Efficient phase estimation for the classification of digitally phase modulated signals using the cross-WVD: a performance evaluation and comparison with the S-transform

1.1. Phase estimation and signal demodulation

Several phase estimation methods are proposed for PSK signal demodulation, interference cancellation, coherent communication over time-varying channels, and direction of arrival estimation [5–12]. Such phase estimation methods can be classified as coherent and non-coherent detections [13]. The coherent detector is often referred to as a maximum likelihood detector [13]. The term non-coherent refers to a detection scheme where the reference signal is not necessary to be in phase with the received signal. One of the earliest contributions for the phase estimation of binary phase shift keying (BPSK) signal is an optimum phase estimator which derives a reference signal from the received data itself using Costas loop [5]. In [6], an open loop phase estimation method for burst transmission is proposed. The phase-locked loop (PLL) method used in conventional time-division multiple access system is inefficient due to the very long acquisition time. This problem is resolved using the new method proposed in this article which yields an identical performance with the PLL method. However, the frequency uncertainty problem degrades the performance of the estimator. In order to overcome this degradation, an improved algorithm which includes the frequency and phase offset is proposed in [7, 8]. By estimating the frequency and phase offset, the performance degradation caused by the frequency offset in [6] is eliminated. The work reported in [9–11] proposed a carrier phase estimator for orthogonal frequency division multiple access systems based on the expectation-maximization algorithm to overcome the computational burden of the likelihood function. This method is actually equivalent to the maximum likelihood phase estimation using an iterative method without any prior knowledge of the phase. Two practical M-PSK phase detector structures for carrier synchronization PLLs were reported in [12]. These two new non-data-aided phase detector structures are known as the self-normalizing modification of the M th-order nonlinearity detector and the adaptive gain detector [12]. Both detectors show improvement in phase error variance due to automatic gain control circuit imperfections.

1.2. Phase estimation and signal classification

All the above-mentioned methods aimed to develop an optimal phase estimator solely for signal demodulation without estimation of instantaneous parameters of the signals. The Costas loop and PLL are crucial for carrier recovery and synchronization in the demodulation of the class of PSK signals [5–8]. However, our applications focused on the analysis and classification of signals for spectrum monitoring. The main objective of such a system [14] is to determine the signal parameters such as the carrier frequency, signal power, modulation type, modulation parameters, symbol rate, and data format which are then used as input to a classifier network. This system is used by the military for intelligence gathering [15] and by the regulatory bodies [16] for verifying conformance to spectrum allocation. Recently, similar requirements were identified for spectrum sensing in cognitive radio [17] to determine channel occupancy and dynamically allocate channels to the various users. Spectrum monitoring systems also use data demodulation [14], but with modems tailored for the specific modulation type and data format.

Since PSK signals are time-varying in phase, time-frequency analysis [[1]8, p. 9] can be used to estimate the signal's instantaneous parameters. The development of signal dependent kernels for time-frequency distribution (TFD) applicable to the class of ASK and FSK signal was proposed in [19]. Further enhancement in [20] improved the time-frequency representation (TFR) by estimating the kernel parameters using the localized lag autocorrelation (LLAC) function. Recent study has proven that the quadratic TFD [21, 22] is capable to analyze and classify the class of ASK and FSK signals at very low signal-to-noise ratio (SNR) conditions (-2 dB). However, the loss of the phase information in the bilinear product computation makes it impractical to completely represent the PSK class of signals. Since PSK signals are characterized by the phase, cross time-frequency distributions (XTFD) based method is proposed as it is capable of representing the signal phase information [23]. Just like the quadratic TFD which suffers from the effect of cross terms, there are unwanted terms known as "duplicated terms"^a which are present in the XTFD. Preliminary work on the XTFD shows that a fixed window is insufficient to generate an accurate IIB-phase estimation [23], thus justifying the need for an adaptive window.

This article presents a time-frequency analysis solution to the optimum phase estimation of PSK class of signals and then evaluates its performance. Signals tested are BPSK, QPSK, and 8PSK signals. The first method is based on the localized adaptive windowed cross Wigner-Ville distribution (AW-XWVD). In this method, the adaptation of the window width is based on the LLAC function of the signals of interest. For comparison, a second method is selected that is based on the S-transform [24]. It is an invertible time-frequency spectral localization technique that combines elements of the Wavelet transform (WT) and the short-time Fourier transform (STFT). This S-transform is selected for comparison as it has the property of preserving the phase of a signal as well as retaining other key characteristics such as energy localization and instantaneous frequency [24].

