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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Radio Frequency Interference (RFI) detection and mitigation algorithms based on a signal's spectrogram (frequency and time domain representation) are presented. The radiometric signal's spectrogram is treated as an image, and therefore image processing techniques are applied to detect and mitigate RFI by two-dimensional filtering. A series of Monte-Carlo simulations have been performed to evaluate the performance of a simple thresholding algorithm and a modified two-dimensional Wiener filter.

Microwave radiometry is today routinely used to remotely measure geophysical parameters. Microwave radiometers are highly sensitive instruments, but measurements may be corrupted by the presence of Radio-Frequency Interference (RFI). Sources of RFI include spurious signals and harmonics from lower frequency bands, spread-spectrum signals overlapping the “protected” band of operation, or out-of-band emissions not properly rejected by pre-detection filters.

The presence of RFI in the pre-detected voltage signal modifies the value of the measured power leading to an erroneous retrieval of the measured geophysical parameters [

Combining the time and frequency domains analysis jointly, the spectrogram stands as a powerful tool which has been previously used in RFI detection in radio-astronomy [

In this work, a simple thresholding algorithm to detect and eliminate interference patterns in radiometric signal spectrograms is developed first, and simulation results are presented. The simulated RFI signals are generically from the sinusoidal and chirp families. By adjusting the signals' parameters, the bandwidth and temporal duration, many different types of signals can be obtained. A two-dimensional (2D) Wiener filter is then applied to the spectrogram in order to try to improve the cancellation of RFI components in the radiometric signal. A brief description of the spectrogram calculation is given in Section 2. Section 3 describes the image processing algorithm proposed in this work, which is called “Smoothing Algorithm”. In Section 4 the Wiener filtering process is described and compared to the “Smoothing Algorithm” proposed. Section 5 shows the results obtained with the application of both algorithms to a thermal noise signal contaminated with a set of RFI chirp signals. Finally, Section 6 summarizes the conclusions of this work.

In this work, radiometric signals are assumed to be sampled at an adequate sampling rate satisfying the Nyquist criterion and the spectrograms are obtained from these discrete time signals.

The spectrogram calculation depends on several parameters which are: the total number of signal samples, the FFT size, the window used for the FFT calculation of the data segments, and the overlapping between these data segments, as shown in

The spectrogram used in this paper is calculated using discrete samples, each one corresponding to a measurement of the pre-detected voltage signal. The number of samples (N) which forms the spectrogram will determine the radiometric resolution obtained by each spectrogram.

The FFT size is the number of samples used to compute each FFT and determines the spectral and temporal resolutions. If the FFT size (L) increases, the spectral resolution increases, while the temporal resolution decreases, since the time lapse between consecutive data segments increases. In contrast, smaller FFT sizes lead to lower frequency resolution, and higher temporal resolution. Therefore, selection of the FFT size must be chosen carefully depending on the RFI present in the radiometric measurements.

The FFT calculation requires the use of an analysis window function in order to minimize unwanted “side lobes” and ringing in the FFT resulting from abrupt truncations at both ends of the data segment. This situation causes an oscillatory behavior in the FFT called the Gibbs phenomenon [

In order to avoid loss of data when using a window in the FFT calculation, some overlapping (O) must exist between consecutive data segments (

The spectrograms calculated in this work have the following parameters:

Number of samples: this parameter must be as high as possible to maximize the radiometric resolution, as a higher number of samples means a longer integration time. On the other hand too large values increase the computation time required to perform the Monte-Carlo simulations. The value chosen in this work is N = 2^{18} samples.

Overlap: for the case of 2^{18} samples, an overlap value of O = 75% is reasonable [

FFT size: The FFT size determines the frequency resolution and the number of FFT segments, determines the temporal resolution. For the simulations performed in this work, and supposing that any information of the RFI is known, the FFT size was chosen to be equal to the number of FFT segments, to have similar spectral and temporal resolutions (both resolutions may not be the optimal ones); thus the FFT size should be the square root of the number of samples (N). Taking into account that there exists a 75% overlap between consecutive Hann windows, the number of FFT segments has to be multiplied by 4. Thus, to equilibrate both resolutions, maintaining the original compromise, the FFT size has been selected to be L = 2^{10}, and now the total number of samples is 2^{20} (accounting redundant points due to overlapping).

