Passive RC Lowpass¶

A simple example for the bilinear transform is the RC lowpass:

Transfer Function

The resulting filter has a low pass characteristic with the following frequency response:

In the Laplace domain its transfer function is defined as:

$$ H(s) = \frac{\omega_c}{\omega_c + s} = \frac{1}{1 + \frac{s}{\omega_c}} $$

$s$ represents the Laplace operator

$$ s = \sigma + j \omega $$

and the cutoff frequency of the passive filter is:

$$ \omega_c = \frac{1}{RC} $$

Substitution

For a transform to the z-domain, the Laplace operator $s$ is subsstituted by the $z$ operator:

$$ s = \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}} $$$$ z = e^{sT} $$

The following steps rearrange the resulting transfer function to be comprised of summands with the factor $z^{-n}$:

$$ \begin{align} % % H(z) = & \frac{\omega_c}{\omega_c + \frac{2}{T} \frac{1-z^{-1}}{1+z^{-1}}} \\ % = & \frac{\frac{T}{2} \omega_c}{ \frac{T}{2} \omega_c + \frac{1-z^{-1}}{1+z^{-1}}} \\ % = & \frac{\frac{T}{2} \omega_c (1+ z^{-1})}{ \frac{T}{2} \omega_c (1+ z^{-1}) + 1-z^{-1}} \\ % = & \frac{\frac{T}{2} \omega_c + \frac{T}{2} \omega_c z^{-1}}{ \frac{T}{2} \omega_c (1+ z^{-1}) + 1-z^{-1}} \\ % = & \frac{\frac{T}{2} \omega_c + \frac{T}{2} \omega_c z^{-1}}{ \frac{T}{2} \omega_c + \frac{T}{2} \omega_c z^{-1} + 1-z^{-1}} \\ % = & \frac{\frac{T}{2} \omega_c + \frac{T}{2} \omega_c z^{-1}}{1 + \frac{T}{2} \omega_c + (\frac{T}{2} \omega_c -1 )z^{-1}} \\ % \end{align} $$

For getting the coefficients of our digital filter, the transfer function needs to meet the following strucutre where $a_0=1$:

$$ H(z) = \frac{b_0 + b_1 z^{-1}}{1+ a_1 z^{-1}} $$

This is achieved by expanding and simplifying:

$$ \begin{align} % % H(z) = & \frac{\frac{T}{2} \omega_c + \frac{T}{2} \omega_c z^{-1}}{ \underbrace{ \mathbf{\color{gray} 1 + \frac{T}{2} \omega_c }}_{\text{must be 1}} + \left(\frac{T}{2} \omega_c -1 \right)z^{-1}} \\ % = & \frac{\frac{\frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} + \frac{ \frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} z^{-1}}{1+ \left(\frac{\frac{T}{2} \omega_c}{1 + \frac{T}{2} \omega_c} - \frac{1}{1 + \frac{T}{2} \omega_c} \right)z^{-1}} \\ % = & \frac{\frac{\frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} + \frac{ \frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} z^{-1}}{1+ \left(\frac{\frac{T}{2} \omega_c -1}{ \frac{T}{2} \omega_c + 1} \right)z^{-1}} \\ \end{align} $$

Coefficients

Following the above rearrangements, the coefficients can be directly extracted from the transfer funcion:

$$ H(z) = \frac{\frac{\frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} + \frac{ \frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} z^{-1}}{1+ \left(\frac{\frac{T}{2} \omega_c -1}{ \frac{T}{2} \omega_c + 1} \right)z^{-1}} = \frac{b_0 + b_1 z^{-1}}{1+ a_1 z^{-1}} $$$$ \begin{align} \mathbf b_0 = & \frac{\frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} \\ \mathbf b_1 = & \frac{\frac{T}{2} \omega_c}{ 1 + \frac{T}{2} \omega_c} \\ \mathbf a_1 = & \frac{\frac{T}{2} \omega_c -1}{ \frac{T}{2} \omega_c + 1} \\ \end{align} $$

Filter Topology

The original dependence of the electrical components R and C is not important at this point. However, with $\omega_c$ = $2 \pi f_0$ we can get the exact coefficients for any cutoff frequency which can be used for the difference equation or applied in the following topology:

Plot Poles and Zeros in Z-Plane¶

Icecast

Henrik von Coler

2020-11-05 13:47

Icecast is a free software solution for creating accessible web-radio streams: https://icecast.org/

It plays nicely with JACK on Linux audio servers, allowing the broadcasting of complex sound synthesis and sonification projects.

