Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

26-30 Passive, Active, and Digital FiltersSignal amplitude420−2−40 500 1000420−2−40 500 1000420−2−40 500 1000420−2−40 500 1000420−2−40 500 1000420−2−40 500 1000Time indexFIGURE 26.12 Power line communications signal enhancement: (a) transmitted signal, (b) observed signalcorrupted by a ¼ 1:25 noise, output of the (c) mean [2:5388]f8:6216g, (d) median [2:4267]f7:4393g, (e) myriad[2:4385]f7:4993g; and (f) meridian filters [2:4256]f7:4412g, where [ ] and fg denotes the mean absolute andsquared error for the corresponding filtering structure.weights, but also the ability of robust nonlinear filters to perform frequency selectivity that is traditionallyaccomplished through linear filtering. Figure 26.13a depicts a two-tone signal with normalized frequencies0.02 and 0.4 Hz. The signal is 1000 samples long, although a cropped version of the original signalf0,200g is shown for presentation purposes. Figure 26.13b shows the multitone signal filtered by a 40-taplinear FIR filter designed by the MATLAB fir1 command with a normalized cutoff frequency of 0.3 Hz.The myriad and median filter weights are optimized in this case utilizing the adaptive methods detailed inthe previous section.* Noting that the median filter is a limiting case of the meridian filter, the meridianfilter weights are set equal to those of the median.The clean multitone input signal is corrupted by additive a-Stable noise with a ¼ 0:4 to form acorrupted observation from which the single high frequency tone must be extracted, Figure 26.13c. Thenoisy multitone signal is processed by the linear, median, myriad, and meridian filters with the resultsgiven in Figures 26.13d through g, respectively. Similarly to the first example, the observation signal inthis case contains very heavy tailed outliers. These outliers cause the linear filter to, once again, breakdown. Each of the nonlinear filters, in contrast, offers more robust processing, with the outputs moreclosely reflecting the range and frequency content of the desired signal. The increasing robustness ofthe operators is again apparent, with the myriad being more robust to outliers than the median and themeridian being the most robust. It should be noted that the nonlinear filters not only minimize theinfluence of outliers, but their flexible weighted structures are able to effectively pass desired frequencycontent while rejecting content outside the desired frequency band. The ability to simultaneously reject* The clean multitone signal and the desired high-frequency signal are utilized as the input and desired signals, respectively.For more detail see Refs. [28,29,31].