Comparison of two cyclostationary detectors for radio frequency interference mitigation in radio astronomy



[1] Radio frequency interference (RFI) mitigation has become a significant issue for current and future radio telescopes. This paper presents a new scheme for removing radio frequency interference from astronomical data. It exploits a priori knowledge of the transmitters, namely, their cyclostationary statistical properties. Two real-time cyclostationary detectors are proposed and compared. Results on both synthetic and real data demonstrate the efficiency of this concept.

1. Introduction

[2] For several years, radio astronomy has had to face two contradictory trends. On the one hand, the exponential expansion of telecommunications has generated a growing demand on the electromagnetic spectrum, reducing the number and the size of bandwidths available for good quality radio astronomical observations. On the other hand, radio astronomical needs in terms of sensitivity and bandwidth have also grown. As a result, radio frequency interference (RFI) mitigation has become a significant issue for current and future radio telescopes. In the short term, the objective is to preserve observation capabilities on previously and newly observed objects. In the medium and long term, the objective is to improve observations of fainter signals, even outside the protected radio astronomy bands.

[3] Various methods have been tried to eliminate RFI depending on the type of interference and the type of instruments. Time, frequency and/or spatial properties are considered in order to find efficient excision processing techniques (see other papers in this special section or Fridman and Baan [2001] and Leshem et al. [2000] for a comprehensive survey of such methods).

[4] The present study focuses on time-frequency blanking on data coming from a single dish. Time-frequency (t-f) blanking consists in removing data blocks detected as polluted in the power t-f plane before integration in order to clean up the final power spectrum. The t-f plane is obtained in real time by a digital filter bank based on FFT or polyphase filter. If possible, the t-f blocks of this plane must be adapted to the temporal and spectral RFI properties. Blanking decisions are generally based on a statistical contrast between the signal of interest (SOI) and the RFI.

[5] The most widely used criterion is power estimation [see, e.g., Ellingson and Hampson, 2003; Weber et al., 2004]. Indeed, its relative simplicity makes real-time implementation possible. However, the effectiveness of such a detector depends on the ability to define the ad hoc threshold, especially when the context is not stationary (i.e., time fluctuations of the instrumental conditions or variability in the source emissions). Even when a running estimate of the threshold is implemented, the difficulty is to make it robust against any RFI contamination. Depending on the temporal and spectral RFI properties, detection can be achieved more or less successfully. Power detectors are actually rather well suited for strong RFI (see example in Figure 1).

Figure 1.

RFI impact on the observation of the mega maser IIIZW35. RFI bursts are due to downlink satellite emissions. (a) Example of t-f plane (2048 frequency channels). The signal has been acquired with a robust digital receiver, but no blanking has been applied. The RFI bursts are clearly visible. As the signal of interest (SOI) is buried in the noise, a spectrum averaging must be carried out to make the SOI profile visible. (b) Averaged spectrum. The averaging time is 13 min. The RFI bursts dominate the spectrum. (c) Zoom on averaged spectrum. Compared with Figure 1b, the filter spectral shape has been removed. Without power blanking, the spectral profile of IIIZW35 (dashed line) is completely scrambled by the RFI bursts (for more detail, see Weber et al. [2004]).

[6] In this paper, we describe a criterion which is power-independent. It is based on the temporal properties of a particular class of RFI, called cyclostationary signals. Two implementations are compared. Section 2 will present the hypotheses and the properties of the signals used in this paper. Section 3 will define cyclostationarity and will compare the two detectors. In section 4, an example of time-frequency blanking is described and discussed.

2. Signal Model

[7] Considering an M channel receiver connected to a single dish antenna, we model the signal, s(t), of one of these frequency channels as the sum of an independent identically distributed Gaussian noise, n(t), and a cyclostationary RFI, b(t),

equation image

Practically, the Gaussian noise n(t) includes all the signal contribution except the RFI (i.e., system noise, sky noise and possibly a part of the astronomical source…). Let us define the interference-to-noise ratio (INR) as the interference power divided by the noise power generated by n(t) in the considered frequency channel.

[8] The problem can be seen as a binary classification of the channel samples (or group of samples) between corrupted ones (case named hypothesis H1) and clean ones (case named hypothesis H0):

equation image

The criterion used to discriminate these two hypotheses is based on the cyclostationary properties of the RFI.

