#### 5.1. Choosing β^{2}, *N*_{blank}, and *N*_{wait}

[27] To investigate choice of β^{2}, data from LISA channel 7 was examined to determine the percent of samples exceeding a specified threshold determined by β. Because the mean measured noise power can be expected to vary significantly throughout a cross-U.S. flight, the mean and variance estimates used to determine the threshold are computed by the running average described in section 3. This running average used *N*_{blank} = 2048 and *N*_{wait} = 0 in removing pulse contributions from the mean and variance computations, as described in section 3.2.2. Figure 3 illustrates the percent of samples out of the 145 16K captures that exceed the threshold specified by β^{2} on the horizontal axis. The dashed line included in Figure 3 represents the percent of exponentially distributed noise that would exceed the same threshold. An exponential, rather than Gaussian, noise distribution is more appropriate here because of the very short integration time of the incoming power for this system. In this case, the presence of RFI in the data set causes the percent blanked to exceed that of the simulated noise for large β^{2} values; however as β^{2} is reduced, the two curves become more identical because the detection of noise dominates both processes. In this case, the turning point with the LISA data appears to be in the range β^{2} ≈ 30 to 40. Further simulations with other LISA channels showed β^{2} values ranging from 40 to 90 to be reasonable; smaller β^{2} values clearly led to excessive blanking as should be expected.

[28] As mentioned previously, the parameter FIFO_LENGTH is fixed to 1024 (i.e., 51.2 μsec of LISA data) in order to model the existing radiometer prototype. Because of this somewhat small size, tests using LISA channels 7–13 indicate that immediate blanking of the FIFO is preferable. For this reason, *N*_{wait} is set to 0 in the remaining simulations. Future work will explore a preferred value of *N*_{wait} for larger FIFO_LENGTH parameters.

[29] A simulation investigating the effect of *N*_{blank} has also been performed. In this study, two threshold levels are defined: a higher “detection threshold” (β^{2} = 90) used for pulse detection as usual while *N*_{blank} is varied. The output of the blanker is then examined to determine the number of samples remaining that exceed a lower “reference threshold” (β^{2} = 30 using the same mean and variance computations as those for the detection threshold). This quantity is labeled *P*_{out} in what follows. The total number of samples exceeding the reference threshold in the original data is also determined, and labeled *P*_{in}. The ratio (*P*_{in} − *P*_{out})/*P*_{in} then provides information on the effectiveness of the higher threshold blanker with a given *N*_{blank} at removing lower-level pulse contributions. Figure 4 illustrates these quantities for a single LISA capture. Note in some cases, samples exceeding the higher threshold are not blanked in this data set; this is because such “trigger” samples may be missed by the subsampled detector if they are less than 4 samples long.

[30] Figure 5 plots (*P*_{in} − *P*_{out})/*P*_{in} as a percentage (solid curve) from the entire channel 7 data set, with *N*_{blank} ≥ 1536 which clearly satisfies equation (5). The curve shows only a modest variation with *N*_{blank}, and an *N*_{blank} value in the range 1536 to 2048 appears appropriate in this case. This can also be interpreted as indicating that pulsed interference longer than 76.8 μsec (i.e., 1536 samples) is not significant in this data. Of course, this *N*_{blank} parameter will vary for RFI environments dominated by different types of sources, but the cross–U.S. flight considered here should be fairly representative of other data sets. Results for smaller *N*_{blank} values may show more sensitivity in Figure 5, but are not explored here because of the limitation of equation (5). Data for smaller *N*_{blank} values could be generated by varying the FIFO_LENGTH and *N*_{wait} parameters, and will be explored in future work.

[31] The saturation of the solid curve at a maximum value of approximately 80% is caused by the presence of spurious internally generated interference in the LISA data set [*Johnson and Ellingson*, 2003]. The dashed curve is computed by redefining *P*_{in} and *P*_{out} to included only consecutive sets of 2 or more samples that exceed the reference threshold. In this case, a removal of approximately 98% of the “low-level” pulse components is achieved.

