Existing sensors for drone surveillance include acoustic sensors, optical sensors, and radar sensors. Acoustic sensors utilize microphones to capture the acoustic signals from the drone and then use acoustic processing to detect the drone [25,26]. However, acoustic sensors have a small operating range, some with a surveillance distance of only 200 m, and they cannot work in bad weather conditions such as rain or snow, and are easily affected by noise in the environment. Optical sensors utilize cameras to track drones and process image and video data for drone detection [27,28]. Optical sensors are limited by the camera’s viewing angle and range of capture, and nighttime, extreme weather conditions, and the presence of obstacles can affect the sensor’s ability to capture. Radar is an important device for drone detection because of its adaptability to weather and lighting conditions, its wide coverage, and its ability to monitor in cluttered environments [7].

There have been researchers using classical machine learning methods combined with radar return signals to carry out drone detection tasks, using machine learning algorithms including SVM [8], DecisionTree [29], CNN [28,30,31], and so on. Most of these methods use Short Time Fourier (STFT) to extract the spectral image dataset and machine learning algorithms then detect and classify the image data [7,23], CNNs can achieve optimal performance [9]. However, the method based on image recognition to detect radar return signals from drones suffers from the problem of high dimensionality of image features. The common datasets for existing image recognition techniques have a minimum image dimension of 28 × 28 [32,33], and the spectrogram feature dimensions of the radar return signals from drones will only be higher, for example, the size of the spectrogram in the paper [24] is 512 × 4, the high-latitude features make the amount of input data processing for machine learning models very large, on the other hand, many of these image features are irrelevant to signal detection, but will instead bring interference to model learning [10]. Therefore, there is a need to find low-latitude features that are sufficient to characterize the detection of radar return signals from drones.

In signal processing, raw signals often have very complex characteristics and structures, which may be very difficult to analyze and process directly. Signal feature extraction can convert the original signal data into statistical quantities or other forms of information that can reflect the characteristics of the data, and the methods include time-domain extraction, frequency-domain extraction, and time-frequency domain extraction. Time domain features are used to extract the statistical characteristics of the signal on the time axis, such as the period of the signal, waveform, etc., which can reflect the fluctuation and transformation of the signal in time. Signal frequency domain feature extraction extracts the spectral information of the signal by frequency, including the amplitude and phase of different frequencies. Time-frequency domain feature extraction combines the time and frequency domains to show the characteristics of the signal in the time and frequency domains at the same time, in which the short-time Fourier transform is used more often to generate spectral maps [34]. Different feature extraction methods need to be selected according to different application scenarios.

Compared with the features extracted from frequency and time-frequency domains, time-domain features are easy to extract. In addition, Compared with image recognition, there are several advantages of time-domain statistical features, firstly, it can better capture the essential properties of the signal, discover the salient laws and statistical characteristics of the signal, and improve the accuracy of analysis; secondly, compared with high-latitude image features, it can accurately focus on the subtle characteristics that are more relevant to signal detection, downsize the signal, reduce the amount of data processing of the model, simplify the processing, and improve the analysis efficiency; thirdly, the extraction of signal time domain statistical features can obtain the stable and invariant features of the signal, and has strong robustness to noise and other interference factors. The advantages of time-domain statistical feature extraction, such as enhanced accuracy, improved analysis efficiency, and high robustness, make it have a wide range of application prospects and potentials in the field of signal processing.

Therefore, we consider extracting time-domain statistical features of the signal, which are selected to have a small feature set size, high accuracy, and stronger stability when there is enough information to distinguish the drone from the noise case. The time-domain feature selection process is described in Experiment 4.4, and the time-domain feature extraction process is described in Section 3.3.

We use the hybrid quantum neural networks with signal feature overlay projection (HQNN-SFOP) to detect the simulated radar return signals generated by real drone parameters, and the links include the generation of radar return signals from drones, signal time-domain statistical feature extraction, and the detection by the hybrid quantum neural networks with signal feature overlay projection (Fig. 1). Next, we take the drone Parrot Disco as an example to show the functions of each link. The remaining four drones, DJI Mavic Air 2, DJI Mavic Mini, DJI Matrice 300 RTK, and DJI Phantom 4, are handled similarly.

Parrot Disco is a drone used in various industries such as mapping, exploration, and environmental protection. It has 2 blades, the distance between the root of the blade and the center of rotation is 0.6 cm, the distance between the tip of the blade and the center of rotation is 5 cm, and the rotational frequency of the blade is 116 Hz.

