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# Secondary Surveillance Radar replies source separation via the Disjoint Component Analysis

Sara Zaghloul, Nicolas Petrochilos, Mamadou Mboup

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Sara Zaghloul, Nicolas Petrochilos, Mamadou Mboup. Secondary Surveillance Radar replies source separation via the Disjoint Component Analysis. IET International Radar Conference (RADAR), 2022, Edimbourg, United Kingdom. pp.483-487, 10.1049/icp.2022.2365. hal-04019576

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# Secondary Surveillance Radar replies source separation via the Disjoint Component Analysis

Sara Zaghloul*, Nicolas Petrochilos, Mamadou Mboup

CReSTIC, University of Reims, BP 1039, 51687 REIMS Cedex2, France

*E-mail: sara.zaghloul@univ-reims.fr

**Keywords:** Secondary Surveillance Radar, source separation, Disjoint Component Analysis, ADS-B

## Abstract

The foreseen increase of Secondary Surveillance Radar (SSR) replies, as well the replies emanating from the ADS-B systems will cause many collisions ending into garbling the received replies. While with a traditional receiver the leading reply might be decoded, the trailing ones will suffer a higher failure rate, and the reply will need to be re-issued and lead to an increase of replies density. To combat this and the loss of data, we propose a new mathematical criterion, the Disjoint Component Analysis (DCA), which allows to separate the replies regardless of the SSR protocol, the power ratio or the delay between replies, as demonstrated by several simulations and real data analysis. As we are still missing a computationally efficient algorithmic solution to minimize the cost function of the DCA criterion, this paper mostly demonstrates the potential of said criterion.

## 1 Introduction

Secondary Surveillance Radar (SSR) was created during the second world war under the acronym Identify Friend or Foe (IFF) and later opened to civil aviation [1], using a communication protocol, the mode A/C, which allowed to exchange information such as altitude and identity. This radar is a cooperative two-way system in which the base-station interrogates the air-plane on the 1030 MHz channel, to which a reply is sent via a device named transponder on the 1090 MHz channel. Later an advanced protocol was implemented, the mode S, which allows longer messaging (see Fig. 1). Recently a new safety system, the Automatic Dependent Surveillance Broadcast (ADS-B) [2], was enabled; it permits to use the 1090 MHz channel to emit spontaneously and periodically a mode S reply, namely a "squitter".

The expansion of the air-traffic and the implementation of the ADS-B increase replies density in the SSR channel, and therefore contribute to situation where two or more replies are impinging at the same time at the SSR Radar, which is named "garbling". The problem is that conventional receiver are inefficient, with usually the trailing reply being dropped.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Real case of a mode A/C reply followed by a mode S.</figcaption></figure>

### *Our proposed architecture and its properties*

In [3], we proposed the following architecture where next to the usual SSR radar, used as an interrogator, we added several receiving station scattered around the station. This architecture solve the garbling problem as the garbling occurs mostly only at one receiving station at a time. And it adds the possibility of performing multi-lateralisation on the received reply from a same air-plane, thus improving the positioning of the plane. It also adds an increase robustness for the message decoding. Furthermore we propose to have at each receiving station an antenna array, which will allows beam-forming and/or source separation.

In this context, the individual receivers, or antenna arrays, are passive therefore without time synchronisation. Given the 1 MHz tolerance accorded by the ICAO to the transponder for the downlink at 1090 MHz, and the 5 MHz bandwidth of the SSR replies, there isn't a frequency synchronisation either. Finally, legacy Mode A/C and Mode S protocols co-exist and have quite different characteristics (see Fig. 1) [4].

### *SSR replies source separation: a quick survey*

In 1994, P. Comon is the first to apply source separation to the mode A/C replies received on a real sum and difference channel of a SSR antenna using ICA [5]. Later AJ. van der Veen proposed a dedicated algorithm, the AZCMA, using the Zero/Constant Modulus properties of the replies to separate two or more mode S replies [6]. In [7], we proposed an extension (MS-ZCMA) and an algorithm based on the Manchester encoding of the data, the MDA, still solving a mixture of only mode S replies. In [8] M. Zhou proposed a new algorithm to overcome the weakness of the MDA for large time delays between the leading and the trailing reply. We proposed a simplified more robust version in [9] (Fold-MDA) for a mixture of mode S.
Finally, we resolved the source separation of a mixture of various modes (A/C +S) via an oblique projection method based on non-totally overlapping replies with the Extended Projection Algorithm (EPA) in [10]. The method estimates time range over which one signal is present, then via a projection method removes this source from the mixture and re-iterate till only one source remains.