As it has been said, the spectrogram operation consists of the power (square of the absolute value) of the STFT magnitude. The spectrogram operation changes the probability density function of the received thermal noise which is a pair of Gaussian distributed random variables (one for the in-phase component and the other one for the quadrature component). The square of the amplitude is equal to the sum of the squares of both Gaussian distributions, which is the definition of a chi-square distribution with two degrees of freedom, which is equivalent to an exponential distribution [

The most obvious way to detect the presence of interference in a radiometric signal is by detecting power peaks in the received signal that are larger than the variance of the measured thermal noise in the absence of RFI. This detection can be performed either in both the time and frequency domains.

This technique can be straight-forwardly extended to the spectrogram. Considering that a power peak is produced by an RFI signal, the threshold value must be a function of the thermal noise variance (power), and must maximize the probability of RFI detection (_{det}) and minimize the probability of false alarm (_{fa}) detection. RFI probability of detection cannot be computed in advance since the presence of RFI is unknown, but the probability of false alarm is easy to obtain as the thermal noise follows a known Gaussian distribution.

All simulations performed in this work have been performed assuming an antenna temperature (_{A}) of 300 K and a radiometer receiver noise temperature of 100 K. In order to make a relationship between the noise power and the RF interfering signal power, the Interference-to-Noise Ratio (INR) is defined as:
^{2}_{RFI} is the power of the interfering signal and ^{2}_{Noise} is the thermal noise power.

The threshold value of this algorithm must be selected to limit the error in the estimated antenna temperature produced by false alarms.

If an RFI-free spectrogram is observed (

The algorithm proposed in this work can be summarized as follows with the aid of

Power spectrogram calculation of a data segment of 2^{18} samples using data segments of 2^{10} samples, with 75% overlap, and a Hann window (

Convolution of the power spectrogram with a 2D low pass filter of a determined size (in this work the Hann window has been chosen, although a Gaussian or a Hamming window would perform similarly (see

Thresholding: a threshold is used in the smoothed power spectrogram, in order to detect clusters of RFI-contaminated pixels (

Antenna noise power is calculated by averaging all spectrogram pixels below the threshold.

Finally, _{A}

As already discussed, the spectrogram of a noise signal with sinusoidal interference signals can be considered as a noisy image, where the noise is the spectrogram of the radiometric signal (the one we want to measure), and the image to be detected is the spectrogram of the interference (the one to be cancelled). Therefore, designing a filter to eliminate the noise from the image is the way to estimate the RFI, for a later removal of the interference without loss of the radiometric data.

The Wiener filter is a well-known adaptive filter used in communications, which provides the best estimation of a signal, equalizes communications channel, and eliminates the noise present in the received signal. In this section the Wiener filter is used to estimate the RFI in the spectrogram image, for a later cancellation and its performance is compared to the smoothing algorithm.

In our case of study, it is not necessary to know the effect of the communications channel in the spectrogram, thus, it will only be necessary to differentiate between the noise and the interfering signal. The way to perform this task is by using a locally adaptive linear minimum mean square error (LLMMSE) [

The LLMMSE algorithm consists of an optimal linear estimator of our interfering signal _{u}, f_{v}

_{u}, f_{v}_{u}, f_{v}_{u}^{th} time point of the spectrogram, _{v}^{th} frequency point and α and β are two parameters chosen to minimize the mean square estimation error criterion

The minimum error (ε) is found for:

From _{s}_{u}, f_{v}_{u}, f_{v}_{s}^{2}(_{u}, f_{v}_{u}, f_{v}_{n}_{n}^{2}(_{u}, f_{v}

The use of different window sizes (_{1} + _{2} + 1 = _{u}, f_{v}_{n}^{2}) is perfectly known. It is observed that the value of

In this section, results obtained with the two image processing algorithms presented (Smoothing and thresholding and 2D Wiener filtering) are shown and discussed.