This correspondence is organized as follows. Section 2 first describes the signal models used in this article and introduces the general representations of the quadratic TFDs, XTFD, and S-transform. Section 3 presents the general equations for cross bilinear product in time-lag domain for both auto-terms and duplicated terms together with the LLAC algorithm for estimating the adaptive window for the PSK class signals. Next, we present the method for IIB-phase using the peak of the AW-XWVD and S-transform. The Cramer-Rao lower bound (CRLB) which is used for bench marking purposes is discussed in the following subsection. Section 4 presents the discrete time implementation of both method and the performance comparison of the AW-XWVD with the S-transform in the presence of noise. The criteria of comparison are based on the TFR, constellation diagram, main-lobe width (MLW) and the phase estimate variance. Then, a comparison in terms of the computational complexity and symbol error rate is given. Conclusions are given in the following section. Throughout this article, we use the following terminology: TFDs represent the mathematical formulations for distributing the signal energy in both time and frequency; the actual representations obtained are called TFRs.

2.1. Signal model

where φ(t) is the IIB-phase which is very crucial in defining digitally phase modulated signals as it contains information of the transmitted data. This article evaluates the comparative performance of the AW-XWVD as an estimator of the IIB-phase for BPSK, QPSK, and 8PSK signals and then compares the results with the S-transform as both methods claimed to provide accurate phase representation. Note that this study does not include the class of quadrature amplitude modulation signals (QAM). Even though this signal has IIB-phase, its time-varying amplitude characteristic is not suitable for the adaptation algorithm described in this article (see Section 3.1.2). The algorithm is developed based on the assumption of constant amplitude signal such as the class of PSK signals.

Figure 1 shows the time representations of the BPSK, QPSK, and 8PSK signals defined in Equation (4). The signal parameters are given in Table 1 and the sampling frequency is assumed to be 1 Hz. The analysis methods proposed in this article is applicable to communication applications in all frequency bands as long as they meet the Nyquist sampling theorem. Due to the frequency dependency of the S-transform, the analysis signals consist of both high- and low-frequency components so as to compare the performance of the AW-XWVD and S-transform for phase estimation.

where z(t) is the noiseless PSK signal and v(t) is the complex-valued additive white Gaussian noise. The noise has independent and identically distributed real and imaginary parts with total variance ${\sigma }_v^{\text{2}}$ and zero mean [[18], p. 437].

2.2. TFDs, cross TFDs, and S-transform

The quadratic TFD is a useful technique to analyze time-varying signals, but the resulting TFR does not represent phase directly. Due to the need to estimate IIB-phase in PSK signals, the XTFD and the S-transform are introduced for this purpose as both can represent phase in the time-frequency domain.

3.1. AW-XWVD

The S-transform can be applied directly to the class of PSK signals to obtain the TFR without any modification in the algorithm. However, this is not the case with the XTFD where interference due to duplicated terms is introduced in the TFR [23]. Previous study defined methods to determine optimum windows for TFDs [38, 39] that can reduce cross terms. A window matching algorithm [39] is used to determine the optimum window for a TFD at all time instant. The algorithm iteratively evaluates the localized energy distribution to minimize the error between successive window estimates. The concept of time-frequency coherence is introduced in [38] where the XWVD and WVD for each signal components are used in its computation. The required window function is estimated based on the autoregressive moving average modeling and Karhunen Loeve expansion. In this PSK communication application, the cross bilinear product has a certain pattern that can be utilized in computing the optimum window. Therefore, the adaptive window is designed based on the characteristics of the cross bilinear product. The resulting distribution, the AW-XWVD, can generate an accurate TFR and the subsequent IIB-phase estimate.

