## What is Fractional Scaling Digital Signal Processing?

Based on Fractional Calculus, sNRL’s advanced, patented Fractional Scaling Digital Signal Processing (FSDSP) algorithms let you quantitatively define and shape the spectrum of any signal.  Demonstrably, FSDSP represents more exact filtering solutions, not approximations, and ​greatly improves upon the performance, accuracy, precision, and efficiency of current digital signal processing (DSP) filters, methods, and algorithms.

Fundamentally, Fractional Scaling Digital Signal Processing (FSDSP) allows fractional calculus, and thus fractional filtering (e.g., fractional scaling, fractional phase shifting, fractional integration, or fractional differentiation) to be performed on a signal.  In fact, FSDSP opens up complete access to a signal allowing each individual frequency the ability to be adjusted to any decimal decibel level in magnitude and/or phase and also the ability to access or alter the time element of the signal or frequency.  FSDSP also encompasses a new class of digital filters (Fractional Scaling Digital Filters or FSDF) and further defines control systems (Fractional Order Control Systems or FOCS) in terms of Fractional Calculus. Representing exact filtering solutions rather than approximations, FSDSP demonstrably exhibits extreme mathematical accuracy, precision, robustness, flexibility, and computational efficiency leading to cleaner signals and more effective control.  And this is just from the equations of fractional calculus, just wait until AI gets involved.

By utilizing a fractional calculus approach to digital signal processing, Fractional Scaling Digital Signal Processing and Fractional Scaling Digital Filters provide sNRL the ability to mathematically define both exact decimal/fractional decibel levels of attenuation or amplification and complete phase modification at each frequency to selectively filter complex data sets. FSDSP can achieve nearly any desired filtering characteristic with a high degree of accuracy from sharp transitions within a narrow bandwidth to complicated structures within the passband, all without introducing the mathematical artifacts of current state-of-the-art filters or resulting in a loss of information in the filtered signal. FSDSP grants sNRL the ability to extract the signal from the noise more effectively with a higher resolution and level of performance than conventional digital signal processing filters and algorithms that are based on traditional integer-based calculus.  In fact, FSDSP allows specific fractional scaling equations to be written for the spectrum of any signal (e.g., radio, data, or sound). These equations can then be used in algorithms as filters to shape, correct, or recognize the spectrum or as models to simulate or reproduce the spectrum.

As the language of dynamic system models, FSDSP also extends to defining, modeling, and filtering of fractional order control systems (FOCS) and improves the response, stability, and machine/deep learning capability of emergent robotic and AI platforms such as Unmanned Vehicle Systems (UAVs), self-driving vehicles, and satellites.   Fractional Order Control Systems invoking FSDSP, such as a fractional order proportional-integral-derivative (PID) controller, provide greater stability and performance under strong perturbations, are more flexible and better able to adapt to dynamic properties of an environment, have more effective damping characteristics, and may recover faster and with greater accuracy from disturbances.

FSDSP also allows for the development of more efficient and accurate ML algorithms for signal processing as there are fewer parameters necessary within the FSDSP equations that would need to be modified or modeled by a ML/DL system when compared to traditional filters and systems allowing AI to come to a solution faster yet more accurately.

FSDSP captures the scaling and frequency behavior of complex time series, systems, and structured data within equations and can yield statistically identical simulations and models of these data sets to allow you to "animate", extrapolate, filter, simulate, and understand these systems giving us the ability to determine "What’s inside the black box?" and how the ML algorithms or AI came to a solution.

This is why FSDSP can emulate natural signals such as a realistic voice as the equations of FSDSP and mathematics of Fractional Calculus more accurately represent how nature works and may bring us closer to a true, cognitively functioning AI.