Inputting these parameters of the Parrot Disco into the MM model allows for the generation of radar return signals that simulate the propeller blades of the drone. In addition, the model can add noise to the signal and also generate a noisy signal. Fig. 2 gives an example of the generation of a radar signal reflected from the drone Parrot Disco at an SNR of 10 dB. Fig. 3 gives an example of the noise signal.

The signal time domain statistical feature extraction is performed on the radar return signal from drones and noise signals. Comparing Figs. 2 and 3, we can see that the drone signal and noise have different time domain statistical features such as maximum value, minimum value, median, peak-to-peak value, and so on. In terms of other subtle features such as mean, variance, peak, margin factor, impulse factor, and other subtle features that are not directly observable by the naked eye or image recognition, as well as the inline relationship of these subtle features, the drone and noise have even more significant differences. Therefore, these extracted statistical features in the time domain can be used as the object of HQNN-SFOP learning to detect drones and noise. For the time-domain statistical features extracted from the real-valued portion and imaginary portion of the signal, we obtain a total of 16-dimensional features as in Table 1, which are used for the subsequent learning of the HQNN-SFOP model.

The extracted time-domain statistical features are encoded from classical data into quantum states and input into the HQNN-SFOP model, where the signal feature superposition projection amplifies the subtle features of the quantum state signal eigenvectors, and the hybrid quantum neural networks model learns the inline relationships of these features to detect drones and noise.

We explore the potential of drone detection tasks through quantum computing. On the one hand, it expands new application areas for quantum computing and explores its advantages in the field of signal analysis; on the other hand, it also explores feasible directions for the iterative technological evolution of signal intelligence analysis and lays the foundation for the application of future disruptive computing technologies. In the following, we will introduce each link in the process in detail.

To compare the results, we use the previous method [24] to generate drone signal and noise datasets by applying the MM model. The MM model combines the application of electromagnetic wave theory, and by inputting different parameters to it, it can simulate the radar return signals of different aircraft propeller blades, which are defined as follows: (1)Ψ(t)=Arej(2πfct−4πλ(R+Vradt))∑n=0N−1(α+βcos⁡(Ωn))e−jL1+L22γnsinc(L2−L12γn)where α=sin⁡(|θ|+Φp)+sin⁡(|θ|−Φp) β=sign(θ)(sin⁡(|θ|+Φp)−sin⁡(|θ|−Φp)) Ωn=2π(frott+nN) γn=4πλcos⁡(θ)sin⁡(Ωn)

The parameters used in the MM model are explained in Table 2. These parameters are categorized by function into radar characterization parameters, aircraft characterization parameters, and aircraft-radar positional relationship characterization parameters. Configuring the two radar characteristic parameters fc and λ identifies the particular radar that one wishes to simulate. By configuring N, L1, L2, and frot these aircraft characteristic parameters can determine the aircraft blade characteristics and thus specify the aircraft type. Configuring θ, Φp, R, and Vrad then specifies the aircraft position.

The simulated radar emission signal is an X-band signal with an emission frequency fc of 10 GHz and a wavelength λ of 3 cm. Five real commercial drones, DJI Mavic Air 2, DJI Mavic Mini, DJI Matrice 300 RTK, DJI Phantom 4, and Parrot Disco, are simulated here to generate samples, each of which is generated by the parameters of one of the five drones were simulated to generate the samples. The parameters of the five drones are shown in Table 3.

To accommodate uncertainty in real-world environments, a set of randomized drones datasets were generated for comparative analysis using the parameters in Table 4. The number of blades N in Table 4 is any integer between 2 and 5, while L1, L2, and frot are random values within a specific range, which are reasonably filtered to accommodate a variety of new and ever-innovating drones. In addition, the value of L2 is an increase from L1 to ensure that the parameters are consistent with the characteristic that the distance between the tip of the blade and the center of rotation of a real drone is greater than the distance between the root of the blade and the center of rotation.

The parameters θ, Φp, and Vrad are configured in Table 5.

In combination with the real-valued scalar Ar, noise can be introduced in the signal. The noise is derived from the single-pulse signal-to-noise ratio, and the equation is defined as follows: (2)SNR=10log10(Ar2δn2)

The noise δn can be solved for according to the above equation, and the noise added to the signal is obtained by sampling in υ∼N(0, δn).

