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2.7 Chebyshev and Butterworth filters

The design of higher-order filters is a specialist area, and mathematically complex, so in this section you will look at gain functions of just two celebrated types. The first is the Butterworth filter. The gain function of a Butterworth filter has the familiar flat passband and roll-off you would expect. However, this filter comes in various orders, such as 2, 4, 6, 8 and 10, depending on how steep the roll-off needs to be.

Figure 11 shows the normalised responses of first-order, second-order and eighth-order low-pass Butterworth filters. As the order of the filter increases, it approximates more closely the ideal brick-wall response.

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Figure 11 Normalised Butterworth magnitude response curves
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This figure is a graph of gain G against normalised frequency. The ideal brick-wall response is shown as a horizontal line at gain 1 to the left of the cut-off frequency, then a vertical line at the cut-off frequency, with the gain above the cut-off frequency being zero. Three graph lines are shown in comparison to this ideal, representing first-order, second-order and eighth-order filters. Each line starts at a gain of 1 for low frequencies, and curves down first gradually and then more steeply to reach a gain of 1 over root 2 at the cut-off frequency. After this, each line continues to descend steeply and then more gradually, until at high frequencies it approaches zero. The higher the order of the filter, the steeper the descent and the closer the graph line comes to the ideal.

Figure 11 Normalised Butterworth magnitude response curves

Steeper roll-offs than those of a Butterworth filter can be had if other desirable features of a filter are compromised. The Chebyshev filter has a steeper roll-off, but at the price of a passband response that is not flat. Figure 12 compares seventh-order normalised Chebyshev and Butterworth filters. Notice the ripple in the Chebyshev filter’s response in the passband. (There is another type of Chebyshev filter that has a flat response in the passband and ripples in the stop band.)

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Figure 12 Comparison of seventh-order Butterworth and Chebyshev gain
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This figure is a graph of gain G against normalised frequency. Two graph lines are shown, representing the Butterworth and Chebyshev filters.

The Butterworth line follows the shape seen in the previous figure: it starts at gain 1 for low frequencies, and curves down first gradually and then more steeply to reach a gain of 1 over root 2 at the cut-off frequency. After this, the line continues to descend steeply and then more gradually, until at high frequencies it approaches zero.

Below the cut-off frequency, the Chebyshev line oscillates between a gain value of 1 and a gain value of halfway between 1 and 1 over root 2. At the cut-off frequency, it drops very steeply and approaches a gain of zero at lower frequencies than the Butterworth line.

Figure 12 Comparison of seventh-order Butterworth and Chebyshev gain

That concludes the introduction to filters and, in particular, analogue filters. In Section 3 you will learn about digital filtering, which is currently a more popular approach to filtering.