# Energy Detection of Multilevel PAM Signals with  Systematic Threshold Mismatch

Authors: Antti Anttonen; Adrian Kotelba; Aarne Mämmelä
Journal: Research Letters in Communications
 (2009)

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## Abstract

We address a symbol decision problem with spectrally efficient energy detected multilevel pulse amplitude modulated (PAM) signals. First, we analytically quantify the relationship between a systematic threshold mismatch and the required increase of the average signal-to-noise ratio to preserve a desired symbol error rate. For the case in which such an increase is not tolerable, we present a novel near-optimal multilevel threshold selection scheme, which is accurate for a wide range of system parameters.


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## Body

## 1. Introduction

The  optimal symbol decision rule in an energy detection system with pulse amplitude modulated
(PAM) symbols reduces to the problem of finding proper decision threshold values
[1, 2]. There are two kinds of sources for a threshold mismatch. Firstly, a
purely random type of threshold mismatch is caused by a possible estimation
error of the signal and noise energies required for the threshold selection. Secondly,
a more predictable systematic threshold mismatch with respect to the received signal-to-noise
ratio (SNR) results from the inability to define the optimal thresholds in a
closed form even if the SNR is perfectly known. The former mismatch can be removed
with a proper averaging whereas the latter mismatch can be more difficult to
compensate.The effect of a systematic threshold
mismatch on the error probability of an energy detected on-off keying (OOK)
system has been studied, for example, in [3]. Gaussian approximations for the distribution
of an energy detected decision variable with different assumptions of the
expected mean and variance are frequently used to reduce the complexity related
to the exact error analysis and threshold selection [4, 5]. These approximations
are based on specific assumptions on the value of degrees of freedom (DOFs) reducing their accuracy and robustness, for
example, in adaptive multiband systems [6] in which the DOFs can change
adaptively or be relatively low. Another approximation for a binary OOK using a
numerical approach is suggested in [6]. However, to the best of our knowledge, none
of the papers address the effect of the threshold mismatch or accurate
threshold selection for a spectrally efficient energy detected PAM (ED-PAM)
system to allow arbitrary selection of the number of modulation levels and DOFs.The novelty of our contribution is two fold.
Firstly, after describing our system in Section2, we analytically quantify the
required increase of the SNR per bit to tolerate a hypothesized threshold mismatch
for a given error probability in Section 3. Secondly, in Section 4 we propose a
new multilevel threshold selection scheme for an ED-PAM system. Since the
symbol decision problem of an energy detector in a slowly fading channel without
interference reduces instantaneously to that of an additive white Gaussian
noise (AWGN) channel [7], for brevity we restrict our channel to be an AWGN
channel. We focus on the systematic threshold mismatch of an ED-PAM system and
assume perfect estimates for the signal and noise energies.


## 2. System Modeling

The decision variable of
a multilevelM-ary ED-PAM system can
be expressed as (1)y(t0)=∫t0t0+Ti[∫−∞+∞r(t−τ)g(τ)dτ]2dt, where r(t)=s(t)+n(t) is the received signal, s(t)=∑kaku(t−kTr) is the transmitted signal, u(t) is the pulse waveform, ak∈{0,E1,2E1,...,(M−1)E1} is the kth
data symbol from the nonnegative and real PAM constellation set, E1=Eb/cb is the received energy corresponding to the minimum nonzero amplitude, Eb is the average energy per
bit, cb=(2M2−3M+1)/(6log2M), M=2b is the number of modulation levels, b is the number of bits per symbol, Tr is the symbol period, n(t) is the zero mean AWGN component with noise power spectral density N0, t0 is the known start time of
the symbol of interest, Ti is the integration time, and g(t) is the impulse response of the
receiver filter with bandwidth B . The
product of the integration time and bandwidth is denoted by D=BTi, and 2D is the DOFs. It is assumed that B and Ti are selected so that D is an integer. The
OOK system is a special case of the nonnegative PAM system with M=2. The received energy of the mth symbol is Em=m2E1(m=0,1,2,…,M−1). We
assume that all symbols are equally probable and that the noise samples
affecting the decision variable are uncorrelated. The average SNR per bit is defined as γ=Eb/N0. We use the maximum
likelihood decision criterion and multiple hypothesis testing approach [8, page
82]. We make a hypothesis Hm(m=0,1,2,…,M−1) that the symbol am was transmitted if (2)p(y∣Hm)>p(y∣Hk)∀k≠m, where in our case p(y∣Hm) is the conditional
central (m=0) or noncentral (m>0) chi-squared probability
density function (pdf) given in [2]. According to the maximum likelihood criterion
(2), the optimal symbol decision threshold values ρm,opt<ρm+1,opt(m=0,1,2,…,M−2) can be found from p(y∣Hm)=p(y∣Hm+1)∣y=ρm,opt, which is not possible to solve in a closed
form.