IIR Filters

Henrik von Coler

2020-05-17 09:39

Polynominals¶

A filter of the order $N$ can be expressed as a polynominal, using the Z-operator $z = e^{sT}$:

$$ h(t) = \frac{\sum\limits_{n=0}^{n=N} b_n z^{-n}}{\sum\limits_{n=0}^{n=N} a_n z^{-n}} $$

Difference Equations¶

$$ \begin{align} y[n] = & b_0 x[n] + b_1 x[n-1] + ... + b_N x[n-N] \\ & -(a_1 y[n-1] + a_2 y[n-2] + ... + a_N y[n-N]) \end{align} $$

Or short:

$$ y[n] = \sum\limits_{i=0}^{i=N} b_i x[n-i] - \sum\limits_{i=1}^{i=N} a_i y[n-i] $$

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-1-c49ca754d0b5> in <module>
----> 1 ax = plt.subplot(111)
      2 ax.axis('equal')
      3 
      4 ax.spines['left'].set_position('center')
      5 ax.spines['bottom'].set_position('center')

NameError: name 'plt' is not defined

FIR Filters

Henrik von Coler

2020-05-17 09:39

Finite Impulse Response (FIR) filters are such digital filters which have - as the name suggests - a finite impulse response. These filters do not have an internal feedback and can be applied simply by convolving their impulse response with an input signal. For FIR filters, the impulse response $h[n]$ is a one-dimensional array containing the $N$ filter coefficients.

The following example shows the coefficients, respectively the impulse response for a filter of the order $N=5$:

The Difference Equation¶

From the impulse response we can get the coefficients $b_n$ to write the difference equation, which is the basis for applying a filter:

$$ y[n] = b_0 x[n] + b_1 x[n-1]+ b_2 x[n-2]+ b_3 x[n-3]+ b_4 x[n-4] + b_5 x[n-5] $$

Another important representation of a filter is the implementation structure. Each $z^{-1}$ block represents a delay by one sample. This case shows the direct form:

Applying a Filter by Convolution¶

Applying the filter with the given impulse response is done by simple convolution, which is the same as passing samples from the input of the implemtation structure above to its output. The above equation is then written as follows:

$$ \begin{eqnarray} y[m] &=& x[n] * h[m]\\ &=& \sum\limits_{m=0}^{M} h[m] x(n-m)\\ &=& \sum\limits_{m=0}^{M} b_m x(n-m)\\ \end{eqnarray} $$

The Transfer Function¶

The difference equation can also be expressed as a sum:

$$ h[n] = \sum\limits_{n=0}^{N} b_n x[n-1] $$

The transfer function of the filter can be obtained via the z-transform:

$$ H[z] = \sum\limits_{n=0}^{N} h[n] z^{-n} $$$$ H(z) = 0.5 z^{-2} + z^{-3} + 0.5 z^{-4} $$

In the next step, all exponents are shifted by expanding with $z^{4}$:

$$ H(z) = \frac{z^2 + 2z + 1}{z^4} $$

Factor the denominator to find its roots:

$$ H(z) = \frac{(z+1)(z+1)}{z^4} $$

There is a double zero at $\mathbf{z=-1}$. As all causal FIR filters it has $N=4$ poles at the origin $z=0$.

With $z=e^{j\omega}$ the complex roots can be obtained:

$$ Z = (e^{-i\omega} +1)(e^{-i\omega} +1) $$

In the Unit Cirlce¶

An LTI system can be visualized in the z-plane by plotting the zeros, respectively the roots of its impulse response.

Zeros = [-1. -1.]

Frequency Response¶

The frequency response of the filter

Plans and Directions

Henrik von Coler

2021-04-07 14:00

The original SPRAWL System was designed as a means for enhanced interaction in Local Area Networks (LAN) (read more). Due to the needs for online teaching and musical performance in 2020, the concept was extended to a Wide Area Network (WAN) solution. In the coming months, the project with be extended in a cooperation with LTU, Sweden and Ritmo, Oslo. The following areas can be tackled in this period.

Multichannel Clients

In this semester we will extend the system to incorporate clients and endpoints with difeerent configurations, including larger loudspeaker setups:

Extended SPRAWL System, allowing Access Points with different configurations.

