Improved Detection Schemes for Non-coherent Pulse-Position Modulation

Abstract

Energy-detection (ED) pulse-position modulation (PPM) receivers exhibit poor performance and low rates. Coherent receivers do not have such problems but their complexity is unacceptable. We propose two detection schemes to increase the performance of non-coherent PPM receivers. Unlike the ED-PPM receiver, the first proposed receiver cubes the absolute value of the received signal before demodulation and achieves a considerable performance gain. This gain is obtained because the absolute-value cubing (AVC) operation reduces the effect of low-SNR samples and increases the effect of high-SNR samples on the decision statistic. To further increase energy efficiency and rate of the non-coherent PPM receivers at almost the same complexity, we use the weighted-transmitted reference (WTR) system instead of the ED-based receiver. The WTR system has adequate robustness to weight coefficients and integration interval variations. To generalize the AVC concept to the WTR-PPM receiver, the reference pulse is first passed through a polarity-invariant squaring (PIS) operation and then is correlated with the data pulses. In this paper, the performance of different receivers employing the binary PPM (BPPM) is investigated at data rates of 2.08 and 9.1 Mbps over in-vehicle channels in the presence of noise, inter-block interference, inter-pulse interference, and inter-symbol interference (ISI). Simulations show that the proposed AVC-BPPM receiver outperforms the ED-based one in the absence of ISI and offers the same performance in the presence of strong ISI, the WTR-BPPM system considerably outperforms the ED-BPPM system (especially at high rates), and the proposed PIS-based WTR-BPPM system considerably outperforms the conventional WTR-BPPM system.

Introduction

Impulse radio (IR) ultra-wideband (UWB) communication systems employ ultra-narrow pulses with less than 1 ns duration for ranging and communications. They have attractive features, such as simplicity, precise ranging, high data rate, low latency, coexistence with current wireless systems, robustness to multipath, high penetration capability, and fine time resolution [1–4]. These features motivate the use of IR-UWB technology in low-latency real-time high-resolution positioning, short-range high-rate wireless connectivity, and tracking applications [5–9].

IR-UWB technology is promising for use in internet of things (IoT) [10], autonomous vehicles [5], navigation [11], robotics [12], smart logistics [2], medical applications [13], wireless body area networks [14], and wireless sensor networks [11]. Currently, Samsung Galaxy Note 20 Ultra uses UWB technology for both device-to-device connection and as a digital key for access control [8]. Also, Apple iPhones are equipped with a UWB chip for spatial awareness to accurately locate one another [8]. UWB ranging has been used for social distancing and contact tracing, especially during the COVID-19 pandemic [15]. Moreover, car manufacturers like Audi and BMW use UWB systems for keyless entry and hands-free access control [8].

Though transmitting short-duration pulses brings many interesting features for the UWB technology, it causes a crucial problem in the receiver design. The received IR-UWB signals contain many multipath components (MPCs) [16]. Estimating such a high number of MPCs makes the receiver implementation infeasible [16]. To exploit multipath diversity with low complexity and collect more energy from MPCs, several non-coherent IR-UWB receivers have been proposed [17–21].

Among the non-coherent IR-UWB receivers, the energy detection (ED) pulse position modulation (PPM) receivers are promising for applications demanding low-power consumption and low complexity [20, 21], such as IoT and robotics. However, the squaring operation in the ED-based receivers makes them vulnerable to noise plus interference and decreases their performance compared with the coherent receivers [20]. Moreover, in an ED-PPM receiver, the time spacing of the PPM modulation (a.k.a PPM parameter) should be greater than the maximum delay spread of the multipath channel, referred to as $T_{\mathit{mds}}$. In this way, inter-symbol interference (ISI) is avoided and high performance is assured. However, keeping the PPM parameter larger than $T_{\mathit{mds}}$ comes at the expense of a severely reduced achievable data rate. Therefore, ED-PPM receivers exhibit poor performance and low data rates.

Coherent receivers such as matched-filter or rake receivers bypass the problems of the ED-PPM receivers. But, coherent receivers need full-channel state information, which makes their complexity impractical. In this paper, we propose two novel detection schemes for non-coherent PPM receivers to promote the performance of the non-coherent PPM receivers existing in the literature. The first scheme, referred to as absolute-value cubing (AVC), increases the performance of reference-free PPM detection schemes. The second scheme, denoted as polarity-invariant squaring (PIS), improves the performance of PPM detection schemes that use a noisy reference (dirty template). Unlike the ED-PPM receiver which squares the received signal, the proposed AVC receiver cubes the absolute value of the received signal before demodulation and achieves a considerable performance gain. This gain is obtained because the AVC operation increases the effect of the high-SNR samples in the decision variable compared to the squaring operation of the ED-based receivers.

