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Lecture 18a, Digital Filtering 


Digital Filtering uses numerical computation rather than analog
components such as resistors, capacitors and inductors to
filter out frequency bands.

A low-pass filter will have a "cutoff frequency" f  such that
frequencies above f will be attenuated and frequencies below f
will be passed. The filter does not produce a sharp dividing
line and all frequencies are changed in both amplitude and
phase angle.

A high-pass filter will have a "cutoff frequency" f such that
frequencies below f will be attenuated and frequencies above f
will be passed. The filter does not produce a sharp dividing
line and all frequencies are changed in both amplitude and
phase angle.

A band-pass filter will pass frequencies between f1 and f2,
attenuating frequencies below f1 and attenuating frequencies
above f2. f1 <= f2. 

The signal  v(t)=sin(2 Pi f t) t=t0, t1, ..tn is a single frequency f.
For digital filtering we assume the digital value of v(ti) is
sampled at uniformly spaced times t0, t1,..., tn.




For an order m filter, with one based subscripts:

 y(i) = ( a(1)*x(i) + a(2)*x(i-1) + a(3)*x(i-2) + ... + a(m+1)*x(i-m)
                    - b(2)*y(i-1) - b(3)*y(i-2) - ... - b(m+1)*y(i-m) )/b(1)




With array x and y in memory, the MatLab code for computing y(1:n) could be:

   for i=1:n
     y(i)=a(1)*x(i);
     for j=1:m
       if j>=i
         break
       end 
       y(i)=y(i) + a(j+1)*x(i-j) - b(j+1)*y(i-j);
     end
     y(i)=y(i)/b(1);  % not needed if b(1) equals 1.0
   end

or, use MatLab   y = filter(a, b, x); % less typing



For an order m filter, with zero based subscripts:

 y[i] = ( a[0]*x(i) + a[1]*x[i-1] + a[2]*x[i-2] + ... + a[m]*x[i-m]
                    - b[1]*y[i-1] - b[2]*y[i-2] + ... - b[m]*y[i-m] )/b[0]

With array x and y in memory, the C code for computing y[0] to y[n-1] could be:

   for(i=0; i<n; i++) {
     y[i]=a[0]*x[i];
     for(j=1; j<=m; j++)
     {
       if(j>i) break;
       y[i]=y[i] + a[j]*x[i-j] - b[j]*y[i-j];
     }
     y[i]=y[i]/b[0];  /* not needed if b[0]==1 */
   }


For reading x samples and writing filtered y values:
  
  read next x sample and compute y and output y
  (the oldest x and y in RAM are deleted and the new x and y saved.
   Typically use a circular buffer [ring buffer] so that the saved x and y
   values do not have to be moved [many uses of modulo in this code].)

Given a periodic input, filters require a number of samples to build up
to a steady state value. There will be amplitude change and phase change.

Typical symbol definitions related to digital filters include:

  ω = 2 Π f

  z = ej ω t  digital frequency

  s = j ω    analog frequency

  z-1 is a unit delay (previous sample)

  IIR Infinite Impulse Response filter or just recursive filter.

  In transfer function form

  yi     a0 + a1 z-1 + a2 z-2 + a3 z-3 + ...
 ---- = -----------------------------------
  xi          b1 z-1 + b2 z-2 + b3 z-3 + ... 
 
  s = c (1-z-1)/(1+z-1)  bilinear transform

  db = 10 log10 (Power_out/Power_in)  decibel for power

  db = 20 log10 (Voltage_out/Voltage_in)  decibel for amplitude

       (note that db is based on a ratio, thus amplitude ratio may
        be either voltage or current. Power may be voltage squared,
        current squared or voltage times current. More on db at end.)


A Java program is shown that computes the a and b coefficients for
Butterworth and Chebyshev digital filters. Options include Low-Pass,
Band-Pass and High-Pass for order 1 through 16. Click on the sequence:
Design, Poles/Zeros, Frequency response, coefficients  to get output
similar to the screen shots shown below.

The Pole/Zero plot shows the complex z-plane with a unit circle.
The poles, x, and zeros o may be multiple.


The frequency response is for the range 0 to 4000 Hz based on
the 8000 Hz sampling. 0 db is at the top, the bottom is the
user selected range, -100 db is the default.


The coefficients are given with zero based subscripts.
(These subscripts may be used directly in C, C++, Java, etc.
 add one to the subscript for Fortran, MatLab and diagrams above.)


Note that MatLab has 'wavread' and 'wavwrite' functions as well
as a 'sound' function to play  .wav  files. 'wavwrite' writes
samples at 8000 Hz using the default.

The files to compile and run the Digital Filter coefficients are:
IIRFilterDesign.java
IIRFilter.java
PoleZeroPlot.java
GraphPlot.java
make_filter.bat
Makefile_filter
mydesign.out

This is enough for you to be able to program and use simple
digital filters. This lecture just scratched the surface.
There are many types of digital filters with many variations.
There are complete textbooks on just the subject of digital filters.

The output from the low pass filter shown above "builds up and settles"
quickly:



A fourth order band pass filter with band 2000 to 2100 requires
more samples for the center frequency 2050 to build up and settle:




Some notes on db
  +3 db is twice the power
  -3 db is half the power

 +10 db is ten times the power
 -10 db is one tenth the power

 +30 db is 1000 times the power, 10^3
 -30 db is 1/1000 of the power, 10^-3

 +60 db is one million times the power, 10^6
 -60 db is very small

Sound is measured to average human hearing threshold,
technically 20 micropascals of pressure by definition 0 db.
Some approximate examples for sound from a close source:

   0 db  threshold of hearing
  20 db  whisper
  60 db  normal conversation
  80 db  vacuum cleaner
  90 db  lawn mover
 110 db  front row at rock concert
 130 db  threshold of pain
 140 db  military jet takeoff, gun shot, fire cracker
 180 db  damage to structures
 194 db  shock waves begin, not normal sound

The sound drops off with distance by the inverse square law.
Each time you move twice as far away from the source,
the sound is 1/4 as strong.

The effective loudness decreases with very short burst of sound.
 160 db for 1/100 second is about a loud as 140 db for 1 second.

Sample db vs power code
dbtopower.c source
dbtopower_c.out results
dbtopower.py source
dbtopower_py.out results
   

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