# Brno University of Technology Faculty of Information

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First found May 22, 2018

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Digital filters
Honza Černocký, ÚPGM
Aliases
•
•
•
•
Numerical filters
Discrete systems
Discrete-time systems
etc.
2
What for ?
Processing of signals
• Emphasizing
• Attenuating
• Detecting
3
Emphasis
• Basses – low frequencies
4
Emphasis
• Trebles – high frequencies
5
Emphasis
• Telephone band 300-3400 Hz
6
Attenuating
• A signal contamined by 1kHz and its
cleaning by a sharp band-stop
7
Detection
• Couple of peaks in noise
8
Detection
• Matched filter
9
• A nice app: RICHTER Jiří. Echo-Based
Distance Measurement on Mobile Phone,
BP FIT, 2014/2015
10
What will be needed ?
• Sampled signal
– At some sampling frequency (most often
8kHz, 16kHz, 44.1kHz (CD), 48kHz, 96kHz)
• For visualization, will show it as
continuous (stem vs. plot)
• For ISS no special formats (MP3, OGG,
WAV), but a plain RAW
– No header, sequence of 16-bit shorts
… wire.c
11
What next ?
• Only 3 simple operations
– Multiplication with a constant
*
+
– Shift
12
Shift ? Only to the past !
x[n]
n
„now“,
„presence“
past
future
13
Shift off-line
x[n-1] x[n]
n-1
n
… just changing the value of the pointer.
14
Shift on-line
• We have only x[n]
• Function with a memory
float filter (float xn) {
static float xn1, xn2;
… some processing …
xn2 = xn1;
xn1 = xn;
return something;
}
15
A simple filter
y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + b3x[n-3]
… difference equation
16
How does it work ?
17
Implementation off-line
• Allocation of space for the output signal.
• Then absolutely verbatim re-writing of the
difference equation
fir_offline.c
18
Implementation on-line
• A function for filtering is called for every
sample
• It must memorize the past samples
fir_online.c
19
What will it do for …
b0
b1
b2
b3
0.25
0.25
1
2
0.25
-0.25
0
0
0.25
0.25
0
0
0.25
-0.25
0
0
0
0
0
1
20
Another possible
implementation…
21
How to represent a filter ?
y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + b3x[n-3]
… difference equation
b0
x[n]
z-1
b1
z-1
b2
z-1
b3
S
y[n]
… scheme
22
Impulse response
• Reaction to unit impulse
0
0
0
0
0
1
0
0
0
0
0
0
23
Finite / infinite impulse
response?
24
Convolution
… demo on our filter
25
It works also the other way round
(convolution is commutative…)
… demo on our filter – homework ?
26
Convolution - summary
• Flip the impulse response
• Shift it to given „n“
• Mutliply
• Write the results
… „paper strip“ method – demo
27
Making use of filter output
• Feedback – recursive filters
28
Diference equation
y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + b3x[n-3]
- a1y[n-1] - a2y[n-2] - a3y[n-3]
… why not a0 ?
… and why do the feed-back coefficients
have negative sign ?
29
Scheme
b0
x[n]
S
y[n]
z-1
b1
-a1
z-1
z-1
b2
-a2
z-1
z-1
b3
-a3
z-1
30
Implementation off-line
• Digging also in the old outputs …
… iir_offline.c
31
Implementation on-line
• The function has to memorize also the
past outputs.
… iir_online.c
32
Impulse response
• For example for a simple, purely recursive
filter (nothing done with the input)
x[n]
S
y[n]
-a1
z-1
• Infinite impulse response - IIR
33
Stability
• Mathematically „Bounded Input Bounded
Output“
• Popularly: “if reasonable stuff on the input
expecting reasonable stuff on the output”
34
Stability of FIR
35
Stability of IIR
36
First order IIR
x[n]
S
y[n]
-a1
z-1
• What must be respected for a1?
• And how about for more complicated IIR
filters ?
37
General filter
38
General filter
• Order of input Q
• Order of output P
Difference equation
y[n] = b0x[n] + b1x[n-1] + … + bQx[n-Q]
- a1y[n-1] + … + aPy[n-P]
39
Implementation off-line
• … still the same, this time with cycles
iirbig_offline.c
40
Implementation on-line
• The cycles must run backward (otherwise
the old values rewrite the new ones)
• The cycle for the output must stop
at index 1.
• „Trash“ memory fields, that will never be
used – make things easier and save us
some „if“s.
iirbig_online.c
41
Frequency responses
• For FIR filters:
• The coefficients of the filter (resp. of its
impulse response) act as a machining
tool – the result will be similar
42
Tests
• 2s of chamber „a“ (440 Hz) plus some
noise
See the Matlab file matlab_filtry.m (section
“filcy atd”)
43
Mason’s float
• 20 equal samples valued 1/20
• Smoothing, less noise
=> Low-pass
44
Mason’s superfloat
• 181 equal samples valued 1/181
• Smoothing, less noise but almost no signal
left 
=> Even more low-pass
45
Cutter with a dented edge
=> High pass
46
Band-pass ?
• We want to select 440 Hz.
• The output will resemble the impulse
response …
… listen to the result also on a speech file.
47
Frequency response more
precisely
• Currently, interested only in magnitude
• i.e. how individual frequencies are
amplified / attenuated
• … not how they are shifted
See the last part of matlab_filtry.m
48
Method 1 – measure !
filtr
49
Method 2 – use impulse
response
• Generate impulse response and perform
its frequency transform
• But the impulse response can be
loooonnnnng.
50
The ultimate solution
z-transform
Basic tools:
x[n] => X(z)
a x[n] => a X(z)
x[n-k] => X(z) z-k
51
Difference equation
=> transfer function
y[n] = b0x[n] + b1x[n-1] + … + bQx[n-Q]
- a1y[n-1] + … + aPy[n-P]
52
Transfer function => frequency
response
z => ej2pf
• f is normalized by the sampling frequency
• Will obtain a complex number, must take
only its magnitude.
• Matlab: by hand or with freqz
53
Relation of time and spectra
y[n] = x[n] * h[n]
(convolution)
Y(f) = X(f) H(f)
(mutliplication)
• Attention, must multiply the complex
numbers in spectra !
54
Summary
Describing the filter by
• Scheme
• Difference equation
• Impulse response
• Transfer function
• Frequency response
55
Summary II.
• Types of filters
– FIR
– IIR
• Implementation
– Mutliplication
– Summation
– Shifts
• Off-line
• On-line
56
Summary III.
• Computing the frequency response
– Ugly:
• „measurement“
• Analysis of h[n]
– Nicely
• Transfer function: replacing z by ej2pf
• f is normalized frequency
57
TODO’s
•
•
•
•
Phase – shifts of signals
Stability for more complex IIR filters
Design of filters
Learning filters on data
58
The END
59