To further increase the energy efficiency and rate of the non-coherent PPM systems at almost the same complexity, we propose to use the weighted-transmitted reference (WTR) system instead of the ED-based one. The WTR system moderates the main deficiencies of the conventional transmitted reference (TR) systems which are [22, 23]: (i) presence of harsh nonlinear ISI (ISI$\times$ISI term) at high rates in the decision variable, (ii) low energy efficiency; (iii) low achievable data rates, and (iv) performance loss because of employing a noisy reference, which leads to the presence of a strong noise$\times$noise term in the decision variable. Moreover, the WTR system has adequate robustness to variations of the weight coefficients and the integration interval. To generalize the AVC concept for the WTR-PPM receiver, the reference pulse is first passed through a PIS operation and then correlated with the data pulses. Unlike the squaring operation that omits the sign of the input, the PIS operation maintains the sign of the input and does not change it. For example, the output of the PIS device for 2 and $-3$ is equal to 4 and $-9$, respectively (the PIS device assigns 4–2 and $-9$ to $-3$). Hence, after correlating the PIS-operated reference pulse with the data pulses, the third power of the received signal is obtained similar to the cubing operation in the AVC approach. This paper shows that the WTR-based receivers, characterized as non-coherent receivers, perform better than the ED-based ones in multipath environments. This paper investigates the performance of different receivers employing the binary PPM (BPPM) at data rates of 2.08 and 9.1 Mbps over measured in-vehicle channels in the presence of noise, inter-block interference (IBI), inter-pulse interference (IPI), and ISI. Simulation results show that (i) the proposed AVC-BPPM receiver outperforms the ED-PPM one in the absence of ISI and offers the same performance in the presence of strong ISI, (ii) the WTR-BPPM system considerably outperforms the ED-BPPM system (especially at high rates), and (iii) the proposed PIS-based WTR-BPPM system considerably outperforms the conventional WTR-BPPM system.

Objective and Contributions

This paper aims to increase the BER performance of the non-coherent detection schemes existing in the literature with a marginal increase in the receiver complexity. The contributions of this paper are two-fold to address this objective: (i) an improved non-coherent receiver denoted as the AVC-PPM receiver is proposed to replace the conventional ED-PPM one to detect PPM signals without using any reference. The AVC-PPM receiver uses the absolute value of the cubed received signal instead of the squared one used in the conventional ED-PPM receiver, and (ii) an improved WTR-PPM receiver, referred to as the PIS-based WTR-PPM receiver, is proposed to generalize the AVC concept to the TR receivers, which use a noisy reference (dirty template) for detection. The PIS-based WTR-PPM receiver squares the reference pulse samples while maintaining their sign, and then correlates the resultant reference pulse with the data pulses.

Literature Survey

Conventional ED-PPM receivers are widely used in the literature [20, 21, 24–30]. An improved pulse position modulation, known as multipulse PPM, is proposed in [31] to achieve a higher bit rate and bandwidth efficiency for PPM communication systems. In [32], this modulation scheme is applied to a UWB communication system using an energy detector, where 1.33 to 2.5 bits are transmitted per pulse (the conventional ED-BPPM which transmits one bit per pulse). The PPM receivers in [33] and [34] use the envelope (absolute value) of the received signal while the PPM receiver in [35] uses the peak value of the envelope of the received signal for detection. Note that using the envelope for detection simplifies the receiver design at the expense of a severe performance loss. To reduce narrowband interference in ED-PPM receivers, [20] proposes the Teager-Kaiser operator, while [21] proposes a nonlinear blind technique with much lower complexity and almost the same performance. In [36], compressed sensing is employed for ED-PPM using the sparsity of the UWB channels at the expense of an increase in receiver complexity. In [17–19], the TR-PPM receiver is proposed which correlates the reference pulse with the data pulses for detection. The maximum ratio combing receiver is used in [37] for detecting BPPM signals in the presence of noise, multi-access interference, and IPI. Other coherent detection schemes for PPM signals are investigated in [38] and [39] without considering path-dependent pulse distortion of UWB channels. CMOS-based UWB transceiver design for PPM is done in [34], which achieves a rate of 1.125 Gbps over a 2 m distance. Since the use of PPM in deep-space optical communication is becoming a hot research topic, [40–42] consider PPM optical communication. Moreover, the multipulse PPM has been recently used for ultraviolet communications [43]. Note that the output of optical or ultraviolet detectors is proportional to the energy of the incident beam. Therefore, the proposed AVC-PPM detection scheme can also be used in optical and ultraviolet communications. Employing the PPM scheme, [19] considers joint communication and radar, and [35] investigates joint information and power transfer. Moreover, covert communication using PPM is proposed in [44], and space-time code design using ED-PPM detectors is investigated in [29, 30]. Table 1 compares different detection methods for PPM in terms of complexity and performance.