FSDSP provides a complete, organized mathematical framework and repository of Fractional Calculus encoded as a library of patented algorithms.  sNRL is actively working towards converting the FSDSP algorithms from MATLAB to a multi-language Software Development Kit (SDK) for edge computing and Application Programming Interface (API) for cloud computing.  Custom interfaces and sample data for each field of use and data type are also planned for inclusion into the FSDSP SDK and API so that customers from a variety of scientific and engineering fields may more readily test, license, and implement the fractional calculus algorithms in both hardware and software products leading to widespread adoption potentially disrupting industry standards in many of these disciplines.  Additionally, our advanced algorithms seamlessly integrate deep learning's ability to learn complex patterns with fractional calculus' precision in modeling fractional order behavior.

The usefulness of fractional calculus and FSDSP including Fractional Scaling Digital Filters and their use in fractional order control systems extends across a multitude of disciplines and industries from control theory, cybernetics, economics, information theory, medicine, neuroscience, neuroengineering, cognitive science, and the human behavioral sciences to the environmental sciences, meteorology, geophysics, aerospace, control systems, robotics, mechanical engineering, mechatronics, sensors, electrical engineering, telecommunications, audio, video, and digital signal processing with numerous applications such as RADAR and SONAR Data Acquisition Systems.  Overall, the integration of sNRL's fractional calculus code libraries into deep learning and AI systems can bring benefits such as improved accuracy, enhanced data analysis capabilities, optimized processes, and cost savings across industries and scientific disciplines ultimately contribute to advancements in research, innovation, and problem-solving. Since Fractional Calculus encompasses and extends mathematics beyond all of Calculus, the current possibilities and the yet undiscovered potential uses are truly limitless.

FSDSP is scalable and can be encoded into a field-programmable gate array (FPGA) device, MicroController, or within a DSP chipset. Compared to conventional DSP, FSDSP requires fewer equations, parameters, and steps to achieve fractional rates of attenuation or amplification of specific frequency regions translating into a reduction in the amount of time necessary for calculations, less error propagation, a reduction in the memory necessary for the calculations, and a reduction in the energy per operation.  Fractional Calculus-based FSDSP thus leads to enhanced sensors with longer battery life.

sNRL is developing FSDSP code libaries for direct integration by Artificial Intelligence to allow AI access to the power of Fractional Calculus.  In addition to AI tuning FSDSP equations, FSDSP may be applied to Machine/Deep Learning for Feature/Pattern Detection, Extraction, Fractional Order Control, and Synthesis and embedded directly into the network architecture. FSDSP captures the scaling behavior of complex time series, control systems, and structured data within equations and can yield statistically identical simulations of these data sets.  AI allows FSDSP to be automated.

FSDSP can produce more accurate and realistic computer models and simulations of natural environments through computer-generated 2D imagery and 3D textures.  From Soundscapes to Landscapes to artificial but statistically identical environments, the equations of fractional calculus-based FSDSP allow hyper-realistic simulations without AI.  When AI is added for tuning or for quick rendering using fractional calculus libraries, FSDSP becomes even more powerful.

FSDSP provides a method to easily filter out or amplify a single distinct frequencies or multiple ranges of frequencies utilizing a fractional equalizer without impacting adjacent frequencies and may be applied in near-real time to live data streams from IoT.  Additionally, unlike signal processing that uses the traditional Fourier Transform (FFT), with sNRL's proprietary complex frequency transform, FSDSP can access more frequencies at a finer resolution in less time giving the one the ability to filter at an unprecedented ultra-high resolution, as if digital data was analog.

FSDSP provides methods to clean up signals and greatly increase signal-to-noise ratio from ubiquitous and noisy sensor/actuators that live at the edge of the network in IoT.  Tests on real-world industrial data have yielded greater than a 100x increase in the signal over the noise, far greater than traditional DSP approaches.  FSDSP can also extract the signal from the noise or generate a statistically identical yet synthetic signal without the noise.

## Fractional Calculus isFrontier Science

#### MACHINE/DEEP LEARNING AND ARTIFICIAL INTELLIGENCE WITH FSDSP

At sNRL, we firmly believe that the key to unlocking the immense potential of artificial intelligence lies in incorporating fractional calculus directly within network architectures to enhance the capabilities of deep learning and AI systems. Fractional calculus, a branch of mathematical analysis that extends the concept of differentiation and integration to non-integer orders can capture the intricate dynamics of complex systems and represents a paradigm shift in algorithmic development. With patented fractional calculus DSP algorithms and cutting-edge code-libraries created in-house, sNRL is positioned at the forefront of transforming the world of deep learning and artificial intelligence, paving the way for innovative advancements in these rapidly evolving fields.

