr ,k )2
k =1
fading, the fast fading is distributed around its mean
value that follows the slow fading distribution
N 2 (2) ∞
1
+
PN
∑ (α (cos(θ )si,k + sin(θ )sr ,k + ni,k )
k =1
2
p (γ ) = ∫ p (γ |γ ) p (γ )d γ (5)
0
Where p(γ ) is the slow fadingdistribution that
Where sr,k,, si,,k,nr,k, ni,k are k-th samples of the is usually approximated by the log-normal
signal and noise corresponding real and imaginary distribution .
parts. By inserting (5) into (4) we have
For high N and independent noisy and signal ∞∞
samples, (2) can be simplified as p ( L) = ∫ ∫ p ( L | γ ). p (γ | γ ). p (γ )d γ )d γ (6)
1 N 0 0
L≃ ∑ (α 2 sk2 + nk2 )........H1
PN k =1 In case of AWGN and a fixed SNR, the decision
(3) variable L has the non-central chi-square
1 N 2
≃ ∑ nk .................H 2
PN k =1
distribution with non-centrality parameter equal to
2γ. We have for the first term in (6)
Where sk2 , nk2 are powers of the k-th complex signal 1 L N −2 L + 2γ
p ( L | γ ) = ( ) 4 e( − ) I N |2 −1 ( 2 Lγ ) (7)
and noise sample. 2 2γ 2
After calculating L, we compare it with a Where Iv(.) is the modified Bessel function of the
threshold λ and vote for one of the hypotheses. In first kind.
order to select λ we need to derive the distribution
of L under the two hypotheses, p(L|H1) and A. Detection probability in Rayleigh-gamma fading
p(L|H0). Obviously, the distribution p(L|H0) is channel
central chi-square with N degrees of freedom. In Assume the detector collects a block of power
what follows we derive the distribution of p(L|H1), samples. Over the block the slow fading value does
hereafter p(L) in the presence of combined fast and not change and is defined by its instantaneous value
slow fading. γ. If the detector moves slowly the fast fading value
does not change either during the block. In the
Rayleigh modeled channel, the fast fading value is
III. DISTRIBUTIONS one particular realization from the exponential
distribution. In order to encompass more complex
Equation (3) shows that under H1, the L is a environments, we consider the case where the fast
function of the signal to noise ratio (SNR,γ). fading takes multiple different values over the block
Therefore the distribution p(L) can be found by this could model an environment where the channel
averaging the distribution of L for a fixed SNR coherence time is less than the total observation
p(L|γ), over all possible γ values time. Alternatively it can also be interpreted as
∞
measurements collected by multiple sensors. Each
p ( L) = ∫ p ( L / γ ) p (γ )d γ (4) sensor has block fading channel with the same slow
0
fading value but the particular fast fading
where p(γ) is the the distribution of SNR due to the fading.
The instantaneous SNR is computed as realizations are different. If we combine n
p rx α 2 . s k2 independent Rayleigh fading samples the signal
γ = = power has chi-square distribution with n degrees of
pn PN
freedom.
In a non line-of-sight channel the fast fading signal γ
1 γ n− 2 − γ
amplitude distribution is commonly modeled by the px2 (γ | γ ) = 2
( ) e (8)
Rayleigh distribution. The Rayleigh distribution n
Γ( )γ γ
depends only on one parameter, the mean signal 2

3. If the fast fading is modeled by the chi-square Where F1 ( ・ , ・ ; ・ ) is the hyper geometric
distribution and the slow fading by the log-normal function of first kind.
distribution, equation (5) has a closed-form solution
in integral form. In order to acquire an analytical
insight into the detector performance we recall that IV. ILLUSTRATIONS
the slow fading can also be described by a gamma We illustrate the usage of (13) by computing the
distribution miss probability of the energy detector. The miss
( α sf −1) γ probability is described by the CDF of the decision
γ . exp( − ) variable L. We compute the CDF by integrating
β sf
p g (γ , α sf , β sf ) = α
(9) (13) numerically. In the computations we assumed
Γ (α sf ).( β sf ) sf that the slow fading has mean SNR µdB = 5 and
By selecting the values of αsf, βsf we can match standard deviation σdB = 3. The size of the block,
the CDF of gamma distribution and the empirical the number of collected power samples, is N =
CDF obtained from measurement records. 1000. The predictions made by the model are
In order to compare the slow fading compared to the simulation results. The simulations
approximation by gamma and log-normal are carried out for Rayleigh/gamma and
distributions we set the moments of those two Rayleigh/log-normal fading environments. Since
distributions to be the same. we do not validate the models against any
measurements, it is difficult to predict whether the
α sf = (eσ − 1) −1 , β sf = e µ +σ / 2 (eσ − 1)
2 2 2
(10) log-normal or the gamma distribution describes
where µ and σ are the scaled mean and standard better the slow fading. Recall that both distributions
deviation of the corresponding log-normal are only approximations to the real fading process.