Gestural Control

All three sites in the project will be equipped with a motion tracking system and multichannel loudspeaker setups, using different configurations. Experiments will aim at sharing spatial sound between the sites and manipulating them with gestural input.

21 channel Ambisonics setup with motion capturing system at BOL.

Clocking and Syncing

Another focus will be experiments with shared clocks and sequences, in cooperation with the RITMO group in Oslo: https://www.uio.no/ritmo/english/projects/dr-squiggles/index.html

We have the possibility to join the Rhythm Production and Perception Workshop for our closing concert on June 22-25.

Back to NSMI Contents

Digital Filters

Henrik von Coler

2020-05-17 09:39

Playing Samples in SuperCollider

Henrik von Coler

2021-05-02 10:40

The Buffer class manages samples in SuperCollider. There are many ways to use samples, based on these buffers. The following example loads a WAV file (find it in the download) and creates a looping node. When running, the playback speed can be changed:

s.boot;

// get and enter the absolute path to a sample
~sample_path = "/some/directory/sala_formanten.wav";

~buffer  = Buffer.read(s,~sample_path);

(
~sampler = {

      |rate= 0.1|

      var out = LoopBuf.ar(1,~buffer.bufnum, BufRateScale.kr(~buffer.bufnum) * rate, 1, 0,0,~buffer.numFrames);

      Out.ar(0, out);

}.play;

)

// set the play rate manually
~sampler.set(\rate,-0.1);

Exercise

Combine the sample looper example with the control bus and mouse input example to create a synth for scratching sound files.

Sending OSC from SuperCollider

Henrik von Coler

2021-04-10 10:40

For sending OSC from SuperCollider, a NetAddr object needs to be generated. It needs an IP address and a port:

~out_address  = NetAddr("127.0.0.1", 6666);

Sending Values Once

This first example sends an OSC message once when the following line is evaluated. The previously created NetAddr object can be used to send OSC messages with its sendMsg method:

~out_address.sendMsg('/test/message', 1);

Sending Values Continuously

Based on the previous example, a routine can be created which continuously reads values from control rate buses to send their instantaneous value via OSC. The osc_routine runs an infinite loop with a short wait interval to limit the send rate and the CPU load:

  ~cBus = Bus.control(s,1);

  ~osc_routine = Routine({

        inf.do({

      // read value from bus
                  var value      = ~cBus.getSynchronous(~nVbap);

      // send value
                  ~out_address.sendMsg('/oscillator/frequency', value);

                  // wait
                  0.05.wait;

          });
});

Once created, the routine can be started and stopped with the methods play() and stop(). While running, bus values can be changed to test the functionality:

~osc_routine.play();

~cBus.set(300);

~cBus.set(700);

~osc_routine.stop();

Exercise

Run the PD patch osc-receive.pd to receive values from SuperCollider via OSC and control the pitch.

Laplace Transform

Henrik von Coler

2021-04-14 16:00

The Laplace transform is a time frequency transform allowing the analysis of LTI systems in terms of their impulse response behavior. This is relevant when designing and investigating filters or other processing units. The transform is defined by the following integral:

$$ F(s) = \int\limits_0^\infty e^{-s t} f(t) dt $$

$s$ is the Laplace operator

$$ s = \sigma + j \omega $$

which can be decomposed into two terms:

$$ F(s) = \int\limits_0^\infty e^{-\sigma t} \ e^{-j \omega t} f(t) dt $$

$e^{-\sigma t}$ = Decay term

$ e^{-j \omega t}$ = Frequency Term

Fourier-Relation¶

For a value of $\sigma=0$, the Laplace transform equals the Fourier transform.

Receiving OSC in SuperCollider

Henrik von Coler

2021-04-10 10:40

By default, a running instance of sclang listens to incoming OSC messsages on the port 57120.

OSCFunc

For listening to a specific OSC message, an OSC function can be defined with the matching path:

 ~osc_receive = OSCFunc(

         { arg msg, time, addr, recvPort;

postln('Revceived message: '+msg[0].string' - '+msg[1].string);

 }, '/test/message');

OSCdef

OSCdef is slightly more flexible and allows to change definitions on the fly, without deleting nodes:

OSCdef(\tester,
              {|msg, time, addr, recvPort|
                      var val;
                      val = msg[1];
                      val.postln;
      },'/test/another', n);