Signals convey information. In nature, this information may be in the form of an analog or continuous signal, such as a physical sound wave or radio wave. With proper sampling and sensors, a microphone or radio in this example, we can numerically represent this physical analog wave as a digital signal which is a discrete sequence of numbers that represents all of the information contained within that waveform.
Often when sampling the environment, our sensors collect unwanted signals, disturbances, or extra information that is not relevant to the signal and obscures the information we want to collect. This irrelevant information we term noise which interferes with our ability to extract meaningful information from the signal and can prevent technologies from working properly.
In order to decode and clean up a signal that contains noise to make the signal useful, we filter or process the signal through electronic circuits or mathematical algorithms to separate and extract the signal from the noise. Collectively, cleaning up and filtering a signal to reduce or eliminate noise is one example of Signal Processing.
Whether we realize it or not, everyday, our lives are influenced by signals and shaped by signal processing. If you drive a car, used navigation maps with GPS, made a cell phone call, used a computer or the Internet, took a digital picture, played a video game, watched a movie on DVD, used a charge card, had a MRI or CAT scan, listened to the radio, or used RADAR or SONAR, you have used signal processing technologies.
Digital Signal Processing (DSP) is often called the Phantom Technology as DSP is in nearly all modern electronics but takes place behind the scenes as algorithms in hardware or software. More specifically, DSP is applying math encoded as algorithms in hardware or software to those numerical values of a digital signal in order to change the signal in some advantageous or useful way, such as in filtering a signal to extract the useful information as in tuning a digital radio.
The power of our patented Fractional Calculus-based technology (FSDSP) to significantly improve performance and to better isolate and filter the signal from the noise has been proven in the following: medical device applications (e.g., biometrics such as heart rate, breathing, and concussion evaluation), electromagnetic borehole telemetry of land-based oil rigs, noise removal from neutron flux density data from nuclear reactors, software defined radio spectrum division, and modeling of natural systems to name a few.
Information is all around us, but is often invisible, either imperceptible or the signal is lost in the noise. As a research, development, and innovation laboratory, the sNoise Research Laboratory (sNRL) is leading the way to develop plug and play Fractional Calculus analytical tools which include advanced digital signal processing libraries, signal-shaping smart filters, advanced machine/deep-learning toolboxes, and algorithmic mathematical solutions for specific fields-of-use in order to better extract the signal from the noise to reveal the hidden reality of information shaping the technologies of our modern world.
The science of noise, or sNoise® (a registered trademark of the sNoise Research Laboratory) is the science and mathematics behind patented Fractional Scaling Digital Signal Processing (FSDSP) filters and algorithms — first introduced in the research and dissertation of Dr. Smigelski, founder of the sNoise Research Laboratory — which are uniquely based on an emerging field of mathematics known as fractional calculus.
Our goal is to integrate sNRL's patented fractional calculus code-libraries and algorithms into deep learning and artificial intelligence systems, enhancing their performance and pushing the boundaries of what is possible in these rapidly evolving fields. By leveraging the power of fractional calculus combined with AI, we aim to revolutionize anomaly detection, predictive modeling, time series analysis, and other critical areas of data analytics.
The sNoise Research Laboratory (sNRL) is building fractional calculus toolbox libraries based on our patented Fractional Scaling Digital Signal Processing (FSDSP) platform technology for both standalone applications (SDK, API, FSDSP Chipsets) and for direct integration into Artificial Intelligence (AI) applications (i.e., machine learning and deep learning combined with fractional calculus-based digital signal processing (DSP) for multiple fields-of-use).
To use an analogy, with FSDSP, sNRL is building a fractional calculus-based AI (FC-AI) that acts like a combination of "ChatGPT" merged with "Shazam" for digital signal processing that may identify, amplify, attenuate, filter, reconstruct, denoise, or synthesize any signal with structure.
“…the mathematical models of reality have been pushed to their limits and beyond, those which were developed and applied so successfully to the explanation and understanding of physical phenomena in the nineteenth and twentieth centuries are no longer adequate to describe the emergent phenomena of the twenty-first century. Herein we propose to adopt a fresh perspective entailed by the use of the fractional calculus.”...
"Tomorrow’s science will be dominated by fractional calculus."
Fractional Calculus View of Complexity:
Tomorrow’s Science, Bruce J. West, 2016.
has worked well for us in the past having been pushed to the limits in
abilities to offer solutions and models of our physical reality. However, as we move towards the future, the
problems we encounter and challenges we face require a paradigm shift for the
whole of science in how we approach algorithmic solutions using all of the
available tools of mathematics. Many of
the limitations of truly describing the mathematics of nature and our physical
reality are solved by applying mathematics and algorithms from the emerging
field of Fractional Calculus which encompasses all of traditional calculus yet extends mathematics beyond and
offers unique solutions unavailable to calculus.
