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

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This document provides an introduction to audio content analysis and convolution. It discusses linear time-invariant (LTI) systems and how convolution describes the output of an LTI system based on the input and impulse response. Examples of simple low-pass filters like moving average and single-pole filters are provided to illustrate convolution. The objectives are to understand LTI systems, convolution operations, and implement basic filters.

Technology

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

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

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

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

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

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

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

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

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

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

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