The use of the FFT implies that the Smoothing Algorithm is implicitly searching for sinusoidal interferences [

_{p}, d_{p}, t_{p}, f_{p}, φ_{p}, β_{p}^{th} chirp signal respectively; _{r}, f_{r}, φ_{r}, v_{r}, η_{r}^{th} sinusoidal signal respectively; rect() is the rectangular function, _{s}

The retrieved error in _{A}_{fa}, the size of the smoothing filter and the power of the interfering signal is presented in _{A}_{A}_{fa} are represented in _{A}_{fa} of the threshold used in our algorithm. The _{A}_{fa} levels and 9 INR levels) was 3 h and 5 min. The actual _{A}_{fa} decreases. As it can be seen, the selection of the _{fa} threshold is crucial, as a low _{fa} values decreases the probability of detection of RFI-contaminated pixels, leading to a retrieved temperature higher than the real antenna temperature. On the other hand, a high _{fa} value will produce a high number of false alarms which will produce a clipping in the probability distribution function of the power spectrogram pixels, leading to a retrieved _{A} lower than the real antenna temperature.

In _{TA}_{fa}, while in the case of RFIs with INR value between −5 dB and −25 dB applying a low threshold leads to an important error produced by the fact that a great part of the RFI contaminated pixels “pass under” the threshold with low _{fa}. The selection of the _{fa} threshold is a compromise between a not too high value to clip the power PDF (_{fa} > 7.24·10^{−4} in _{fa} < 7.24·10^{−4} in

Combining different simulations with RFIs of different powers (INR from 5 dB to −30 dB compared to the noise power) it is observed that the threshold with optimum performance has a _{fa} ∼ 7.24 ×·1 0^{−4} (_{fa}|_{Opt} in _{TA}_{TA}_{RFI=0dB} in _{fa}|_{Opt} is equal to 1.37·^{2}_{n}

Three additional Monte Carlo sets of simulations have been performed with a size of the Hann window smoothing filter of 35 × 35, 25 × 25, and 5 × 5. Similar results have been obtained (_{A} RMS error independently of the RFI power are presented, in addition to the _{fa} associated with this threshold and the _{A} RMS error obtained with this threshold in the absence of RFI. For all cases, it is important to have a low _{A} RMS error value for the most suitable threshold in the black dotted curve, as absence of RFI should be the most probable case. In _{A} RMS error value decreases; in addition this value also decreases in absence of RFI.

The best performance is obtained with the largest window (35 × 35 Hann window). However, large smoothing windows exhibit a poorer radiometric resolution, as explained below. The radiometric resolution of an ideal total power radiometer is inversely proportional to the square root of the product of the noise bandwidth and the integrating time,
_{rec}

_{SA}_{SA}_{el}

A Hann window between 15 × 15 and 25 × 25 is recommended, as for higher values, retrieved _{A} error shows little improvement, while radiometric resolution may decrease excessively leading to a worsening of the retrieved geophysical parameters.

As broadband communication systems are increasing exponentially, it is likely to find broadband RFI added to the measured signal. Therefore, the Smoothing Algorithm has been tested with Pseudo-Random Noise (PRN) and Orthogonal Frequency Division Multiplex (OFDM) RFI signals defined in

_{PRN}, T_{SPRN}, f_{PRN}, φ_{PRN}_{Rad}

_{OFDM}_{c}_{m}_{c}_{m}^{th} subcarrier of the OFDM signal, defined as:
_{SOFDM}

The retrieved error in _{A} as a function of the threshold's _{fa}, the size of the smoothing filter and the power of the interfering signal has been calculated in the same way that in the previous section, and it is presented in

In _{TA}_{fa} must be used to detect a PRN interfering signal. In this case, the compromise of the _{fa} threshold selection leads to a minimum error of retrieved antenna temperature of 14.39 K, (_{fa} = 0.0384) for an INR value of 5 dB.

On the other hand, _{fa}, thus leading to a minimum error of retrieved antenna temperature of 9.07 K, (_{fa} = 0.02).

Following the RFI study in the same way as in the previous section, three additional Monte-Carlo simulations have been performed for different Hann window smoothing filter sizes (35 × 35, 25 × 25, and 5 × 5) for both broadband RFI cases; which are presented in _{fa}|_{Opt} for all INR parameter values as in _{A}; therefore higher INR parameter values leads to higher error in the retrieved _{A} values.

In contrast, for the OFDM RFI case, there exists a _{fa}|_{Opt} for all INR parameter values as in the sinusoidal RFI case (_{fa} and threshold values and the maximum error in the retrieved _{A} for the OFDM RFI case; this table is similar to _{A} values.