Table 1.

Comparison of different PPM detection schemes

Reference(s) Number	Detection method	Complexity	Performance
[33, 34]	Using the absolute value of the received signal	Very low	Very low
[35]	Using the peak value of the received signal envelope	Very low	Very low
[36]	Energy detection + compressed sensing	Moderate	Moderate
[24–30]	Energy detection for conventional PPM scheme	Low	Moderate
[32]	Energy detection for enhanced PPM scheme	Moderate	Moderate (better than conventional ED-BPPM)
[17–19]	Correlating the reference pulse with data pulses (TR)	High	Moderate (better than ED-PPM)
[38, 39]	Coherent detection	Very high	Best
This paper (AVC-PPM)	Cubing the absolute value of the received signal	Low	Better than ED-PPM
This paper (PIS-based WTR-PPM)	Squaring the transmitted-reference pulse without losing its sign, then correlating the result with data pulses	High	Better than the conventional WTR-PPM

Simulation results show that the proposed AVC-PPM and the PIS-based WTR-PPM receivers offer much better BER performance over the ED-PPM and the conventional WTR-PPM receivers, respectively, over realistic in-vehicle UWB channels measured in [45].

Following this introduction, Sect. 2 models the ED-BPPM, the proposed AVC-BPPM, the conventional WTR-BPPM, and the proposed PIS-based WTR-BPPM systems. The origin of the performance improvement of the AVC operation in non-coherent PPM detection is intuitively analyzed and numerically validated in Sect. 3. Section 4 evaluates the uncoded BER performance for different receivers through simulations over measured line-of-sight (LOS) and non-LOS (NLOS) in-vehicle UWB channels. Finally, Sect. 5 concludes this paper and provides future directions. Table 2 describes the mathematical symbols used throughout this paper.

Table 2.

Definition of mathematical symbols

Notation	Definition
$\mu$	Mean of a random variable
$\sigma$	The standard deviation of a random variable
$w_{\mathit{tr}}(t)$	Transmitted UWB pulse
C(t)	The impulse response of the multipath channel
f(t)	The impulse response of the receiver front-end bandpass filter
F(f)	Fourier transform of f(t)
$\ast$	Linear convolution operator
g(t)	Received noise-free bandpass filtered signal when $w_{\mathit{tr}}(t)$ is sent, i.e., $g(t)=w_{\mathit{tr}}(t)\ast C(t)\ast f(t)$
s(t)	Received noise-free bandpass filtered signal
n(t)	Bandpass filtered additive white Gaussian noise at the output of the receiver front-end bandpass filter
b	Transmitted bit
$\widehat{b}$	Estimate of the transmitted bit b
$N_0$	One-sided power spectral density of the noise in the communication channel
$T_s$	Symbol dusration in ED-PPM and AVC-PPM systems/
	Time spacing between consecutive data pulses in WTR-PPM systems
$T_i$	The length of the integration interval
$|x|$	The absolute value of x
$\alpha$	Weight of the reference pulse in WTR systems
$\beta$	Weight of the data pulses in WTR systems
$N_s$	Number of data pulses per block in WTR systems
$T_d$	Guard time between the reference pulse and data pulses in WTR systems
$T_B$	Block length in WTR systems
$T_{\mathit{mds}}$	Maximum delay spread of the channel
u(t)	Unit step function defined as: $u(t)=\left\{\begin{array}{c}1,t\ge 0\\ 0,t<0\end{array}\right)$
sign(t)	Modified signum function defined as: $sign(t)=\left\{\begin{array}{c}+1,t\ge 0\\ -1,t<0\end{array}\right)$

System Models

To simplify notation, multiple access schemes are neglected throughout this paper. Moreover, it is assumed that each bit is transmitted by a single UWB pulse, and receivers are perfectly synchronized.