Our feature extraction method is different from other methods, which use the Fourier transform to obtain spectrograms and then learn image features to detect them by classical machine learning methods. For the radar signal generated by the MM model, we extract the time-domain statistical features with strong relevance to drone detection, including the maximum, minimum, median, mean, variance, peak, peak-to-peak, margin factor, impulse factor, etc., which are used to describe the statistical properties of the signal. HQNN-SFOP is used to learn the inline relationship of the 16-dimensional features to detect the signals. These features are defined and described as follows:

Max: Maximum value of the sampling result of each sampling point of the signal. (3)SC1=max(xi)

Minimum: Minimum value of the sampling result of each sampling point of the signal. (4)SC2=min(xi)

Median: The sampling value in the signal is in the middle of each sampling point. (5)SC3={xn+12n is an odd numberxn2+xn2+12n is an even number

Mean: Expectation, is the average value of each sample of the signal. (6)SC4=1n∑i=0n−1xi

Variance: The mean of the squares of the difference between the sampled values and the average value of the signal, which reflects the degree of dispersion between the data. (7)SC5=1n∑i=0n−1(xi−SC4)2

Peak: Maximum value of absolute value of each sample of signal. (8)SC6=max(|SC1|, |SC2|)

Peak-to-peak: Difference between the maximum and minimum of the signal, describes the transform range of the signal. (9)SC7=SC1−SC2

Margin: The ratio of the maximum value of the signal to the square of the square root of the expectation of the sampling result, which describes the transform margin of the signal. (10)SC8=SC1(1n∑i=0n−1|xi|)2

Pulse: The ratio of the maximum value of the signal to the expected absolute value of the sampling result, which describes the pulse characteristic of the signal. (11)SC9=SC11n∑i=0n−1|xi|

Due to the characteristics of the signal itself, Max, Minimum, Median, Mean, Peak-to-peak, Margin, and Pulse include real and imaginary values, thus forming 16-dimensional features. We analyze the correlation between these features and the labels through the feature selection of Experiment 4.4, and obtained the 16-dimensional feature selection scheme.

Compared with image detection, this method of extracting the statistical features of the signal in the time domain and then detecting them can more accurately obtain the subtle features of the signal, reduce the degree of redundancy of the feature data, and improve the detection ability of the signal processing, as well as effectively reduce the complexity of the HQNN-SFOP model, and improve the efficiency of the training and stability, which is effectively demonstrated in the later Experiment 4.5 and Experiment 4.6.

We use the MM model to generate the dataset, which is defined as follows: (16){S1:ψ(t)+υ(t)S0:υ(t)

S1 represents the case of the existence of a drone, S0 represents the case of no drone only noise, in the formula ψ(t) is the drone without noise, υ(t) is the noise. This is the theory of hypothesis testing in mathematical theory, according to the verdict for the existence and non-existence of the target signal, which is a binary hypothesis testing problem. We generate a sample dataset based on this definition, and the length of each sample is 0.4 s. For this dataset, the machine learning method is used to do binary classification after extracting time domain statistical features.

The number of samples for each training dataset is 900, of which 450 are sample data generated from the parameters for five types of drones and 450 are noise sample data. The number of samples for the validation dataset is 300, with half of the sample data from the parameters for five types and half of the sample data from noise. There are three test datasets so that the standard deviation of the measurement results can be generated, and the number of sample entries for each test dataset is 300, with half each of the five drones parameter sample data and the noise sample data as well. The ratio of training set, validation set, and test set is 6:2:2.

To improve the compatibility of the model, drone sample data were also generated using random parameters, with the same number and proportion of training, validation, and test sets as above, with the difference that the five drones parameter sample data were replaced with random drones sample data.

We generated sample datasets at SNRs of 0, 5, 10, 15, and 20 dB, respectively, which are convenient for subsequent experimental analysis and comparison. In the real task, a negative SNR indicates that the noise is stronger than the signal, and this extreme case generally requires noise reduction processing to strip off the noise to optimize the signal quality before analysis, which is not in our current discussion.

We use Tensorflow to build a neural network framework and Tensorcircuit to simulate quantum circuits. Classical machine learning algorithms such as SVM, LogisticRegression, DecisionTree, GaussianNB and BernoulliNB come from Scikit-learn. All implementations are based on the Python language.

We use 16 qubits to execute the HQNN-SFOP circuits. The training batch_size is 16 and the epoch is 6. We used the binary cross-entropy function to calculate the loss value, and the Adam algorithm with a learning rate of 0.03 was used for optimization.

We use a baseline of metrics commonly used in machine learning, including ACC and F1-score, to provide an all-encompassing assessment of classifier results.

A confusion matrix is shown in Table 6, which contains the true and predicted classifications of the sample dataset with Positive and Negative results in the true and predicted classifications, respectively.

