In recent years, several techniques have been proposed in the literature in order to attempt the emulation of nonlinear electro-acoustic devices, such as compressors, distortions, and preamplifiers. Among them, the dynamic convolution technique is one of the most common approaches used to perform this task. In this paper an exhaustive objective and subjective analysis of a dynamic convolution operation based on principal components analysis has been performed. Taking into consideration real nonlinear systems, such as bass preamplifier, distortion, and compressor, comparisons with the existing techniques of the state of the art have been carried out in order to prove the effectiveness of the proposed approach.

1. Paper ID 84
Approximation of Dynamic Convolution
Exploiting Principal Component Analysis:
Objective and Subjective Quality Evaluation
A. Primavera1
, S. Cecchi1
, L. Romoli1
M. Gasparini1
, and F. Piazza1
1
A3Lab - DII - Universit`a Politecnica delle Marche
Via Brecce Bianche 1, 60131 Ancona Italy
www.a3lab.dibet.univpm.it
Abstract
In the recent years, several techniques have been proposed in the literature in order
to attempt the emulation of nonlinear electro-acoustic devices, such as compres-
sors, distortions, and pre-amplifiers. Among them, the dynamic convolution tech-
nique is one of the most common approaches used to perform this task. In this
paper, an exhaustive objective and subjective analysis of a dynamic convolution op-
eration based on principal components analysis has been performed. Taking into
consideration real nonlinear systems, such as bass pre-amplifier, distortion, and
compressor, comparisons with the existing techniques of the state of the art have
been carried out in order to prove the effectiveness of the proposed approach.

2. Introduction
Dynamic convolution technique is one of the most common approaches used to perform a nonlinear
convolution (emulation of compressors, limiters and pre-amps).
DYNAMIC
CONVOLUTION
COMPUTATIONAL
COST MINIMIZATION
Problem
An efficient approach to approximate dynamic convolution [1] [2] has been proposed [3].
• Lowering the computational required to perform the operation.
• Maintaining the same perceived audio quality.
Proposed Algorithm

3. Dynamic Convolution
For discrete-time signals x and impulse response h with a finite length N, the linear convolution
results:
y[n] = x[n] ∗ h[n] =
N−1
m=0
x(n − m)h(m) (1)
This operation cannot be used in nonlinear case.
To cope with this problem the dynamic convolution operation has been introduced.
y[n] =
N−1
m=0
x[n − m]H [m, S(x[n − m])] , (2)
where:
• H is the matrix of impulse responses obtained through the system analysis procedure.
• S(x[n]) = 1 + {|x[n]| /(fs/M)} represents the selector function.
Dynamic Convolution

4. Proposed Algorithm
A simplified model of dynamic convolution procedure has been developed. It allows a faithful repro-
duction of the convolution operation lowering the computational cost required:
Block diagram of the proposed algorithm.
MAIN PHASES
• System Identification (Offline)
• Preprocessing based on PCA (Offline)
• Emulation (Real-Time)

5. Proposed Algorithm (system identification)
A MMLS technique has been used in order to obtain M IRs related to M input signals of different
amplitudes.
MMLS signal used in the system identification
procedure.
Dataset of IRs obtained analyzing the BOSS DS-2
Turbo Distortion stomp box.
Parameters configuration:
• Amplitude decreasing of 1dB for step.
• Levels used: M = 64.
• Sample rate of 48 kHz.

6. Proposed Algorithm (PCA based processing)
PCA is applied to the matrix H obtained through the system analysis procedure [4].
Basis vectors (i), principal components (ii), percentage of cumulative variance (iii) obtained applying the PCA analysis
with the BOSS DS-2 Turbo Distortion stomp box IRs dataset.
PCA offers a mechanism for performing lossy data compression: high compression rate is provided
by discarding the last principal components, (i.e., those exhibiting the lowest variance). More in
detail:
ˆH = V · W, (3)
where:
• the basis vectors V is computed as the eigenvector of the covariance matrix C = (H − H)(H −
H)T with H representing a L × M matrix with the averages of hk(n).
• the principal components W are obtained as follows: W = V T · H
Going Deeper into PCA

7. Proposed Algorithm (real-time emulation)
The proposed approach allows one to approximate the dynamic convolution operation using pairs of
amplitude waveshapers [5] and FIR filters.
The coefficients of these structures are set as:
• Amplitude waveshapers → basis vectors Vi
• FIR filters → principal components Wi
The number N of Waveshaper and FIR filters used during the emulation depends on the desired
value of cumulative percentage of variance (higher is the percentage value better is the approxima-
tion).
Several tests have been carried out to evaluate the effectiveness of the PCA based approach:
• Three real nonlinear systems (bass pre-amplifier [6], distortion [7], and compressor [8]) have
been taken into account.
• Objective measure (percentage of comulative variance and MSE) have been compared.
• Subjective listening tests according to the ITU-RBS.1534 (MUSHRA) have been performed.
Tests

8. Results (objective analysis)
Several objective measures have been reported in order to show the effectiveness of the PCA based
approach.
Evaluation of the percentage of cumulative variance as a function of the number of employed principal components.
Evaluation of the MSE between dynamic convolution and PCA based approach as a function of the number of
employed principal components.

9. Results (subjective analysis)
A MUSHRA listening test [9] has been performed involving 10 subjects (7 male and 3 females).
Pre-amplifier listening test results for a light (i), medium (ii), and heavy (iii) nonlinearity strength.
Compressor listening test results for a light (i), medium (ii), and heavy (iii) nonlinearity strength.
Distortion stomp-box listening test results for a light (i), medium (ii), and heavy (iii) nonlinearity strength.

10. Conclusions
• A complete analysis of a dynamic convolution operation based on principal components analysis (PCA) has been
provided;
• The analysis aims to demonstrate the utility of the PCA approach applied to dynamic convolution:
– Reducing the computational cost required to perform the convolution operation.
– Maintaining the same perceived audio quality.
• Different tests have been carried out according to objective (MSE evaluation) and subjective measures (MUSHRA
listening tests), proving the effectiveness of the approach.
• Future works will be oriented toward the refinement of the dynamic convolution exploiting PCA approach through
the introduction of an adaptive structure.
References
[1] M. Kemp, “Analysis and Simulation of Non-Linear Audio Processes using Finite Impulse Responses Derived at Multiple Impulse Amplitudes,” in Proc. 106th AES Convention, Munich, Germany.
[2] A. Farina and E. Armelloni, “Emulation of Not-Linear, Time-Variant Devices by the Convolution Technique,” in Congresso AES Italia, Como.
[3] A. Primavera, S. Cecchi, L. Romoli, M. Gasparini, and F. Piazza, “An Efficient DSP-Based Implementation of a Dynamic Convolution Approach Exploiting Principal Component Analysis,” in Proc. 5th
European DSP In Education And Research Conference, Amsterdam, The Netherlands, Sep. 2012, pp. 30–34.
[4] S. Haykin, Neural networks. Macmillan College Publishing Company, Inc., 1994.
[5] M. L. Brun, “Digital Waveshaping Synthesis,” Journal of the Audio Engineering Society, vol. 27, no. 4, pp. 250–266, 1979.
[6] “Aguilar,” 2012. [Online]. Available: www.aguilaramp.com
[7] “BOSS U.S,” 2012. [Online]. Available: www.bossus.com
[8] “dbx,” 2012. [Online]. Available: www.dbxpro.com
[9] ITU-R BS. 1534, “Method for subjective listening tests of intermediate audio quality,” Geneva.