# Differentiable Signal Processing With Black-Box Audio Effects

###### Abstract

We present a data-driven approach to automate audio signal processing by
incorporating stateful third-party, audio effects as layers within a deep
neural network. We then train a deep encoder to analyze input audio and
control effect parameters to perform the desired signal manipulation,
requiring only input-target paired audio data as supervision. To train our
network with non-differentiable black-box effects layers, we use a fast,
parallel stochastic gradient approximation scheme within a standard auto
differentiation graph, yielding efficient end-to-end backpropagation. We
demonstrate the power of our approach with three separate automatic audio
production applications: tube amplifier emulation, automatic removal of
breaths and pops from voice recordings, and automatic music mastering. We
validate our results with a subjective listening test, showing our approach
not only can enable new automatic audio effects tasks, but can yield results
comparable to a specialized, state-of-the-art commercial solution for music
mastering.

Index Terms—  audio effects, deep learning, differentiable signal processing,
black-box optimization, gradient approximation.

##  1 Introduction

Audio signal processing effects (Fx) are ubiquitous and used to manipulate
different sound characteristics such as loudness, dynamics, frequency, and
timbre across a variety of media. Many Fx, however, can be difficult to use or
are simply not powerful enough to achieve a desired task, motivating new work.
Past methods to address this include audio effects circuit modeling [1],
analytical methods [2], and intelligent audio effects that dynamically change
their parameter settings by exploiting sound engineering best practices [3].
The most common approach for the latter is adaptive audio effects or signal
processing systems based on the modeling and automation of traditional
processors [4]. More recent deep learning methods for audio effects modeling
and intelligent audio effects include 1) end-to-end direct transformation
methods [5, 6, 7], where a neural proxy learns and applies the transformation
of an audio effect target 2) parameter estimators, where a deep neural network
(DNN) predicts the parameter settings of an audio effect [8, 9, 10] and 3)
differentiable digital signal processing (DDSP) [11, 12], where signal
processing structures are implemented within a deep learning auto-
differentiation framework and trained via backpropagation.

While these past methods are very promising, they are limited in several ways.
First, direct transform approaches can require special, custom modeling
strategies per effect (e.g. distortion), are often based on large and
expensive networks, and/or use models with limited or no editable parameter
control [5, 6]. Second, methods with parameter control commonly require
expensive human-labeled data from experts (e.g. inputs, control parameters,
and target outputs) [7]. They are also typically optimized to minimize
parameter prediction error and not audio quality directly [10], which can
reduce performance. Third, DDSP approaches require a differentiable
implementation for learning with backpropagation, re-implementation of each
Fx, and in-depth knowledge, limiting use to known differentiable effects and
causing high engineering effort [11, 12].

Fig. 1: Our DeepAFx method consists of a deep encoder that analyzes audio and
predicts the parameters of one or more black-box audio effects (Fx) to achieve
a desired audio processing task. At training time, gradients for black-box
audio effects are approximated via a stochastic gradient method.

In this work, we introduce DeepAFx, a new approach to differentiable signal
processing that enables the use of _arbitrary_ stateful, third-party black-box
processing effects (plugins) within any deep neural network as shown in Figure
1. Using our approach, we can mix-and-match traditional deep learning
operators and black-box effects and learn to perform a variety of intelligent
audio effects tasks without the need of neural proxies, re-implementation of
Fx plugins, or expensive parameter labels (only paired input-target audio data
is required). To do so, we present 1) a deep learning architecture in which an
encoder analyzes input audio and learns to control Fx black-boxes that
themselves perform signal manipulation 2) an end-to-end backpropagation method
that allows differentiation through non-differentiable black-box audio effects
layers via a fast, parallel stochastic gradient approximation scheme used
within a standard auto differentiation graph 3) a training scheme that can
support stateful black-box processors, 4) a delay-invariant loss to make our
model more robust to effects layer group delay, 5) application of our approach
to three audio production tasks including tube amplifier emulation, automatic
non-speech sounds removal, and automatic music post-production or mastering,
and 6) listening test results that shows our approach can not only enable new
automatic audio production tasks, but is high-quality and comparable to a
state-of-the-art proprietary commercial solution for automatic music
mastering. Audio samples and code are online at
https://mchijmma.github.io/DeepAFx/.

