1. III Digital Audio
III.8 (Wed Oct 24) Filters and EQ (= Equalizing)

2. input signal filter output signal
Definition: In music technology, a filter is a function that alters an audio signal on the basis of its Fourier spectrum
input signal filter output signal
Fourier analysis of input signal

3. Recall this distinction of realities!
mental
reality
physical
sender
message
receiver
synthesis
analysis
construction
decomposition

4. f(t) F(f)∙F(g) ∫ ±∞ f(x)∙g(t-x)dx
This distinction of realities also applies to filtering!
Not every mathematical construction can be realized on the technological level!
Fourier analysis of input signal
f(t) F(f)∙F(g) ∫ ±∞ f(x)∙g(t-x)dx
mental
reality
physical

5. P → ∞ period P The mathematical construction
For Fourier analysis, we construct a periodic function:
f(t)
time
period P
This is artificial. Mathematicians can solve this: Let P tend to infinity!
And therefore the fundamental frequency tends to zero! Fourier then looks like this:
amplitude
frequency
discrete Fourier decomposition of f(t):
One amplitude for each multiple of the fundamental frequency
amplitude
frequency
amplitude
frequency
continuous Fourier decomposition of f(t) = Fourier transform
F(f) = function of frequency!
P → ∞

6. F F* The Fourier Transform is an isomorphism
i.e. a 1-1 function between these spaces of functions:
space of all frequency functions
amplitude
frequency ν
F(f)
space of all time functions
f(t)
amplitude
time F F*
F(sin)(ν)
amplitude
frequency ν
ν = ω
sin(2πωt)
amplitude
time

7. big problem for real-time technology!
Convolution Theorem: Generalized Calculation of Partials
space of all time functions
space of all frequency functions
f(t)
amplitude
time
frequency ν
F(f) F F*
big problem for real-time technology!
product F∙G of frequency functions F and G:
F∙G(ν) = F(ν) ∙G(ν)
convolution f∗g of time functions f and g:
f∗g(t) = ∫ ±∞ f(x)∙g(t-x)dx
F(f)∙F(g) = F(f∗g) f g
F = F(f)
G = F(g)
F(f)∙F(g)
f∗g

8. F(f)∙F(sin) = spike at frequency ν = ω = ω-partial amplitude of f
Convolution Theorem: Generalized Calculation of Partials
F(f)∙F(g) = F(f∗g) f g
F = F(f)
G = F(g)
F(f)∙F(g)
f∗g
g(t) = sin(2πωt)
sin(2πωt)
amplitude
time
F(sin)(ν)
frequency ν
ν = ω
F(f)∙F(sin) = spike at frequency ν = ω = ω-partial amplitude of f

9. filter function Filter construction F = F(f) F∙G G cutoff frequency
amplitude
frequency ν
F = F(f)
amplitude
frequency ν
F∙G
amplitude
frequency ν G
cutoff frequency
filter function

10. Filter types low pass high pass low shelf high shelf band pass notch
amplitude
frequency ν
cutoff frequency
low pass
amplitude
cutoff frequency
high pass
amplitude
cutoff frequency
low shelf
amplitude
high shelf
cutoff frequency
amplitude
cutoff frequency
band pass
amplitude
cutoff frequency
notch

11. Graphic and Parametric EQ (=Equalization) ~ Complex Filtering
Graphic: many small frequency bands
defining the filter
function G
20Hz
20kHz
Parametric: a small parameter set
defining the filter
function G
cutoff frequency

12. wall electric circuit Hardware filter low pass cutoff frequency
amplitude
frequency ν
cutoff frequency
low pass
wall
electric circuit
cutoff frequency
= 1/2πRC