# Cut Filters

August 20, 2008 11:11:11 AM PDT

### A: Cambridge includes the following cut-filter types: coincident-pole, Bessel, Butterworth, and Elliptic. Let's examine each of these types.

#### Coincident-pole filter

The coincident-pole filter is made by placing one or more single-pole filters in cascade or series. A cascade arrangement is achieved by using the output of one filter as the input for the following filter in the cascade. Figure 1 shows four filters in cascade or series.

Figure 1

A one-pole filter is defined in terms of its cutoff frequency, or minus 3dB point. Below the cutoff frequency, the filter has a relatively flat frequency response. Above this frequency, the filter response begins to be attenuated at a rate of 6 dB per octave, or 20 dB per decade. When filters are placed in cascade, their magnitude-transfer functions multiply; if plotted in log-space, the magnitude-transfer functions "add." Placing multiple single-pole filters in cascade results in cut filters with rejection slopes at multiples of 6 dB per octave. Figure 2 shows magnitude-transfer functions for coincident-pole filters of order one, two, three and four. These filters have rejection slopes of 6, 12, 18, and 24 db/octave.

Figure 2

#### Bessel filter

The Bessel filter is a form of optimal filter. For a given filter order, the Bessel filter is the all-pole lowpass filter which has a phase response that is as close as possible to linear phase at low frequencies. For a given Bessel filter, two degrees of freedom are used to make the low-frequency magnitude response equal to one, and to choose the cutoff frequency for the filter. The rest of the degrees of freedom are spent making the group delay as close as possible to constant. The mechanics of filter design are carried out by using n degrees of freedom to set the first n derivatives of the group delay to zero. The (nearly constant) value of the group delay at low frequencies depends on the cutoff frequency for the filter.

Group delay represents the delay encountered by the envelope of a narrow-band signal when processed by the filter. It is calculated as the derivative with respect to frequency of the filter's phase response. Phase delay is the delay, in time, encountered by a single-frequency sinusoid when it is processed by the filter. Phase delay is calculated as the filter's phase response (in radians) divided by the frequency (in radians per second) at which the response was measured. Figures 3 and 4 are graphical depictions of phase and group delay. Group delay can be important when envelopes of signals must be preserved. Phase delay is relevant when multiple signals are to be mixed together, as coherence can be affected by phase delay.

Figure 3

Group delay represents the delay encountered by the envelope of a narrow-band signal
when processed by the filter.

A linear-phase filter has constant phase delay, which means that the filter's phase response is proportional to frequency. This results in a group delay that is constant and equal to the phase delay, since the group delay is simply the derivative of phase with respect to frequency. The Bessel filter approximates linear phase in its pass band, which produces nearly constant group delay and phase delay.

#### Butterworth filter

The Butterworth filter is another form of optimal filter. The optimization carried out for the Butterworth filter is to make the magnitude response of the filter as flat as possible at low frequencies. This is achieved by setting as many derivatives as possible of the magnitude response to zero at DC. The Butterworth filter has a flatter pass band than the Bessel filter, but has a phase response that is not as linear in the pass band. At very high frequencies, the Bessel and Butterworth filters have responses that asymptotically approach each other, as well as the coincident-pole filter of the same order. All three filters roll off at multiples of 6dB per octave, depending on filter order.

#### Elliptic filter

Yet another form of optimal filter, the Elliptic filter generates the smallest maximum gain-error in the pass band, the smallest maximum gain in the stop band, and the smallest transition region for a given filter order. Elliptic filters are equiripple filters in both the pass band and the stop band, which means that the magnitude of the transfer function "oscillates" between two limit values, hitting each of the two limits repeatedly in sequence. In the stop band, the response magnitude will achieve zero at discrete points, and grow to a predetermined maximum allowed value at discrete points in between. Unlike the Bessel, Butterworth, and coincident-pole filters, the Elliptic filter does not continue to roll off monotonically in the stop band; there is no rejection slope associated with the Elliptic filter. Instead, the Elliptic filter is guaranteed to have a response magnitude less than a given value at every point in the stop band. At very high frequencies, the response does not decay towards zero, but "oscillates" between zero and the prescribed value. Similarly, a "maximum allowed error" is defined for the pass band. If ripple can be tolerated in the pass band and stop band, the Elliptic filter produces a very sharp transition between pass band and stop band. It should be noted that Elliptic filters can have high group delay near the cutoff frequency.

Figure 5
Figure 6

Figures 5, 6, 7, and 8 show magnitude and phase responses for coincident-pole, Bessel, Butterworth, and Elliptic filters with a cutoff frequency of 100 Hz. It can be seen from Figure 5 that the Elliptic filter has the sharpest transition, but that it does not continue to roll off at high frequencies. Figure 6 shows the phase response for the filters over the whole audio band, but the frequency axis is logarithmic, so that linearity of phase is not easily recognized. Due to a zero in the transfer function, the Elliptic filter has a phase jump of 180 degrees between 2 and 3 kHz. This jump is not perceptually important, since it occurs in the stop band of the filter.

Figure 7

— Dr. Dave Berners