Introduction to Computer Music: Volume One

3. Filters

Filters are normally used to remove specific frequency components from a complex sound, hence the technique is often called subtractive synthesis, mentioned previously . This is not an entirely accurate description, since filters may also add energy to portions of the spectrum. Most analog synthesizers of the '70's and '80's came with the following four basic filter types (they are carried forward today on digital instruments with many variations): Lowpass, Highpass, Bandpass and Notch (sometimes called Band Reject). Lowpass filters cut off high frequencies, pass low frequencies. Highpass filters cut off low frequencies and pass high frequencies. Bandpass filters pass a certain width or band of frequencies and cut off frequencies on either side of the band.  And Notch filters cut out a notch or band in the center and passes frequencies to either side of the notch or band.

Looking at the graphs above, you will notice that most filters do not suddenly cut off sound at a specific frequency. Rather, they "roll off" the frequencies gradually, for example 12 dB's per octave. We specify a cutoff frequency (c.o.f.) at the point a specific frequency component would have lost approximately half the power (-3 dB) of unaffected frequencies, often referred to as the half-power point. A common synthesis technique is to sweep the cutoff frequency up or down to provide a 'spectral shape' to a sound over time. Cutoff frequencies are usually controlled by an envelope generator or an oscillator (timbre modulation).  The portion of the sound not reduced by 3 dB power or more is called the passband, while the rolled-off portion from -3 dB down to -infinity is called the stopband.

Additional filter types became available, some in analog form, but most in digital form due to the complex calculations. These include:

Low Shelf and High Shelf
A low shelf will either cut or boost signals below a c.o.f. in a manner resembling a shelf, or an equal strength amplitude band past the rolloff. A high shelf filter will either cut or boost signals above a c.o.f. similarly.

Peak/Notch Filter
Peak/Notch filters are frequently used for the mid-band frequency controls of multi-band equalizers where they are connected in series (see next page). Pictured below is MOTU Digital Performer's Masterwork EQ. On the low end and high end (pictured in gray) are the highpass and lowpass filters we saw above, but the four mid-bands (pictured in colors) are peak/notch filters with alternating cut and boost.

Controls are similar to the bandpass and notch filters shown above, with the degree of boost or cut and the width of the band of notch also controllable (with the 'Q' setting, described on the next page). A positive or negative gain value determines whether it is a peak or notch or flat in a single control. A key difference between a peak/notch and the other filters is that outside of the boost- or cut-band, it leaves the surrounding frequencies unmodified. With the filter gain parameter set to a negative value creating a notch, there is no difference between this filter and the classic notch filter above. However, by dynamically changing the gain over time, the peak/notch filter has the ability to change to a peaking filter, while a straight notch filter above does not have that capability.

Comb Filter
A comb filter creates a series of evenly spaced bands (or notches) that resemble the 'teeth' of a comb when excited by a resonant frequency. The teeth are created by constructive and destructive interference.

A group of comb filters is often used in parallel to resonate multiple frequencies of a signal. A short delay with feedback can create a comb filter. Acoustically, we may experience a comb effect if the same signal arrives at our ears at a slightly different time. Similarly, placing microphone at a different distance from a source can also create an unwanted comb effect.

Allpass Filter
An allpass filter may sound like a contradiction in nomenclature, in that it passes all frequencies at equal amplitude, and therefore does not seem to 'filter' in the traditional sense of frequency boost or cut. What it actually does is change the phase of various frequencies via time delay. In digital or electronic circuits, these may be used to "fix" certain phase displacements, but for our purposes, parallel banks of allpass filters plus comb filters are often used to create reverb effects and other constructive/destructive interference techniques that require both time and frequency alteration.

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