We understand that traditional methods often fall short when it comes to capturing the complexities of real-world data. That's why we are developing a unique approach that combines Fractional Calculus and deep learning to provide unparalleled insights and predictive capabilities. Our data-driven approach allows us to learn fractional calculus operators directly from your data, eliminating the need for predefined fractional order operators. By integrating deep learning techniques, we empower our models to learn complex patterns and representations from your data, enabling us to uncover hidden insights and make accurate forecasts. This flexibility enables us to adapt to diverse datasets, tackling complex and noisy datasets, where traditional methods fall short.

Our proprietary and patented Fractional Calculus algorithms, as API/SDK code libraries, are designed to enhance the performance and effectiveness of deep learning algorithms and to seamlessly integrate with existing deep learning frameworks, providing a powerful toolset for researchers, developers, and data scientists. By incorporating our code-libraries into their projects, they gain access to the unparalleled capabilities of fractional calculus, especially when applied to digital signal processing (DSP), enabling them to unlock new insights and achieve superior performance. With our algorithms, developers can more readily harness the power of fractional calculus to enhance their deep learning models and AI systems.

By operating on non-integer order data and equations, these algorithms enable more accurate modeling of complex patterns and systems, better capturing long-range dependencies and memory effects, and improving the efficiency and convergence rates of optimization processes. Through the integration of our fractional calculus code-libraries, we aim to empower AI developers and data scientists to push the boundaries of what is possible. Imagine having a predictive model that not only captures intricate temporal dependencies but also accurately represents the fractional order dynamics of your system. Whether it's financial forecasting, anomaly detection, or optimizing control strategies, sNRL empowers you to make smarter decisions and achieve superior performance.

As a computational tool, which would you rather have?

Fractional Calculus includes all of Traditional Calculus, offers unique solutions unavailable to Calculus, and is more reflective of our physical reality and how the universe works encoded within digital signals.  Mathematically speaking, Fractional Calculus is a superset of Calculus and Calculus is a subset of Fractional Calculus.

### The concepts of Fractional Scaling Digital Signal Processing, encompassing Fractional Scaling Digital Filters and their use in fractional order control systems, extends across a multitude of disciplines and industries.

 Control theory Neuroscience Environmental sciences Radio Astronomy Mechanical engineering Telecommunications Cybernetics Neuroengineering Meteorology Aerospace Mechatronics Audio/Music Information theory Cognitive science Geophysics Control systems Sensor​s/Sensing Digital signal processing (DSP) Medicine Human behavioral sciences Physics Robotics Electrical engineering Artificial Intelligence LIDAR Finance GPS broadcasting (Radio, Television) RADAR Data Acquisition Systems Nuclear/Directed Energy Marketing digital optics (Telescopes) Video SONAR Autonomous Vehicles

Identified Potential Applications of Fractional Calculus and FSDSP

 Quantum Computing Imaging (Medical Devices) Fiber Optics Adaptive/ Augmented hearing Electromagnetic Borehole Telemetry Oscillator-Based Computers (OBCs) First Responder (XRAY) Telescopes (Theoretical Limit) Music Synthesis / Processing/Tuning Neutron Flux Density Analog Computers Heart Rate ID Security Internet of Things (IoT) Medical Data (EEG, MEG, fMRI) Nystagmus Events Telemedicine (Sensors) RF Communications / Software Defined Radio (SDR) / Spectrum Congestion Robotics (UAVs, Autopilot) Economic Data (stock market algorithms) Deep Learning / Scalable Deep Data-Mining Spectroscopy Bluetooth 3D Audio / FSDSP HRTF Environmental Data (temperature) Vibrational Analysis Hyperspectral Imaging WiFi Speech Synthesis / Modification Bionics/Prosthetics Fractional Order Control Systems