3.1.1. The cross bilinear product

The cross bilinear product consists of auto-terms and duplicated terms. The duplicated terms carry the same information as the auto-terms but shifted in time and lag domain; therefore, it can cause interference to the auto-terms [23]. In order to obtain an accurate XTFR, the auto-terms must be preserved and the duplicated terms must be removed or attenuated. The auto-terms and duplicated terms for any PSK class signal can be expressed as

$K_{zr,\text{auto}}=\sum _{k=1}{\left|A\right|}^2\text{exp}\left(j\left(2\pi f_k+{\varphi }_k\right)\right)K_{\mathrm{\Pi }}\left(t-\left(k-1\right)T_b,\tau \right)$

(16)

$K_{zr,\text{duplicated}}=\sum _{k=1,k\ne l}^N\sum _{l=1}^N{\left|A\right|}^2\text{exp}\left(j\left(2\pi f_k\tau \right)+{\varphi }_k\right)K_{\mathrm{\Pi }}\left(t-\frac{\left(k+l-2\right)T_b}{2},\tau \left(l-k\right)T_b\right)$

(17)

where K_∏(t, τ) is the instantaneous autocorrelation function of the box function given as

$K_{\mathrm{\Pi }}\left(t,\tau \right)=\mathrm{\Pi }\left(t+\frac{\tau }{2}\right)\mathrm{\Pi }\left(t-\frac{\tau }{2}\right)$

(18)

The proofs for Equations (16) and (17) are given in Appendix 1.

In practical digital communication applications, the amplitude of the signal might not be ideally constant due to channel impairments such as multipath fading, attenuation by the propagation channel and any kind of amplification performed by the circuits at both the transmitter and receiver [13]. Therefore, the variable A is retained throughout the derivation of the cross bilinear product. Other than that, signals that combine amplitude and phase modulations such as QAM can also be used provided a suitable adaptation algorithm for the XTFD is designed. The variation in the amplitude, A, caused by the transmitted binary data will correspond to the variation in the energy represented in the XTFR.

Figure 2 shows the graphical representation of the above cross bilinear product. All the auto-terms lie along the τ = 0 axis, whereas the duplicated terms are shifted in both time and lag. Therefore, the terms labeled as K_{zr 1,2}, K_{zr 1,3}, and K_{zr 1,4}are the duplicated terms for the auto-term, K_zr1,1, which are centered at τ = 0 axis. The same label applies to the rest of the auto-terms and duplicated terms. The following examples illustrate the problem caused by the duplicated terms to the estimation of IIB-phase. There is no interference observed in the IIB-phase estimate if there are only auto-terms present. For instance, at time t = T _b /2 the cross bilinear product evaluated is given as

${\left.K_{zr}\left(t,\tau \right)\right|}_{t=T_b/2}=A^2\text{exp}\left(j\left(2\pi f_0\tau +{\phi }_1\right)\right)\mathrm{\Pi }\left(\tau +T_b\right)$

(19)

Only the auto-terms with the IIB-phase of φ₁ exist. When both the auto-terms and duplicated terms are present, there will be more than one phase term. This is observed at t = 3T _b /2 where the cross bilinear product is represented as

$\begin{array}{cc}\hfill {\left.K_{zr}\left(t,\tau \right)\right|}_{t=3T_b/2}=& A^2[\text{exp}\left(j\left(2\pi f_0\tau +{\varphi }_3\right)\right)\mathrm{\Pi }\left(\tau +3T_b\right)+\text{exp}\left(j\left(2\pi f_0\tau +{\varphi }_2\right)\right)\mathrm{\Pi }\left(\tau +T_b\right)\hfill \\ +\text{exp}\left(j\left(2\pi f_0\tau +{\varphi }_1\right)\right)\mathrm{\Pi }\left(\tau -T_b\right)]\hfill \end{array}$

(20)

The interaction of the auto-terms and duplicated terms can be visualized as the addition of multiple vector components which result in a new vector component with different magnitude and phase. Instead of IIB-phase of φ₂ which is caused by the auto-terms, the resulting IIB-phase consists of the interaction between all the phase terms φ₁ and φ₃ caused by the duplicated terms.

Since all auto-terms lie along the τ = 0 axis, a fixed width lag window was used in [23] to preserve the auto-terms and partially remove the duplicated terms that cause distortion in the IIB-phase represented on the XTFR. However, success is limited because the duplicated terms are not completely removed resulting in a distorted IIB-phase estimate. To resolve this problem, the fixed lag window w (τ) in Equation (11) is replaced with a time-dependent window function w (t, τ) and the resulting new TFD, known as the AW-XWVD, is given as

${\rho }_{zr,\text{AWXWVD}}\left(t,f\right)=\underset{-\infty }{\overset{\infty }{\int }}{K}_{zr}\left(t,\tau \right)w\left(t,\tau \right)\text{exp}\left(-j2\pi f\tau \right)d\tau$

(21)

