# MSP Filter Tutorial 1: Simple Filters

Click here to open the tutorial patch: 01fSimpleFilters.maxpat

## Contents |

### Introduction

This group of tutorials look at different ways to use *filters* in MSP.
This includes the basic uses of filters for the equalization and shaping of a
sound and using filters to create timbres in subtractive synthesis. Along the
way, we'll look at some of the theory behind filters and how they work.

### So what is a filter, anyway?

Filters are just algorithms that alter the frequency spectrum of
a sound. When working with digital audio in the time domain (i.e. as a
stream of samples representing the amplitude of a wave), filters are
implemented as equations that use short *delays* to shape an
incoming waveform.

As an example, let's say we wanted to roll off the treble on an audio signal. If we plot a waveform, we can intuit the visual difference between low frequency and high frequency content:

*Two waveforms, one with a low frequency, one with lots of high frequencies*

As we can see, the top waveform (stored in the buffer~
named `lowfrequency`

, contains a sine wave at 50 Hz. The bottom
waveform (in the `highfrequency`

buffer~) contains a
complex FM tone with lots of high frequencies. If we wanted to roll
off the treble on the bottom waveform, we could think of how it looks:
high frequencies look like sharper angles when plotted in time. In
order to cut high frequencies, we could *smooth* this waveform.
One way to smooth a signal is to *average* it over time.

Let's say that we take a much simpler signal, that of a single sample
of `1`

in a sea of `0`

's. This is called an *impulse*:

An impulse has a frequency response equivalent to pure noise... hypothetically, all frequencies are present at equal volume (think of a 'click' in a digital audio signal or any other short burst of sound). So it contains plenty of high frequencies. If we wanted to smooth this signal, we could average each sample with the previous sample in this signal:

*Our impulse, smoothed over one sample*

This has the result of smearing the energy of the impulse
across two samples. As a result, its frequency response will
contain much less high-frequency energy; in fact, it's almost
as if we've lowered the sampling rate: a click that lasts one sample
at 44,100 Hz contains energy all the way up to 22,050 Hz; by derivation,
a click that lasts *two* samples at that sampling rate is the same
as a one-sample click at half that rate, i.e. it only has energy up
to 11,025 Hz.

### Some filter definitions

If we were to generalize what we just did to our impusle when we smoothed it, we could say this:

y_{n} = 0.5x_{n} + 0.5x_{n-1}

where **x** represents *incoming* samples, **y** represents
outgoing samples, and **n** represents the current *time* on
the sample clock (i.e. *now*). This equation defines the filter:
we're averaging (multiplying by 0.5) the current and previous incoming
samples to generate the outgoing samples.

To put a name on this filter, we could call it a first-order non-recursive
lowpass filter. The *order* of a filter refers to how many samples of
delay it contains: because we're only looking at one previous input, it's a
first-order filter. Because the filter only uses incoming samples in its
equation, it's non-recursive. As for what it does, it *passes* low
frequencies (and cuts high ones): hence the term *lowpass*.

Now consider this equation:

y_{n} = 0.5x_{n} + 0.5y_{n-1}

This filter uses the previous *outgoing* sample from the filter
as part of the filter itself; by implementing feedback in the filter,
we get a much stronger effect:

*Our impulse, averaged with the previous output sample*

This equation defines a *recursive* filter; as a result, the effect
of the filter is dissipated beyond the order of the filter. While our
first equation spread the energy of our one-sample click over two samples,
this new equation spreads the energy over many, because of the averaging.
Consider how the click interacts with the equation:

(x_{n} + y_{n-1}) / 2 = y_{n}

(1.0 + 0.0) / 2 = 0.5

(0.0 + 0.5) / 2 = 0.25

(0.0 + 0.25) / 2 = 0.125

(0.0 + 0.125) / 2 = 0.075

and so on...

In the filter described above, the energy of the click, hypothetically,
will *never* fully dissipate. Another term for this kind of filter
is an *IIR*, or infinite impulse response, filter; our first filter,
which only uses incoming samples in its terms, has a finite impulse
response (an *FIR* filter).

In a later tutorial, we'll revisit some more filter theory. For now, it's simply important to understand that filters are made by manipulating very short (often single sample) delays (either with or without feedback) and mixing them with the current sample.

### Our first filter: lores~

Take a look at the tutorial patcher. Patcher area `1`

contains
a simple sampler, playing the *sacre.aiff* sound (loaded into
a buffer~ named `chords`

) using the groove~ object.
The circuit shown in this patch allows us to "play" the sample at any
pitch with the kslider.

labeled 'Dry volume' and play some of the notes on the kslider. You should hear the sample play at different notes.

- Turn down the 'Dry volume' and turn up the next number box,

labeled 'Lowpass volume'. Notice the change in sound. Turn the dial object
at the top of patcher region `2`

. As you move the dial to a higher
value, more of the high frequencies from the sample are audible.