Energy Detection BPPM Receiver

The system structure of an ED-PPM receiver is shown in Fig. 1. First, a low-noise amplifier amplifies the received signal, and then, a bandpass filter (BPF) removes the out-of-band noise and interference. Next, the received bandpass filtered signal, r(t), passes through the ED device followed by a sampler and information detector.

Fig. 1.

Structure of the ED-PPM receiver

The ED-based receivers detect the information by comparing the energy of the received signal in predefined intervals. To have no IPI and ISI, the duration of the intervals should be greater than or equal to $T_{\mathit{mds}}$. We express the signal at the output of the BPF by:

$$\begin{array}{c}\hfill r(t)=\sum _{j=-\infty }^{\infty }g(t-j{T}_s-b_j\delta )+n(t)=s(t)+n(t)\end{array}$$ 1

where $g(t)=w_{\mathit{tr}}(t)\ast C(t)\ast f(t)$ is the received noise-free pulse ($w_{\mathit{tr}}(t)$ is the transmitted bandpass UWB pulse with a bandwidth W and a center frequency $f_c$, C(t) is the multipath channel impulse response, and f(t) shows the BPF impulse response). In (1), s(t) is the received noise-free bandpass filtered UWB signal and n(t) shows the bandpass filtered additive white Gaussian noise (AWGN) with a two-sided power spectral density $\frac{N_0}{2}{|F(f)|}^2$, where $|x|$ indicates the absolute value of x and F(f) denotes the Fourier transform of f(t). Moreover, $T_s$ denotes the symbol duration; $b_j\in \{0,+1\}$ represents the jth information bit; and $\delta$ is the PPM parameter. To demodulate the jth information bit, the received signal is squared and integrated from $j{T}_s$ to $j{T}_s+T_i$ and from $j{T}_s+\delta$ to $j{T}_s+T_i+\delta$. Then, the output of the integrator is sampled and the former sample is subtracted from the latter one. Finally, the subtraction result is applied to a unit step function. Thus, in the conventional ED-BPPM receiver, the integration output for the jth information bit is expressed as:

$$\begin{array}{c}\hfill z_i[j]={\int }_0^{T_i}{r}^2(t+j{T}_s+i\delta )dt,i\in \{0,1\}\end{array}$$ 2

and finally, the jth information bit is detected as follows:

$$\begin{array}{c}\hfill {\widehat{b}}_j=u(z_1[j]-z_0[j]).\end{array}$$ 3

where u(t) indicates the unit step function.

Proposed Absolute-Value Cubing BPPM Receiver

The structure of an absolute-value cubing (AVC) PPM receiver is illustrated in Fig. 2. Note that the structure of the AVC-BPPM receiver is the same as that of the ED-BPPM one except that the filtered received signal is fed to the AVC device instead of the squaring device.

Fig. 2.

Structure of the proposed AVC-PPM receiver

Since high-SNR samples are very likely to have a larger amplitude than the low-SNR ones, the AVC device increases the contribution of the high-SNR samples to the decision variable compared with the squaring device. This is intuitively and numerically shown in Sect. 3. Hence, the AVC receiver shows more capability for small-sample removal than the ED receiver and consequently provides better performance than the ED receiver. Note that the calculation burden and complexity of the AVC-BPPM receiver are almost the same as those of the ED-BPPM one.

Conventional WTR-BPPM Receiver

A WTR system transmits data block-wise. If each data pulse carries one information bit, each block consists of $N_s+1$ pulses. In this case, the first pulse in each block is the reference pulse which is followed by $N_s$ data pulses closely spaced in time as shown in Fig. 3. To prevent IBI and have an interference-free reference, a guard time $T_d$ is inserted between the reference pulse and the data pulses as seen in Fig. 3. Moreover, by assigning a larger amplitude to the reference pulse than to the data pulses, the SNR of the reference pulse which plays an important role in detection increases. Thus, we can express the transmitted signal of the WTR-PPM system by:

$$\begin{array}{c}\hfill s_{\mathit{tr}}(t)=\sum _{j=-\infty }^{\infty }\{\alpha w_{\mathit{tr}}(t-j{T}_B)+\beta \sum _{i=0}^{N_s-1}{w}_{\mathit{tr}}(t-j{T}_B-i{T}_s-b_{j,i}\delta -T_d)\}\end{array}$$ 4