Detection and elimination of a PRN signal results in a high error on the retrieved _{A} as applying an FFT to a PRN RFI signal does not concentrate the RFI signal, as it happens with a sinusoidal signal. PRN signal behaves like noise.

In contrast, error on the retrieved _{A} produced by the contamination of an OFDM interfering signal is lower than in the PRN signal's case, as this signal is based in a frequency modulation, and the FFT process can concentrate the energy of the interfering signal.

The main problem of the presence of broadband RFI is that, even if it is correctly detected, radiometric resolution of the measurements will be degraded due to the minimum introduced _{A} error and the loss of radiometric resolution due to the high number of eliminated pixels, as it can be seen in

Threshold used in

On the other hand, threshold used in

Results obtained with the LLMMSE filter show that this algorithm is suitable for denoising signals (

For an optimal performance, it is necessary to accurately estimate in advance the power of the thermal noise (_{A}). Error in the estimation of the thermal noise power, introduces an error in the denoising process which leads to an error in the retrieved _{A} itself as it can be seen in

The reason for performing RFI extraction is to accurately estimate _{A}, which in fact is the thermal noise power. However, the LLMMSE algorithm does not improve the accuracy in the estimation of the thermal noise as it can be seen in

Two new RFI detection and mitigation algorithms have been presented. Both are based on processing of the radiometric signal's spectrogram and thus operate in the time and frequency domains simultaneously.

The Smoothing (and thresholding) Algorithm is studied and its performance is estimated using Monte Carlo simulations. The threshold value in the Smoothing Algorithm is the most critical parameter, as it minimizes the retrieved _{A} error depending on the filter size and the RFI power, which is a priori unknown. For a determined filter size, the best threshold can be calculated varying the RFI signal power and keeping the noise power constant. In case that the interference is sinusoidal, it is found that there is an optimal threshold value which minimizes the retrieved _{A} error for any RFI power. This threshold value diminishes with the filter size used in the Smoothing Algorithm. For a sinusoidal RFI, with a threshold value of 1.37·^{2}_{n}_{A} error is 2 K for a filter size of 25 × 25 pixels, and any INR value.

In case that the RFI is broadband, two cases have been studied, which are a PRN RFI, and an OFDM RFI. The PRN RFI behaves like noise and it is found that there is not an optimal threshold value, as increasing the RFI power always increases the retrieved _{A} error. The OFDM RFI behaves like a sinusoidal RFI, so there also exists an optimal threshold value, although the maximum retrieved error in the _{A} is higher than in the sinusoidal case. In addition, broadband RFI's contaminate extense areas of the spectrogram, resulting in a poorer radiometric resolution as many more pixels of the spectrogram have been eliminated.

A 2D de noising filter based on the optimum Wiener filter (LLMMSE) is also studied to estimate the RFI in the signal's spectrogram and to substract it from the contaminated one. However, it has been found that the Wiener filter has an acceptable performance applied to the spectrogram to detect RFI signals for subsequent substraction only if the noise power of the received signal is precisely known, which is actually the magnitude to be determined. Therefore although the Wiener filter is optimal for signal denoising in signal processing, the accuracy required in the estimation of the noise power is much higher in microwave radiometry than in typical telecommunications applications, and therefore it does not prove to be suitable for RFI mitigation.

These two algorithms have been tested only with sinusoidal, chirp, PRN and OFDM like signals, testing other types of RFI signals with these algorithms will be performed in the future as well as processing measured data.

Process of obtaining a Spectrogram from a sampled signal of thermal noise.

Transformation of the zero-mean Gaussian noise distributions (in-phase and quadrature components) into an exponential distribution with same standard deviation and mean.

Smoothing algorithm diagram.

Smoothing Algorithm applied to a simulated Radio Frequency Interference (RFI) contaminated radiometric signal. (^{2}_{RFI} = ^{2}_{n}_{fa} = 10^{−4}: black pixels are considered RFI; (

RFI mitigation technique: an estimation _{u}, f_{v}

Error in the estimation of the antenna temperature as a function of the Interference-to-Noise Ratio (INR).

