Comparison of Kurtosis and Fourth Power Detectors with Applications to IR-UWB OOK Systems

On-off keying (OOK) is one of the modulation schemes for non-coherent impulse radio Ultra-wideband systems. In this paper, the utilization of the kurtosis detector (KD) and fourth power detector (FD) receivers for OOK signaling is introduced. We investigate the effect of integration interval and the optimum threshold on the performance of energy detector (ED), KD and FD receivers. The semi analytic expression of BER is obtained by using generalized extreme value distribution function for KD and FD receivers. From performance point of view, the simulation results show that FD receiver outperforms KD and ED receivers. In contrast, the sensitivity to the optimum threshold is greatly reduced in KD receiver compared to ED and FD receivers.

• Ultra-Wideband (UWB); Non-Coherent Receiver; OOK; Energy Detector; Kurtosis Detector; Fourth Power Detector

1. Introduction

Impulse Radio Ultra-wideband (IR-UWB) systems are based on the transmission of pulses with very short duration [1,2]. Coherent and non-coherent receivers are commonly used in IR-UWB systems. The non-coherent receivers have low complexity implementation and are used in low cost applications. In this paper, non-coherent receivers are investigated for IR-UWB on-off keying (OOK) scheme.

Energy detector (ED) is one of the non-coherent receivers for the IR-UWB signal reception [3]. ED receivers usually are used for pulse position modulation (PPM) and OOK signaling in IR-UWB systems. The decision mechanism in PPM is made by sign detector, and in OOK scheme, by comparing the output of energy integrator with a threshold value. Threshold value in OOK signaling is investigated in [4-6]. Comparing the various transmission schemes in [3], it is confirmed that OOK modulation with ED outperforms the PPM scheme. So in this paper, OOK signaling is used for IR-UWB non-coherent system.

Kurtosis detector (KD) [7] and fourth power detector (FD) [8] are recently proposed non-coherent detectors to enhance the performance of energy detector. Simulation results in [7] and [8] show that the KD and FD receivers has a lower bit error rate than ED receiver for a IR-UWB PPM system using IEEE 802.15 CM1 channel model.

In this paper, we propose utilizing the KD and FD receivers for IR-UWB OOK signalling scheme. The approximation of optimum threshold value for symbol decision and the semi analytic BER expression are calculated from GEV distribution function for IR-UWB OOK scheme. We show that, FD receiver outperforms KD and ED receivers, and the KD receiver outperforms in high integration intervals compare to ED receiver. We also show that, KD receiver does not require optimizing integration interval, and the KD receiver has a very low sensitivity to the optimum threshold value variations compare to ED receiver.

The rest of the paper is organized as follows. In Section 2, the system model of IR-UWB OOK is presented. Sections 3 and 4 describe the conventional ED structure and the proposed KD and FD receivers’ structures for OOK scheme, respectively, for the detection of IR-UWB signals. The performance evaluation and the results are discussed in Section 5. Finally, concluding remarks are presented in Section 6.

2. System Model

The transmitted signal in OOK scheme can be expressed as follows

where w(t) is the UWB pulse, E_w is the energy of w(t), T_b is the symbol time and b_i{0,1} is the binary information bits.

Signal s(t) propagates through a multipath channel with impulse response

where L is the number of multipath components, α_k and τ_k are the gain and delay associated with the kth multipath component according to IEEE802.15.4 channel model [9], and δ(.) is the Dirac delta function. Then the received signal can be expressed as

where n(t) is the white Gaussian noise with power spectral density N₀/2, and g(t) = w(t) * h(t) is the channel response to w(t).

3. Energy Detector

An energy detector employs a square device, an energy integrator and a threshold decision mechanism which are shown in Figure 1. The decision variable in ED is obtained as follows

where T_i is the integration interval and r(t) is the received signal passing through a band pass filter.

In OOK scheme, the demodulation stage has two hypotheses

where g(t) and n(t) are the received desired signal and noise respectively. The symbol decision in receiver is made by comparing z_ED with a threshold value Th. If the received signal energy is lower than a threshold value, the detector decides that the transmission bit is 0. If the received signal energy is larger than a threshold value, the detector decides that the transmission bit is 1.

Hypotheses 0 and 1 have the probability density functions (PDF) p₀(x) and p₁(x), respectively. The optimum threshold value Th_opt is obtained by the solution of p₀(x) = p₁(x). The PDFs of p₀(x) and p₁(x) are shown to be

central and non-central chi square distribution (X²) respectively [4]:

where M = BT_i, Г(.) denote Euler function, B is the signal bandwidth and I_n is the nth Bessel function of the first kind.

4. Proposed Kurtosis Detector and Fourth Power Detector for OOK Scheme

In this section we propose two non-coherent receivers for IR-UWB OOK signalling scheme, by using the fourth order statistics of received signal.

4.1. Kurtosis Detector

The Kurtosis for random variable x is defined as

where E{} denotes the expected value of the variable. If x is a Gaussian random variable, its kurtosis is zero. If x has a subgaussian distribution, it means that the distribution of x has flatness and shorter tails relative to Gaussian distribution, its kurtosis has a negative value. If x has a supergaussian distribution, it means that the distribution of x has peakedness and longer tails relative to the Gaussian distribution, its kurtosis has a positive value. In impulse radio UWB, the received signal has a supergaussian distribution in general; therefore, its kurtosis value is too larger than zero.

