The article was updated on 04-06-25

The Allure of Digital Emulations in Audio Production

In the ever-evolving landscape of audio production, digital plugins — especially digital emulations of analog hardware —have become increasingly popular. These software tools promise to replicate the much sought-after sound of analog hardware without the physical limitations and high costs associated with the real gear. Musicians, producers, and audio engineers are drawn to plugins for their convenience, versatility, and the ability to have multiple instances across numerous tracks. Unlike their hardware counterparts, which are often bulky, expensive, and singular in use, digital emulations provide an accessible and scalable solution for achieving professional-grade audio quality.

Analog emulation plugins offer the promise of analog warmth, tube saturation, and the unique tonal qualities of classic gear, all within a compact and affordable digital package. This charm is amplified by the ease of integration into digital audio workstations (DAWs), enabling seamless workflows and unprecedented creative flexibility. As the technology behind these emulations continues to advance, the gap between the sound of digital plugins and their analog inspirations seems to be narrowing.

However, despite these advancements, the question remains: can digital emulations truly replicate the exact sound and nuanced behavior of analog hardware? While plugins deliver remarkable approximations and boast features that hardware cannot match, there are fundamental reasons—rooted in computational theory and the nature of digital systems—that suggest they may never fully capture the essence of their analog counterparts. A 2022 paper by Boche and Pohl examines the fundamental limitations of digital computers, demonstrating that even a simple analog circuit, such as a passive 1-pole low-pass filter, can produce output values where the accuracy of the result cannot be guaranteed [1,2].

Analog vs. Digital Signals

Fundamental Differences

At the heart of audio processing lies the distinction between analog and digital signals, a fundamental concept that influences the entire landscape of sound engineering. Analog signals are continuous waveforms that vary smoothly over time, representing the original sound within the limitations of the recording and playback systems. These signals are an electrical analog of the variations in air pressure that create sound waves.

Digital signals, on the other hand, are discrete. Instead of a continuous waveform, digital audio is represented by a series of snapshots, or samples, taken at regular intervals. Each sample is quantified into a finite set of values, a process known as quantization. This transformation from a continuous to a discrete signal allows sound to be stored, manipulated, and transmitted by digital systems like computers and DAWs.

Continuous vs. Discrete

The continuous nature of analog signals means they can capture a wide range of detail within the physical limitations of the recording medium and the analog circuit, which it is processed by. When you play an analog tape, for example, the signal is not just a continuous representation of the original performance but it also includes all the imperfections of the tape electronics, the play and record heads, and the physical characteristics of the tape itself. Many subtle inflections, dynamic shifts, and tonal qualities are integrated into the signal and it will add the audio signal a unique character.

In contrast, digital signals are inherently limited by their sampling rate and bit depth. The sampling rate, measured in Hertz (Hz), determines how many times per second the analog signal is measured. Common sampling rates include 44.1 kHz, 48 kHz, and higher rates such as 88.2 kHz, 96 kHz or 192 kHz used in audiophile audio settings. The bit depth, measured in bits, determines the precision of each sample, affecting the dynamic range and noise floor of the recording. Standard bit depths include 16-bit, 24-bit and 32-bit.

Digital signals can accurately represent the original analog waveform when properly sampled according to the Nyquist-Shannon theorem. With adequate sampling, discretization does not necessarily lead to a loss of fine details. However, if is there are higher frequencies in the signal which are above half the sampling frequency, aliasing can occur, meaning they got fold-back into the audio band and can distort the signal or lead to artifacts.

Analog vs. Digital Processing

The comparison between analog and digital audio processing reveals distinct advantages and limitations for each approach. In an analog device, such as an equalizer, the audio signal is processed continuously by the physical characteristics of the circuit components—resistors, capacitors, and transistors. These components interact with the signal in real-time, providing high resolution and a direct response as the signal passes through the circuit, though physical limitations and component imperfections can introduce noise and distortion. The imperfections are mainly caused by value mismatch of the components, noise introduced by the components, and most important, the non-linearities of the components, and the analog circuit over all, what makes emulating the analog circuit digitally so challenging.

On the other hand, digital equalizers process audio signals by employing algorithms that perform the equalization function in the digital domain. The signal is first converted into a digital format, where it is represented as a series of numerical values. The algorithm then manipulates these values to achieve the desired equalization effect. While digital processing offers advantages such as greater flexibility, precision in controlling specific frequencies, and the ability to easily replicate and modify settings, due to the non-linearities in analog circuits it is very hard to capture every aspect with a digital emulation.

However, it is important to note that advancements in digital technology have significantly reduced the gap between analog and digital processing. High sampling rates and advanced algorithms can now produce results that are nearly indistinguishable from analog counterparts, even to trained listeners. Yet, some audiophiles argue that analog devices still provide a warmth and naturalness that digital systems struggle to fully replicate. The choice between analog and digital often comes down to personal preference, the specific application, and the desired characteristics of the sound.

But how close can digital emulations of analog hardware really come? Are there inherent mathematical limitations that prevent a perfect match?