The descriptions of TP, FP, FN, and TN in the confusion matrix are as follows:

TP: represents the case where the true classification value is Positive and the predicted classification value is also Positive.

FP: represents the case where the true classification value is Negative and the predicted classification value is Positive.

FN: represents the case where the true classification value is Positive and the predicted classification value is Negative.

TN: represents the case where the true classification value is Negative and the predicted classification value is Negative.

Based on the confusion matrix above, Accuracy, Recall, Precision, and F1-score can be defined.

ACC: is the proportion of the total sample size that is correctly categorized. (17)ACC=TP+TNTP+FP+TN+FN

In general, the larger that accuracy is, the better. In the case of unbalanced label distribution, especially in extreme cases, such as 99 out of 100 sample data are negatively labeled, then the classifier just needs to give all the predictions negative to get 99% accuracy, and this kind of classifier is not what we want, so other metrics need to be introduced.

Recall: the proportion of results actually categorized as positive samples that are predicted to be categorized as positive samples. (18)Recall=TPTP+FN

Precision: the proportion of results predicted to be categorized as positive samples that are actually categorized as positive samples. (19)Precision=TPTP+FP

Recall and precision represent the ability of the classifier to recognize positive and negative samples, respectively; the higher the recall, the better the ability of the classifier to recognize negative samples, and the higher the precision, the better the ability of the classifier to recognize positive samples. The use of recall or precision alone does not fully represent the performance of the classifier, precision and recall affect each other, although both are high is a desired ideal situation, however, in practice, it is often the case that the precision is high and the recall is low, or the recall is low, but the precision is high, so the F1-score is introduced to take into account the two.

F1-score: the harmonic mean of precision and recall. (20)F1−score=2∗Precision∗RecallPrecision+Recall

The features of the signal sample dataset after feature extraction are not uniformly distributed. To avoid the learning results being biased towards some of the features with obvious eigenvalues, it is necessary to standardize the sample dataset to make the various features of the sample dataset more comparable with each other. We use the classical algorithm of StandardScaler for the process, which standardizes the features by removing the mean and scaling by standard variance to facilitate the subsequent quantum coding.

Researchers’ experiments have shown that choosing a finite feature set that accurately indicates the characteristics of the data outperforms a large-scale feature set, because a larger feature set introduces more noise and may also make the model overly focused on the training data [10]. Therefore, the feature set needs to be selected.

We use the Pearson correlation coefficient method [44] to analyze the strength of linear correlation between time-domain features extracted from the drones dataset and the labels in order to filter the feature set. The Pearson correlation coefficient method is defined as follows:

(21)r=∑(x−x¯)(y−y¯)∑(x−x¯)2∑(y−y¯)2where x is the time domain feature, y is the label, and r is the correlation coefficient between x and y. r > 0 is a positive correlation, r < 0 is a negative correlation, and r = 0 means there is no correlation between the two. Features positively correlated with labels are shown in Fig. 5.

Analyzing the correlation coefficients of these time-domain features in Fig. 5, the correlation coefficients of two features, Margin-real and Pulse-real, are greater than 60%, and the correlation coefficients of four features, Max-real, Minimum-real, Peak, and Peak-to-peak-real, are greater than 40%. We choose time domain features with correlation coefficients greater than 5%. In addition, there are some features such as Max, Margin, and Pulse, whose correlation coefficients are more than 5% for their real values and less than 5% for their imaginary values, and some features such as Median and Mean, whose correlation coefficients are more than 5% for their imaginary values and less than 5% for their real values. To ensure the completeness of the signals, both the real and the imaginary parts of these features are retained, resulting in a 16-dimensional feature.

Finally, the 16-dimensional time-domain features are used for the feature extraction of radar return signals from drones, and the sample dataset generated from them was fed into the hybrid quantum neural networks for computation.

We use the classical machine learning algorithms SVM [45], LogisticRegression [46], DecisionTree [47], GaussianNB [48], BernoulliNB [49], NN [50], and the hybrid quantum machine learning algorithm HQNN-SFOP to perform experiments on the 16-dimensional signal time-domain statistical features extracted from the radar return signal dataset with an SNR of 10 dB. Both the training and test sets are radar return signal datasets generated from five drones parameters. The difference between NN and HQNN-SFOP is that the quantum feature learning module of HQNN-SFOP is replaced with the fully connected layer of the classical neural networks, while the other parts such as the classical optimization module and the training parameters are kept the same.