[9] A signal s(t) is said to be cyclostationary if its autocorrelation function, Rs(t, τ), is periodic in t [Gardner, 1988]. This periodicity is not visible on the signal waveform but can be seen on higher orders (see an example in Figure 2). The inverse of this hidden period, Tc, is called the cyclic frequency αc. The Fourier coefficients of Rs(t, τ) are Rsα(τ), called the cyclic autocorrelation function:

equation image

where E[.] represents the expectation. The interesting point is that Rsα(τ) is nonzero at {α = k/Tc, kN}, if s(t) is cyclostationary. Besides, for noncyclostationary signals, such as n(t), the corresponding information is concentrated in α = 0. Thus the presence of significant power at the discrete frequencies {α = k/Tc, kN*} is a discriminating indicator of a cyclostationary RFI (see example in Figure 3).

Figure 2.

Cyclostationary signal, b(t). The signal is a random succession of binary pulses with Tc = 8. The pulse shape is given by a Tukey window with the ratio of taper to constant sections equal to 0.1. (a) Part of the waveform. (b) Spectra of the signal (curve i) and the signal square (curve ii). No spectral line is present in curve i (i.e., the random sign assigned to each pulse hides the periodicity due to these pulse repetitions). On the other hand, several spectral lines at frequencies that are multiples of 1/Tc are present in curve ii; the hidden periodicity is recovered in the signal square.

Figure 3.

2D-representations of Fourier coefficients Rsα(τ) defined in equation (3) (a) for the cyclostationary signal defined in Figure 2a and (b) for a white Gaussian noise. Only the cyclostationary signal presents nonzero values of Rsα(τ) at {α = k/Tc, kN*}.

3. Cyclostationary Detectors

[10] Two cyclostationary detectors for RFI mitigation are presented in this section. They both work on the same basic two-step principle. The first step consists in transforming a cyclostationary signal of hidden period Tc into a Tc-periodic signal. The second step is a periodicity detector. The result is then compared to a threshold level in order to obtain a binary decision for blanking or not the received data.

[11] In our real-time context, we assume that Tc is known. It has been extracted either from the telecommunication system specifications or from preliminary RFI observations. For example, depending on the modulation type, the baud rate or the carrier frequency could be used as a cyclic frequency parameter.

[12] The transformation of a cyclostationary signal into a periodic signal is obtained by a quadratic time invariant transformation. A very simple one is the square modulus [Gardner, 1988]. This transformation is all the simpler as it is usually done by the radio astronomy spectral receiver in order to compute the power spectra. Thus, in each frequency channel, just by looking at any periodicity in the time fluctuation of the power, the presence of a cyclostationary RFI will be detected. We will therefore focus only on the second step of the method. Two different ways of detecting a periodic signal are compared. The first one uses a synchronized averaging while the second one is based on Fourier analysis.

3.1. Synchronized Averaging

[13] The oldest known technique for extracting periodicity is synchronized averaging. This method was used to detect a cyclostationary signal [Gardner, 1988; Weber and Faye, 1998]; the latter reference is in the RFI mitigation context. The basic idea is that, if a signal is Tc periodic, it can be partitioned into Tc-length adjacent segments. These segments are then averaged to reduce random noise.

[14] Under hypothesis H1, the result of synchronized averaging will give a noiseless segment, with non-null energy, whereas the energy obtained under hypothesis H0 tends to be zero. The time variant mean of a Tc-periodic signal x(t), x(t) = ∣s(t)∣2 in our case, with N samples is

equation image

We can generalize this technique by considering the filtering of x(t) with a comb filter h1/Tc. Equation (4) is just the filtering by a comb filter with the following coefficients:

equation image

By choosing other values for h1/Tc(m.Tc), the spectral shape of the filter can be modified. In particular, the spectral line at α = 0, which is present both under H0 and H1 can be removed. The new coefficients are

equation image

By including all these considerations, the decision criterion of the cyclostationary detector can be written as

equation image

where ⊗ is the convolution operator and Ps is a running power estimation of the received signal s(t). The block diagram description of this algorithm is shown in Figure 4.

Figure 4.

Functional description of the cyclostationary detector using synchronized averaging. The mean extraction removes the spectral line at α = 0.

[15] This method is efficient and simple but has the disadvantage of working optimally only when Tc is a multiple of the channel sampling period Ts. If not, a signal resampling can be envisaged to make the phase error negligible over the detection window. The counterpart is a larger use of hardware resources which may compromise the real-time capabilities of the system. This is all the more true as the initial Tc/Ts ratio will in practice be small. Indeed, the maximization of the INR requires the channel frequency selectivity (i.e., 1/Tc) to match the RFI main bandwidth, closely related to the cyclic parameter Tc. Another point is that only the first spectral lines are of interest either because the Tc/Ts ratio is small or because distant harmonic contributions are negligible in equation (7). In the next section, we propose a criterion which uses only the main spectral line. Thus the constraint on Tc can be removed.