Kurtosis detector is based on kurtosis value of the received signal [7]. In [7] the KD receiver is proposed for PPM signaling scheme. The KD receiver structure is shown in Figure 2.

In this paper we used KD receiver for OOK signaling in IR-UWB systems. In this case, the kurtosis value of the received signal is calculated in receiver as follows

and then, two hypotheses in KD receiver are defined as follows

In KD receiver case similar to ED receiver, the symbol decision is made by comparing z_KD with a threshold value Th_K:

where the optimum threshold value Th_Kopt is obtained by the solution of p_K0(x) = p_K1(x), and the functions of p_K0 (x) and p_K1(x) are the probability density functions (PDF) of H_K0 and H_K1 respectively.

By using Maximum likelihood (ML) parameter estimation in simulations the PDFs of p_K0(x) and p_K1(x) can be fitted by Generalized Extreme Value (GEV) distribution density function. The GEV distribution function defined as follows

(12)

where ζ, σ, μ are the parameters of GEV distribution function that obtained from ML parameter estimation.

The semi analytic expression for BER is obtained by using GEV distribution parameters. The parameters of GEV distribution can be obtained from numerical methods in simulations. The approximation of threshold value is obtained by solving the following equation,

where (ζ₀, σ₀, μ₀) and (ζ₁, σ₁, μ₁) are the parameters of GEV distribution for hypothesis 0 and hypothesis 1 respectively. By using the approximation of threshold value (Th_gev), the BER expression of bit 0 can be evaluated as

and, the BER expression of bit 1 can be evaluated as

Finally, the BER can be expressed as

4.2. Fourth Power Detector

In [8], improved ED receiver is proposed by replacing the squaring operation in ED receiver with an arbitrary positive power operation. In this paper the fourth power operation is used for OOK IR-UWB signalling scheme, which has been called fourth power detector (FD). We used FD receiver because it is more practical and it has better performance.

The structure of FD receiver, which is shown in Figure 3, is similar to the ED receiver except that the FD receiver employs two square devices. The decision variable in FD is obtained using the following expression

and the two hypotheses are defined as follow

Similar to ED and KD receivers, the symbol decision in FD receiver is made by comparing z_FD with a threshold value Th_F:

where the optimum threshold value Th_Fopt is obtained by solving p_F0(x) = p_F1(x). p_F0(x) and p_F1(x) are the probability density functions of H_F0 and H_F1, respectively.

In [8], the PDFs of p_F0(x) and p_F1(x) are approximated by using Gamma distribution function. In this paper, these PDFs are approximated by GEV distribution function which have the same relations as p_K0(x) and p_K1(x). Results of simulation in section V show that the GEV distribution function has higher accuracy than Gamma distribution function.

The semi analytic expression of BER for FD receiver can be calculated from Equation (16). In this equation, parameters of GEV distribution for H_F0 and H_F1 are obtained by ML parameter estimation.

5. Simulation Results

Simulations are done in IEEE 802.15.4a CM1 channel model [9] with maximum delay spread (T_mds) truncated to 200 nsec. The second derivative of Gaussian pulse is used with pulse duration T_p = 1.5 nsec, and the symbol duration is T_b = 400 nsec. The energy of the channel impulse response is normalized to have unit power gaini.e.. We also assume perfect synchronization.

Figures 4 and 5 show the accurate cumulative density functions (CDFs) and fitted GEV CDFs for H_K0 and H_K1. Figures 6 and 7 show the accurate CDFs, fitted GEV, and fitted Gamma CDFs of H_F0 and H_F1, respectively. The accurate CDFs are obtained by using the histogram method and GEV and Gamma approximate CDFs are obtained by using ML estimation of distribution parameters.

According to the above-mentioned figures, GEV CDFs fitted to H_K0, H_K1, H_F0 and H_F1 CDFs have high accuracy for different amounts of E_b/N₀ and integration intervals. In FD receiver the fitted GEV and Gamma distributions of H_F0 have almost the same accuracy, while in the case of H_F1, the fitted GEV CDF has a better accuracy than the fitted Gamma CDF.

Figure 8 shows the bit error rate (BER) performance of ED, KD and FD receivers as a function of integration interval (T_i) for sample amounts of E_b/N₀ = 14 dB and E_b/N₀ = 16 dB. In ED and FD receivers, there is an optimum integration interval that minimizes the BER. Increasing the amount of E_b/N₀ causes this optimum value

to increase. In KD receiver, short integration intervals have a negative effect on the performance, and BER is almost constant for large values of integration intervals. For large integration intervals, the KD receiver does not require optimization of integration interval. The FD receiver outperforms ED and KD receivers for almost all amounts of integration intervals.

Figure 9 shows the BER performance of ED, FD and KD receivers for integration interval T_i = T_mds = 200 nsec. This figure also demonstrates the BER of ED and FD for optimum integration intervals. For T_i = 200 nsec and BER = 10^–3, the KD receiver has a 1.4 dB better performance than the ED receiver and the FD receiver has a 2 dB better performance than the ED receiver. The ED receiver with optimized integration interval has a 0.2 dB better performance than the KD receiver for high values of E_b/N₀. The FD receiver with the optimum integration intervals has a better performance than ED and KD receivers in all values of E_b/N₀. For BER = 10^–4, the FD

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