## 3. Performance with Systematic Threshold Mismatch

The symbol error rate (SER)
of the target system can be expressed as [2] (3)PM(D,γ,ρ0′,ρ1′,…,ρM−2′)=1M[M−1+QD(0,2ρ0′)−QD(2(M−1)2cb−1γ,2ρM−2′)−∑m=1M−2QD(2m2cb−1γ,2ρm−1′)+∑m=1M−2QD(2m2cb−1γ,2ρm′)], where ρm′ is the selected mth threshold (m=0,1,2,…,M−2) normalized by N0 and Qv(⋅,⋅) is the generalized vth order Marcum Q-function defined in [9]. To theoretically evaluate the relation
between a threshold mismatch and the corresponding margin in the required SNR
per bit to preserve a desired SER, we set (4)PM(D,γopt,ρ0,opt′,ρ1,opt′,…,ρM−2,opt′)=PM(D,γmis,ρ0,mis′,ρ1,mis′,…,ρM−2,mis′), where now γopt is the SNR per bit to
achieve the desired PM with the normalized optimal thresholds ρm,opt′ and γmis=γoptΔγ is the required SNR per bit to achieve the desired PM with the mismatched
thresholds ρm,mis′=ρm,opt′,(1+Δρm). In other words, Δγ denotes the required SNR margin per bit resulted from the
hypothesized relative threshold offset Δρm compared to the corresponding optimal thresholds. We define the above
relationship formally as  (5)Δγ=f(Δρm),Δγ≥1,|Δρm|<1,m=0,1,2,…,M−2, where f(⋅) denotes a functional relation
between Δγ and Δρm and it depends on selected values of M, D,
and γopt. It is convenient
to express Δγ in decibels by Δγ=10log10(γmis/γopt) dB and Δρm in percentages by Δρm=100⋅(ρ′m,mis/ρ′m,opt−1)%.
There is no closed form solution for (5) from (4). We use the numerical bisection
method described in [10, page 261] to find the relationship and some examples are
given in Figure 1. Assuming the offset Δρ=Δρm in percentage is the
same for all m, we suggest that a
simple quadratic function with a single parameter can be used to approximate (5)
with a good accuracy for relatively low values of Δρ and Δγ . The parameter
of the quadratic function depends on the selected values of M, D,
and γopt and it can be
found numerically.Figure 1
Required SNR margin versus threshold mismatch (M=4).