In this paper, we explore the usage of the solution presented by J. Annemüller [11] to separate source with disjoint support, i.e non-overlapping sources, namely the Disjoint Component Analysis (DCA). The DCA criterion proposed by Annemüller has been developed later for disjoint and quasi-disjoint signals as in [12] and in several other areas such as the acoustic source localization [13], low rank decomposition [14] where DCA allowed the separation between the horizontal and vertical components of seismic images where the other algorithms could not solve this problem. The "disjointness" constraint was also used in the non-negative matrix factorization (NMF) [15] in its time-frequency form [16], and in [17] for convolutive signals.

Next section presents the data model and the DCA method. We then compare the performance of the algorithms with DCA with several simulations before validating the results with real data obtained from recording nearby Roma airport [18].

## 2 Model and algorithms

Upon emission, a SSR transponder mode A/C emits $n = 2 \times 6$ bits of informations separated by a bit equal to 0 and surrendered by bracket, i.e. 2 bits with value 1. A Mode S transponder emits $n = 56$ or $n = 112$ bits of information encoded by a Manchester code: a bit $b_i = 0$ is encoded with a raising front ($b_i = [0, 1]$), and $b_i = 1$ is encoded by a downward front ($b_i = [1, 0]$); altogether preceding a preamble for time synchronisation at reception.

$$ \mathbf{b}_{AC} = [1, 0, 0, b_1, 0, 0, \dots, b_6, 0, 0, 1, 0, 0, b_7, \dots, b_{12}, 0, 0, 1] $$

$$ \mathbf{b}_S = [1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, \mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n] $$

for a total length of $N = \{43, 128, 240\}$ bits. The Mode A/C preamble bracket also serve to time synchronize the reply at reception. The pulse to modulate the signal is a quasi-rectangular pulse in form, $p_S(t)$; And the signal emitted by the transponder is:

$$ b_S(t) = \sum_{n=0}^{127/239} \mathbf{b}[n] p_S(t - nT) \quad (1) $$

where $\mathbf{b}[n]$ is the n-th element of $\mathbf{b}_S$ or $\mathbf{b}_{AC}$, $T = 0.5 \mu s$ is the data time period.

To transmit the signal, $b(t)$ is up-converted to the frequency band $f_e$, but the International Civil Aviation Organisation [1] allows a shift up to $\pm 1$ MHz around $f_c = 1090$ MHz, thus $f_e \neq f_c$. Due to this carrier frequency mismatch, a residual

frequency $f$ remains after down-conversion by $f_c$ to baseband. This residual carrier adds a significant phase rotation to the transmitted symbols: up to a complex gain, the received baseband signal is described as:

$$ s(t) = g b(t) \exp (2\pi j ft) $$

where $g$ contains the received power and a phase which is the bulk delay at reception of the first symbol.

We extend this model to the reception of $d$ independent source signals on an $m$-element antenna array, assuming the multipath is negligible. The replies, $s_i(t)$, occur with times of arrival $\tau_i$, gains $g_i$. We assume that the narrow-band hypothesis is verified, therefore the times delays between elements are equivalent to phase-shifts. The element noise is Gaussian, identically distributed, independent, and all have the same variance $\sigma^2$.

Let's consider a time interval $L$ such that all received replies are totally included within.

$$ \mathbf{S} = \begin{bmatrix} s_1 & 0 \\ \vdots & \vdots \\ 0 & s_d \end{bmatrix} \quad (2) $$

We have the classical source separation model [19]:

$$ \mathbf{X} = \mathbf{M} \cdot \mathbf{S} + \mathbf{N} \quad (3) $$

where $\mathbf{X}$ is the $m \times L$ matrix containing the antenna received signals. $\mathbf{M}$ is the mixing matrix containing antenna array response and the replies gains. $\mathbf{N}$ is the noise matrix.

### 2.1 EPA and DCA criterion

Extended Projection Algorithm (EPA) [10] is an source extraction iterative algorithm: since the source have a limited support, the assumption is that there is always a time range with only one source present, see Eq. 2. Over this time range, the array signature is estimated for the source $i$: $\mathbf{m}_i$, the received matrix is projected on this vector to estimate the $i$-th source: $\hat{\mathbf{s}}_i = \mathbf{m}_i^H \mathbf{X}$, next via a Gram-Schmidt orthonormalization this source is removed from the received matrix $\mathbf{X}$. We stop the iteration when only noise remains.