##  2 Method

###  2.1 Architecture

Our model is a sequence of two parts: a deep encoder and a black-box layer
consisting of one or more audio effects. The encoder analyzes the input audio
and predicts a set of parameters, which is fed into the audio effect(s) along
with the input audio for transformation.

The deep encoder can be any deep learning architecture for learning
representations of audio. To allow the deep encoder to learn long temporal
dependencies, the input $\tilde{x}$ consists of the current audio frame $x$
centered within a larger audio frame containing previous and subsequent
context samples. The input to the encoder consists of a log-scaled mel-
spectrogram non-trainable layer followed by a batch normalization layer. The
last layer of the encoder is a dense layer with $P$ units and sigmoid
activation, where $P$ is the total number of parameters. The output is the
predicted parameters $\hat{\theta}$ for the current input frame $x$.

The audio Fx layer is a stateful black-box consisting on one or more connected
audio effects. This layer uses the input audio $x$ and $\hat{\theta}$ to
produce an output waveform $\bar{y}=f(x,\hat{\theta})$. We calculate a loss
between the transformed audio and targeted output audio.

###  2.2 Gradient approximation

In order to train our model via backpropagation, we calculate the gradients of
the loss function with respect to the audio Fx layer and the deep encoder. For
the latter, we compute the gradient using standard automatic differentiation.
For the former, gradient approximation has been addressed in various ways such
as finite differences (FD) [13], FD together with evolution strategies [14],
reinforcement learning [15], and by employing neural proxies [16]. For our
work, we use a stochastic gradient approximation method called simultaneous
permutation stochastic approximation (SPSA) [17], which is rooted in standard
FD gradient methods and has low computational cost [13]. In our formulation,
we ignore the gradient of the input signal $x$ and only approximate the
gradient of the parameters $\hat{\theta}$.

FD can be used to approximate the gradient of the audio Fx layer at a current
set of parameters $\hat{\theta}_{0}$, where the gradient $\tilde{\nabla}$ is
defined by the vector of partial derivatives characterized by
$\tilde{\nabla}f(\hat{\theta}_{0})_{i}=\partial
f(\hat{\theta}_{0})/\partial\theta_{i}$ for $i=1,\dots,P$. The two-side FD
approximation $\tilde{\nabla}^{FD}$ [17] is based on the backward and forward
measurements of $f()$ perturbed by a constant $\epsilon$. The $i$th component
of $\tilde{\nabla}^{FD}$ is

$\tilde{\nabla}^{FD}f(\hat{\theta}_{0})_{i}=\frac{f(\hat{\theta}_{0}+\epsilon\hat{d}^{P}_{i})-f(\hat{\theta}_{0}-\epsilon\hat{d}^{P}_{i})}{2\epsilon},$
(1)

where $0<\epsilon\ll 1$ and $\hat{d}^{P}_{i}$ denotes a standard basis vector
with $1$ in the $i$th place and dimension $P$. From eq. 1, each parameter
$\theta_{i}$ is perturbed one at a time and requires 2$P$ function
evaluations. This can be computationally expensive for even small $P$, given
that $f()$ represents one or more audio effects. Even worse, when $f()$ is
stateful, FD perturbation requires $2P$ instantiations of $f()$ to avoid state
corruption, resulting in high memory costs.