where $N_s$ is the number of data pulses per block, $T_B=2T_d+(N_s-1)T_s+\delta$ shows the block length, $T_d$ shows the guard time between the reference pulse and the preceding/succeeding data pulses, and $T_s$ is the time spacing between two successive data pulses. Also, $\alpha$ and $\beta$ respectively denote the weights of the reference pulse and data pulses. Finally, $b_{j,i}\in \{0,+1\}$ shows the ith information bit of the jth block. To have a fair comparison between weighted and unweighted systems, the following equation should be satisfied:

$$\begin{array}{c}\hfill {\alpha }^2+N_s{\beta }^2=N_s.\end{array}$$ 5

Fig. 3.

A typical transmitted signal of the WTR-BPPM system

The structure of the conventional WTR-BPPM receiver is depicted in Fig. 4. First, the received signal passes through the BPF. The signal at the output of the BPF is written as:

$$r(t)=\sum _{j=-\infty }^{\infty }\left\{\alpha g(t-j{T}_B)+\beta \sum _{i=0}^{N_s-1}g(t-j{T}_B-i{T}_s-b_{j,i}\delta -T_d)\right\}+n(t);=s(t)+n(t),$$ 6

Without loss of generality, we focus on the demodulation of the first block. To detect the kth bit of the block, we multiply the received signal r(t) by $r(t+k{T}_s+T_d)$ and $r(t+k{T}_s+T_d+\delta )$. Then, we integrate these two multiplications and sample the output of the integrators. Next, as shown in Fig. 4, the upper sample is subtracted from the lower one and the result is applied to a unit step function. The output of the integrators for the kth bit of the block in the conventional WTR-PPM receiver is:

$$\begin{array}{c}\hfill z_i[k]={\int }_0^{T_i}r(t)r(t+k{T}_s+T_d+i\delta )dt,i\in \{0,1\}\end{array}$$ 7

Finally, the kth bit of the block is detected by a unit step function as follows:

$$\begin{array}{c}\hfill {\widehat{b}}_k=u(z_1[k]-z_0[k]).\end{array}$$ 8

Fig. 4.

The receiver diagram of the conventional WTR-BPPM system

Proposed Polarity-Invariant Squaring-Based WTR-BPPM Receiver

The structure of the PIS-based WTR-BPPM receiver is illustrated in Fig. 5. Comparing Fig. 4 with Fig. 5 shows that the structure of the PIS-based WTR-BPPM receiver is the same as that of the conventional WTR-PPM one except that the reference pulse is first passed through the PIS device and then is used for demodulation. The PIS device squares the reference pulse and keeps its sign/polarity. In other words, the PIS device squares the reference pulse without altering its sign. For instance, if 2 and $-3$ pass through the PIS device, it outputs 4 and $-9$, respectively.

Fig. 5.

The receiver diagram of the proposed PIS-based WTR-BPPM system

Squaring operation decreases the effect of the low-SNR (noise-dominant) samples and increases the effect of high-SNR samples of the reference pulse on the decision variable. Thus, the PIS operation shows small (low-SNR) sample removal capability and thereby offers superior performance. Note that the calculation required by the PIS device can be done with a square-law device, a sign detector, and a mixer (multiplier). Hence, the complexity of the new receiver is almost the same as the original one.

How the AVC Operation Improves the Performance: Intuitive Analysis and Numerical Validation

In this section, we answer the following question through intuitive analysis and numerical calculations: how does absolute-value cubing (AVC) improve the BER performance in non-coherent PPM detection? To answer this question, first, we show that correlating the PIS-operated reference pulse with the data pulses is almost equivalent to the AVC operation. Denoting the received bandpass filtered signal by r(t) and stating from the term under integral in the PIS-based WTR-PPM receiver, we end up with the term under integral in the AVC-PPM receiver as follows:

$$\begin{array}{cc}\hfill & \stackrel{\begin{array}{c}\text{PIS-Operated}\\ \text{Reference Pulse}\end{array}}{\overbrace{sign(r(t))r^2(t)}}\times r(t)=sign(r(t))r^3(t)=sign(r(t))sign(r(t)){|r(t)|}^3\hfill \\ \hfill & =sig{n}^2{(r(t))|r(t)|}^3={|r(t)|}^3\hfill \end{array}$$ 9

where sign(x) indicates the modified signum function. The last equality in (9) is obtained because $sig{n}^2(x)=1$.