The adjustment of this time-dependent window width is based on the computation of the LLAC function at every time instant to separate the auto terms and duplicated terms. This is equivalent to use a separable kernel to reduce all cross terms as shown in [21, 22]. An analysis window centered at τ = 0 is used as a reference to perform the similarity test using the LLAC function. This similarity test detects the variation of the signal in the lag direction at every time instant and estimates the window width. The time-dependent window function can be implemented using one of the common windows used in digital filter design and spectrum estimation. In this application, a rectangular window is used and it can be defined as

$w\left(t,\tau \right)=\left\{\begin{array}{c}1\begin{array}{}\end{array}-{\tau }_g\left(t\right)\le t\le {\tau }_g\left(t\right)\hfill \\ 0\begin{array}{}\end{array}\text{elsewhere}\hfill \end{array}\right.$

(22)

where τ _g (t) is the time-dependent window width defined within 0 ≤ t ≤ T, and T is the signal duration. Since the cross bilinear product is asymmetric, the window width in the positive lag and negative lag must be estimated accordingly. The desired τ _g (t) in the positive lag (or in the negative lag direction) is selected if

${\tau }_g\left(t\right)=\underset{\varsigma }{\text{min}}\left(\left|R_{KK}\left(t,\varsigma \right)\right|\right)\begin{array}{c}\hfill 0<\varsigma \le T,\hfill \\ \hfill -T\ge \varsigma >0,\hfill \end{array}$

(23)

where ς is the time instant in lag and |R _KK (t, ς)| is the amplitude of the LLAC which will be discussed in the following section. Note that the rectangular window was used for simplicity as we observed that the proposed methodology performance is not affected significantly by the choice of the window shape.

3.1.2. Adaptation algorithm

The LLAC [20] of the kernel, K, is a function of time and lag and it can be defined as

$R_{KK}\left(t,\varsigma \right)=\underset{-T}{\overset{T}{\int }}{w}_a{\left(\tau \right)}^2{K}_{zr}\left(t,\tau \right)K_{zr}\left(t,\tau -\varsigma \right)d\tau$

(24)

where w _a (τ) is the analysis window, τ is the lag instant, and ς is the lag running variable. The possible range for the normalized LLAC amplitude is

$0\le \left|R_{KK}\left(t,\varsigma \right)\right|\le 1$

(25)

A higher value of the amplitude of the LLAC function implies that the similarity is high and vice versa. The miscorrelation in the signal is indicated by a drastic drop in the amplitude of the LLAC function. The LLAC function will give a value approaching unity at lag instant, ς = 0.

The analysis window is a parameter of the LLAC. Its selection is important to ensure that the LLAC can detect the variation along the lag axis based on the condition specified in Equation (23) as to estimate the time-dependent window width. The analysis window is defined as

$w_a\left(\tau \right)=\frac{1}{{\tau }_a}0<{\tau }_a<<T$

(26)

where τ _a is the analysis window width. In this article, the analysis window width is chosen experimentally as τ _a = 10 s based on the sampling frequency of 1 Hz. The LLAC is applied to the signal x(t) and evaluated for the normalized frequencies of 1/32, 1/16, 1/8, and1/4 Hz. In this evaluation, the signal is defined as follows (with similar characteristic to the cross bilinear product in lag)

$\begin{array}{cc}\hfill x\left(t\right)& =\text{exp}\left(j2\pi f_{1}t\right)0\le t\le T_b\hfill \\ =\text{exp}\left(j2\pi f_1\left(t-T_b\right)\right)\text{exp}\left(j\pi \right)T_b\le t\le 2T_b\hfill \end{array}$

(27)

Table 2 shows the minimum values of the LLAC evaluated in time. An analysis window of τ _a = 10 s is sufficient to determine the time-dependent window width for frequencies ranged from 1/32 to 1/4 Hz. Thus, the analysis width is valid for the test signals as specified in Section 2. In the case of PSK signals, two consecutives symbols may be different from each other depending on the transmitted data. By applying the LLAC on the cross bilinear product, the resulting adaptive window resembles the shape of a parallelogram as shown in Figure 3.

Number	Signal frequency (Hz)	Min |R KK (t, ς)|
1	1/32	0.567
2	1/16	0.251
3	1/8	0.121
4	1/4	0.089

3.2. IIB-phase estimation from the peak of TFDs

By extending the approach used for IF estimation from the peak of WVD presented in [26], the IIB-phase is estimated frrm the peak of the AW-XWVD and S-transform as outlined in the following sections.

