Audio Filtering

Filtering digital audio can mean many things, but most commonly refers to attenuating (reducing) or boosting (increasing) the gain (volume level) at some frequencies but not at other frequencies. The classic filter response types are

• lowpass: low frequencies pass through, high frequencies are eliminated
• highpass: only high frequencies pass through
• bandpass: some frequencies in the middle pass through, lows and highs are attenuated)
• bandstop or notch: frequencies in the middle are attenuated, leaving only the lows and highs.

Black Box Filters

Max offers several all-in-one filter objects. These objects hide the gory details of how the filters work and simply provide a couple of high-level attributes for controlling the filter. While not as flexible as the "open box" solution described below, these provide the easiest way to perform rudimentary filtering on audio signals.

• onepole~ : the simplest lowpass filter, with a gradual transition
• lores~ : a lowpass pass filter with a resonant peak, and a steeper transition
• reson~ : a simple bandpass filter

All of these filters may be used to produce the 'opposite' response using the -~ object. This means you can achieve a highpass response from a lowpass filter, or bandstop response from a bandpass filter. This was the subject of the Did You Know series #5.

The fffb~ object is simply a group of reson~ objects internally. You can visualize the fffb~ object as being somewhat like a band-separator for a graphic equalizer.

"Open Box" Filters

More flexibility is afforded where the basic filtering process is separated from the mechanism used to create the internal filtering parameters. The biquad~ and cascade~ objects both filter audio using a set of low-level coefficients. The coefficients are simply numbers generated using mathematical algorithms to make the filter work in some prescribed way.

The number of coefficients required for a filter is related to it's order. A simplification of filter order might say that for a higher order of filter, more complex or dramatic the filter responses may be attained. For a first or second order filter, you can use the biquad~ object. For higher order filters you must use the cascade~ object.

The challenge for these filters is generating the correct coefficients to attain a desired result. Max provides several objects to calculate the coefficients for various filter responses.

• filtergraph~: a user interface object/widget that provides an interactive means to control the filter specification, usually for 2nd-order filters
• filtercoeff~: an object that generates the coefficients for 2nd-order filters (biquad~) at audio signal rate
• filterdesign: generates the coefficients for filters up to 200th-order, which can be realized with the cascade~ object (or the biquad~ object in the case of 2nd-order filters).

Of these options, the filtergraph~ is the most easily accessible object, whereas the filterdesign object offers the most control and flexibility.

Using the filterdesign object

The simplest way to use a filterdesign object is to simply create it and connect it to a cascade~ object. Quite often, however, it is useful to attain the characteristics of a the filter you've created, and to visualize the impact of the filter. By connecting the output of a filterdesign object to the input of a filterdetail object, we are able to access the said characteristics. This includes the filter's magnitude (gain) response and group delay (phase distortion).

The output of the filterdetail outlets are simply lists of numbers. To visualize these, the lists can be sent to a multislider or plot~ object. The plot~ object offers a number of features that make it ideally suited to visualizing the output of filterdetail.

The filter is designed as a function of its attributes.

The response attribute defines the response type of the filter which may be lowpass, highpass, bandpass, or bandstop.

The frequency attribute sets the cutoff frequency. By default this value is expressed in Hertz, based upon Max's audio sample rate. The sample rate may be over-ridden with a sample rate of your choosing with the samplerate attribute. The object maybe define the cutoff frequency in units other than hertz by setting the units attribute. If the filter is a bandpass or bandstop filter, then there will be two cutoff frequencies defining the two edges of the band. In this case the attribute is set with a list of two numbers.

The order attribute will define the order of the filter. A higher order can achieve a steeper transition from the passband to the stopband, greater attenuation in the stopband, and a flatter passband, depending upon the filter topology. However, this higher order also comes at a cost: the CPU usage for the cascade~ object will be higher with a higher-order filter, the phase distortion will be greater, there could be more latency in the filter, and the filterdesign object itself will demand more cpu resources to calculate the coefficients.

