However, in some geographical areas, RFI is so persistent in time that is not possible to acquire RFI-free radiometric data. Current mitigation techniques are mostly based on blanking in the time and/or frequency domains where RFI has been detected. They include time- and/or frequency domain analyses, or statistical analysis of the received signal which, in the absence of RFI, must be a zero-mean Gaussian process. In recent years, techniques to detect the presence of RFI have been developed. 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 the pre-detection filters due to the finite rejection modify the detected power and the estimated antenna temperature from which the geophysical parameters will be retrieved. The performance of microwave radiometers can be seriously degraded by the presence of radio-frequency interference (RFI). The new, modified signal is obtained by applying the inverse Gabor transform. This is easily achieved by multiplying the Gabor function with some other function α ( τ, ω ) that has necessary characteristics. To do both simultaneously in the Gabor domain, one simply suppresses high or low frequencies in G ( τ, ω ) depending on the τ variable. In the Fourier domain, blurring is achieved by suppressing the high (spacial) frequencies, while sharpening is achieved by suppressing the low frequencies. For nonstationary filtering of a photographic image, one might imagine blurring one part of an image, while simultaneously sharpening another part of the image. Here, we want to simplify by working with images. Again, understanding the filtering process for seismic data assumes the student already understands a lot about the physics of the processes generating a seismic signal. In a seismic signal, one may wish to perform different levels of whitening at different parts of the data, or design a deconvolution method that changes the frequency content of a signal by different amounts, at different times in the signal. The key idea behind nonstationary filtering is to somehow apply different filters (or filters with different characteristics) to different parts of a signal. Their product then forms the partition of unity. A useful compromise which localizes both the Gabor transform and the reconstruction is to choose both w ( t ) and w ( t ) to be the square root of the raised cosine window. The roles could be reversed, so w ( t ) is the constant one, and w ( t ) is raised cosine window. In these examples, the window function w ( t ) is carefully chosen so that the dual window can simply be taken as the constant one. Summing many such windows gives a flat plateau as large as necessary. If we take nine of these windows, prop- erly spaced, we see they sum to a constant plateau in the middle, as shown in Figure 7. A better choice in two dimensions would be the product of two raised cosine functions, w ( t ) = (cos( t 1 ) − 1)(cos( t 2 ) − 1), as shown in Figure 6. However, no matter how one translates these, they do not sum to one exactly (although good approximations are possible). Gabor proposed using multidimensional Gaussians as the window functions, w ( t ) = e − t 2, as shown in in Figure 5.
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