3.3. Comparison to CRLB

This section compares the performance of both AW-XWVD and S-transform as a phase estimator with the CRLB which is often used as a benchmark [41], as it gives the theoretical lower limit to the variance of any unbiased parameter estimator [42]. The CRLB derived in [43, 44] uses a likelihood function on a known signal in the presence of additive white noise for the digitally phase modulated signal.

where N is the total number of samples, ${\widehat{\varphi }}_n$ is the estimated phase at every time sample n, and $\bar{\varphi }$ is the actual IIB-phase.

3.4. PSK signal detection algorithm

4.1. Discrete-time formulation and implementation

The discrete time representation of the S-transform is similar to the spectrogram. However, there is a tradeoff between the time and frequency resolution for the S-transform as the window width is frequency dependent.

4.2. Results

Figures 4 and 5 show the TFR, TFR slice, IIB-phase, instantaneous energy, and constellation plots for QPSK2 signals at SNR of 10 dB using the AW-XWVD and S-transform, respectively. The two TFRs show at which frequency the signal exists, but for the S-transform there are distortions in the TFR at every symbol transition. The high contrast region in the TFR of the S-transform indicates that there are low-density components other than the signal component. However, this is not present in the TFR of the AW-XWVD. The TFR slice is normalized to the peak value of the TFR and observed in frequency for time n = 100 samples. From the TFR slice, it is shown that the AW-XWVD gives better frequency concentration compared to the S-Transform. This is because the MLW of the TFR slice for the S-transform appears to be much wider than AW-XWVD. As shown in Equation (12), the S-transform's window width is frequency dependent where the window is wider for low-frequency signal and narrower for high-frequency signal. This implies that the S-transform at higher frequency gives worse frequency resolution and wider MLW. Results confirming this statement are presented in Table 3. Besides the MLW of the TFR slice, there is also a difference in the average side lobe level. The side lobe level is higher for the S-transform at about 0.18, while this level is lower at 0.05 for the AW-XWVD. This explains the appearance of the high contrast region on the TFR of the S-transform.

The IIB-phase plot shows that the AW-XWVD gives better accuracy for the IIB-phase estimate. For the S-transform, distortion is observed in the IIB-phase estimate at the phase transition regions which is absent in the AW-XWVD. The sliding window in the S-transform causes distortion in the IIB-phase at the symbol transition region due to the interaction between adjacent symbols. Since digitally phase modulated signals have constant amplitude, their instantaneous energy should also be constant at all times. However, due to noise, the amplitude of the signal appears to vary. This is reflected as variation in the magnitude of the instantaneous energy for AW-XWVD and S-transform. A significant drop is observed in the instantaneous energy for the S-transform at every symbol transition. Similar to the phase, this drop is caused by the interactions between the adjacent symbols within the sliding window. Since the AW-XWVD produces accurate instantaneous energy and IIB-phase estimates, the constellation diagram generated shows almost no variation from the original points and is better compared to the S-transform. Table 3 shows the MLW estimated at SNR of 6and 10 dB using both methods. In general, the SNR has no significant effect in the MLW obtained for both methods. However, the effect of signal frequency is more significant for the S-transform compared to the AW-XWVD. For instance, the MLW for both BPSK1 and BPSK2 with the AW-XWVD based estimate is the same at 0.012 Hz. However, for the S-transform the MLW is larger for BPSK2 than BPSK1 with a difference of 0.028 Hz. The scaling of the Gaussian window results in a broader MLW for higher-frequency signal. The signal modulation level has no significant effect on the MLW. This is shown by the MLW measured for BPSK, QPSK, and 8PSK signals where there are only minor differences. These results imply that the AW-XWVD gives better IIB-phase estimation results compared to the S-transform as the performance of an estimator is associated with the MLW [[46], p. 50].