3.2. Fourier Analysis

[16] The second technique is based on a cyclostationary indicator recently proposed [Raad et al., 2003]. Its principle is to integrate the cyclic information of a signal through the cyclic autocorrelation function at zero lag:

equation image

As the cyclic period Tc is known, the indicator is simplified by computing just for the cyclic frequency αc = 1/Tc, instead of summing on every α. Thus the new indicator is

equation image

Its stochastic version is given by

equation image

In fact, the principle of the indicator of equation (10) is to obtain the energy of the signal ∣s(t)∣2 at the frequency 1/Tc by computing its discrete Fourier transform at this frequency. If s(t) is cyclostationary, only its first regenerated spectral line will modify the criterion value. The block diagram description of this algorithm is shown in Figure 5.

Figure 5.

Functional description of the cyclostationary detector using Fourier analysis.

3.3. Performance Comparison

[17] The objective is to compare the two cyclostationary detectors presented in the previous section. On one hand, equation image uses all the spectral lines regenerated by the quadratic transformation but the cyclic parameter Tc must be a multiple of Ts, on the other hand equation image uses less spectral information (only the main spectral line) but there is no constraint on Tc.

[18] With a view to evaluating these detectors' performances, the Fisher criterion [Fisher, 1938] is computed according to the formula

equation image

where C is one of the tested criteria equation image or equation image, E[.] the expectation and Var[.] the variance under hypotheses H0 and H1. The Fisher criterion, F, represents the ratio of the distance between hypotheses H0 and H1 in relation to their dispersion. This performance measurement is global and evades any threshold issue.

[19] The Fisher criterion is determined by Monte Carlo simulations for the detectors according to two parameters; the INR and N, the number of samples used in each determination of the value equation image or equation image. For the simulations, N varies between 128 and 4096 samples and the INR between 0 and −5 dB. Besides, the RFI, b(t), is based on real RFI specifications. The studied system is a telecommunication satellites constellation (the iridium system) using a combination of time division multiple access (TDMA) and frequency division multiple access (FDMA) techniques. In other words, the t-f plane is divided into small t-f slots (see Figure 1a), which are used, or not, depending on the communication activity. t-f blanking is well adapted for this kind of system. In each slot, a quadratic phase shift keying (QPSK) modulation is performed with a baud rate 1/Tc (i.e., every two bits, the phase of the carrier frequency takes one of the following values ±pi/4, ±3pi/4). In all simulations, we have assumed that the frequency channels receiver maps the RFI FDMA channels. As a result, the QPSK carrier frequency is in the center of the receiver frequency channel. For both detectors, two cases have been studied: when the ratio Tc/Ts is an integer (Tc/Ts = 4) and when Tc/Ts is not an integer (Tc/Ts = 1/0.23 = 4.3478…).

[20] The results are shown in Figure 6. For each point on these curves, 100 runs have been performed. The Fisher criterion increases with N and INR as expected. The methods presented here are both quite efficient when Tc/Ts is an integer. equation image is even a little better than equation image for high INR (more spectral lines are involved in the criterion value). Nevertheless, their performances are quite equivalent when INR is low (only the main spectral line contribution is significant). However, it appears clearly that equation image is noticeably better when Tc/Ts is not an integer. The results of equation image in this case are poor.

Figure 6.

Comparison of the two detectors. In each case, the Fisher criterion is plotted for several values of the INR and N. Two values of Tc are tested in which (a and c) Tc is an integer and (b and d) Tc is not an integer. As expected, the Fourier analysis method works well even if Tc is not an integer.

[21] Actually, the sampling frequency of the radio telescope is fixed, so the ratio Tc/Ts is rarely an integer unless the radio telescope receiver can be specifically reconfigured. This is generally difficult. Therefore the cyclostationary detector using the Fourier analysis has been preferred for our RFI mitigation work.

4. Examples of RFI Mitigation

[22] In order to illustrate the performances of the chosen cyclostationary detector, two RFI mitigation contexts have been processed: the first is simulated and the second uses real data. In each case, a similar digital processing has been applied: a signal with bandwidth B is downconverted to baseband [−B/2, B/2] (the samples are complex values), then a polyphase filter bank provides M frequency channels. The cyclostationary detector is applied on each channel of the t-f plane with a sliding window of length N = 512, which corresponds to the size of the RFI time slots. These N samples are blanked when the criterion is over the threshold.