The lores~ object implements a *lowpass* filter on an incoming audio
signal (in our case, the output of the groove~ object. A lowpass filter,
as we saw in the tutorial introduction, passes the low frequencies and attenuates
the high frequencies of the incoming signal. The two parameters that the filter
takes are the *cutoff frequency* (specified in the middle inlet or as the
first argument to the object) and the *resonance* (specified in the right
inlet or as the second argument).

The cutoff frequency of a lowpass filter determines the frequency at which the
audio is attenuated 6 dB. The resonance amount, when greater than `0.`

,
controls a peak of resonation (boosted frequencies) immediately below the cutoff.
If we plot the response of the filter on a graph with the *X* axis
representing frequency and the *Y* axis representing gain, it would look
like this:

*A lowpass filter with and without resonance: A and B are the*
cutoff frequencies;

**C**shows the resonance peak

- With the sound going, adjust the number box labeled 'Resonance' in patcher

area `2`

. Notice how as the resonation approaches `1`

the ringing at the
resonance frequency becomes very loud. Adjust the cutoff frequency with the
resonance set to a high number. Notice how you can now audibly "sweep" the filter
based on hearing the resonation.

### Bandpass filters: the reson~ object

- Turn down the volume of the lowpass filter, and look at section
`3`

in the

tutorial. Turn up the number box labeled 'Bandpass volume'. Sweep the dial labeled 'Center frequency' and listen to the result.

Just as a lowpass filter passes low frequency, a *bandpass* filter passes
a **band** of frequencies, attenuating anything lower or higher than a center
frequency. The MSP reson~ object implements a bandpass filter with three
parameters (controllable as inlets or arguments): the filter's gain, the center
frequency, and something called the *Q*.

- With the dial controlling the center frequency at 12 o'clock, click in

the number box labeled 'Q' in patcher area `3`

. Type the number `3`

and hit return. Listen to the results. Try other positive numbers, such as `6`

,
`10`

, or `0.5`

. Notice that the higher the number is, the less
frequencies make it through the filter. If necessary, adjust the 'Gain' with
the number box to the left.

Q is a measure of the *width* of a bandpass filter, and is expressed as the
ratio of the center frequency divided by the *bandwidth* of the filter, i.e.
the distance (in Hz) between the lower and higher -6 dB points around the center
frequency. Higher Q values mean narrower filters (less bandwidth relative to the
center frequency):

A bandpass filter with Q values of `0.5`

, `1.0`

, `3.0`

, and `30.0`

, respectively.

### The state-variable filter: svf~

- Turn down the 'Bandpass volume' and look at patcher area
`4`

. Turn up

the number box labeled 'Lowpass', adjust the 'Cutoff/Center Freq.' dial, and set the 'Resonance' number box to something that sounds good to you. Now, turn down the 'Lowpass' control and raise the 'Highpass'. Notice the difference. Do the same with the <link type="refpage" name="number">number box</link> objects labeled 'Bandpass' and 'Notch'. Play with different combinations of volumes and settings.

The MSP svf~ object simulates an analogue *state-variable* filter.
Because of the way in which filters are wired using electronic components,
the difference between one type of filter and another is often simply a
matter of how you wire (or where you 'tap') the circuit. A state-variable
filter is a filter that allows you to 'tap' energy from several places in
the filter, getting four simultaneous filters for the price of one.
The svf~ object gives you four filtered sounds: a lowpass output
(which cuts frequencies above the cutoff frequency), a *highpass* output
(which cuts frequencies *below* the cutoff), a bandpass output
(cutting frequencies *around* the center frequency), and a *notch* output.
The notch output cuts the area directly around the center frequency and should
mirror the response of the equivalent bandpass filter. Notch filters are often
called *bandstop* or *bandreject* filters. A plot of these possibilities
shows their frequency responses:

The outputs of a state-variable filter: lowpass, highpass, bandpass, notch.

- Play with different combinations of the filters in the tutorial, mixing

them in different ways with different settings. In the next tutorial, we'll look at building more complex filter arrangements.

### Summary

In digital signal processing, *filters* refer to equations which
modify the frequency response of a signal. Filters are constructed by
mixing small amounts of delayed signal with the original, smoothing or
sharpening the waveform to accentuate or attenuate different frequencies.
Common filter types include lowpass, highpass, bandpass, and notch. Lowpass
filters can be created with the lores~ object, bandpass filters with
the reson~ object, and all four with the svf~ object. Filters
commonly have controls for their center or cutoff frequency and their Q or
resonance.

### See Also

lores~ - Resonant lowpass filter

reson~ - Resonant bandpass filter

svf~ - State-variable filter with simultaneous outputs