4.3. Variance comparison with the CRLB

In order to evaluate the performance of the AW-XWVD method, we compare it with the S-transform. Both methods are benchmarked with the CRLB for phase estimate. It is assumed that there is perfect synchronization between the received and reference signals. We consider the signal is corrupted by a zero mean additive white Gaussian noise channel with variance σ². In simulations, the received signal is generated by adding the noiseless signal, z(t), and the additive white Gaussian noise, v(t), as given in Equation (6) where the SNR is varied at 1 dB step from 0 to 10 dB. Monte Carlo simulations based on 1,000 realizations of the predefined signals are conducted for each value of SNR. The estimated IIB-phase is obtained from the peak of the XTFD as described in Section 3.2. Assuming the actual IIB-phase of the signal is known, the MSE is computed using Equation (34). Figures 6, 7, and 8 show the results of the estimated IIB-phase variance for BPSK, QPSK, and 8PSK signals, respectively. In general, for both BPSK and QPSK signals, the variance of IIB-phase estimate using the AW-XWVD lies close to the CRLB. The AW-XWVD estimate meets the CRLB at SNR ≥ 5 dB for both BPSK1 and BPSK2 signals. Figure 7 shows that the cutoff point for both QPSK1 and QPSK2 signals are the same as the BPSK signals, at SNR of 5 dB. However, the variances in the IIB-phase estimate for QPSK signals are higher compared to the BPSK signals at the same SNR. A higher cutoff point is recorded for the 8PSK estimates where both 8PSK1 and 8PSK2 signals meet the CRLB at SNR ≥ 7 dB. In general, the AW-XWVD outperformed the S-transform as a phase estimator and the S-transform estimates never meet the CRLB for all signals even at SNR of 10 dB. As expected, the performance of the S-transform deteriorates greatly for higher frequency signals due to broader MLW. This is in contrast with the AW-XWVD where it maintains reasonably insignificant changes in the variance of IIB-phase estimates for all range of frequency. Thus, the AW-XWVD is more robust to noise for phase estimation compared to the S-transform. The result obtained in this article is in line with [30] where the XWVD is shown to be more robust to noise compared to the WVD.

4.4. Symbol error rate performance

where P_E(M) is the symbol error rate, the function Q(x) is the complementary error function, E_s is the energy per symbol, N₀ is the noise power, and M is the size of symbol set. Detection of the PSK signals is done based on the IIB-phase estimate and the sets of rules defined in Section 3.4. All the defined test signals of 10,000 symbols are evaluated under AWGN channel. The results presented in Figures 9, 10, and 11 show the symbol error rate performance as a function of SNR for BPSK, QPSK, and 8PSK signals. BPSK signal required a SNR of about 5 dB to achieve a symbol error rate of 10^-3 for the AW-XWVD method. Using the same method, a higher SNR is observed for QPSK signal, approximately 8 dB. To achieve the same performance, the conventional detector needs an SNR of 7 and 8 dB for BPSK and QPSK signals, respectively. For 8PSK signal, it is shown that to achieve a symbol error rate of 10^-3, SNR of 10 dB is required for AW-XWVD. As for the conventional detector, it gives the same performance at SNR of 11 dB. From the symbol error rate plot, for all the test signals, it is observed that the advantage of the AW-XWVD over the conventional detector is at the low SNR range where it gives lower error rate. In general, the symbol error rate for the AW-XWVD is much lower compared to the S-transform. The S-transform could not provide symbol error rate ≤ 10^-3 even at SNR of 12 dB for all signals. Thus, it is impractical to use it as a detector for the class of PSK signals.

4.5. Computation complexity

Therefore, the total of multiplication required to compute the AW-XWVD is N(N _τ + N _A N _τ + N _w + 0.5N _τ log₂ N _τ ).

Thus, the total number of multiplication required to implement the S-transform is N(1 + 0.5N _τ log₂ N _τ ).

Therefore, the total number of multiplications required to implement the conventional detector is 4N.

In terms of the number of multiplications, the AW-XWVD requires approximately 4 times more computations compared to the S-transform and 2,000 times more for the conventional detector. Although there is a significant additional number of a computation for the AW-XWVD, recent advances in digital electronics as well as decimation procedures can take care of them; in addition, the performance in terms of the IIB-phase estimates enables more efficient signal parameters estimation in the proposed area of applications. These parameters can be used to classify a signal from a set of reference parameters. If necessary, we can use the IIB-phase estimate to detect PSK signals at low SNR conditions where the conventional detector failed. For higher SNR conditions, it is not necessary to use a technique which is computationally intensive when the symbol error rate is low. Thus, we can setup the conventional detector using the parameters estimated from the IIB-phase to detect PSK signal. So, we conclude that, with the enhancement of current computer processing combined with appropriate decimation procedures, the real-time implementation of AW-XWVD is feasible with the use of multiple processors or parallel processing [48] and the design proposed in [49].