The topology attribute of the filter will select between several different ways of organizing the poles and zeroes of the filter. Each topology has slightly different characteristics which are traded-off to achieve a particular goal.

Butterworth

The Butterworth topology provides a flat passband. Once the transition band begins to attenuate the signal it continues to decrease until it gets to zero (DSP nerds say that this is "monotonically decreasing"). There is some phase distortion with this filter, but less than for either of the Chebyshev Types. The easiest way to visualize the distortion is to look at the group delay output of the filterdetail object.

Chebyshev Type I

The Chebyshev Type I topology offers the steepest transition band of any of the current options in the filterdesign object. Like the Butterworth filter, it is monotonically decreasing in the stopband. However, the passband is not flat. This means that not all frequencies in the passband will have the same gain applied to them.

The peak-to-peak magnitude of the ripples can be controlled in this topology using the passband_ripple attribute, defined in decibels. By default this is set to 0.5 db. All other things held equal, greater amounts of ripple will give a steeper transition from passband to stopband.

Generally speaking, there is also more phase distortion in the passband of the Chebyshev Type I filter than the Butterworth of Chebyshev Type II. This may or may not matter depending on the application. This is the tradeoff for having the steepest filter at the minimum cost.

Chebyshev Type II

The Chebyshev Type II topology offers a steeper transition than the Butterworth, but not generally not as steep as the Chebyshev Type I. The passband is flat, like Butterworth but contrasting with Chebyshev Type I, but there are ripples in the stopband.

Like the Chebyshev Type I, the magnitude of the ripples may be traded-off against the steepness of the transition. This is controlled by setting the stopband_attenuation attribute to the value in decibels that defines the minimum amount of attenuation required. All frequencies in the stopband will then be attenuated by this amount or greater. Less stopband attenuation will result in a filter with a steeper transition if all other variables are held equal.

For musical purposes, the ripples in the stopband are often considered innocuous because they are attenuated to a point where they are not audible. The Chebyshev Type II provides the steepest filter while keeping the passband flat. For this reason, it is the default filter topology for the filterdesign object.

One caveat when working with the Chebyshev Type II filter is that the cutoff frequency is not defined in the same manner as for most classical filters. Typically, filters are designed where a cutoff frequency is defined to be the point where the magnitude has decreased to 3 decibels lower than the passband magnitude. In constrast, the Chebyshev Type II is designed for its stopband. The frequency attribute in this case defines the point at which the stopband_attenuation is first reached.

Additional Filter Responses

Allpass Filters

Allpass filters have a flat magnitude response, meaning that all frequencies pass through with unaltered gain. The phase, however, is altered. It is possible to define your own allpass filter for biquad~ or cascade~ by simply providing the feedforward (numerator) coefficients in the reverse order of the feedback (denominator) coefficients. Max also offers two pre-built allpass filters.

The allpass~ object implements a generic allpass filter. With a delay time of 1 sample, it is a canonical first-order allpass filter. With higher delay times it takes on some of the response characteristics of higher-ordered allpass filters, but still applies only a single coefficient. This yields a stair-stepped phase response with the number of steps numbering roughly half the number of samples of delay. The steepness of the steps (the amount of phase distortion at the equally-spaced intervals) is determined by the amount of feedback.

The phaseshift~ object implements a 2nd-order allpass filter. Rather than specifying the raw coefficients to the filter, you specify the frequency at which you want phase shift to be centered, and the steepness of the transition. The object then calculates the coefficients internally.

Comb Filters

Max's comb~ and teeth~ objects implement comb filters. In their simplest form, comb filters are merely delays with feedback, using smallish delay times to yield a rich and resonant effect. It is also possible to re-create this effect using using the tapin~ and tapout~ objects, provided that you have a small enough signal vector size set in Max's preferences.

The magnitude response of comb filters is composed of many, equally-spaced, peaks and a valleys repeated across the spectrum. The help patcher for the filterdetail object demonstrates some of the possibilities.