distribution. Because in (10) we express the log- However, we stress that unlike the lognormal
normal parameters in dB the scaling factor is approximation our model allows analytical
[log(10)/10] : µ = [log(10)/10] µdB and σ = treatment. Because of that, it is easier to study for
[log(10)/10] σdB where µdB and σdB are the mean instance the impact of fast/slow fading parameters
and standard deviation of 10 log10 γ . to the detection performance.
By inserting (8) and (9) into (5) we have
4γ
1 n
(α sf + )
(γ / β sf ) 2 2
.K n / 2 −α sf ( )
β sf
p (γ ) = 2. (11)
γ .Γ (n / 2).Γ (α sf )
where Kν( ・ ) is the modified Bessel function of
second kind. Unfortunately, as far as we know, by
using (11) and (7), equation (4) does not lead to a
closed-form solution. To overcome this problem we
notice that (11) can be closely approximated by a
gamma distribution. Again we do the
approximation by matching the two first moments
of (11). The gamma distribution approximating (11)
has parameters
α sf .n / 2
αγ = , βγ = β sf (1 + α sf + (n / 2)) (12)
1 + α sf + (n / 2) Fig.1. Miss Probability if fast fading has exponential
Using pg(γ,αγ, βγ) and (7) in (4) we have distribution, n = 2.
( N / 2 − 1) −L/2
L .e N L/2
p(L) = F (α , ,
γ ) (13)
2 1 + 1/ βγ
1
N α
2N /2 Γ( )(1 + β γ ) γ
2

4. not provide much improvement (compare fast/slow
fading calculated curves). In general, if we increase
the number of independent fast fading samples we
notice that the slow fading starts to dominate the
decision variable distribution and thus the detection
performance. This result agrees with intuition: the
impact of fast fading is averaged out as n increases.
For high n the calculated and the simulated results
with Rayleigh/gamma modeled distribution match
quite well. If the n increases the approximation
becomes tighter. Difference between the
Rayleigh/gamma and Rayleigh/lognormal curves is
due to the different tails of gamma and log-normal
distributions. However, both the gamma and
Fig.2. Miss Probability if fast fading has chi-square lognormal distributions are only approximating
distribution, n = 20 models of the real slow fading process. In this paper
Below, we investigate the impact of slow fading we matched the first two moments of gamma
with respect to the number of independent fast distribution to the corresponding log-normal
fading samples over a block of received samples. distribution. In practice the gamma distribution
Equation (13) allows us to select the number of moments have to be matched to the measurement
independent fast fading samples n per measured data.
block. An interesting extreme case is the block
fading channel where the fast and slow fading each
V. CONCLUSION
take a single value over the block. For a single fast
fading value we have n = 2. However, during the In this paper we proposed a model that
block there are actually N/n signal samples with the describes the signal power distribution in fast/slow
same power. By summing the power samples as in fading environment. The model allowed deriving
(3) we increase the mean of the observed the distribution of the decision variable in energy
distribution. The increase of the mean can be detection. With this distribution at hand, we could
incorporated into the equation by scaling up the easily predict the detector performance. The
mean of the slow fading: µdB → µdB + 10log10 proposed model is a useful tool for studying the
(N/n). It is interesting to note that for n = 2 the detection performance in different fading
impact of slow fading is relatively small. The signal environments. We illustrated that by comparing the
model that accounts solely for the fast fading detection performance in the fast/slow fading
describes the detector performance quite well (Fig. environment with the fast fading environment.
1). The situation is reversed when the block We found that if the mean signal power does not
contains multiple fast fading values. This case is change during the measured block of the signal
illustrated in Fig. 2 where we combine 10 samples, the simple fast fading model describes the
independent fading blocks with 50 complex detector performance quite well. The slow fading
samples each, n = 20: the cdf for the fast/slow becomes more important if multiple blocks with
fading model and the model that contains only fast different fast fading values are combined.
fading are significantly different.
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