SPSA offers an alternative and is an algorithm for stochastic optimization of
multivariate systems. It has proven to be an efficient gradient approximation
method that can be used with gradient descent for optimization [17]. The SPSA
gradient estimator $\tilde{\nabla}^{SPSA}$ is based on the random perturbation
of all the parameters $\hat{\theta}$ at the same time. The $i$th element of
$\tilde{\nabla}^{SPSA}$ is

$\tilde{\nabla}^{SPSA}f(\hat{\theta}_{0})_{i}=\frac{f(\hat{\theta}_{0}+\epsilon\hat{\Delta}^{P})-f(\hat{\theta}_{0}-\epsilon\hat{\Delta}^{P})}{2\epsilon\Delta^{P}_{i}},$
(2)

where $\hat{\Delta}_{P}$ is a $P$-dimensional random perturbation vector
sampled from a symmetric Bernoulli distribution, i.e. $\Delta^{P}_{i}=\pm 1$
with probability of 0.5 [18]. In each iteration, the total number of function
evaluations $f()$ is two, since the numerator in eq. 2 is identical for all
the elements of $\tilde{\nabla}^{SPSA}$. This is especially valuable when $P$
is large and/or when $f()$ is stateful. While the random search directions do
not follow the steepest descent, over many iterations, errors tend to average
to the sample mean of the stochastic process [17].

###  2.3 Training with stateful black-boxes

While training samples typically should be generated from an independent and
identically distributed (i.i.d.) process, we note that audio effects are
_stateful_ systems. This means output samples depend on previous input samples
or internal states [19]. Therefore, audio Fx layers should not be fed
i.i.d.-sampled audio during training as this will corrupt state and yield
unrealistic, transient outputs. As such, we feed consecutive non-overlapping
frames of size $N$ to the audio Fx layer, i.e. audio frames with a hop size of
$N$ samples, where the internal block size of each audio effects is set to a
divisor of $N$. This ensures that there are no discrepancies between the
behavior of the audio effects during training and inference time.

To train the model using mini-batches and maintain state, we use a separate Fx
processor for each item in the batch. Therefore, for a batch size of $M$, we
instantiate $M$ independent audio effects for the forward pass of
backpropogation. Each of the $M$ plugins receives training samples from random
audio clips, however across minibatches, each instance is fed samples from the
same input until swapped with a new sample. When we consider the two-side
function evaluations for SPSA, we meet the same stateful constraints by using
two additional Fx per batch item. This gives a total of 3$M$ audio effects
when optimizing with SPSA gradients, whereas FD requires an unmanageable
$(\textrm{2}P+\textrm{1})M$ effects for a large number of parameters $P$ or
batch size $M$. Finally, we parallelize our gradient operator per item in a
minibatch, similar to recent distributed training strategies [20], but using a
single one-GPU, multi-CPU setup.

###  2.4 Delay-invariant loss

Multiband audio effects correspond to audio processors that split the input
signal into various frequency bands via different types of filters [19]. These
filters can introduce group delay, which is a frequency-dependent time delay
of the sinusoidal components of the input [21]. These types of effects and
similar can also apply a $None$ phase shift, i.e. to invert the sign of the
input. While such properties are critical to the functioning of Fx, it means
that difficulties can arise when directly applying a loss function with
inexact inputs, either in time or frequency domain. Thus, we propose a delay-
invariant loss function designed to mitigate these issues.

We first compute the time delay $\tau=\operatorname*{argmax}(\bar{y}\star y)$
between the output $\bar{y}$ and target $y$ audio frames via cross-correlation
($\star$). Then, the loss in the time domain

$L_{time}=\operatorname{min}(||\bar{y}_{\tau}-y_{\tau}||_{1},||\bar{y}_{\tau}+y_{\tau}||_{1}),$
(3)

corresponds to the minimum $L1$ distance between the time-aligned target
${y}_{\tau}$ and both a $None$ phase shift and $None$ phase shift time-aligned
output $\bar{y}_{\tau}$. We compute $\bar{Y}_{\tau}$ and $Y_{\tau}$: a
1024-point Fast Fourier Transform (FFT) magnitude of $\bar{y}_{\tau}$ and
$y_{\tau}$, respectively. The loss in the frequency domain $L_{freq}$ is
defined as

