0A-02-ACA-Fundamentals-Convolution.pdf

1. Introduction to Audio Content Analysis module A.2: fundamentals — convolution alexander lerch

2. overview systems convolution filter examples summary introduction overview corresponding textbook section appendix A.2 lecture content • LTI systems • convolution • filter examples learning objectives • basic understanding of linearity and time-invariance • basic understanding of the convolution operation • ability to implement simple filters module A.2: fundamentals — convolution 1 / 9

3. overview systems convolution filter examples summary introduction overview corresponding textbook section appendix A.2 lecture content • LTI systems • convolution • filter examples learning objectives • basic understanding of linearity and time-invariance • basic understanding of the convolution operation • ability to implement simple filters module A.2: fundamentals — convolution 1 / 9

4. overview systems convolution filter examples summary systems introduction a system: any process producing an output signal in response to an input signal System x(t) y(t) name examples for systems in signal processing module A.2: fundamentals — convolution 2 / 9

5. overview systems convolution filter examples summary systems introduction a system: any process producing an output signal in response to an input signal System x(t) y(t) name examples for systems in signal processing filters, effects vocal tract room (audio) cable . . . module A.2: fundamentals — convolution 2 / 9

6. overview systems convolution filter examples summary systems LTI systems LTI: Linear Time-Invariant Systems are a great model for many real-world systems linearity 1 homogeneity: f (ax) = af (x) 2 superposition (additivity): f (x + y) = f (x) + f (y) time invariance: f (x(t − τ)) = f (x)(t − τ) module A.2: fundamentals — convolution 3 / 9

7. overview systems convolution filter examples summary convolution introduction convolution convolution operation describes the output of an LTI system: y(t) = (x ∗ h)(t) := ∞ Z −∞ x(τ)h(t − τ)dτ y(i) = (x ∗ h)(i) := ∞ X j=−∞ x(j)h(i − j) module A.2: fundamentals — convolution 4 / 9

8. overview systems convolution filter examples summary convolution animation module A.2: fundamentals — convolution 5 / 9 matlab source: matlab/animateConvolution.m

9. overview systems convolution filter examples summary convolution properties identity: x(i) = δ(i) ∗ x(i) commutativity: h(i) ∗ x(i) = x(i) ∗ h(i) associativity: g(i) ∗ h(i) ∗ x(i) = g(i) ∗ h(i) ∗ x(i) distributivity: g(i) ∗ h(i) + x(i) = g(i) ∗ h(i) + g(i) ∗ x(i) circularity: h(i) periodic ⇒ y(i) = h(i) ∗ x(i) periodic module A.2: fundamentals — convolution 6 / 9

14. overview systems convolution filter examples summary filter example 1: Moving Average y(i) = J −1 X j=0 b(j) · x(i − j) replaces current sample with average of J samples smooths a signal (low pass) IR: rectangular linear phase, but inefficient for many coefficients Finite Impulse Response (FIR) module A.2: fundamentals — convolution 7 / 9

15. overview systems convolution filter examples summary filter example 2: Single-Pole y(i) = (1 − α) · x(i) + α · y(i − 1) recursive system: output depends on previous output the larger alpha, the less the current input is taken into account (low pass) alpha from 0. . . 1 efficient, but non-linear phase Infinite Impulse Response (IIR) module A.2: fundamentals — convolution 8 / 9

16. overview systems convolution filter examples summary summary lecture content LTI system • good model for many real-world system • linear (homogeneity, superposition) and time-invariant • impulse response reflects all system properties convolution • operation that computes the output of an LTI system from the input low pass filters • often used to smooth a signal • typical examples are moving average (FIR) and single pole (IIR) module A.2: fundamentals — convolution 9 / 9

0A-02-ACA-Fundamentals-Convolution.pdf

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