However, it appears to have a slight ripple in the passband, and to descend slightly more rapidly around the cut-off frequency. In screenshot (b), the actual response is very similar to that in screenshot (a). However, it does not drop vertically between one and the other at the cut-off frequency of 5000, but descends diagonally between frequencies of approximately 42. In screenshot (a), the actual response seems to follow the ideal response exactly in the passband and stop band – that is, it does not oscillate but remains horizontal. The actual response is shown as a second line. In each case, the ideal filter response is shown as a horizontal line at 1 below the cut-off frequency and a horizontal line at 0 above the cut-off frequency, with the cut-off frequency represented as a vertical line at 5000. This figure consists of two parts, each showing a screenshot from Signal Wizard. You can see that both have reduced the ripples in the passband. The design implemented in Figure 29(a) uses a Blackman window, while the design implemented in Figure 29(b) uses a Hamming window. The next set of designs, in Figure 29, keeps the number of taps at 127 and varies the window function used. Figure 28 Digital filter design with a rectangular window: (a) 15 taps (b) 63 taps (c) 127 taps The oscillations die away more quickly than in screenshot (b). In screenshot (c), the actual response oscillates very rapidly around 1, then descends diagonally between frequencies of approximately 49, then oscillates very rapidly around 0. The oscillations die away more quickly than in screenshot (a). In screenshot (b), the actual response oscillates more rapidly around 1, then descends diagonally between frequencies of approximately 47, then oscillates more rapidly around 0. In screenshot (a), the actual response oscillates slowly around 1, then descends diagonally between frequencies of approximately 37, then oscillates slowly around 0. This figure consists of three parts, each showing a screenshot from Signal Wizard. However, the amplitude of the ripples in the passband and the stop band remains unchanged, although the frequency increases as the number of taps increases. As the number of taps in the design increases from the top image to the bottom, the transition zone narrows and the designed filter more closely matches the filter specification. The first design uses 15 taps (Figure 28(a)), the second uses 63 taps (Figure 28(b)) and the third uses 127 taps (Figure 28(c)). These designs all use a rectangular window, which gives an abruptly truncated sinc function. The graphical interface windows in Figure 28 all show the ‘brick-wall’ specification in red with the implementation in black. The specification of an FIR low-pass filter with a gain of 1 and a cut-off frequency of 5000 Hz is entered into the filter design interface. Figure 27 Characteristics of the input signal The overall shape of the frequency spectrum is a diagonal line that descends steadily from minus 40 at a frequency of 0 to minus 100 at a frequency of just below 20 thousand, but again it fluctuates randomly by a small amount around this line. Most of the time it stays within the range minus 0.25 to plus 0.25, only occasionally going outside this. The time waveform fluctuates randomly around a value of 0. It shows two graphs, one labelled Time Waveform and one labelled Frequency Spectrum. It is typical for filter design to be an iterative “guess and check” process until the exact desired weights or frequency response is obtained.This figure is a screenshot from Signal Wizard. The filter lengths are not exact but are reasonably close to the length L=21 the filter was designed with. Similarly from Figure 7, the second filter with and gives a filter length approximation ofĪnd the second filter with and gives a filter length approximation of Using the parameters from Figure 7, =39 and, the filter length for a transition bandwidth of is approximated to be However the cut-off frequency has no impact which reinforces the earlier filters from Figures 4 – 8. The approximation in ( 2) shows that changes to the transition bandwidth will directly impact the sidelobe attenuation when the filter length is held constant. The frequencies and can be in any units as long as they are consistent. Where is the transition bandwidth, is the sampling frequency and is the sidelobe attenuation in dB. Fred harris’ filter length approximation is
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