$L_{freq}=||\bar{Y}_{\tau}-Y_{\tau}||_{2}+||\log\bar{Y}_{\tau}-\log
Y_{\tau}||_{2}.$ (4)

The final loss function is $L=\alpha_{1}L_{time}+\alpha_{2}L_{freq}$, where we
empirically tune the weighting and use $\alpha_{1}=10$ and $\alpha_{2}=1$.

|  |
---|---|---
Fig. 2: (Left) Tube amplifier emulation. (Center) Automatic non-speech sounds
removal. (Right) Automatic music mastering. From top to bottom: input, target,
our result, and selected predicted parameters. th, and gain-c mean threshold
and compressor makeup gain, respectively.

##  3 Experiments

We apply our approach to three applications including tube amplifier
emulation, automatic non-speech sounds removal, and automatic music mastering.
In each case, we use one or more different effects and different datasets to
illustrate the generality of our approach. We also explore two deep encoder
variants; an Inception network [22] and MobileNetV2 [23]. The number of
parameters for each encoder is approx. $2.8$M and $2.2$M, respectively. The
input context $\tilde{x}$ and current audio frame $x$ are 40960 and 1024
samples (1.85 sec and 46 ms) at a 22,050Hz sampling rate. The log-scaled mel-
spectrogram input layer has a 46 ms window size, 25% overlap, and 128 mel-
bands. We use effects from the LV2 audio plugin open standard [24] and
experiment with continuous parameters scaled between 0 and 1. At inference we
apply a lowpass filter to $\hat{\theta}$ to ensure smoothness. For intuition,
our final optimization scheme with the Inception encoder and parametric
equalizer Fx (EQ) [25] takes 3 minutes per epoch (1000 steps) to train with
batch size $M=100$ on a Tesla-V100 GPU. We use Adam optimization and early
stop on validation loss.

###  3.1 Tube amplifier emulation

The emulation of distortion effects is a highly-researched field [1, 2] whose
goal is to recreate the sound of analog reference devices (e.g. tube
amplifiers or distortion circuits). Amplifier emulation using deep learning
has been explored in the context of deep neural proxies [26, 27] and been
shown to achieve virtually indistinguishable emulation under certain
conditions [5, 6]. Such approaches, however, are _specifically_ designed for
this task, making it an interesting benchmark. We ask – is it possible to
emulate distortion using a simple multiband dynamic range compressor?

A compressor modifies the amplitude dynamics of audio by applying a time-
varying gain. Compressors are commonly used for loudness control and typically
introduce little harmonic distortion in contrast to tube amplifiers [19].
Nonetheless, we apply our approach and learn 21 parameters; the threshold,
makeup gain, ratio, and knee for each of the 4 frequency bands; the 3
frequency splits; and the input and output gains on a multiband compressor
[28]. For training, we use the subset of the IDMT-SMT-Audio-Effects dataset
[29] used by Martínez et al. [5] consisting of 1250 raw notes from various
electric guitars and bass guitars and processed through a Universal Audio 6176
Vintage Channel Strip tube preamplifier. The train, validation, and test
dataset sizes are 31.6, 4.0, and 4.0 minutes, respectively. For all tasks, the
non-trainable Fx parameters are set to defaults with the exception of the
attack and release gates, which are set to their minimum (10 ms) due to the
fast control rate of our model (46 ms).

###  3.2 Automatic non-speech sound removal

Removing non-speech vocal sounds, such as breaths and lip smacks, is a common
task performed by sound engineers [30]. This task can be done by manually
editing the audio waveform or by using a noise gate, to reduce signal below a
certain threshold via a reduction gain and ratio setting [19]. Both approaches
can be time consuming, require expert knowledge, and, to the best of our
knowledge, the automation of this task has not been investigated with the
tangential exception of an intelligent noise gate for drum recordings [31] and
automatic detection of breaths for speaker recognition [32, 33].