Retrieved _{A}_{A}^{ideal} = 300 K (gray line). _{A} is represented in Kelvin (_{fa} (

Error obtained by the Smoothing Algorithm in the retrieved _{A} using a 15 × 15 Hann window as smoothing filter. Error is represented in Kelvin (_{fa} (

Error obtained by the Smoothing Algorithm in the retrieved _{A} using a Hann window as smoothing filter. (

Radiometric sensitivity degradation due to pixel elimination as a function of the filter size. Eight RFI signals have been used, with total INR value labeled for each case. Thresholds used are the most suitable ones for each window size (

Error obtained by the Smoothing Algorithm in the retrieved _{A} using a 15 × 15 Hann window as smoothing filter. Error is represented in Kelvin (_{fa} (

Error obtained by the Smoothing Algorithm in the retrieved _{A} when the RFI is a broadband PRN like signal, using a Hann window as smoothing filter. (

Error obtained by the Smoothing Algorithm in the retrieved _{A} when the RFI is a broadband OFDM signal, using a Hann window as smoothing filter. (

Smoothing Algorithm applied to a simulated PRN RFI contaminated radiometric signal. (^{2}_{RFI} = ^{2}_{n}

Smoothing Algorithm applied to a simulated OFDM RFI contaminated radiometric signal. (^{2}_{RFI} = ^{2}_{n}

Locally adaptive linear minimum mean square error (LLMMSE) Algorithm applied to an RFI contaminated radiometric signal. (^{2}_{RFI} = ^{2}_{n}

Error in the estimation of the retrieved temperature using the LLMMSE denoising algorithmin as a function of the error in the a-priori estimation of the thermal noise power. Both errors are represented in Kelvin. Colored lines represent RFI contaminated radiometric signals with an INR determined by its label. Black dotted line represents a radiometric signal in abscense of RFI.

Maximum retrieved _{A} error for the best threshold _{fa} for four different cases of Smoothing Algorithm filtering, including the case of not applying the algorithm. Maximum retrieved _{A} error in absence of RFI for the most suitable threshold is also shown.

_{fa} |
3.52·× 10^{−3} |
2.09·× 10^{−3} |
7.24·× 10^{−4} |
7.06·× 10^{−5} |
2.35·× 10^{−3} |

Threshold value | 1.24·^{2}_{n} |
1.37·^{2}_{n} |
1.72·^{2}_{n} |
4.04·^{2}_{n} |
6.08·^{2}_{n} |

Max ε_{TA} |
2.05 K | 2.09 K | 2.33 K | 3.71 K | 5.89 K |

Max ε_{TA} |
1.16 K | 1.41 K | 1.84 K | 3.71 K | 5.89 K |

Maximum retrieved _{A} error for the best threshold _{fa} for four different cases of Smoothing Algorithm filtering, including the case of not applying the algorithm; for the OFDM interfering signal case. Maximum retrieved _{A} error in absence of RFI for the most suitable threshold is also shown.

_{fa} |
2.16 × 10^{−2} |
2.4 × 10^{−2} |
2 × 10^{−2} |
1.76 × 10^{−2} |
3.02 × 10^{−2} |

Threshold value | 1.18·^{2}_{n} |
1.24·^{2}_{n} |
1.42·^{2}_{n} |
2.21·^{2}_{n} |
3.5·^{2}_{n} |

Max ε_{TA} |
3.8 K | 5.88 K | 9.07 K | 21.97 K | 44.5 K |

Max ε_{TA} |
3.51 K | 5.57 K | 9.01 K | 21.97 K | 44.5 K |

Retrieved _{A} error for the best threshold _{fa} for the four different cases of Smoothing Algorithm filtering, shown in

_{fa} |
3.68 × 10^{−2} |
2.79 × 10^{−2} |
2.1 × 10^{−2} |
1.7 × 10^{−2} |

Threshold value | 1.15·^{2}_{n} |
1.23·^{2}_{n} |
1.42·^{2}_{n} |
2.22·^{2}_{n} |

ε_{TA} |
5.15 K | 6.35 K | 9.12 K | 21.7 K |

Retrieved _{A} error for the threshold used in

_{fa} |
2.16 × 10^{−2} |
2.4 × 10^{−2} |
2 × 10^{−2} |
1.76 × 10^{−2} |

Threshold Value used | 1.18·^{2}_{n} |
1.24·^{2}_{n} |
1.42·^{2}_{n} |
2.21·^{2}_{n} |

ε_{TA} |
2.7 K | 4.48 K | 1.45 K | 22.3 K |

This work, conducted as part of the award “Passive Advanced Unit (PAU): A Hybrid