In moderate to high-SNR range, high-SNR samples are very likely to have larger amplitudes than the low-SNR samples. Therefore, cubing the received noisy signal amplifies the high-SNR samples much more than the low-SNR samples. Thus, cubing the received signal or its absolute value magnifies the amplitude difference between the high-SNR samples and low-SNR samples. As a result, the proposed detection schemes reduce the effect of small (noise-dominant) samples and increase the effect of high-SNR samples on the decision variable. In the ED-based detection of PPM signals, the energy difference between the signal plus noise interval and the pure-noise interval affects the receiver performance. The more energy difference (distance) is, the better performance is obtained. Therefore, cubing the signal or its absolute value offers better performance than squaring the signal because it magnifies the distance between decision hypotheses. Figure 6 intuitively shows the effect of squaring and absolute-value cubing on the amplitude of pure-noise, low-SNR, and high-SNR samples. In this figure, we consider a zero-mean AWGN with a standard deviation (SD) of 2/3. We remark that the value of a normal random variable with mean $\mu$ and SD $\sigma$ is between $\mu -3\sigma$ and $\mu +3\sigma$ with a chance of 99.73$\%$. Hence, the value of the considered normal noise is between $-2$ and 2 with a chance of 99.73$\%$. Similarly, if the value of the signal is a, the value of the considered noise plus signal is between a-2 and a+2 with a chance of 99.73$\%$. In Fig. 6, this range of values for the noisy signal (signal + noise) is depicted by a red line on the left side of the noise-free sample which is shown by a blue dot. In this figure, $R_1$ denotes the amplitude ratio of the high-SNR sample to the low-SNR sample and $R_2$ indicates the amplitude ratio of the high-SNR sample to the pure-noise sample in the worst case (i.e., considering the minimum value for the high-SNR sample and the maximum value for the low-SNR and pure-noise samples). It is seen that $R_1$ and $R_2$ increase with the signal exponent. Since the noise-free sample is not located in the center of the noisy-signal range and is closer to the minimum value of the noisy signal, the skewness of the noisy signal is to the right which is seen in Fig. 7. To further validate the observations drawn from Fig. 6, the statistical mean of different types of samples are compared in Fig. 7 using a zero-mean AWGN vector of length 10000000 with an SD 2/3. This figure also illustrates the SD and histogram of different types of samples. Henceforth, we show the ratio of the high-SNR sample mean to the low-SNR sample mean by $R_{\mathit{hl}}$, the ratio of the high-SNR sample mean to the pure-noise sample mean by $R_{\mathit{hn}}$, and the ratio of the low-SNR sample mean to the pure-noise sample mean by $R_{\mathit{ln}}$. Table 3 compares the values of $R_{\mathit{hl}}$, $R_{\mathit{hn}}$, and $R_{\mathit{ln}}$ for different exponents by using the values given in Fig. 7. It is seen that $R_{\mathit{hl}}$, $R_{\mathit{hn}}$, and $R_{\mathit{ln}}$ increase with the signal exponent, and $R_{\mathit{hn}}$, $R_{\mathit{hl}}$, and $R_{\mathit{ln}}$ have the highest increment rate in descending order. From Fig. 7, it is also seen that the skewness of the noisy sample is to the right which increases with the signal exponent and decreases with SNR. One may conclude that increasing the signal exponent will improve the performance endlessly. However, increasing the signal exponent lessens the contribution of low-SNR samples and in the extreme case, the detection will rely on only one sample (the highest-SNR sample). In this case, the information in other samples is not taken into account and time diversity is not utilized. Therefore, theoretically and practically there is an end to the performance improvement caused by increasing the signal exponent in non-coherent detection schemes. There are other limitations for the proposed schemes: (i) when coherent detection schemes are used or (ii) when the detection is based on only one sample or (iii) when the amplitude of the received signal is lost due to monobit quantization, or (iv) when the SNR is below a certain threshold (the signal is buried in the noise), the proposed schemes do not offer any performance gain.

Fig. 6.

The effect of squaring and absolute-value cubing on three types of samples, i.e., pure-noise (left column), low-SNR (center column), and high-SNR (right column) samples

Fig. 7.

The effect of squaring and absolute-value cubing on the mean, SD, and histogram of three types of samples, i.e., pure-noise (left column), low-SNR (center column), and high-SNR (right column) samples

Table 3.