From large scale physics to the quantum level, Fractional Calculus offers a more nuanced mathematics to define the structure, system dynamics, and resolution of both natural and artificial worlds. Based on fractional Calculus, Fractional Scaling Digital Signal Processing captures this structure allowing one to quantitatively define (model) and surgically shape (filter) any structure or signal (in both the time and frequency domains) at individual frequency resolutions leading to greater than current state-of-the-art solutions.
Integration of Artificial Intelligence with Fractional Calculus is a powerful combination and a necessary step in the evolution of AI. Any data or signal with structure can be learned by Artificial Intelligence. In order to best manipulate and filter that data structure, AI requires access to the most advanced mathematical algorithms. In other words, AI requires access to fractional calculus. Combining machine learning and fractional calculus, AI can quickly fine tune fractional calculus equations to arrive at digital signal processing solutions faster with greater accuracy than AI using traditional mathematics. Additionally, a Fractional Calculus approach to AI will provide greater stability and performance under strong perturbations, is more flexible and better able to adapt to dynamic properties of an environment, and has more effective damping characteristics leading to faster recovery with greater accuracy from disturbances.
AI is currently limited in that there are no fractional calculus toolboxes or algorithmic libraries available for machine learning to directly integrate the power of fractional calculus. sNRL is working to change this limitation by developing Fractional Calculus toolbox libraries tailored for machine-learning and deep-learning applications merging Artificial Intelligence with advanced digital signal processing across a variety of fields-of-use within current and future technologies.
The deep learning and AI market is experiencing exponential growth, with organizations across industries adopting these technologies for improved decision-making, automation, and optimization. However, current algorithms have limitations in capturing long-range dependencies and accurately modeling complex systems. This is where fractional calculus provides a significant advantage, enabling more precise modeling and enhanced performance. Our target market includes AI developers, data scientists, research institutions, and companies looking to harness the full potential of deep learning and FC-AI.
sNRL is developing code-libraries that integrate our patented fractional calculus DSP algorithms into existing deep learning frameworks and also coding our own frameworks across multiple fields-of-use. These code-libraries will allow practitioners to effortlessly incorporate fractional calculus principles into their models thus lowering the bar of entry to use this advanced technology, improving accuracy, convergence rates, and efficiency. Our libraries will support multiple programming languages and frameworks, ensuring compatibility and ease of use for a wide range of users.
Artificial Intelligence and Machine/Deep Learning Algorithms often rely on the brute force of computational power to process a signal and arrive at a solution. By building fractional calculus code toolboxes for AI, sNRL is providing AI an expanded mathematical library leading to more advanced solutions and the ability to tackle the toughest problems in signal processing.
Integration of Artificial Intelligence with Fractional Calculus is a Powerful Combination that refines signal processing through these fractional calculus equations (FSDSP) rather than brute forcing the data. Additionally, the equations of FSDSP are computationally more efficient and can save energy, battery life, and memory used by AI.
Fractional calculus can provide a more accurate representation of complex systems by capturing long-term dependencies and memory effects in data, while deep learning excels at learning complex patterns and representations. By incorporating AI techniques, such as deep learning, into the modeling and analysis process powered by fractional calculus, we can develop more accurate and robust models for complex systems. Deep learning can learn the underlying patterns and relationships in the data and adaptively adjust the fractional calculus models to better capture and provide a more accurate representation the underlying dynamics of the system, especially when dealing with non-integer order and fractal-like behavior.
Fractional calculus is well-suited for analyzing and modeling time series data with long-term dependencies. Deep learning, particularly recurrent neural networks (RNNs), can effectively capture temporal dependencies in sequential data. By integrating fractional calculus with RNNs or other deep learning architectures, we can develop more powerful models for time series analysis, forecasting, and anomaly detection. This can have applications in finance, weather prediction, health monitoring, and more.
Fractional calculus can be used to describe and control systems with fractional order dynamics, while deep learning can optimize complex functions and learn control policies. By combining the two, deep reinforcement learning or evolutionary algorithms can learn control policies that optimize system performance, while fractional calculus can provide a more accurate representation of the system's dynamics to develop intelligent control strategies that actively adapt to the fractional order characteristics of the system. This can lead to improved control performance and optimization in various domains, such as robotics, autonomous systems, and process control.
Fractional calculus can capture long-term dependencies and memory effects in time series data. By combining AI techniques, such as deep learning or recurrent neural networks, with fractional calculus, we can develop predictive models that can forecast future behavior more accurately. This can have applications in areas like financial forecasting, weather prediction, or stock market analysis.
Deep learning can be used to learn fractional calculus operators directly from data. Instead of relying on predefined fractional order operators, deep learning algorithms can learn the fractional derivatives or integrals directly from the available data. This data-driven approach can be particularly useful when dealing with complex or noisy data where the underlying fractional order behavior is not well-known or easily defined.
Fractional calculus can help in detecting anomalies or abnormal behavior in complex systems. By integrating AI algorithms, such as anomaly detection or pattern recognition, with fractional calculus, we can develop intelligent systems that can identify and diagnose faults or anomalies in real-time. This can be valuable in areas like fault detection in industrial processes, cybersecurity, or medical diagnosis.