## 4. Proposed Near-Optimal Threshold Selection Scheme

To solve the decision
threshold selection problem we propose to find an error function between the optimal
and asymptotically approximated threshold values, that is, when the SNR
approaches infinity. A similar approach was first proposed in [6] without
detailed evaluation for an energy detection system with binary OOK. However,
the problem of finding an error function for a multilevel ED-PAM system is
evidently more complicated. First, the asymptotically optimal values ρm,asy should be
found and they are derived differently for the threshold values ρ0,asy and ρm,asy with m>0. It is shown in [6] that ρ0,asy=E1/4 when the SNR approaches infinity. However, we
observed that the corresponding result for ρm,asy with m>0 is much more
complicated (the result is not shown here due to space limitations). We alternatively
assume that for a high SNR, the mean value of the decision variable is
unaffected by the noise and the decision variable follows a symmetrical
distribution, which directly results in a simple approximation ρm,sym=(Em+Em+1)/2,  m>0. The overall  scheme is referred to as a semiasymptotically optimal scheme since
only ρ0,asy is
asymptotically unbiased whereas there will be some bias in ρm,sym with m>0 even if the SNR approaches infinity. The second step is introduced
to find an error function em between the optimal and semiasymptotically optimal threshold value set mentioned
above. After some manipulations we propose to minimize (6)em={min{1ND∑i=0ND−1ρ0,optα−E1/4−d2,0(i)d1,0(i)},m=0,min{1ND×∑i=0ND−1ρm,optβ−(Em+Em+1)/2−d2,m(i)d1,m(i)},m>0, where
α
denotes [0,E1/N0,RD(i)] and
β
denotes [Em/N0,Em+1/N0,RD(i)] and RD(i)=i+1(i=0,1,…,ND−1) is  the ND-element
value set to be supported for the parameter D, d1,m, and d2,m are the first-order coefficients, and ρm,opt(⋅,⋅,⋅) are the optimal
thresholds depending on the particular value of D and the SNRs of the symbols. An analytical closed form solution
to the above problem cannot be found. In order to make the error function
parameterized by D, its relation to the
error function is found experimentally with the help of the first-order
coefficients d1,0(i)=N0RD(i)−1 if m=0,d1,m(i)=N0 if  m>0, and d2,m(i)=RD(i)N0 for m≥0. The remaining error is
averaged over the selected range of the expected values of D. We select a polynomial-based error function with the aim to
maintain low complexity. Starting from (6), we obtain, after some
manipulations, a novel data-fitting threshold selection scheme for an M-ary ED-PAM system as (7)ρm,dat={E1/4+N0(D−1K0+D),m=0,(Em+Em+1)/2+N0(Km+D),m>0,where Km=km,VmxmVm+km,Vm−1xmVm−1+⋯+km,2xm2+km,1xm+km,0 is the polynomial with order Vm for the mth threshold, xm=Em+1/N0 is the received SNR of the
(m+1)th symbol, and km,v are the coefficients of
the polynomial. The coefficients, which minimize (6), are generated by applying
the well-known least squares curve-fitting method [10, page 528] for the target
SNR range. In Figure 2, we plot the SER as a function of the SNR per bit for
the quaternary ED-PAM system using (3) now with the actual threshold values
determined by four different threshold selection schemes. In this example, V0=2,V1=1,V2=1 are adopted in (7). For a comparison we apply the Gaussian approximation
scheme presented in [4] to determine each mth
threshold (m=0,1,2) separately.
We found that the polynomial order has the highest effect on the first
threshold and that the
selection of D has a significant
impact on the relative performance of the schemes. However, the data-fitting
scheme performs well for a wide value range of D. More detailed reasoning is given in [2].Figure 2
SER comparison of
different threshold selection schemes (M=4).


## 5. Conclusion

We have presented some useful guidelines between
a systematic threshold mismatch and the corresponding margin in the average SNR
per bit to preserve a desired error probability in a multilevel ED-PAM system. In
the case where the increase of the error probability or average SNR is not tolerable,
we have proposed a novel near-optimal multilevel threshold selection scheme
based on a closed form polynomial data-fitting approach. We have shown that the
proposed scheme is robust for a wide range of system parameters whereas other suboptimal
schemes assume a more restricted value range for the DOFs to work properly. The results can be extended to
slowly fading channels and they can be applied, for example, for a high-speed
impulse radio.



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*Source: 457342-2009-03-04.xml*