Finding the Interference Karhunen-Loève Transform as an Instrument to Detect Weak RF signals

T he signal-processing methods currently used to detect interfering signals have reached their practical limits and have little more to offer in dealing with problem of very weak signal detection. The most popular of these include: • the Fast Fourier Transform (FFT), which directly transforms a signal into its spectral representation • the Short Time Fourier Transform (STFT), based on the Fourier Transform (FT), which gives information about frequency and time of the signal • the Wigner-Ville probability distribution , which also provides frequency and time information. This column presents a very competitive alternative to FT, namely the Karhunen-Loève Transform (KLT). In contrast to the FT, the KLT offers a solution to problems that are today still intractable. The KLT has indeed been proposed for applications to process the signals collected by astronomers worldwide in the framework of the SETI program (Search for Extra Terrestrial Intelligence). In a 2010 paper (cited in the Additional Resources section at the end of this column), Dr. Claudio Maccone presented his most recent discoveries concerning the KLT theory and its application to the detection of very weak signals hidden in noise. He also presented an example of the successful detection of an unknown sinusoidal signal with a signal-to-noise ratio (SNR) of as low as-23 decibels. He concluded that an even lower SNR could be mastered by the KLT. Following the ideas of the SETI scientists , this column will first discuss the advantages and limitations of the KLT, especially with respect to its application to detect typical interferences of GNSS signals. After providing some theoretical insight into the KLT approach, we will present examples of a successful detection of a very weak wideband signal, including a performance comparison of the KLT to the FFT, the STFT, and the Wigner-Ville methods. The KLT was not chosen for weak signal detection by accident. Several factors explain why KLT could be an appropriate mathematical tool for that purpose. As the Fourier Transform is of paramount importance in signal processing nowadays, this article uses it as the reference transform against which the KLT properties are compared. The main advantages of the KLT compared to the FT are the following: 1. The KLT works equally well for nar-rowband and wideband signals, GnSS signals, like other RF signals in general, are exposed to interference and jamming. In GnSS however, the problem is far more challenging and difficult to solve. The …