II Acoustic Reality II.6 (W Oct 29) Filters and EQ (= Equalizing)

1. II Acoustic Reality II.6 (W Oct 29) Filters and EQ (= Equalizing)

2. 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! construction decomposition mental reality physical reality synthesis analysis sender message receiver

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

5. 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 amplitude amplitude frequency frequency frequency discrete Fourierdecomposition of f(t): One amplitude for each multiple of the fundamental frequency continuous Fourierdecomposition of f(t)= Fourier transform F(f) = function of frequency! P → ∞

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

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

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

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

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

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

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