Unfortunately, The obvious shortcoming of the EPA is when two sources almost totally overlap. Moreover, it is a sensitive algorithm that requires several setting to estimate the total number of source, the number of source on a given time range, and the choice of best time support to estimate $\mathbf{m}_i$; So the whole algorithm lacks robustness.

On the other hand, the Disjoint Component Analysis (DCA) criterion is non-supervised, therefore is immune to these later problems. Assuming a de-mixing matrix $\mathbf{W}$ giving the estimated sources as:

$$ \hat{\mathbf{s}}[k] = \mathbf{W}^H \mathbf{x}[k] \quad (4) $$

so that the “avelapp” defined in [11] between two outputs $\hat{\mathbf{s}}_i$ et $\hat{\mathbf{s}}_j$ be minimal:

$$ o_{ij} = E(|\hat{\mathbf{s}}_i| \cdot |\hat{\mathbf{s}}_j|) \quad (5) $$
where $E(.)$ is the mathematical expectation (which is replaced by a time summation for practicality). If two sources have a totally disjoint support, then $o_{ij}$ is equal 0. the criterion to minimize is the sum of all couples $o_{ij}$ for $1 \le i \ne j \le m$:

$$H(\mathbf{W}) = \frac{1}{2} \sum_{i \ne j} o_{ij} = \frac{1}{2} \sum_{i \ne j} E(|\hat{s}_i| | \hat{s}_j|) \quad (6)$$

under a constraint to avoid the null solution; In the original [11], this was ensured by keeping the row of W unit-norm.

Note that the DCA criterion is independent of time ordering (unlike for instance Fourier Transform), so if we consider the case of two overlapping SSR replies as in Fig. 2, on the left side, the replies are represented as they are received, and on the right side the time axis has been reshuffled so that cases where one source is zero are pushed on the side. The main observation here is that the value of the criterion (6) is exactly the same, but the right side shows us a partially overlapping sources for which the DCA criterion is performing well. This explain why the DCA can solve the case of fully overlapping SSR replies, that MDA cannot.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Case with totally overlapping replies.</figcaption></figure>

The article [11] propose an algorithm for received signal with real value, but not for complex-valued data; therefore we had to propose an ad-hoc method: after a second-order whiten- ing (SVD), we minimize (6) in the P. Comon way [20]: by searching for the minimum only for two received signals while maintaining the other constant, doing it sequentially over all the couple $(i, j)$. The rotation between two received signal is performed with the complex version of the Givens matrix [20], which contains two independent variables to optimize $c \in [0, 1]$ and $\phi \in [0, 2\pi]$; the best couple minimizing $H(\mathbf{W})$ is decided by an exhaustive search on the $(c, \phi)$ space.

This technique is of course computationally intensive and can- not be used in real-life radar yet, but the numerous advantages that we will demonstrate next in the simulation and real-life data sections easily explain why we focus our effort in finding an analytic solution.

## 3 Simulations

For these two short series of simulations, we consider the mix- ture of two short mode S which residual frequencies are $\pm 50$ kHz, received onto a $m = 4$ elements array, with Direction- Of-Arrival (DOA) equal to $\{70, 90\}^0$, which result in a con- ditioning number of M of 1.54. In one case, we vary the

Signal-to-Noise-Ratio (SNR), and in the other we vary the time-lag between the two sources.

We consider that the result of the separation is a failure if the output Signal-to-Interference-and-Noise-Ratio (SINR) is less than 6 dB

<figure><img src="image_4.png" /><figcaption>Fig. 3. Failure rate as a function of SNR.</figcaption></figure>

While varying the input SNR of the sources, we kept the time delay equal to $30\mu s$, and derive the failure rate presented in Fig. 3 for the MSZCMA, the MDA and fold-MDA, the EPA, the DCA, the FastICA [21] and the DCA. Knowing exactly the matrix M can be used to derive the zero-forcing solu- tion, which ensure no output interference and can be used as a reference. We simulate only 10000 independent runs due to computational time. We first note that the best algorithm here are the EPA and the DCA with failure rate similar to the zero- forcing method ($\mathbf{W} = (\mathbf{M}^H\mathbf{M})^{-1}\mathbf{M}^H$). Note that when the line disappear, it means that not enough simulations have been done to detect one failure. All the other competitors have high failure rate for input SNR below 5dB, we also note that MDA, fold-MDA and fast-ICA have a floor effect a high SNR.