4.1. RFI Mitigation With Synthetic Data

[23] The t-f plane of the simulated signal is shown in Figure 7a. It represents the M = 128 frequency channels. Figures 7b and 7c represent the time integration of this t-f plane without and with blanking, respectively.

Figure 7.

RFI mitigation of simulated iridium. Besides the iridium pulses, several signals of interest (SOI) have been added: a sinusoidal signal on channel 6, some iridium-like Gaussian pulses on channel 15, a strong Gaussian pulse centered at sample 2300 between channels 40 and 50, and a continuous weak Gaussian noise located between channels 70 and 80. (a) Time-frequency representation of the polluted signal. (b) Time integration without blanking. (c) Time integration after blanking. The threshold has been set to provide a 100% probability of detection. All the SOI, even the strongest, have been preserved. For the whole set of data, the probability of false alarm is 14%. It is interesting to note that the iridium-like pulses are not mixed up with RFI.

[24] The RFI signals are QPSK modulations with a baud rate equal to Tc/Ts = 0.23. They are centered on 8 different channels (channels 12, 25, 33, 58, 75, 88, 94 and 111) for several spaces of time. Their INR are different for each occupied channel and vary between −1 to −5 dB. Besides the white Gaussian noise, several signals of interest (SOI) have been added: a sinusoidal signal on channel 6, iridium-like Gaussian pulses on channel 15, a strong Gaussian pulse centered at sample 2300 between channels 40 and 50, and a continuous weak Gaussian noise located between channels 70 and 80. To reach the full detection probability (Pd), the threshold has been set empirically at 0.01. The corresponding false alarm probability (Pfa) over the whole t-f plane is 14%. This rather high value is understandable because detection of the lowest RFI (INR = −5 dB) is in the criterion confusion area. Indeed, the corresponding Fisher criterion value, F, is 0.89, which means that the detector contrast (equation image[C] − equation image[C])2 is close to its uncertainty (equation image[C] + equation image[C]). Despite its high level, the sine wave has not been mixed up with the RFI (Pfa in this channel is null). The same result is found for the iridium-like pulses. The measured Pfa values for the strong Gaussian pulse and the continuous week Gaussian noise are 4.7% and 7.3%.

4.2. RFI Mitigation With Real Data

[25] A B = 3.5 MHz bandwidth has been acquired with a digital receiver at Nançay observatory (France) in the iridium frequency band. The complex value baseband waveform has been sampled at 3.5 MHz during 570 ms. Off-line, a polyphase filter bank critically decimated with M = 84 frequency channels has been applied. As a result, the channel sample frequency is 1/Ts = 3.5/84 = 41.66 kHz which corresponds to an iridium channel bandwidth. In this context, the cyclic parameter Tc/Ts to be used is 1/0.3 = 0.33333… The t-f plane of the real data is shown in Figure 8a. Its time integration without blanking is shown in Figure 8b. The detector threshold has been set empirically to 0.07. The resulting blanked areas are represented in Figure 8c, that corresponds to 1% of the whole set of data. Figure 8d represents the 'clean' time integrated spectrum: identified RFIs have been removed.

Figure 8.

RFI mitigation of iridium signals on real data. The t-f plane maps the iridium frequency channels. (a) Time-frequency representation of the polluted signal. (b) Time integration without blanking. (c) Time-frequency representation of the blanked t-f slots (black lines). The data loss is 1%. (d) Time integration after blanking; identified RFIs have been removed.

5. Conclusion

[26] This paper deals with RFI mitigation in the radio astronomy context by using the properties of cyclostationary RFI. Two detectors are compared. They are based on the same signal conditioning, which is a square modulus of the time-frequency plane. This quantity is easy to obtain in real time from a radio astronomical receiver. Then, the detectors perform different filtering to detect the presence of RFI characteristic spectral lines. Finally, the most robust method has been used to perform a realistic simulation of time-frequency blanking. The results are a total detection of the cyclostationary RFI and a very good immunity to any power fluctuations, even strong, of the SOI. The latter result could be very interesting in the case of bright and nonstationary SOI such as solar or Jovian emissions. The next stage will be the real-time implementation and a theoretical analysis of the probability of false alarm.


[27] The Nançay Radio Observatory is the Unité Scientifique de Nançay of the Observatoire de Paris, associated as Unité de Service et de Recherche 704 to the French Centre National de la Recherche Scientifique (CNRS). The Nançay Observatory also gratefully acknowledges the financial support of the Conseil Régional de la Région Centre in France. The authors gratefully acknowledge Elizabeth Jolivet for the English support.