#### 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.

#### 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 Section 2, 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 multilevel *M*-ary ED-PAM system can
be expressed as where is the received signal, is the transmitted signal, is the pulse waveform, is the *k*th
data symbol from the nonnegative and real PAM constellation set, is the received energy corresponding to the minimum nonzero amplitude, is the average energy per
bit, , is the number of modulation levels, *b* is the number of bits per symbol, is the symbol period, is the zero mean AWGN component with noise power spectral density , is the known start time of
the symbol of interest, is the integration time, and is the impulse response of the
receiver filter with bandwidth *B* . The
product of the integration time and bandwidth is denoted by , and 2*D* is the DOFs. It is assumed that *B* and are selected so that *D* is an integer. The
OOK system is a special case of the nonnegative PAM system with . The received energy of the *m*th symbol is . 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 . We use the maximum
likelihood decision criterion and multiple hypothesis testing approach [8, page
82]. We make a hypothesis that the symbol was transmitted if where in our case is the conditional
central or noncentral chi-squared probability
density function (pdf) given in [2]. According to the maximum likelihood criterion
(2), the optimal symbol decision threshold values can be found from , 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] where is the selected *m*th threshold normalized by and is the generalized *v*th 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 where now is the SNR per bit to
achieve the desired with the normalized optimal thresholds and is the required SNR per bit to achieve the desired with the mismatched
thresholds In other words, denotes the required SNR margin per bit resulted from the
hypothesized relative threshold offset compared to the corresponding optimal thresholds. We define the above
relationship formally as where denotes a functional relation
between and and it depends on selected values of *M*, *D*,
and . It is convenient
to express in decibels by dB and in percentages by .
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 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 and . The parameter
of the quadratic function depends on the selected values of *M*, *D*,
and and it can be
found numerically.

#### 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 should be
found and they are derived differently for the threshold values and with . It is shown in [6] that when the SNR approaches infinity. However, we
observed that the corresponding result for with 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 , . The overall scheme is referred to as a semiasymptotically optimal scheme since
only is
asymptotically unbiased whereas there will be some bias in with even if the SNR approaches infinity. The second step is introduced
to find an error function between the optimal and semiasymptotically optimal threshold value set mentioned
above. After some manipulations we propose to minimize where
denotes and
denotes and is the -element
value set to be supported for the parameter *D*, , and are the first-order coefficients, and 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 if if , and for . 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 where is the polynomial with order for the *m*th threshold, is the received SNR of the
()th symbol, and 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, are adopted in (7). For a comparison we apply the Gaussian approximation
scheme presented in [4] to determine each *m*th
threshold () 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].

#### 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.

#### Acknowledgments

This work has been mainly carried out in the projects Pervasive Ultra-wideband Low Spectral Energy Radio Systems (PULSERS) and Coexisting Short Range Radio by Advanced Ultra-Wideband Radio Technology (EUWB) which have been partly funded by the European Union. The authors are thankful to Stephane Paquelet and Alexis Bisiaux for the discussions on the energy detection principle. This work is based on the work by Anttonen et al., which appeared in ICUWB '08, 2008IEEE. The paper received the Best Student Paper Award.