In the second experiment, the input SNR is fixed at 10 dB and the time delay between two sources varies. Fig. 4 presents the failure rate as a function of the time delay; First, we ver- ify that the point with a SNR equal to 10 dB is consistent with Fig. 3. We confirm that the Fold-MDA version improves over the MDA, but stay short compared to the other algorithms. Fast-ICA has a failure rate of 1 for zero time delay, which is consistent with the findings in [22]: SSR mode S replies have quasi-Gaussian statistics, and therefore cannot be separated by High-Order Statistic methods. The rate improves as the time delay increase due to the inclusion of several more data equal to 0 therefore changing their statistics. EPA cannot work at zero time delay, since there is no time-range to estimate the array signature vectors. Albeit a slight failure rate of 2% for a zero time delay, the DCA criterion is the best. With next to it the MSZCMA, which has similar performance with a small peak
<figure><img src="image_5.png" /><figcaption>Fig. 4. Failure rate as a function of SNR.</figcaption></figure>

of failure around 50μS.

The output SINR were also derived, but did not show significant results as the curve were all similar and almost all yielded the expected 6 dB gain.

In overall, EPA and DCA have the best simulation results. Fold-MDA does indeed improves sensibly over MDA. The HOS methods weakness has been demonstrated and for a mixture of mode S only, the MSZCMA is also a good contender.

## 4 Real data

<figure><img src="image_6.png" /><figcaption>Fig. 5. The TDR antenna</figcaption></figure>

Courtesy of the RADARLAB from University of Roma-Tor Vergata, we can use the real data obtained with the Transponder Data Recorder (TDR [18], Figure 5). The TDR is a 6-elements planar linear antenna array, which the central 4 elements are linked to a linear receiver; the last 2 elements being there for coupling reasons. The received signal are down-converted to an intermediary frequency of 21.5 MHz, then sampled at a rate of

100 MHz. The TDR was located on the roof of RADARLAB, the main airport being at 30 km, and the second at 10 km. From the measurement campaign made on october 2016, the RADARLAB team gathered 419 cases above noise level of at least overlapping mode S over ten days of recording. All these cases have been confirmed with an operator and allow for evaluation of Fold-MDA, EPA, and DCA. Due to the inability to process mode A/C, we do not evaluate MSZCMA; moreover Fold-MDA is only used as a benchmark for when there are no mode A/C.

<table><caption>Table 1 Success rate for mode S and mode A/C replies.</caption><thead><tr><th>Case</th><th>n<sub>i</sub></th><th>F-MDA S</th><th>EPA S||A/C</th><th>DCA S||A/C</th></tr></thead><tbody><tr><td>Global</td><td>419</td><td>0.78</td><td>0.79||0.26</td><td>0.89||0.81</td></tr><tr><td>2 mode S</td><td>335</td><td>0.76</td><td>0.80</td><td>0.89</td></tr><tr><td>2 S + 1 A/C</td><td>58</td><td>0.76</td><td>0.84||0.22</td><td>0.90||0.79</td></tr><tr><td>2 S + 2 A/C</td><td>7</td><td>0.71</td><td>0.57||0.43</td><td>0.57||0.86</td></tr><tr><td>3 mode S</td><td>13</td><td>0.00</td><td>0.69</td><td>0.54</td></tr><tr><td>3 S + 1 A/C</td><td>6</td><td>0.00</td><td>0.50||0.18</td><td>0.50||0.82</td></tr></tbody></table>

Considering the table, we can read that either for mode S, DCA has 10% more success into detecting and decoding, but furthermore the mode A/C detection success rate is multiplied by 3, which makes it a much more robust solution than existing ones. As a reference, Fold-MDA is below the other algorithm/solution under investigation or barely equal in term of success rate, but of course, it is blind to the mode A/C.

The prevalence of DCA over EPA can be explained by the fact that in order to detect a mode A/C reply, EPA needs that this mode A/C reply has a similar total energy than an overlapping mode S, when it is actually quite more sparse. In the case of DCA, because even at low energy, the source would be push out from the mode S due to the successive rotation of the Ad-hoc algorithm, it gets separated from the mode S replies.

## 5 Conclusion

Though simulations and real world data, we have shown that the DCA criterion is better performing than its competitor, including the direct one, the EPA. Furthermore, as a non-supervised algorithm, it is quite more robust. It can also work with any kind of mixture: mode A/C and mode S. We even have observed a few cases where it did managed more sources than sensors. We need to make it more efficient in term of computation in order to reach to an operative algorithm, hence a technical solution.

## 6 Acknowledgements

We thank Prof. G. Galati of university of Tor Vergata, Roma, and the university of Calabria for sharing their data. We also thank Q. Zhang and F. Belloir for their help with preparing the data.
# 7 References

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