Limitations of Digital Algorithms

A scientific paper by Boche and Pohl from 2022 shows that there is a mathematical barrier that limits the accuracy of digital algorithms [1]. To understand the fundamental result in this paper, it's helpful to start with a basic concept in computer science: the Turing machine. A Turing machine is a theoretical model that mathematicians and computer scientists use to understand the limits of what can be computed. It’s an abstract computational model that can read and write symbols on an infinite tape, move the tape head left and right, and follow a set of instructions (an algorithm) to decide what to do next. This model helps explore the boundaries of computation and the concept of computability. 

Now, when we talk about algorithms, we’re essentially talking about a step-by-step procedure or set of rules that a computer follows to solve a problem or perform a task. This could be anything from adding two numbers to running a complex simulation. The important thing to note is that these algorithms can be executed by a Turing machine. So, in a very fundamental sense, any algorithm that a real computer runs can be thought of as something that a Turing machine could also perform, given enough time and resources.

In other words, a Turing machine is a simplified model of what computers do, and if something can’t be computed by a Turing machine, then it can’t be computed by any digital computer either. This is where we start to touch on the concept of computability—whether or not a problem can be solved by any algorithm running on a Turing machine, and thus on a real computer.

The paper by Boche and Pohl delves into a deeper aspect of this idea. It explores situations where the problems we want to solve, specifically those related to circuits and wave equations (which describe things like electrical circuits and the behavior of waves), might be non-computable. This means that no algorithm—no matter how powerful—can solve these problems exactly. The paper investigates how these non-computable problems can be undetectable by digital computers and examines the degree of non-computability involved.

Non-Computability and Complex Behaviors

The Concept of Non-Computable Functions

In the realm of computer science and mathematics, a function is considered computable if there exists an algorithm that can produce the correct output for any given input within a finite amount of time. However, not all functions meet this criterion. Non-computable functions are those for which no such algorithm exists. These functions defy exact calculation and prediction by digital computers, regardless of processing power or algorithmic sophistication [1,2].

This concept of non-computability is highly relevant when examining the limitations of digital emulations of analog hardware. While digital systems excel at handling linear, predictable behaviors, they struggle with the complex, often non-linear dynamics inherent in many analog circuits.

Figure 1: 1-pole low-pass filter

For example, even a simple analog circuit like a passive 1-pole low-pass filter (see figure 1) can have output values where digital emulations can't be sure if the result is correct. This non-computability presents a fundamental limitation in achieving perfect digital emulation [1].

Moreover, non-computable functions highlight a fundamental limitation: some aspects of analog behavior may simply be beyond the reach of digital algorithms. Even with advanced modeling techniques, such as machine learning and neural networks, there remain aspects of analog circuitry that resist precise digital emulation [3].

Conclusion and Outlook

In the quest to bridge the gap between analog warmth and digital convenience, digital emulations of analog hardware have made remarkable progress. These software tools offer an accessible, scalable solution that provides musicians, producers, and engineers with the flexibility to recreate the sought-after characteristics of analog gear without the associated physical and financial constraints. However, the allure of digital emulations must be tempered with an understanding of their inherent limitations. While digital algorithms can closely approximate analog behavior, fundamental differences between continuous analog processing and discrete digital processing introduce challenges that may prevent a perfect match.

The work of Boche and Pohl in 2022 sheds light on these challenges, particularly highlighting the concept of non-computability. Their research underscores that certain complex behaviors of analog circuits may be inherently non-computable, meaning no digital algorithm can fully capture every nuance of these systems. This finding suggests that while digital emulations can get remarkably close, they may never entirely replicate the intricate and often nonlinear dynamics of analog hardware.

The ongoing advancements in digital signal processing, including higher sampling rates (by oversampling), more sophisticated algorithms, and the integration of machine learning, will likely continue to narrow the gap between digital emulations and their analog counterparts. However, the recognition of these fundamental limits also opens new avenues for innovation. Future research may explore hybrid approaches that combine the strengths of both analog and digital systems, or even new computational paradigms that transcend the current boundaries of digital processing such as quantum computing.

Ultimately, the choice between analog and digital will remain a matter of personal preference, application, and desired sound characteristics. As technology evolves, the industry will continue to grapple with the balance between the convenience of digital systems and the irreplaceable warmth and character of analog sound.

References

[1] H. Boche and V. Pohl, "On Non-Detectability of Non-Computability and the Degree of Non-Computability of Solutions of Circuit and Wave Equations on Digital Computers," in IEEE Transactions on Information Theory, vol. 68, no. 8, pp. 5561-5578, Aug. 2022, doi: 10.1109/TIT.2022.3172837.

[2] J. Myhill "A recursive function, defined on a compact interval and having a continuous derivative that is not recursive.," Michigan Mathematical Journal, Michigan Math. J. 18(2), 97-98, (May 1971)

[3] Boche, H., Fono, A., & Kutyniok, G. (2023). Limitations of Deep Learning for Inverse Problems on Digital Hardware. IEEE Transactions on Information Theory, 69(12), 7887-7908. https://doi.org/10.1109/tit.2023.3326879

Author publications