In addition, we introduce the results of the CNN method mentioned in the paper [24] for comparison, which also uses the same radar return signal dataset, extracts time-frequency domain features on the signals to generate spectrograms, and achieves the target detection by image recognition of the spectrograms with the dimension of 512 × 4 by CNN. The differences in signal extraction methods and input feature dimensions used by the various algorithms are shown in Table 7, and the experimental results are shown in Table 8.

Analyzing the experimental results, it can be seen that the F1-scores of SVM, LogisticRegression, DecisionTree, GaussianNB, BernoulliNB, NN, and HQNN-SFOP for time-domain statistical features are better than the F1-score of the CNN method for spectrogram identification, which is 0.859 ± 0.013, and it indicates the effectiveness of our extracted time-domain statistical features for drone detection using radar return signals. This effectiveness proves that the extracted time-domain features are sufficient to express the signal characteristics, while the precise selection of the features avoids the noise interference that may result from too many features.

On the other hand, compared to other classical methods, HQNN-SFOP has the highest ACC value and F1-score of 0.941 ± 0.007 and 0.938 ± 0.007, respectively, which are also higher than those of 0.93 ± 0.012 and 0.925 ± 0.015 for NN, which are improved by 0.011 and 0.013, respectively. This validates that the introduction of quantum circuits in the HQNN-SFOP method improves the neural network’s ability to acquire feature inline relations and improves the accuracy of drone detection using radar return signals.

We use 5-fold cross-validation to calculate the ACC value and F1-score of each model. The SNR of the sample data set is 10 dB, the extracted signal time-domain statistical features are 16-dimensional, and the number of samples is 1200. 5-fold cross-validation randomly divides the sample data set into five parts, each time randomly selects four parts as the training set and one part as the test set. The ratio of the training set and test set is 4:1, and the ACC value and F1-score of the model are calculated using this training set and test set. This operation is repeated five times, and the average of the five test results is calculated, which is used to evaluate the accuracy of the model. The experimental results are shown in Table 9.

By analyzing the experimental results of the 5-fold cross-validation, it can be found that compared with other methods, HQNN-SFOP has the highest ACC value and F1-score, which are 0.944 ± 0.015 and 0.941 ± 0.015, respectively, thus verifying the performance advantage of HQNN-SFOP from a statistical point of view.

This set of experiments illustrates that the HQNN-SFOP method has significant advantages on the radar return signal dataset, both in terms of statistical feature extraction in the time domain and terms of signal detection capability, realizing the extraction of a smaller set of features yet obtaining the highest drone detection accuracy.

We trained and tested the five types of drones datasets and the random drones datasets with SNRs of 0, 5, 10, 15, and 20 dB, respectively, and conducted cross-testing experiments between these datasets, that is, the model generated by the training of the five types of drones dataset was tested not only with the five types of drones dataset, but also tested with the random drones dataset, and the model trained by the random drones dataset training was also the same, so as to verify its migration generalization ability, and the results are shown in Fig. 6. where 5Dro is the dataset generated from the five types of drones parameters, and RDro is the dataset generated from the random drones parameters.

Analyzing Fig. 6, both HQNN-SFOPs perform well and undergo a linearly increasing change with increasing SNR. The F1-score for both self-validation and cross-validation exceeds 0.85 for SNR greater than or equal to 5 dB. The F1-score for both self-validation and cross-validation exceeds 0.90 for SNR greater than or equal to 10 dB. And the F1-score for both self-validation and cross-validation exceeds 0.94 for SNR greater than or equal to 15 dB. Even though the SNR is 0 dB, the F1-score for both self-validation and cross-validation of HQNN-SFOP exceeds 0.80. In contrast, compared to the classical image detection CNN algorithm [24], the F1-score for both self-validation and cross-validation for an SNR of 0 dB is only as high as 0.5. The F1-score for both self-validation and cross-validation for an SNR equal to 5 dB is around 0.60. When SNR is equal to 10 dB, its F1-score for self-validation and cross-validation is around 0.85. When SNR is equal to 15 dB, only one set of its F1-score for self-validation and cross-validation reaches 0.9 and the rest of them are below 0.9.

The F1-score of HQNN-SFOP outperforms the results of CNN at SNRs of 0, 5, 10, 15, and 20 dB, respectively, and the F1-scores generalized to other data type datasets at the same SNR are all stable. The experiments show that HQNN-SFOP has a strong migration generalization ability, and compared with CNN, HQNN-SFOP has a stronger feature expression ability and migration generalization ability.