#### 5.2. Output χ^{2} Test

[32] Now that means for choosing the APB parameters have been established, it is of interest to quantify the quality of the output data. Because thermal noise power should approach a Gaussian distribution when integrated sufficiently, the χ^{2} test against a Gaussian distribution [*Press et al.*, 1992] can be utilized to evaluate if the output of the blanker satisfies this expectation. However, the entire data set for a specific channel cannot be applied in this test, as the mean noise power in a channel will vary as different locations are observed. The test was instead performed using sets of consecutive 5 16K captures, all of which were measured within 5 seconds. The χ^{2} statistic using 4 degrees of freedom was computed using data power integrated over 128 samples in order to approach the Gaussian distribution.

[33] Original data from one set of LISA channel 7 data is illustrated in the top plot of Figure 6 before and after the APB algorithm is applied with β^{2} = 40 and *N*_{blank} = 2048. The presence of interference in the original data set results in a high χ^{2} value of 262.53, indicating that the data are not likely to be from a Gaussian distribution. The bottom plot illustrates the χ^{2} statistic of the nonblanked data after blanking with β^{2} = 40 versus *N*_{blank}, and shows a greatly reduced value compared to the preblanking case. Critical values based on α = 1% and 10% (the probability of incorrectly classifying a true Gaussian distribution as non-Gaussian) are also illustrated in Figure 6. Clearly for this example, the APB output data is much more Gaussian than the input data, particularly for *N*_{blank} exceeding 1366. The poor performance for *N*_{blank} = 1024 is not surprising, since this case does not satisfy equation (5), allowing the possibility that some detected samples remain unblanked.

[34] Simulations from other LISA data subsets show similar results, with a few exceptions. In particular, LISA channel 6 (1310–1330 MHz) sometimes contains very strong interference from multiple aviation radars, and a large value of χ^{2} remains even after blanking with *N*_{blank} up to 4096. The limitation of the fixed FIFO_LENGTH parameter is an issue here, and future work will examine if increasing this parameter can improve these problematic data sets. However it should certainly be expected that there are cases with exceptional RFI corruption for which the APB algorithm cannot retrieve the original noise power.

#### 5.3. Effect of Blanking on Integrated Spectra

[35] The ideal pulse-blanking algorithm would remove only RFI information, without changing properties (particularly the mean power level) of the remaining noise information. One questionable issue of the APB algorithm is the impact of forcing data to zero when pulses are detected. This introduces discontinuities into the signal which may lead to undesired effects on the final output, as well as calibration uncertainties. Note after the APB operation an FFT is performed in the interference-suppressing radiometer of section 2; clearly the impact of blanked samples on the FFT output should be investigated.

[36] To examine these effects, APB outputs were processed through FFT and integration operations. Each 16K LISA capture after blanking was first separated into 32 512 sample “frames” (i.e., a 512 point FFT operation is used). Prior to the FFT, each frame can be categorized as either BLANK (contains no nonzero samples), NO BLANK (contains no blanked samples), or PARTIAL BLANK (some samples are blanked), as shown in Figure 7. The FFT is performed on each frame, the power computed in each FFT bin, and all results in each FFT bin are averaged.

[37] It is clear that the only effect of the BLANK category is to decrease the noise power level of the final average. It is trivial to correct for this effect simply by counting the total number of frames and the number of BLANK frames. However, the effect of the PARTIAL BLANK frames, which contain discontinuities, is more complex. An FFT operation on such a frame clearly will produce a distorted spectrum and a reduced noise power level, with the degree of distortion and power reduction related to the number of blanked samples within the frame. Clearly, narrowband noise sources may experience some distortion in this process; however, noise sources with bandwidths larger than a few MHz, which are the subject of this work, are not considerably affected. An example PARTIAL BLANK spectrum is compared to the corresponding NO BLANK average spectrum in Figure 8. The reduction in power level is clearly visible, along with a moderate distortion in the overall shape of the spectrum. Note the large power levels observed near the center of this spectrum are due to an internal DC component of the measured power, not due to external interference.

[38] Various means for coping with the PARTIAL BLANK issue can be conceived. A simple strategy (called method 1) is to eliminate such frames from the averaging operation; however this approach may also eliminate a large fraction of the incoming data in high-RFI environments. A second approach to retain these frames while correcting for the power level reduction is called “method 2: instantaneous scaling.” By Parseval's theorem, the effect of blanking on total average power of the frame can be corrected simply by increasing the power of the computed spectrum by *N*/*N*_{rem}, where *N*_{rem} is the number of nonblanked samples in the frame. This correction is applied to the power level of each frame before including the frame in the average computation. A final approach to is to included all (unscaled) frames in the spectral average operation, and to maintain a separate count of the total number of nonblanked time domain samples included in the average, labeled *N*_{tot,≠0}. Only the final average power is scaled by *N*_{tot}/*N*_{tot,≠0}, where *N*_{tot} is the total number of time domain samples that make up the average operation. This approach is termed “method 3: slow scaling”.