We train a model with a multiband noise gate [28] Fx layer with 17 parameters
(threshold, reduction gain and ratio for each of the 4 frequency bands; the 3
frequency splits; and the input and output gains). We use the DAPS dataset
[34] which contains 100 raw and clean speech recordings with manually removed
breaths and lip smacks. The train, validation, and test dataset size is 213.5,
30.2, and 23.8 minutes, respectively.

###  3.3 Automatic music mastering

Music post-production or mastering is the process of enhancing a recording by
manipulating its dynamics and frequency content [3]. This manipulation is
typically done by an experienced mastering engineer and is carried out using
dynamic range effects, such as a compressor and limiter; and frequency-based
processors, such as EQ. Automatic mastering has been explored using adaptive
audio effects [35, 36], using DNNs to predict dynamic range compression gain
coefficients [37], and even commercialized as proprietary online mastering
software (OMS) [38].

We train a model with multiple audio effects in series, similar to the way
mastering engineers perform this task, consisting of a multiband compressor
[28], a graphic EQ [25] and mono limiter [39]. A limiter is a dynamic range
processor that more aggressively adjusts any input above the threshold, e.g.
by using a ratio of $\infty$ [40]. We train our model to learn 50 parameters –
16 parameters for the multiband compressor (threshold, makeup gain and ratio
for each of the 4 frequency bands, the 3 frequency splits, and the input
gain), 33 parameters for the graphic EQ (the gain for each of the 32 frequency
bands and the output gain), and 1 parameter for the limiter (threshold). For
training data, we collected 138 unmastered and mastered music tracks from The
‘Mixing Secrets’ Free Multitrack Download Library [41], and as preprocessing
step, we perform time-alignment using cross-correlation and loudness normalize
each unmastered track to -25dBFS. The train, validation, and test dataset size
is 429.3, 51.1, and 50.3 minutes, and respectively.

##  4 Results & Analysis

###  4.1 Qualitative Evaluation

We show qualitative analysis by visual inspection via Figure 2, in which we
depict input, target and predicted spectrograms for all tasks. For the tube
amplifier task, it can be seen that the model emulates very closely the analog
target. Also, the multiband compressor parameters move over time along the
attack and decay of the guitar note to emulate the nonlinear hysteresis
behavior of the tube amplifier. For the non-speech sound removal task, the
model successfully removes breath and lip smacking sounds. It can be seen how
the different thresholds vary according to the non-speech sounds present at
the first and third second of the input recording. The music mastering example
also successfully matches the sound engineer mastering target and the
parameters of the three audio effects gradually change based on the input
content of the unmastered music track.

| Model | Epochs | Time | $\tilde{d}_{MFCC}$
---|---|---|---|---
Tube amplifier | Inception | 97 | 9.07 | 0.2596
emulation | MobileNetV2 | 63 | 6.4 | 0.2186
| CAFx | 723 | 5.5 | 0.0826
Non-speech | Inception | 89 | 7.4 | 0.0186
sounds removal | MobileNetV2 | 60 | 4.8 | 0.0231
Music | Inception | 202 | 19.8 | 0.0282
mastering | MobileNetV2 | 178 | 17.5 | 0.0542
| OMS | - | - | 0.0157

Table 1: Training epochs & time (hours), and test MFCC distance.

###  4.2 Quantitative Evaluation

We show quantitative evaluation in Table 1, including number of training
epochs until early stopping, training time in hours, and the mean cosine
distance of the mel-frequency cepstral coefficients or $\tilde{d}_{MFCC}$ as a
proxy for a perceptual metric, following past work [5]. For this, we compute
13 MFCCs from a log-power mel-spectrogram using a window size of 1024 samples,
25% hop size and 128 bands. As a baseline for the tube amplifier emulation
task, we use the Convolutional Audio Effects Modeling Network (CAFx) [5], and
for the music mastering task, we use an online mastering software (OMS) [38].
We see that the tube amplifier emulation distance for our approach is higher
than the other two tasks, likely caused by using a compressor to achieve
distortion. We see both encoders achieved similar performance, although the
Inception model tends to perform slightly better, and all training times are
under a day. We also find the CAFx and OMS models have lower distance, but
yield to perceptual listening tests for further conclusions.