Comparison of $R_{\mathit{hl}}$, $R_{\mathit{hn}}$, and $R_{\mathit{ln}}$ for Different Exponents

Exponent	$R_{\mathit{hl}}$	$R_{\mathit{hn}}$	$R_{\mathit{ln}}$
1 (original)	$8/2=4$	$8/0.53=15.01$	$4/0.53=7.55$
2 (squaring)	$64.45/4.44=14.52$	$64.45/0.44=146.48$	$4.44/0.44=10.09$
3 (AVC)	$522.68/10.67=48.96$	$522.68/0.47=1112.1$	$10.67/0.47=22.7$

Simulation Results and Discussions

This section investigates the uncoded BER performance for the conventional ED-PPM, the AVC-PPM, the conventional WTR-PPM, and the PIS-based WTR-PPM receivers in measured LOS and NLOS in-vehicle channels [45] through computer simulations employing binary PPM (BPPM) modulation, taking into account noise, IPI, ISI, and IBI.

The transmitted pulse is a bandpass UWB pulse whose center frequency and 10 dB bandwidth are 3.35 GHz and 500 MHz, respectively. Simulations show that the estimate of the optimal value for $\beta /\alpha$ is about 0.625 for the WTR-PPM receivers. Therefore, we use $\beta /\alpha =0.625$ in this paper. Simulation results are obtained by assuming perfect synchronization for receivers. Table 4 summarizes the simulation parameters.

Table 4.

Simulation parameters

Parameter	Value
$T_d$	200 ns
$T_s$	200 ns or 15 ns
$\mathit{\delta }$	100 ns or 7.5 ns
$T_B$	2400 ns or 550 ns
$T_i$	200 ns or 15 ns
$R_b$	2.08 Mbps or 9.1 Mbps
$\beta /\alpha$	0.625

Figures 8 and 9 compare the BER performance of different receivers for a low-rate case ($R_b=2.08$ Mbps) in LOS and NLOS scenarios, respectively. Note that the AVC-PPM and PIS-based WTR-PPM receivers outperform the conventional ED-PPM and conventional WTR-PPM receivers, respectively. The PIS-based WTR-PPM, conventional WTR-PPM, AVC-PPM, and ED-PPM receivers have the best performance in descending order. In other words, the PIS-based WTR-PPM receiver offers the best performance and the ED-PPM receiver has the worst performance. Moreover, Figs. 8 and 9 show that at $BER={10}^{-3}$, the proposed AVC-PPM and PIS-based WTR-PPM receivers outperform their corresponding conventional receivers (ED-PPM and conventional WTR-PPM) by about 1 dB and 0.8 dB in LOS and NLOS scenarios, respectively. Note that the performance gain brought by the AVC-PPM and the PIS-based WTR-PPM decreases in NLOS channels because these receivers reduce the effect of the low-SNR samples and the number of such samples is larger in the LOS channels compared to the NLOS ones. As seen in Figs. 8 and 9, the WTR-PPM receivers considerably outperform the AVC-PPM and the ED-PPM receivers in both LOS and NLOS scenarios. This is because WTR-PPM receivers are more capable to distinguish between noise and the desired signal by employing a reference pulse despite this reference pulse is noisy.

Fig. 8.

Performance comparison of different binary receivers in LOS scenario for a low-rate case, i.e. $R_b=2.08$ Mbps

Fig. 9.

Performance comparison of different binary receivers in NLOS scenario for a low-rate case, i.e. $R_b=2.08$ Mbps

Figures 10 and 11 illustrate the BER performance of different receivers for a high-rate case ($R_b=9.1$ Mbps) for LOS and NLOS scenarios, respectively. Note that the ED-PPM and AVC-PPM receivers fail to detect information bits because the ISI is no longer negligible at high rates and these receivers cannot distinguish the desired signal from noise plus interference. However, the WTR-PPM receivers can recognize the desired signal from noise plus interference by using a noisy reference pulse and consequently, provide superior performance. It is seen that at BER=${10}^{-3}$, the PIS-based WTR-PPM receiver outperforms the conventional WTR-PPM one by 1.5 dB and 1 dB in LOS and NLOS channels, respectively. Again, because of the larger number of low-SNR samples in LOS channels, the performance gain of the proposed PIS-based WTR-BPPM receiver is higher in such channels. Comparing Figs. 8 and 9 with Figs. 10 and 11 shows that the PIS-based WTR-PPM receiver offers higher improvement in higher rates because (i) the PIS-based WTR-PPM receiver reduces the effect of the low-SNR samples and increases the effect of high-SNR samples on the decision statistic and thereby greatly reduces noise and interference, (ii) the amplitude of the reference pulse is larger than that of the data pulses in the WTR systems. Hence, the WTR-PPM receivers specifically the PIS-based one assures higher SNR for the reference pulse as the cornerstone of detection and thereby provides reliable performance even in channels with strong ISI.