[39] The top plot of Figure 9 compares average spectra from a single Channel 6 LISA 16K capture before any blanking operations with the averages obtained from methods 1 and 2. APB parameters β^{2} = 90 and *N*_{blank} = 2048 were used in the APB algorithm. Both blanking methods are seen to be highly effective in removing contributions from a pulsed interferer centered at approximately 1315 MHz in this example. Results from Methods 1 and 2 are also observed to be very similar, demonstrating that the instantaneous scaling approach is correcting for the reduced noise power level due to blanking effects. To highlight the differences between methods 1 and 2, the bottom plot of Figure 9 illustrates the difference (subtracted decibel values) between the method 2 (instantaneous scaling) and 1 (NO BLANK only) average spectra. Differences are generally within 1.5 dB in all cases, and appear noise-like, indicating that further averaging would likely make these differences less significant.

[40] The difference between averaged spectra for methods 3 (slow scaled) and 1 (NO BLANK only) is also illustrated in the bottom plot of Figure 9; errors from method 3 are observed to be somewhat smaller than those from method 2 on average. Clearly method 3 is a simpler operation than that of method 2 (favorable for hardware implementation) since corrections are required at a much slower rate. In addition, method 3 should be preferable to method 2 because method 2 allows PARTIAL BLANK frames with a great deal of blanking to be weighted equally in terms of the power averaging computation. However, these frames also have the largest degree of spectral distortion, so reducing their weight should be advantageous. Figure 10 illustrates the mean error for methods 2 and 3 (compared to method 1) for LISA channels 7 through 13 when averaging results over the entire data set and the entire frequency spectrum. The mean error is of interest because it indicates the degree to which the average power level is not being corrected properly. Results clearly show the method 3 error (slow scaling) generally to be smaller than that of method 2. A hardware implementation of method 3 is currently in progress for the digital radiometer prototype.

#### 5.4. Frequency Domain Blanking

[41] Although the digital radiometer prototype does not implement RFI mitigation strategies in hardware after the FFT operation at present, it is of interest to simulate the expected performance of such approaches. The “channelization” of the FFT should allow an improved signal-to-noise ratio in detecting pulsed interference within a single FFT bin. A hardware blanking algorithm could conceivably operate on each FFT bin in real time by using a strategy identical to that of the APB processor. Such an approach would allow lower-level, rapidly pulsed RFI to be removed if missed by the original APB.

[42] This algorithm was simulated using the LISA data of 145 captures in channel 6. After passing each capture through the time domain APB algorithm with β^{2} = 40, and *N*_{blank} = 4028, each 16K capture was split into 32 512-point frames. An FFT operation was then applied to each frame, resulting in a total of 32 × 145 = 4640 temporal samples for each FFT bin. For each bin, a second APB algorithm with β^{2} = 90, *N*_{blank} = 4, and FIFO_LENGTH − N_{wait} = 2 was then applied to these 4640 samples, with the mean and variance computed by the averaging filter process used in the original APB algorithm. Figure 11 illustrates the average spectra before and after the frequency domain blanking operation, and shows a slight change in results near the center of the spectrum. The effect of the blanker is more obvious in the “max hold” spectra also illustrated in the plot; “max hold” refers to the maximum value of the 4640 temporal samples. Clearly a significant degree of RFI is included in this data set near the center frequency 1320 MHz; the frequency domain blanking operation reduces this interference so that corruption of the average spectrum is less significant.

[43] An alternative approach, “χ^{2} blanking” can also be considered if blanking at a slower temporal rate is deemed acceptable. In this method, the χ^{2} test is performed on data from each bin. If the χ^{2} value exceeds a specified critical value, samples in the data set exceeding a power threshold are removed and χ^{2} then re-evaluated. This iteration is repeated until the distribution satisfies the χ^{2} test for a specified critical value. The requirement for a slower temporal rate here is due to the complexity and iterative nature of this algorithm, which is not suited for integration in hardware. However, the algorithm could be applied to already integrated data as a postprocessing step in software.