###  4.3 Perceptual Evaluation

We performed a subjective listening test to evaluate the perceptual quality of
our results using the multiple stimulus hidden reference (MUSHRA) protocol
[42]. Seventeen participants took part in the experiment which was conducted
using The Web Audio Evaluation Tool [43]. Five participants identified as
female and twelve participants identified as male. Participants were among
musicians, sound engineers, or experienced in critical listening.

Fig. 3: Listening test violins plots. (Left) Tube amplifier emulation results
show our approach does not outperform CAFx [5], but still yields good quality.
(Center) Non-speech sounds removal results show near human quality. (Right)
Music mastering results show our method performs equivalent to the state-of-
the-art (OMS).

Five listening samples were used from each of the three test subsets. Each
sample consisted of a 4-second segment with the exception of the tube
amplifier emulation samples, where each note is 2 seconds long. Participants
were asked to rate samples according to similarity relative to a reference
sound, i.e. a sample processed by a sound engineer or a recording from an
analog device.

The samples consisted of an unprocessed sample as low-anchor (LA), a mid-
anchor (MA), a hidden reference as high-anchor (HA), and the output of our
DeepAFx method with the Inception encoder (Ours). The mid-anchor corresponds
to a tube distortion [44] with drive of +8 dB; a gate [45] with default
settings; and peak normalization to -1dBFS for each task, respectively. All
samples for the first two tasks were loudness normalised to -23dBFS. The
samples for the music mastering task were monophonic and not loudness
normalized so as to evaluate loudness as part of the task.

The results of the listening test can be seen in Figure 3 as a violin plot.
For tube amplifier emulation, the median scores in figure order are (0.0,
0.24, 0.96, 0.63, 0.93), respectively. The CAFx method ranks above our model,
which is reasonable considering it corresponds to a custom, state-of-the-art
network. Our result for this task is still remarkable, however, since we are
only using a multiband compressor in a field where complex models or deep
neural proxies have predominated. For the non-speech sounds removal task, the
median scores in figure order are (0.39, 0.11, 0.89, 0.96), respectively. Our
model is rated almost as high as the hidden-anchor, which is again remarkable
given the typical expertise and time required for the task. Also note, 1) the
mid-anchor is rated lower than the low-anchor because of agitating gating
artifacts from Fx processing and 2) the high variance of the low anchor, which
contains different speech content (breaths and voice pops) and appears to
confuse listeners when rating similarity. For the automatic music mastering
task, the median scores in figure order are (0.15, 0.24, 0.95, 0.88, 0.79).
Our model is rated above OMS, which we consider state of the art. We further
conducted multiple post-hoc paired t-tests with Bonferroni correction [46] for
each task and condition vs. our method. We found the rank ordering of the
results are significant for $p<0.01$, with the exception of our approach
compared to OMS ($p\approx 0.0378$).

##  5 Conclusion

We present a new approach to differentiable signal processing that enables the
use of stateful, third-party black-box audio effects (plugins) within any deep
neural network. The main advantage of our approach is that it can be used with
general, unknown plugins, and can be retrained for a wide variety of tasks by
substituting in new audio training data and/or customizing with specific Fx.
We believe that this framework has the potential for broader use, including
optimizing legacy signal processing pipelines, style matching, and generative
modeling for audio synthesis. For future work, we hope to address limitations
including optimization of discrete parameters, gradient approx. accuracy,
removing the need for paired training data, and robustness to black-box
software brittleness.

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