In general, an OFDM signal can be defined as follows: x t = ∑ k = 1 N A k ∙ e j 2 π f k t + φ k t , (1) where N represents the number of OFDM subcarriers and A k , f k , φ k are the amplitude, frequency, and phase of the k t h subcarrier. For an OFDM signal with BPSK subcarriers, the phase can be defined by: φ k = ∑ i = 1 M k b i , k ∙ u t − i − 1 T s − u t − i T s ∙ π , (2) where M k represents the total number of symbols transmitted through the k t h subcarrier, T s is the symbol period, u t stands for the Heaviside step function, and b i , k = 0,1 depending on the transmitted bit.

The phase diagram is a way of analyzing nonlinear data based on a new representation domain of an analyzed time series. In this new representation space, different characteristics can be highlighted, bringing new information about the analyzed time series (Marwan et al., 2007). The transition to this new representation space of the time series is performed based on the phase space vectors, according to Eq. 3: v i → = ∑ k = 1 m x i + k − 1 d ∙ e k → , i = 1,2 , … , M . (3) In the construction of the phase space vectors, we can observe the introduction of two new parameters: the time delay d between the samples of a signal’s vector and encapsulation dimension m —the number of samples in the vectors; that is, the vectors’ projections on the axis are the samples of the signal. Most often, these two parameters can be determined through the mutual information method and the false nearest neighbor method, respectively (Digulescu et al., 2016).

One of the major advantages of this analysis method is that each analyzed signal has a specific representation in the phase diagram. Figure 1 shows the phase diagram representation of a BPSK signal, defined as in Eq. 4: x B P S K t = A 0 ∙ e j 2 π f 0 t + φ t , (4) where A 0 , f 0 , φ 0 are the amplitude, frequency, and phase of the signal.

Figure 1 shows that the phase diagram of a BPSK signal consists of one ellipse with two distinct branches. These branches correspond to the moments when the transition from one phase to another happens in the analyzed signal.

In this paper, we use the angle extracted from the phase diagram for the OFDM signal detection and its subcarrier characterizations. The extraction of this angle is based on the analysis of the points that build the trajectory of the series analyzed in the phase diagram (Scripcaru et al., 2020).

Inspired by Figure 2, we consider any three points ( A , B , and C ) of the representation. Thus, we aim to determine the angle between the plane determined by these three points and the x O z plane based on the following scalar product: α = N A B C → ∙ N x O z → N A B C → ∙ N x O z → , (5) where N A B C → = A B → × A C → is the normal to the plane formed by the three points and N x O z → = 0,1,0 . The lower image in Figure 2 shows the variation of the α angle for the BPSK analyzed signal presented in the upper image in Figure 2. As can be seen, the variation of the extracted angle highlights the phase changes in the BPSK signal.

We consider the case of an OFDM signal with a guard interval of 20% of the total length of the signal shown in Figure 3 composed of two orthogonal BPSK subcarriers with the following frequencies: f 1 = 10   k H z , f 2 = 11   k H z , and the sampling frequency of the signal f e = 55   k H z .

Figure 4 shows the spectrum and spectrogram of the OFDM signal from Figure 3. The individual characteristics of the subcarriers are not visible in these representation domains, and we cannot find any information on the type of modulation of the subcarriers.

To decompose the OFDM signal band, we use the set of orthogonal bandpass Chebyshev type II filters, as shown in Figure 5. The pass band of the filter is ∆ f i , k = [ f min + k − 1 B 2 i − 1 , f min + k B 2 i − 1 ] , where B = f max − f min , i is the decomposition level, and k is the number of sub-band frequencies.

Three situations can occur in the filtered frequency domain: the presence of a single BPSK subcarrier, the presence of more than one BPSK carrier, or the absence of any subcarrier.

To illustrate these examples, we have used the following filtering bands: ∆ f 2,1 = 10.5   k H z , 11.5   k H z , ∆ f 1,1 = 9.5   k H z , 11.5   k H z , and ∆ f 2,0 = 8.5   k H z , 9.5   k H z . The signals obtained after applying the filters are presented in Figure 6, where each signal (from left to right) represents the signal obtained after applying the bandpass filter at the i t h level in the j t h band.

On the filtered signals, we extract the angle from the phase diagram. Its variation can be observed in Figure 7.

