iir-filter.rst

Single Pole Lowpass IIR filter

In this example we'll use these:

>>> from audiolazy import lowpass, pi, z, saw_table, sHz
>>> from pyscanprev import enable_scan, prepend
>>> from hipsterplot import plot

Say we want an IIR (Infinite Impulse Response) LTI (Linear Time Invariant) digital filter, e.g. a lowpass filter whose cutoff frequency is about pi/30 rad/sample (i.e., 735 Hz when the sample rate is at 44100 samples per second). After some math, an approximation for a single pole filter would be:

>>> radius = 0.9

This filter can be described by its Z-Transform equation:

>>> lowpass(pi / 30)
     0.0993371
-------------------
1 - 0.900663 * z^-1

Let's store this filter with our radius for further comparison:

>>> lzfilt = (1 - radius) / (1 - radius * z ** -1)

In the time domain, it can be expressed as:

y[n] = (1 - radius) * x[n] + radius * y[n - 1]

Let's create the filter using PyScanPrev:

>>> @enable_scan("yn_1")
... def pspfilt(data):
...     return [(1 - radius) * xn + radius * yn_1 for xn in data]

Say we have some square way signal at 315 Hz (pi/70 rad/sample), let's get 6.35 milliseconds (280 samples) from that signal:

>>> data = [0] * 70 + [1] * 70 + [0] * 70 + [1] * 70
>>> plot(data, num_x_chars=56)
    0.9667               ##############              ##############
    0.9000
    0.8333
    0.7667
    0.7000
    0.6333
    0.5667
    0.5000
    0.4333
    0.3667
    0.3000
    0.2333
    0.1667
    0.1000
    0.0333 ##############              ##############

Comparing the results for both filters:

>>> lzresult = list(lzfilt(data))
>>> pspresult = pspfilt(prepend(0, data))[1:]
>>> lzresult == pspresult
True
>>> plot(pspresult, num_x_chars=56)
    0.9661                    #########                   #########
    0.8994                  .#         .                .#
    0.8328                  |          .                |
    0.7662                 |                           |
    0.6996                 :           .               :
    0.6329                :            .              :
    0.5663                .            .              .
    0.4997                :             :             :
    0.4331               .              .            .
    0.3664               .              :            .
    0.2998               .               :           .
    0.2332                               |
    0.1666               .                |          .
    0.0999               .                .#         .
    0.0333 ##############                   #########

Applying it on a 150 Hz (pi/147 rad/sample) sawtooth wave:

>>> s, Hz = sHz(rate=44100)
>>> saw_data = saw_table(150*Hz).take(0.01*s)
>>> plot(saw_data, num_x_chars=56)
    0.9268                                   .##|
    0.7939                                 ###
    0.6610                              .##:
    0.5281                            ###
    0.3953                         .##:
    0.2624                       ###
    0.1295                    .##|
   -0.0034                  ###                                  :#
   -0.1363                ##|                                  ###
   -0.2691             ###                                  .##|
   -0.4020           ##|                                  ###
   -0.5349        |##                                  .##|
   -0.6678      ##|                                  |##
   -0.8007   |##                                  .##|
   -0.9336 ##|                                  ###
>>> plot(pspfilt(saw_data), num_x_chars=56)
    0.8676                                    ##|
    0.7388                                 |##  .
    0.6100                               ##|    .
    0.4812                            :##.
    0.3524                          ###         .
    0.2236                       .##:           .
    0.0948                     |##              ..
   -0.0340                   ##|                 .                #
   -0.1628                |##                    .              ##|
   -0.2916              ###                      :           |##
   -0.4204           :##.                        |         ###
   -0.5492         ###                            |     :##.
   -0.6780      .##|                              #:  ###
   -0.8068    |##                                  ###.
   -0.9356 ###|

Other linear filters can be created with the combination of IIR/FIR filters and their results. FIR (Finite Impulse Response) filters don't require knowledge about the previous inputs, these would be done with common list comprehensions or AudioLazy ZFilter objects like 1 - z ** -1.

The State-space model PyScanPrev example has more signal processing / control theory models, including another LTI filtering example.