Fig. 10.

Performance comparison of different binary receivers in LOS scenario for a high-rate case, i.e. $R_b=9.1$ Mbps

Fig. 11.

Performance comparison of different binary receivers in NLOS scenario for a high-rate case, i.e. $R_b=9.1$ Mbps

Conclusions and Future Works

This paper proposed two novel non-coherent detection schemes to improve the BER performance of non-coherent PPM receivers existing in the literature. The proposed receivers achieve better performance by decreasing the effect of low-SNR samples and increasing the effect of high-SNR samples on the decision statistic. The first proposed scheme, denoted as AVC, cubes the absolute value of the received signal instead of squaring it (squaring is done in ED-PPM receivers). The second proposed technique, referred to as PIS, is applied to the WTR-PPM receiver. This scheme squares the reference pulse while maintaining its sign, and then correlates it with data pulses. The performance of the ED-PPM, AVC-PPM, WTR-PPM, and PIS-based WTR-PPM receivers have been investigated in measured LOS and NLOS in-vehicle channels for low-rate and high-rate modes of operation through simulations, taking into account noise, ISI, and IBI for binary PPM (BPPM). Among all receivers, the PIS-based WTR-PPM receiver offers the best performance and the ED-PPM receiver has the worst performance. Since the number of low-SNR samples is larger in LOS channels, the performance gain of the proposed receivers is larger in LOS channels. Moreover, the ED-PPM and AVC-PPM receivers fail to detect information bits at high rates because they cannot distinguish the desired signal from noise plus interference. However, the PIS-based WTR-PPM receiver offers a larger gain in higher rates because on one hand, it reduces the effect of the low-SNR samples and increases the effect of high-SNR samples on the decision statistic. As a result, it greatly reduces noise and interference. Hence, the PIS-based WTR-PPM receivers increase the SNR of the reference pulse as the golden key for detection and thereby provide reliable performance even in the presence of strong ISI. Future research includes applying the proposed non-coherent techniques to non-UWB systems, non-binary PPM systems, and other non-coherent demodulation schemes that use no reference (e.g. on-off keying) or use noisy reference (dirty template). The AVC approach can be applied to the multipulse PPM system proposed in [31] or the multi-level PPM systems proposed in [25, 27, 28] for nanocommunications in Terahertz technology to achieve better BER performance. Moreover, using the proposed AVC-PPM instead of the ED-PPM is promising to increase the energy efficiency and positioning accuracy of the UWB ranging and localization systems.

Hassan Khani

received the B.Sc. degree from the Ferdowsi University of Mashhad, Mashhad, Iran, in 2002, and the M.Sc. and Ph.D. degrees from the Tarbiat Modares University, Tehran, Iran, in 2005 and 2010, respectively, all in electrical engineering. From 2003 to 2005, he was with the Signal Processing Research Group at the Iran Telecommunication Research Center, Tehran, Iran. From 2006 to 2007, he was with the Advanced Communications Research Institute, Sharif University of Technology, Tehran, Iran. From 2008 to 2010, he was with the Electrical and Computer Engineering (ECE) Department, Quchan University of Technology (QUT), Quchan, Iran. From 2011 to 2013, he was a Postdoctoral Fellow/Research Scientist at the Department of Industrial Technology, University of Northern Iowa, Cedar Falls, IA, USA. He is currently an Assistant Professor with the ECE Department of QUT. His current research interests include modulation and coding techniques, signal processing, and ultra-wideband communications.

Author Contributions

Not applicable.

Funding

This work is supported by the Quchan University of Technology under Grant No. 95/5012.

Data Availability

Data and materials of this article are available upon request (hassan.khani@gmail.com)

Code availability

Matlab codes of this article are available upon request (hassan.khani@gmail.com).

Declarations

Conflict of interest

The author has no conflict of interest/competing interests to declare that are relevant to the content of this article.

Ethical approval

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Consent to participate

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Consent for publication

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