As it can be seen, depending on the case, the variation of the angle α may highlight a different number of peaks. To be able to decide about the existence of the BPSK subcarrier, we quantify these peaks using the information provided by the angle variation histogram, as shown in Figure 8. Therefore, we define the percentage of spikes ( P o S ) as in Eq. 6 to see if the number of samples, which are in the same bins as the spike’s samples, represent less than 3 % of the total number of samples (Scripcaru et al., 2021). If P o S is higher than 3 % , we conclude that the filtered signal does not represent any BPSK subcarrier.

The definition of the P o S is as follows: P o S = ∑ i = 1 N s p i k e − b i n s s i z e b i n i N ∙ 100 % , (6) where N is the total number of samples and N s p i k e − b i n s is the number of bins that contain at least one spike.

The P o S values for the three filtered signals analyzed are: P o S 11 = 0.75 % , P o S m = 1.12 % , and P o S 0 = 88.39 % . P o S 11 corresponds to a single subcarrier present, P o S m corresponds to multiple subcarriers present, and P o S 0 corresponds to the no subcarriers case.

Depending on the studied case, we have several possible interpretations of the histogram results. For a single carrier, the values of the samples are concentrated in a main bin or near a certain value. In the case of two subcarriers, the values are distributed in multiple bins, and in the case of no carrier, false peaks are introduced, and their distribution is similar to the previous case. Consequently, the algorithm must be completed with another condition.

It is necessary to check if the number of samples from the two most-populated bins represents more than 85 % of the total number of samples. The two most-populated bins contain values close to the average value of the angle. We introduce the percentage of averages ( P o A ) parameter defined by Eq. 7. P o A = s i z e b i n 1 + b i n 2 N ∙ 100 % . (7)

If P o A is smaller than 85 % , then we decide that the filtered signal is not a BPSK subcarrier (Scripcaru et al., 2021). The P o A values for the three filtered signals analyzed are the following: P o A 11 = 89.89 % , P o A m = 54.68 % , and P o A 0 = 86.89 % .

Therefore, the steps of the presented algorithm for the OFDM signal carrier detection and characterization are proposed in the diagram form shown in Figure 9.

Next, we aim to verify the efficiency of the proposed algorithm in the presence of noise. In addition, we have considered the cases with four BPSK subcarriers and eight BPSK subcarriers. The data are analyzed by reporting at two levels of the signal-to-noise ratio (SNR): S N R = 10   d B and S N R = 5   d B . The obtained results are shown in Table 1, Figure 10.

We notice that the proposed algorithm successfully detects a single subcarrier of the multi-carrier signal for the two SNR values of the additive white Gaussian noise. Thus, the noise presence in the analyzed signal still allows the detection of the OFDM signal.

In this case, we start from the characteristics of the IEEE 802.11a standard for WLAN [Wireless Local Access Network—(IEEE Std 802.11a-1999)] having a bandwidth B = 20   M H z and the subcarrier spacing ∆ f = 312.5   k H z , modulation BPSK, and four subcarriers (Figure 11).

Hereinafter, we present the evolution of the algorithm in the three possible situations discussed previously (one carrier in the filtered signal, more than one carrier in the filtered signal, or no carrier in the filtered signal). Figure 12 highlights the corresponding filtered signal for a single carrier present case.

The corresponding angle variation and histogram of this signal with a single subcarrier are displayed in Figure 12. The results obtained for the two parameters required for detection are: P o S 1 = 2.56 % and P o A 1 = 95.27 % for the bandwidth ∆ W = 2.4106 , 2.4109   M H z .

The results obtained for the case in which we have two subcarriers present are shown in Figure 13. The results obtained for the two parameters required for detection are: P o S m = 15.42 % and P o A m = 46.28 % for the bandwidth ∆ W = 2.4106 , 2.4112   M H z .

The results obtained for the case in which no subcarrier is present are shown in Figure 14. The results obtained for the two parameters required for detection are: P o S 0 = 28.25 % and P o A 0 = 43.69 % for the bandwidth ∆ W = 2.4102 , 2.4106 M H z .

Based on these results, we can observe that, in the case of a single carrier present, the proposed algorithm provides the results in the imposed limits; see Figure 9. Furthermore, in the cases in which we do not have any subcarrier present or multiple carriers are present, the proposed algorithm corresponds to the chosen restrictions.